An ecosystem-scale model for the spread of a host-specific forest pathogen in the Greater Yellowstone Ecosystem.
Ecological Applications (2011)
- PubMed: 21774419
Available from
Michael Dietze's profile on Mendeley.
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Abstract
The introduction of nonnative pathogens is altering the scale, magnitude, and persistence of forest disturbance regimes in the western United States. In the high-altitude whitebark pine (Pinus albicaulis) forests of the Greater Yellowstone Ecosystem (GYE), white pine blister rust (Cronartium ribicola) is an introduced fungal pathogen that is now the principal cause of tree mortality in many locations. Although blister rust eradication has failed in the past, there is nonetheless substantial interest in monitoring the disease and its rate of progression in order to predict the future impact of forest disturbances within this critical ecosystem. This study integrates data from five different field-monitoring campaigns from 1968 to 2008 to create a blister rust infection model for sites located throughout the GYE. Our model parameterizes the past rates of blister rust spread in order to project its future impact on high-altitude whitebark pine forests. Because the process of blister rust infection and mortality of individuals occurs over the time frame of many years, the model in this paper operates on a yearly time step and defines a series of whitebark pine infection classes: susceptible, slightly infected, moderately infected, and dead. In our analysis, we evaluate four different infection models that compare local vs. global density dependence on the dynamics of blister rust infection. We compare models in which blister rust infection is: (1) independent of the density of infected trees, (2) locally density-dependent, (3) locally density-dependent with a static global infection rate among all sites, and (4) both locally and globally density-dependent. Model evaluation through the predictive loss criterion for Bayesian analysis supports the model that is both locally and globally density-dependent. Using this best-fit model, we predicted the average residence times for the four stages of blister rust infection in our model, and we found that, on average, whitebark pine trees within the GYE remain susceptible for 6.7 years, take 10.9 years to transition from slightly infected to moderately infected, and take 9.4 years to transition from moderately infected to dead. Using our best-fit model, we project the future levels of blister rust infestation in the GYE at critical sites over the next 20 years.
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An ecosystem-scale model for the spread of a host-specific forest pathogen in the Greater Yellowstone Ecosystem.
AN ECOSYSTEM-SCALE MODEL FOR THE SPREAD OF A HOST-SPECIFIC FOREST PATHOGEN IN THE 1
GREATER YELLOWSTONE ECOSYSTEM 2
3
4
JACLYN A. HATALA1,7, MICHAEL C. DIETZE2, ROBERT L. CRABTREE3, KATHERINE KENDALL4, 5
DIANA SIX5, PAUL R. MOORCROFT6 6
7
1 Department of Environmental Science, Policy, and Management, University of California 8
Berkeley, Berkeley, California 94720 USA 9
2 Department of Plant Science, University of Illinois at Urbana-Champaign, Urbana, Illinois 10
61801, USA 11
3 Yellowstone Ecological Research Center, Bozeman, Montana 59718, USA 12
4 Northern Rocky Mountain Science Center, U.S. Geological Survey, West Glacier, Montana 13
59936, USA 14
5 College of Forestry and Conservation, The University of Montana, Missoula, Montana 59812, 15
USA 16
6 Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, 17
Massachusetts 02138 USA 18
19
7 Corresponding author email: jhatala@berkeley.edu 20
21
22
23
GREATER YELLOWSTONE ECOSYSTEM 2
3
4
JACLYN A. HATALA1,7, MICHAEL C. DIETZE2, ROBERT L. CRABTREE3, KATHERINE KENDALL4, 5
DIANA SIX5, PAUL R. MOORCROFT6 6
7
1 Department of Environmental Science, Policy, and Management, University of California 8
Berkeley, Berkeley, California 94720 USA 9
2 Department of Plant Science, University of Illinois at Urbana-Champaign, Urbana, Illinois 10
61801, USA 11
3 Yellowstone Ecological Research Center, Bozeman, Montana 59718, USA 12
4 Northern Rocky Mountain Science Center, U.S. Geological Survey, West Glacier, Montana 13
59936, USA 14
5 College of Forestry and Conservation, The University of Montana, Missoula, Montana 59812, 15
USA 16
6 Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, 17
Massachusetts 02138 USA 18
19
7 Corresponding author email: jhatala@berkeley.edu 20
21
22
23
Page 2
2
ABSTRACT 24
25
The introduction of non-native pathogens is altering the scale, magnitude and persistence 26
of forest disturbance regimes in the western United States. In the high altitude whitebark pine 27
forests of the Greater Yellowstone Ecosystem (GYE), blister rust is an introduced fungal 28
pathogen that is now a primary source of tree mortality. Although blister rust eradication efforts 29
have failed in the past, there is nonetheless large interest in monitoring the disease in order to 30
predict the impacts of forest disturbances on this important ecosystem in the future. 31
This study integrates data from five different field monitoring campaigns over forty years 32
from 1968-2008 to create a blister rust infection model for sites located throughout the GYE. Our 33
model parameterizes the rate of blister rust spread through the past in order to project its impact 34
on the future for high altitude whitebark pine sites. We use four infection classes in our model: 35
susceptible, slightly infected, moderately infected and dead. In our analysis, we evaluate four 36
possible dynamic blister rust infection models in order to assess local versus global density 37
dependence on the dynamics of blister rust infection. We compare models with no density 38
dependence, local density dependence, local density dependence with a static global infection 39
rate, and both local and global infection density dependence. Model evaluation through the 40
predictive loss criterion for Bayesian analysis supports the model with both local and global 41
infected whitebark pine density dependence. Using this best-fit model, we predict the average 42
residence times for the four stages of blister rust infection in our model, and find that on average 43
within the GYE, whitebark pine trees remain susceptible for 6.7 years, take 10.9 years to 44
transition from infected to slightly infected, and take 9.4 years to transition from infected to 45
dead. Using our best-fit model, we project the future levels of blister rust infestation in the GYE 46
over the next twenty years. 47
ABSTRACT 24
25
The introduction of non-native pathogens is altering the scale, magnitude and persistence 26
of forest disturbance regimes in the western United States. In the high altitude whitebark pine 27
forests of the Greater Yellowstone Ecosystem (GYE), blister rust is an introduced fungal 28
pathogen that is now a primary source of tree mortality. Although blister rust eradication efforts 29
have failed in the past, there is nonetheless large interest in monitoring the disease in order to 30
predict the impacts of forest disturbances on this important ecosystem in the future. 31
This study integrates data from five different field monitoring campaigns over forty years 32
from 1968-2008 to create a blister rust infection model for sites located throughout the GYE. Our 33
model parameterizes the rate of blister rust spread through the past in order to project its impact 34
on the future for high altitude whitebark pine sites. We use four infection classes in our model: 35
susceptible, slightly infected, moderately infected and dead. In our analysis, we evaluate four 36
possible dynamic blister rust infection models in order to assess local versus global density 37
dependence on the dynamics of blister rust infection. We compare models with no density 38
dependence, local density dependence, local density dependence with a static global infection 39
rate, and both local and global infection density dependence. Model evaluation through the 40
predictive loss criterion for Bayesian analysis supports the model with both local and global 41
infected whitebark pine density dependence. Using this best-fit model, we predict the average 42
residence times for the four stages of blister rust infection in our model, and find that on average 43
within the GYE, whitebark pine trees remain susceptible for 6.7 years, take 10.9 years to 44
transition from infected to slightly infected, and take 9.4 years to transition from infected to 45
dead. Using our best-fit model, we project the future levels of blister rust infestation in the GYE 46
over the next twenty years. 47
Page 3
3
INTRODUCTION 48
49
Anthropogenic disturbances are playing an increasingly important role in shaping 50
ecosystems, especially in the western United States where introduced pathogens are creating 51
novel disturbance regimes (Ellison et al. 2005, Logan et al. 2003). During the past 100 years, the 52
economic and ecological consequences resulting from the introduction of non-native forest 53
pathogens have presented a tremendous challenge for forest managers (Brown and Hovmoller 54
2002). An important example of this is white pine blister rust (Cronartium ribicola), which is a 55
now a principal cause of mortality within whitebark pine (Pinus albicaulis) populations at high 56
altitudes in the Greater Yellowstone Ecosystem (GYE) (Kendall and Arno 1990). White pine 57
blister rust is a non-native fungal pathogen introduced to North America in 1910 near British 58
Columbia on a stock of white pine imported for a plantation (Spaulding 1922). It has two 59
obligate alternate hosts: five-needled pines and shrubs belonging to the genus Ribes. Despite a 60
continental-scale federal program to eradicate white pine blister rust from the landscape that 61
began in the 1930s and ran until 1965, white pine blister rust still pervades high altitude five-62
needled pine populations within much of the western United States (Smith and Hoffman 2000). 63
Pathogens operate through mechanisms that are biologically species-specific, yet they 64
produce disturbance patterns with cascading effects that scale up to measurable changes at the 65
landscape level (Castello et al. 1995). Pathogen disturbances in forests can produce dramatic 66
ecosystem-scale effects through the reorganization of community structure (Frelich and Reich 67
1999) in addition to the alteration of abiotic processes such as hydrology and soil composition 68
(Ellison et al. 2005). Consequently, ecologists have identified many links between host and 69
pathogen dynamics that vary with biotic heterogeneity such as landscape connectivity patterns, 70
as well as with abiotic heterogeneity, such as topography (Holdenrieder et al. 2004). 71
INTRODUCTION 48
49
Anthropogenic disturbances are playing an increasingly important role in shaping 50
ecosystems, especially in the western United States where introduced pathogens are creating 51
novel disturbance regimes (Ellison et al. 2005, Logan et al. 2003). During the past 100 years, the 52
economic and ecological consequences resulting from the introduction of non-native forest 53
pathogens have presented a tremendous challenge for forest managers (Brown and Hovmoller 54
2002). An important example of this is white pine blister rust (Cronartium ribicola), which is a 55
now a principal cause of mortality within whitebark pine (Pinus albicaulis) populations at high 56
altitudes in the Greater Yellowstone Ecosystem (GYE) (Kendall and Arno 1990). White pine 57
blister rust is a non-native fungal pathogen introduced to North America in 1910 near British 58
Columbia on a stock of white pine imported for a plantation (Spaulding 1922). It has two 59
obligate alternate hosts: five-needled pines and shrubs belonging to the genus Ribes. Despite a 60
continental-scale federal program to eradicate white pine blister rust from the landscape that 61
began in the 1930s and ran until 1965, white pine blister rust still pervades high altitude five-62
needled pine populations within much of the western United States (Smith and Hoffman 2000). 63
Pathogens operate through mechanisms that are biologically species-specific, yet they 64
produce disturbance patterns with cascading effects that scale up to measurable changes at the 65
landscape level (Castello et al. 1995). Pathogen disturbances in forests can produce dramatic 66
ecosystem-scale effects through the reorganization of community structure (Frelich and Reich 67
1999) in addition to the alteration of abiotic processes such as hydrology and soil composition 68
(Ellison et al. 2005). Consequently, ecologists have identified many links between host and 69
pathogen dynamics that vary with biotic heterogeneity such as landscape connectivity patterns, 70
as well as with abiotic heterogeneity, such as topography (Holdenrieder et al. 2004). 71
Page 4
4
Epidemiology models at the landscape scale have indicated that spatial extent (O’Neill et al. 72
1992) and rate (Park et al. 2002) of pathogenic disturbance interacts through feedbacks with 73
spatially explicit landscape patterns at both the local and regional scales (Park et al. 2001). 74
Because whitebark pine has such an ecologically significant role within high-altitude 75
ecosystems and because pathogens often operate in ways that are spatially explicit, we expected 76
the existence of key landscape drivers to describing and predicting the spatial spread of blister 77
rust in whitebark pine throughout the GYE. In this analysis, we built a meta-population model 78
for the spread of blister rust in whitebark pine, based on field surveys collected between 1968-79
2008. This analysis used a dataset aggregated from five different blister rust monitoring 80
campaigns to estimate transitions between four stages of infection: susceptible, slight infection, 81
moderate infection and dead, at 121 sites throughout the GYE from 1968-2008. We created our 82
model using a Bayesian statistical framework, which permitted the modeling of 95% confidence 83
intervals for the four levels of infection in 1968-2008 where data exists, and also in our future 84
projections of blister rust infection. Furthermore, the Bayesian framework of our model 85
permitted the inclusion of informed priors, where we used the field experience of forest 86
managers in the GYE to set an informed starting point for our parameters. 87
Since whitebark pine trees exist only at high elevations throughout the GYE, across the 88
ecosystem, patches of whitebark pine occur at distances where it becomes insightful to analyze 89
both local and global infection dynamics in a metapopulation structure. Using the Bayesian 90
infection model, we test four hypotheses for the mechanistic infection dynamic of blister rust 91
operating at the ecosystem scale: blister rust infection is independent of infected tree density, 92
blister rust is proportional to the local infected tree density, blister rust infection is proportional 93
to the local infected tree density with a static background global infection rate, or blister rust is 94
Epidemiology models at the landscape scale have indicated that spatial extent (O’Neill et al. 72
1992) and rate (Park et al. 2002) of pathogenic disturbance interacts through feedbacks with 73
spatially explicit landscape patterns at both the local and regional scales (Park et al. 2001). 74
Because whitebark pine has such an ecologically significant role within high-altitude 75
ecosystems and because pathogens often operate in ways that are spatially explicit, we expected 76
the existence of key landscape drivers to describing and predicting the spatial spread of blister 77
rust in whitebark pine throughout the GYE. In this analysis, we built a meta-population model 78
for the spread of blister rust in whitebark pine, based on field surveys collected between 1968-79
2008. This analysis used a dataset aggregated from five different blister rust monitoring 80
campaigns to estimate transitions between four stages of infection: susceptible, slight infection, 81
moderate infection and dead, at 121 sites throughout the GYE from 1968-2008. We created our 82
model using a Bayesian statistical framework, which permitted the modeling of 95% confidence 83
intervals for the four levels of infection in 1968-2008 where data exists, and also in our future 84
projections of blister rust infection. Furthermore, the Bayesian framework of our model 85
permitted the inclusion of informed priors, where we used the field experience of forest 86
managers in the GYE to set an informed starting point for our parameters. 87
Since whitebark pine trees exist only at high elevations throughout the GYE, across the 88
ecosystem, patches of whitebark pine occur at distances where it becomes insightful to analyze 89
both local and global infection dynamics in a metapopulation structure. Using the Bayesian 90
infection model, we test four hypotheses for the mechanistic infection dynamic of blister rust 91
operating at the ecosystem scale: blister rust infection is independent of infected tree density, 92
blister rust is proportional to the local infected tree density, blister rust infection is proportional 93
to the local infected tree density with a static background global infection rate, or blister rust is 94
Page 5
5
proportional to both the local and global infected tree density. 95
96
METHODS 97
Data 98
White pine blister rust has a complex life history with five spore stages and two obligate 99
alternate hosts (Arthur 1934), outlined in Figure 1. White pine blister rust does not spread 100
directly from one whitebark pine tree to another, but indirectly through its alternate host, various 101
species of currants and gooseberries (members of the genus Ribes). Explicitly monitoring and 102
modeling the intra-annual spread of white pine blister rust between its alternating hosts within is 103
difficult since there are over seventeen species of Ribes within the GYE that would require intra-104
annual white pine blister rust surveys. Within the context of our study, we therefore focus the 105
inter-annual progress of white pine blister rust infection. As a result, we do not model Ribes 106
populations or infection levels explicitly, since white pine blister rust can only overwinter within 107
the woody tissue of whitebark pine. 108
As is with many tree pathogens, it is difficult to observe the physical presence of the 109
pathogen itself, and instead field data is collected on the symptoms exhibited by host individuals. 110
The field data used here is from five separate white pine blister rust field campaigns during the 111
period 1996-2008: the Interagency Whitebark Pine Monitoring Group (IWPMG), the National 112
Park Service (NPS), the U.S. Geological Survey (USGS) Northern Rocky Mountain Science 113
Center, the University of Montana (UM) School of Forestry, and the Yellowstone Ecological 114
Research Center (YERC). The sampling protocols varied by study, and are summarized in Table 115
1. In all cases, plots are located within 20 meters to collect data at a plot or transect that was a 116
truly representative whitebark pine stand for the surrounding area, and thus the combined dataset 117
proportional to both the local and global infected tree density. 95
96
METHODS 97
Data 98
White pine blister rust has a complex life history with five spore stages and two obligate 99
alternate hosts (Arthur 1934), outlined in Figure 1. White pine blister rust does not spread 100
directly from one whitebark pine tree to another, but indirectly through its alternate host, various 101
species of currants and gooseberries (members of the genus Ribes). Explicitly monitoring and 102
modeling the intra-annual spread of white pine blister rust between its alternating hosts within is 103
difficult since there are over seventeen species of Ribes within the GYE that would require intra-104
annual white pine blister rust surveys. Within the context of our study, we therefore focus the 105
inter-annual progress of white pine blister rust infection. As a result, we do not model Ribes 106
populations or infection levels explicitly, since white pine blister rust can only overwinter within 107
the woody tissue of whitebark pine. 108
As is with many tree pathogens, it is difficult to observe the physical presence of the 109
pathogen itself, and instead field data is collected on the symptoms exhibited by host individuals. 110
The field data used here is from five separate white pine blister rust field campaigns during the 111
period 1996-2008: the Interagency Whitebark Pine Monitoring Group (IWPMG), the National 112
Park Service (NPS), the U.S. Geological Survey (USGS) Northern Rocky Mountain Science 113
Center, the University of Montana (UM) School of Forestry, and the Yellowstone Ecological 114
Research Center (YERC). The sampling protocols varied by study, and are summarized in Table 115
1. In all cases, plots are located within 20 meters to collect data at a plot or transect that was a 116
truly representative whitebark pine stand for the surrounding area, and thus the combined dataset 117
Page 6
6
could overall be biased towards sampling trees at the center of stands. Because many of the plots 118
were located on within the national park boundary, plot locations could not be permanently 119
marked, and trees could not be tagged, precluding the use of simpler demographic models that 120
track the fate of individual trees. Accordingly, plot re-censuses were therefore treated as repeated 121
samples from a local population rather than re-measurement in the strict sense. 122
Tree evaluation protocols varied between study groups, but all protocols recorded the 123
presence or absence of white pine blister rust symptoms including bole cankers, fruiting bodies 124
and rodent chewing (indicating consumption of a canker) at the tree level. The YERC and USGS 125
protocols evaluated trees into four classes of white pine blister rust infection that reflect exactly 126
the four classes used in the modeling. The UM protocol recorded a more detailed 1-18 infection 127
score for each tree (Six and Newcomb 2005), and these infection classes were translated into the 128
same four classes as the YERC and USGS protocol using a decision tree that first separated trees 129
uninfected (uninfected class) from those infected, and then separated scores of less than 9 130
(slightly infected) from those with scores between 9-15 (moderately infected) and from those 131
with scores between 16-18 (dead). The IWPMG protocol evaluated trees based on the number of 132
cankers in each third of the total tree branch area and also within in each third of the total tree 133
bole. Therefore, these data were also translated into four classes by a decision tree that first 134
separated uninfected trees (uninfected class) from infected trees, and next separated trees with 135
cankers in one-third or less of the tree (slightly infected) from trees with cankers in two-thirds or 136
greater of the bole or branches (moderately infected). Dead trees were given a separate mortality 137
code within this protocol, and dead trees with white pine blister rust symptoms were placed into 138
the dead class. The National Park Service dataset recorded trees with no infection, light 139
infection, moderate infection, and fatal infection, and thus fell naturally within four classes of 140
could overall be biased towards sampling trees at the center of stands. Because many of the plots 118
were located on within the national park boundary, plot locations could not be permanently 119
marked, and trees could not be tagged, precluding the use of simpler demographic models that 120
track the fate of individual trees. Accordingly, plot re-censuses were therefore treated as repeated 121
samples from a local population rather than re-measurement in the strict sense. 122
Tree evaluation protocols varied between study groups, but all protocols recorded the 123
presence or absence of white pine blister rust symptoms including bole cankers, fruiting bodies 124
and rodent chewing (indicating consumption of a canker) at the tree level. The YERC and USGS 125
protocols evaluated trees into four classes of white pine blister rust infection that reflect exactly 126
the four classes used in the modeling. The UM protocol recorded a more detailed 1-18 infection 127
score for each tree (Six and Newcomb 2005), and these infection classes were translated into the 128
same four classes as the YERC and USGS protocol using a decision tree that first separated trees 129
uninfected (uninfected class) from those infected, and then separated scores of less than 9 130
(slightly infected) from those with scores between 9-15 (moderately infected) and from those 131
with scores between 16-18 (dead). The IWPMG protocol evaluated trees based on the number of 132
cankers in each third of the total tree branch area and also within in each third of the total tree 133
bole. Therefore, these data were also translated into four classes by a decision tree that first 134
separated uninfected trees (uninfected class) from infected trees, and next separated trees with 135
cankers in one-third or less of the tree (slightly infected) from trees with cankers in two-thirds or 136
greater of the bole or branches (moderately infected). Dead trees were given a separate mortality 137
code within this protocol, and dead trees with white pine blister rust symptoms were placed into 138
the dead class. The National Park Service dataset recorded trees with no infection, light 139
infection, moderate infection, and fatal infection, and thus fell naturally within four classes of 140
Page 7
7
blister rust infection (National Park Service 1968-71). Coercing the UM and IWPMG data into 141
these four classes by decision trees yields a large simplified dataset that is ecologically 142
informative for population-level study, and from a computation standpoint is more efficient to 143
model. Bayesian modeling accommodates the differences in sampling methods between datasets, 144
since observation error is modeled as a posterior distribution of values, rather than as a single- 145
value parameter. 146
Model formulation 147
Our model parameterizes the transition rates between four classes of blister rust infection 148
outlined at each site k as: . The hierarchical Bayesian formulation (sensu 149
Clark 2007) consists of three sub-models: the data model, the process model, and the parameter 150
model (Figure 3). The data model estimates the proportion of trees in each blister rust infection 151
class at each site and year based on the field data, which describes the number of trees within 152
each white pine blister rust infection class at each field census. In doing so, it accounts for the 153
sampling error, and the different levels of confidence about infection rates that result from 154
different sampling schemes. Observation errors associated with tree misclassification are 155
assumed to be sufficiently small, and not to differ systematically among datasets. The field data 156
at each site k in each year where data exists y is modeled as a multinomial distribution: 157
158
and 159
160
where the four elements of the vectors N and P respectively represent the raw tree count data and 161
infection rates in each of the four blister rust infection classes. The multinomial distribution is a 162
generalization of the binomial that describes the probability of observing counts given N 163
blister rust infection (National Park Service 1968-71). Coercing the UM and IWPMG data into 141
these four classes by decision trees yields a large simplified dataset that is ecologically 142
informative for population-level study, and from a computation standpoint is more efficient to 143
model. Bayesian modeling accommodates the differences in sampling methods between datasets, 144
since observation error is modeled as a posterior distribution of values, rather than as a single- 145
value parameter. 146
Model formulation 147
Our model parameterizes the transition rates between four classes of blister rust infection 148
outlined at each site k as: . The hierarchical Bayesian formulation (sensu 149
Clark 2007) consists of three sub-models: the data model, the process model, and the parameter 150
model (Figure 3). The data model estimates the proportion of trees in each blister rust infection 151
class at each site and year based on the field data, which describes the number of trees within 152
each white pine blister rust infection class at each field census. In doing so, it accounts for the 153
sampling error, and the different levels of confidence about infection rates that result from 154
different sampling schemes. Observation errors associated with tree misclassification are 155
assumed to be sufficiently small, and not to differ systematically among datasets. The field data 156
at each site k in each year where data exists y is modeled as a multinomial distribution: 157
158
and 159
160
where the four elements of the vectors N and P respectively represent the raw tree count data and 161
infection rates in each of the four blister rust infection classes. The multinomial distribution is a 162
generalization of the binomial that describes the probability of observing counts given N 163
Page 8
8
independent draws. 164
The process model describes the temporal dynamics of blister rust infection (P) at each 165
site k at an annual timestep t. Disease progression is modeled by the transitions between the four 166
classes at a yearly timestep using a simple matrix model: 167
168
which indicates that the proportions within the infection classes Pt,k of each site k in each year t 169
are determined by the proportions in each infection class in the previous year multiplied by the 170
transition matrix : 171
172
where the elements of A represent the transition rates between the four blister rust infection 173
classes. As described bellow, the elements of matrix A are comprised of parameters both across 174
all sites and specific to individual sites, and thus A varies by site. In all initial runs of our model, 175
the value of , the probability of remaining in the dead class, converged to one, so we 176
constrained this value to be one for the rest of the analysis, since removal from the dead class 177
through decay occurs on a much slower timescale than the scope of our current model. Similarly, 178
we assume that recruitment is also negligible on the time scale of the model, and thus the total 179
population size remains constant. The above considerations allow us to simplify the model, so 180
that the column sums of equal one and we only need model the off-diagonal matrix elements: 181
182
where the elements of are the transition rates described above. The transition rates are not 183
assumed to be constant, but vary over time t and space k: 184
185
independent draws. 164
The process model describes the temporal dynamics of blister rust infection (P) at each 165
site k at an annual timestep t. Disease progression is modeled by the transitions between the four 166
classes at a yearly timestep using a simple matrix model: 167
168
which indicates that the proportions within the infection classes Pt,k of each site k in each year t 169
are determined by the proportions in each infection class in the previous year multiplied by the 170
transition matrix : 171
172
where the elements of A represent the transition rates between the four blister rust infection 173
classes. As described bellow, the elements of matrix A are comprised of parameters both across 174
all sites and specific to individual sites, and thus A varies by site. In all initial runs of our model, 175
the value of , the probability of remaining in the dead class, converged to one, so we 176
constrained this value to be one for the rest of the analysis, since removal from the dead class 177
through decay occurs on a much slower timescale than the scope of our current model. Similarly, 178
we assume that recruitment is also negligible on the time scale of the model, and thus the total 179
population size remains constant. The above considerations allow us to simplify the model, so 180
that the column sums of equal one and we only need model the off-diagonal matrix elements: 181
182
where the elements of are the transition rates described above. The transition rates are not 183
assumed to be constant, but vary over time t and space k: 184
185
Page 9
9
where the vector describes the global mean transition rate across all sites between each of the 186
four blister rust infection classes: 187
188
where the subscripts on correspond to the transition rates . The elements of are assumed to 189
have multivariate normal priors with mean a0 and variance Va. The process variance on is 190
modeled as normal with variance: 191
192
193
where again, the subscripts match those of the transition rates , and the parameter is fitted to the 194
data across all sites. Each of the elements of is assumed to have inverse gamma priors with 195
parameters s0 and s1. Thus, the parameter can be interpreted as the process of transition 196
between each of the four classes across all sites, and can be interpreted as the random 197
variation on that process. The parameter describes the site level variation in the rate of disease 198
progression: 199
200
where the subscripts match the transition rates . is modeled as a site-specific influence on the 201
transition rate chosen within our model from the normal distribution with mean zero and 202
variance . 203
With these three parameters influencing the transition rates, the state transitions for each 204
class of blister rust infection at each site k in each year y+1 are given as: 205
206
where the vector describes the global mean transition rate across all sites between each of the 186
four blister rust infection classes: 187
188
where the subscripts on correspond to the transition rates . The elements of are assumed to 189
have multivariate normal priors with mean a0 and variance Va. The process variance on is 190
modeled as normal with variance: 191
192
193
where again, the subscripts match those of the transition rates , and the parameter is fitted to the 194
data across all sites. Each of the elements of is assumed to have inverse gamma priors with 195
parameters s0 and s1. Thus, the parameter can be interpreted as the process of transition 196
between each of the four classes across all sites, and can be interpreted as the random 197
variation on that process. The parameter describes the site level variation in the rate of disease 198
progression: 199
200
where the subscripts match the transition rates . is modeled as a site-specific influence on the 201
transition rate chosen within our model from the normal distribution with mean zero and 202
variance . 203
With these three parameters influencing the transition rates, the state transitions for each 204
class of blister rust infection at each site k in each year y+1 are given as: 205
206
Page 10
10
As before, the proportions in each infection class at the current timestep multiplied by the 207
transition rates, comprised of parameters , and implicitly , determine the proportions in 208
each infection class at the next timestep. The three sources of process variability , and 209
are outlined schematically as Figure 4a, and the improved performance of the model with the 210
inclusion of all three sources of process variability is plotted as Figure 4b. Note that while the 211
process is computed at an annual timestep, the parameters are assumed to be time invariant. 212
Since the population is not being observed every year, the posterior estimates of Pk,t, 213
which are a latent, unobserved quantity in the model, are constrained by the population state at 214
both the previous and following timesteps, Pk,t-1 and Pk,t+1, and the process model: 215
216
Model formulation is simplified by the fact that each plot has only been recensused once, and 217
thus that first and last census were used as the starting and ending points of modeling each plot. 218
This introduces two additional cases, for the first and last model steps: 219
220
221
For the initial time step there is also a prior on the initial model state, modeled by the Dirichlet 222
distribution with shape parameters Dir(0.9,0.9,0.9,0.9). 223
Metapopulation formulation 224
The model above describes the simple case of local population dynamics at each site with 225
a static infection rate. To evaluate the dynamics of blister rust at the local versus global spatial 226
scales, we create four dynamic models: two in which there is only local infection, and two meta-227
population models, in which the infection occurs both locally and globally. The differences 228
between the models are in the terms describing the transition between the susceptible and slightly 229
As before, the proportions in each infection class at the current timestep multiplied by the 207
transition rates, comprised of parameters , and implicitly , determine the proportions in 208
each infection class at the next timestep. The three sources of process variability , and 209
are outlined schematically as Figure 4a, and the improved performance of the model with the 210
inclusion of all three sources of process variability is plotted as Figure 4b. Note that while the 211
process is computed at an annual timestep, the parameters are assumed to be time invariant. 212
Since the population is not being observed every year, the posterior estimates of Pk,t, 213
which are a latent, unobserved quantity in the model, are constrained by the population state at 214
both the previous and following timesteps, Pk,t-1 and Pk,t+1, and the process model: 215
216
Model formulation is simplified by the fact that each plot has only been recensused once, and 217
thus that first and last census were used as the starting and ending points of modeling each plot. 218
This introduces two additional cases, for the first and last model steps: 219
220
221
For the initial time step there is also a prior on the initial model state, modeled by the Dirichlet 222
distribution with shape parameters Dir(0.9,0.9,0.9,0.9). 223
Metapopulation formulation 224
The model above describes the simple case of local population dynamics at each site with 225
a static infection rate. To evaluate the dynamics of blister rust at the local versus global spatial 226
scales, we create four dynamic models: two in which there is only local infection, and two meta-227
population models, in which the infection occurs both locally and globally. The differences 228
between the models are in the terms describing the transition between the susceptible and slightly 229
Page 11
11
infected class: 230
231
232
where indexing is as before except for the models with global infection, which introduce the new 233
parameter , a global rate of infection. In the model with dynamic global infection, this 234
parameter is proportional to the global mean proportion of whitebark pine trees in the infective 235
classes (blister rust spore-producing classes) at year y, , which is approximated as 236
the sample mean of Iy+My integrating over their posterior uncertainty for all sites. In the models 237
with dynamic local infection, we make the transition between the susceptible and infected classes 238
proportional to the proportion within the infective classes at that site k, . Note that 239
with our parameter formulation, the combination of static local and dynamic global infection is 240
impossible, since when local infection does not vary with the proportion of locally infected trees, 241
the parameters , global mean transitions, and , global infection rate, become redundant and the 242
model collapses to the simpler model with static local infection and no global infection. 243
244
infected class: 230
231
232
where indexing is as before except for the models with global infection, which introduce the new 233
parameter , a global rate of infection. In the model with dynamic global infection, this 234
parameter is proportional to the global mean proportion of whitebark pine trees in the infective 235
classes (blister rust spore-producing classes) at year y, , which is approximated as 236
the sample mean of Iy+My integrating over their posterior uncertainty for all sites. In the models 237
with dynamic local infection, we make the transition between the susceptible and infected classes 238
proportional to the proportion within the infective classes at that site k, . Note that 239
with our parameter formulation, the combination of static local and dynamic global infection is 240
impossible, since when local infection does not vary with the proportion of locally infected trees, 241
the parameters , global mean transitions, and , global infection rate, become redundant and the 242
model collapses to the simpler model with static local infection and no global infection. 243
244
Page 12
12
Model fitting 245
The models were implemented through a Markov chain Monte Carlo (MCMC) scheme 246
with 100,000 iterations per model to assure convergence of all parameters. Convergence of the 247
parameters within the four models was determined by visually inspecting the parameter values 248
plotted against the MCMC iterations to ensure both parameter mixing over the MCMC routine as 249
well as parameter convergence to a stable mean by the end of the 100,000 iterations. 250
The fit of the four dynamic models is compared using the predictive loss criterion, which 251
evaluates model performance by minimizing the predictive loss of the posterior distribution of 252
the parameters (Gelfand and Ghosh 1998, Clark 2007). We chose the predictive loss criterion 253
over other metrics of Bayesian model selection because its emphasis on the performance of 254
prediction capabilities matched with our goal of predicting the future progress of blister rust at 255
different sites throughout the GYE. 256
The predictive loss value Dm for each model m, is calculated as the sum of two terms: Gm 257
+ Pm. Gm is the error sum of squares, which is the cost for selecting the wrong model: 258
259
Pm is the penalty term, which is the predictive variance: 260
261
The model with the lowest value of Dm is considered the best fit (Clark 2007). The units of Dm 262
are absolute values since both Gm and Pm scale with sample size, but are only compared between 263
models fitted to the same datasets. 264
RESULTS 265
Each of the four dynamic blister rust infection models was parameterized with the dataset 266
Model fitting 245
The models were implemented through a Markov chain Monte Carlo (MCMC) scheme 246
with 100,000 iterations per model to assure convergence of all parameters. Convergence of the 247
parameters within the four models was determined by visually inspecting the parameter values 248
plotted against the MCMC iterations to ensure both parameter mixing over the MCMC routine as 249
well as parameter convergence to a stable mean by the end of the 100,000 iterations. 250
The fit of the four dynamic models is compared using the predictive loss criterion, which 251
evaluates model performance by minimizing the predictive loss of the posterior distribution of 252
the parameters (Gelfand and Ghosh 1998, Clark 2007). We chose the predictive loss criterion 253
over other metrics of Bayesian model selection because its emphasis on the performance of 254
prediction capabilities matched with our goal of predicting the future progress of blister rust at 255
different sites throughout the GYE. 256
The predictive loss value Dm for each model m, is calculated as the sum of two terms: Gm 257
+ Pm. Gm is the error sum of squares, which is the cost for selecting the wrong model: 258
259
Pm is the penalty term, which is the predictive variance: 260
261
The model with the lowest value of Dm is considered the best fit (Clark 2007). The units of Dm 262
are absolute values since both Gm and Pm scale with sample size, but are only compared between 263
models fitted to the same datasets. 264
RESULTS 265
Each of the four dynamic blister rust infection models was parameterized with the dataset 266
Page 13
13
comprising 121 blister rust infection sites that span 1968-2008 through a Markov chain Monte 267
Carlo routine with 100,000 iterations to ensure parameter convergence. With the predictive loss 268
criterion, a lower score indicates a better-fit model, and the scores for the four models are 269
included as Table 2. 270
The predictive loss scores indicate that the three models that include a dynamic local 271
infection term substantially outperform the model with static infection. Model 4 is best fit by the 272
predictive loss criterion, although the predictive loss score of Model 2 is close enough to that of 273
Model 4 to also merit interpretation. 274
The model outputs for each site can be compared to further evaluate the relative 275
performance of the four models. Six representative site-level outputs are included in Figure 5. As 276
Figure 5 illustrates, the relative performance of the four models varies from site-to-site; however, 277
as Figure 5 illustrates the differences between six different sites. As can be seen in Figure 5 Site 278
A, which included data from 1968 and 2008, bridging the largest time span in the dataset, models 279
2 and 4 clearly outperform models 1 and 3, following the conclusions of the predictive loss 280
criterion. As can be seen in Figure 5 Site B, a site with the shortest time span in the dataset from 281
2004 to 2007, the models are generally indistinguishable. This could result from the short four-282
year time span of this site’s data, which may not allow for the more complex dynamics of density 283
dependence in models 2, 3 and 4 to make much of an impact on either the parameterization or 284
the actual ecological dynamics, as infection levels did not change much during this short time 285
period, indicated by the large overlap of the data’s 95% confidence intervals between 2004 and 286
2007 for this site. Figure 5 Sites C and D illustrate that at sites encompassing a moderate time 287
scale of 13-14 years in which the blister rust infection levels were either low or non-existent in 288
the first census, Models 2 and 4, in some cases also Model 3, captured this infection dynamic 289
comprising 121 blister rust infection sites that span 1968-2008 through a Markov chain Monte 267
Carlo routine with 100,000 iterations to ensure parameter convergence. With the predictive loss 268
criterion, a lower score indicates a better-fit model, and the scores for the four models are 269
included as Table 2. 270
The predictive loss scores indicate that the three models that include a dynamic local 271
infection term substantially outperform the model with static infection. Model 4 is best fit by the 272
predictive loss criterion, although the predictive loss score of Model 2 is close enough to that of 273
Model 4 to also merit interpretation. 274
The model outputs for each site can be compared to further evaluate the relative 275
performance of the four models. Six representative site-level outputs are included in Figure 5. As 276
Figure 5 illustrates, the relative performance of the four models varies from site-to-site; however, 277
as Figure 5 illustrates the differences between six different sites. As can be seen in Figure 5 Site 278
A, which included data from 1968 and 2008, bridging the largest time span in the dataset, models 279
2 and 4 clearly outperform models 1 and 3, following the conclusions of the predictive loss 280
criterion. As can be seen in Figure 5 Site B, a site with the shortest time span in the dataset from 281
2004 to 2007, the models are generally indistinguishable. This could result from the short four-282
year time span of this site’s data, which may not allow for the more complex dynamics of density 283
dependence in models 2, 3 and 4 to make much of an impact on either the parameterization or 284
the actual ecological dynamics, as infection levels did not change much during this short time 285
period, indicated by the large overlap of the data’s 95% confidence intervals between 2004 and 286
2007 for this site. Figure 5 Sites C and D illustrate that at sites encompassing a moderate time 287
scale of 13-14 years in which the blister rust infection levels were either low or non-existent in 288
the first census, Models 2 and 4, in some cases also Model 3, captured this infection dynamic 289
Page 14
14
particularly well. It is evident from these plots that Model 1, which does not include effect of 290
infection density, was unable to accurately represent these sites with low or nonexistent levels of 291
initial infection. These results also support the general conclusion of the predictive loss model 292
evaluation, where Models 2 and 4 performed best, followed by Model 3, wherein Model 1 scored 293
particularly low at capturing the blister rust dynamics. The model output in Figure 5 Sites E and 294
F demonstrate that in sites with moderate time spans and a dramatic rise in the rate of blister rust 295
infection during the time period, the models generally perform equally at capturing the dynamic. 296
Figure 6 shows the averaged dynamic of infection across all sites for the time period 297
1968-2008. Consistent with the predictive loss scores (Table 2), Models 2 and 4 strongly 298
outperform Model 1. Model 3 also generally performs well, but with a higher tendency to 299
overpredict the proportion of uninfected individuals and underpredict the proportions of infected 300
individuals in 1988, compared to Models 1 and 2. The jump in the level of infection in the mid-301
nineties reflects the portion of our model that is data-rich, since most datasets fall within the 302
period 1995-2008. 303
Model Predictions 304
Models 2, 3 and 4 all indicate a rapidly accelerating transition rate between the 305
uninfected and slight infection classes, especially within the past ten years. This implies that 306
blister rust infestation will continue to accelerate along this trajectory into the future. Figure 7 307
plots the future predicted levels of blister rust infestation over the next ten years for Models 2, 3 308
and 4 (Model 1 was excluded in this analysis due to its high predictive loss score). In the 309
globally averaged future predictions in Figure 7, the three models again perform similarly, and 310
only differ slightly in the rate of the transition between the uninfected and slightly uninfected 311
class. Model 4 exhibits the fastest decline to an average of 90% infection rate at all sites, by the 312
particularly well. It is evident from these plots that Model 1, which does not include effect of 290
infection density, was unable to accurately represent these sites with low or nonexistent levels of 291
initial infection. These results also support the general conclusion of the predictive loss model 292
evaluation, where Models 2 and 4 performed best, followed by Model 3, wherein Model 1 scored 293
particularly low at capturing the blister rust dynamics. The model output in Figure 5 Sites E and 294
F demonstrate that in sites with moderate time spans and a dramatic rise in the rate of blister rust 295
infection during the time period, the models generally perform equally at capturing the dynamic. 296
Figure 6 shows the averaged dynamic of infection across all sites for the time period 297
1968-2008. Consistent with the predictive loss scores (Table 2), Models 2 and 4 strongly 298
outperform Model 1. Model 3 also generally performs well, but with a higher tendency to 299
overpredict the proportion of uninfected individuals and underpredict the proportions of infected 300
individuals in 1988, compared to Models 1 and 2. The jump in the level of infection in the mid-301
nineties reflects the portion of our model that is data-rich, since most datasets fall within the 302
period 1995-2008. 303
Model Predictions 304
Models 2, 3 and 4 all indicate a rapidly accelerating transition rate between the 305
uninfected and slight infection classes, especially within the past ten years. This implies that 306
blister rust infestation will continue to accelerate along this trajectory into the future. Figure 7 307
plots the future predicted levels of blister rust infestation over the next ten years for Models 2, 3 308
and 4 (Model 1 was excluded in this analysis due to its high predictive loss score). In the 309
globally averaged future predictions in Figure 7, the three models again perform similarly, and 310
only differ slightly in the rate of the transition between the uninfected and slightly uninfected 311
class. Model 4 exhibits the fastest decline to an average of 90% infection rate at all sites, by the 312
Page 15
15
year 2013. This fast decline compared with the other two models is due to the density 313
dependence in the rate of global infection within Model 4, which does not exist in the other two 314
models. Model 2 predicts a 90% global infection level in 2026 and Model 3 predicts the same for 315
2033. These two models exhibit a slower decline to 90% global infection at all sites due to either 316
the absent global infection in Model 2 and the constant rate of global infection in Model 3. 317
In our model formulations, we included a site-specific random effect, the parameter, 318
in order to capture environmental heterogeneity between sites that in the blister rust infection 319
process. The inclusion of this the parameter is significant for model performance (see Figure 320
4). Within our model formulation, the site effects parameter is modeled as a random effect 321
for each site, which we then compared to possible sources of environmental heterogeneity that 322
could explain differences in the rate of blister rust infection at the site level. While the analysis 323
indicates that model performance is clearly enhanced by the inclusion of the parameter, we 324
were not able to identify an obvious cause: a preliminary analysis showed that differences in 325
parameter between sites were not related to slope, aspect or elevation. This lack of significant 326
correlation to site variables was also exhibited in blister rust infected whitebark pine trees in 327
British Columbia (Campbell and Antos 2000). 328
The computed model parameters , , and g (when applicable) for all four models are 329
included as Table 3. The values for the hierarchical variance on the process of transition 330
between infection stages converged to nearly the same values for Models 2, 3 and 4, 331
demonstrating that the amount of process error is well constrained between all models. 332
Ecologically, this indicates that between the three dynamic infection scenario models, the 333
amount of error in our understanding of the infection transition process in our model is relatively 334
small and constant between the three models. The values for the hierarchical variance on the 335
year 2013. This fast decline compared with the other two models is due to the density 313
dependence in the rate of global infection within Model 4, which does not exist in the other two 314
models. Model 2 predicts a 90% global infection level in 2026 and Model 3 predicts the same for 315
2033. These two models exhibit a slower decline to 90% global infection at all sites due to either 316
the absent global infection in Model 2 and the constant rate of global infection in Model 3. 317
In our model formulations, we included a site-specific random effect, the parameter, 318
in order to capture environmental heterogeneity between sites that in the blister rust infection 319
process. The inclusion of this the parameter is significant for model performance (see Figure 320
4). Within our model formulation, the site effects parameter is modeled as a random effect 321
for each site, which we then compared to possible sources of environmental heterogeneity that 322
could explain differences in the rate of blister rust infection at the site level. While the analysis 323
indicates that model performance is clearly enhanced by the inclusion of the parameter, we 324
were not able to identify an obvious cause: a preliminary analysis showed that differences in 325
parameter between sites were not related to slope, aspect or elevation. This lack of significant 326
correlation to site variables was also exhibited in blister rust infected whitebark pine trees in 327
British Columbia (Campbell and Antos 2000). 328
The computed model parameters , , and g (when applicable) for all four models are 329
included as Table 3. The values for the hierarchical variance on the process of transition 330
between infection stages converged to nearly the same values for Models 2, 3 and 4, 331
demonstrating that the amount of process error is well constrained between all models. 332
Ecologically, this indicates that between the three dynamic infection scenario models, the 333
amount of error in our understanding of the infection transition process in our model is relatively 334
small and constant between the three models. The values for the hierarchical variance on the 335
Page 16
16
site effects varied between all models, indicating that the relative contribution of the site 336
effects parameter varies from model to model. The variance is highest for Model 4, the best fit 337
model. Since the site effects parameter is selected from a random normal distribution, a 338
higher value of variance indicates that the model parameterization expresses higher site-level 339
environmental variability between sites. Because Model 4 was the best fit, this indicates that 340
variability between sites is relatively more important than process error variance in 341
determining the overall transition rates. 342
343
DISCUSSION 344
By comparing the performance of the four dynamic model formulations, we can compare 345
different hypotheses about the dynamics of blister rust infection in whitebark pine at the 346
ecosystem scale. The model with static local and absent global infection tests whether there is 347
and a constant per-capita rate of blister rust infection, i.e. that the rate of blister rust infection 348
within each stand is independent of the proportion of infected individuals at either the local or 349
global scale. Support for this model would indicate that white pine blister rust infection occurs 350
through a static background rate of spores proportional only to the proportion of susceptible trees 351
at each site. 352
The model with dynamic local infection evaluates whether per capita blister rust infection 353
increases with the proportion of individuals that are infected within a site. Although blister rust 354
infection cannot spread from tree-to-tree, this model formulation tests whether stands are located 355
close enough to the alternate host Ribes in order to complete the annual life cycle and infect 356
other sites within the same site at a rate dependent on the proportion infected trees at that site. 357
The model with dynamic local and static global infection evaluates whether in addition to per 358
site effects varied between all models, indicating that the relative contribution of the site 336
effects parameter varies from model to model. The variance is highest for Model 4, the best fit 337
model. Since the site effects parameter is selected from a random normal distribution, a 338
higher value of variance indicates that the model parameterization expresses higher site-level 339
environmental variability between sites. Because Model 4 was the best fit, this indicates that 340
variability between sites is relatively more important than process error variance in 341
determining the overall transition rates. 342
343
DISCUSSION 344
By comparing the performance of the four dynamic model formulations, we can compare 345
different hypotheses about the dynamics of blister rust infection in whitebark pine at the 346
ecosystem scale. The model with static local and absent global infection tests whether there is 347
and a constant per-capita rate of blister rust infection, i.e. that the rate of blister rust infection 348
within each stand is independent of the proportion of infected individuals at either the local or 349
global scale. Support for this model would indicate that white pine blister rust infection occurs 350
through a static background rate of spores proportional only to the proportion of susceptible trees 351
at each site. 352
The model with dynamic local infection evaluates whether per capita blister rust infection 353
increases with the proportion of individuals that are infected within a site. Although blister rust 354
infection cannot spread from tree-to-tree, this model formulation tests whether stands are located 355
close enough to the alternate host Ribes in order to complete the annual life cycle and infect 356
other sites within the same site at a rate dependent on the proportion infected trees at that site. 357
The model with dynamic local and static global infection evaluates whether in addition to per 358
Page 17
17
capita infection rates being influenced by the proportion of locally infected trees there is also a 359
constant rate of global spore influx. Finally, the dynamic local and dynamic global model 360
evaluates whether the background rate of global spore influx is proportional to the proportion 361
infected trees averaged across all sites. 362
The performance of the models that included local density dependence in infection rate 363
(Models 2, 3 and 4) compared to the density independent model (Model 1), indicate that blister 364
rust infestation in whitebark pine spreads through mechanisms that are dependent on the amount 365
of local infestation already at that site, despite the fact that blister rust does not spread directly 366
from tree to tree. The importance of the local density dependence term indicates that most likely 367
pathway for the spread of blister rust spores is local dispersion from local Ribes to local 368
whitebark pines within a single growing season. Our results reached no clear conclusion 369
however, regarding the existence of regional scale density-dependence in blister rust infection 370
rates. 371
The fitted parameter values for the transition rates between the four stages of blister rust 372
infection can be transformed into mean residence times for each infection stage. Using the 373
parameters from Model 4, the best fit model, the mean residence times across all sites (parameter 374
α), indicate that whitebark pine trees take on average 6.7 years to transition from uninfected to 375
infected, 10.9 years to transition from infected to slightly infected, and 9.4 years to transition 376
from infected to dead. This is the first known study of the rate of blister rust progression in the 377
high altitude whitebark pine species, and serves as an informative temporal parameter for forest 378
managers within the GYE. The residence times of the slightly infected and moderately infected 379
stages indicate that on average in our dataset, it trees live for approximately 20 years following 380
infection, a longer infection period than other lower elevation pine species infected by blister rust 381
capita infection rates being influenced by the proportion of locally infected trees there is also a 359
constant rate of global spore influx. Finally, the dynamic local and dynamic global model 360
evaluates whether the background rate of global spore influx is proportional to the proportion 361
infected trees averaged across all sites. 362
The performance of the models that included local density dependence in infection rate 363
(Models 2, 3 and 4) compared to the density independent model (Model 1), indicate that blister 364
rust infestation in whitebark pine spreads through mechanisms that are dependent on the amount 365
of local infestation already at that site, despite the fact that blister rust does not spread directly 366
from tree to tree. The importance of the local density dependence term indicates that most likely 367
pathway for the spread of blister rust spores is local dispersion from local Ribes to local 368
whitebark pines within a single growing season. Our results reached no clear conclusion 369
however, regarding the existence of regional scale density-dependence in blister rust infection 370
rates. 371
The fitted parameter values for the transition rates between the four stages of blister rust 372
infection can be transformed into mean residence times for each infection stage. Using the 373
parameters from Model 4, the best fit model, the mean residence times across all sites (parameter 374
α), indicate that whitebark pine trees take on average 6.7 years to transition from uninfected to 375
infected, 10.9 years to transition from infected to slightly infected, and 9.4 years to transition 376
from infected to dead. This is the first known study of the rate of blister rust progression in the 377
high altitude whitebark pine species, and serves as an informative temporal parameter for forest 378
managers within the GYE. The residence times of the slightly infected and moderately infected 379
stages indicate that on average in our dataset, it trees live for approximately 20 years following 380
infection, a longer infection period than other lower elevation pine species infected by blister rust 381
Page 18
18
such as sugar pine and western white pine (Smith and Hoffman 2000). 382
The comparison of Models 2, 3 and 4 would be aided by additional data, as well as data 383
that tracks not only proportions of the infected population but also the rate of infection within 384
individual trees. The modeling framework we establish in this study would benefit not only from 385
the inclusion of additional field data, but could also be combined with remote sensing analyses of 386
tree infection (Hatala et al. 2009). This combined dataset from the Greater Yellowstone 387
Ecosystem could be combined with blister rust datasets from other ecosystems to help deduce 388
spatial environmental drivers of blister rust spread. 389
Our current meta-population model predicts well at individual sites, and we are currently 390
limited to extending our model to one where space is treated explicitly. Limiting our model to be 391
spatially implicit presents some challenges to the interpretation of blister rust dynamics in the 392
GYE, particularly when considering the spread of blister rust into other uninfected areas. We are 393
constrained in our approach due to the lack of a spatial pattern or correlation with environmental 394
variables within the initial dataset from which to project a continuous surface of initial conditions 395
from which to run a spatially explicit model. Additionally, our spatially implicit meta-population 396
model revealed no correlation between the site effects parameter with any possible source of 397
environmental heterogeneity. In this analysis we treat space implicitly due to uncertainty in the 398
initial conditions of blister rust distribution throughout the landscape at the start of our model, 399
patchy infestation at a small scale and lack of spatial autocorrelation in the infection data. The 400
inclusion of additional data types would aid in developing a spatially explicit model of disease 401
spread. In particular, obtaining data across the entire landscape from remote sensing imagery in 402
order to determine the spatial extent and spatial patchiness in disease incidence would provide a 403
key initially conditions for a spatially explicit model. In addition, remotely sensed imagery 404
such as sugar pine and western white pine (Smith and Hoffman 2000). 382
The comparison of Models 2, 3 and 4 would be aided by additional data, as well as data 383
that tracks not only proportions of the infected population but also the rate of infection within 384
individual trees. The modeling framework we establish in this study would benefit not only from 385
the inclusion of additional field data, but could also be combined with remote sensing analyses of 386
tree infection (Hatala et al. 2009). This combined dataset from the Greater Yellowstone 387
Ecosystem could be combined with blister rust datasets from other ecosystems to help deduce 388
spatial environmental drivers of blister rust spread. 389
Our current meta-population model predicts well at individual sites, and we are currently 390
limited to extending our model to one where space is treated explicitly. Limiting our model to be 391
spatially implicit presents some challenges to the interpretation of blister rust dynamics in the 392
GYE, particularly when considering the spread of blister rust into other uninfected areas. We are 393
constrained in our approach due to the lack of a spatial pattern or correlation with environmental 394
variables within the initial dataset from which to project a continuous surface of initial conditions 395
from which to run a spatially explicit model. Additionally, our spatially implicit meta-population 396
model revealed no correlation between the site effects parameter with any possible source of 397
environmental heterogeneity. In this analysis we treat space implicitly due to uncertainty in the 398
initial conditions of blister rust distribution throughout the landscape at the start of our model, 399
patchy infestation at a small scale and lack of spatial autocorrelation in the infection data. The 400
inclusion of additional data types would aid in developing a spatially explicit model of disease 401
spread. In particular, obtaining data across the entire landscape from remote sensing imagery in 402
order to determine the spatial extent and spatial patchiness in disease incidence would provide a 403
key initially conditions for a spatially explicit model. In addition, remotely sensed imagery 404
Page 19
19
would also provide a greater opportunity to identify the environmental factors responsible for the 405
observed site-level variation in the blister rust infection process identified in this analysis. 406
Another important source of information would the inclusion of data at finer scales (tagging 407
individuals), in particular to document the spatially spread of the disease into areas that are 408
currently uninfected. 409
The predictions of Model 4, the best fit model, indicate a 90% average infection rate 410
across all sites by the year 2013, and the other two models indicate a 90% infection level by the 411
years 2026 and 2033. This indicates that blister rust will continue to spread within whitebark 412
pine in 10-20 years to a level where nearly all trees at the sites will be impacted. The rapid 413
progression of blister rust spread in whitebark pine of the GYE described in this analysis should 414
spark interest in community-level ecological studies, which could examine the species-level 415
dynamics that might ensue with the decline of whitebark pine. This time frame might also be 416
used to explore the possibility of planting blister rust resistant whitebark pines (Hoff 1980) in 417
areas of the GYE. However, due to the slow growth of whitebark pine, this measure may not be 418
enough to combat the immediate effects of blister rust. 419
420
CONCLUSION 421
The formulation of our spatially implicit meta-population model predicts the future levels 422
of blister rust infection in whitebark pine at sites throughout the GYE by parameterizing a 423
dataset that spans 1968-2008. By conducting an analysis that compares four possible blister rust 424
dynamic infection models, we conclude that blister rust operates through mechanisms that are 425
density dependent to the amounts of both local and global infected trees. The results from our 426
model are used to calculate the residence times for each of the infective stages of blister rust, as 427
would also provide a greater opportunity to identify the environmental factors responsible for the 405
observed site-level variation in the blister rust infection process identified in this analysis. 406
Another important source of information would the inclusion of data at finer scales (tagging 407
individuals), in particular to document the spatially spread of the disease into areas that are 408
currently uninfected. 409
The predictions of Model 4, the best fit model, indicate a 90% average infection rate 410
across all sites by the year 2013, and the other two models indicate a 90% infection level by the 411
years 2026 and 2033. This indicates that blister rust will continue to spread within whitebark 412
pine in 10-20 years to a level where nearly all trees at the sites will be impacted. The rapid 413
progression of blister rust spread in whitebark pine of the GYE described in this analysis should 414
spark interest in community-level ecological studies, which could examine the species-level 415
dynamics that might ensue with the decline of whitebark pine. This time frame might also be 416
used to explore the possibility of planting blister rust resistant whitebark pines (Hoff 1980) in 417
areas of the GYE. However, due to the slow growth of whitebark pine, this measure may not be 418
enough to combat the immediate effects of blister rust. 419
420
CONCLUSION 421
The formulation of our spatially implicit meta-population model predicts the future levels 422
of blister rust infection in whitebark pine at sites throughout the GYE by parameterizing a 423
dataset that spans 1968-2008. By conducting an analysis that compares four possible blister rust 424
dynamic infection models, we conclude that blister rust operates through mechanisms that are 425
density dependent to the amounts of both local and global infected trees. The results from our 426
model are used to calculate the residence times for each of the infective stages of blister rust, as 427
Page 20
20
well as the time to a global average of 90% infection across all sites. Although we were not able 428
to find any environmental explanations for variation in blister rust infection at the site level, our 429
analysis reinforces the fact that blister rust has pervaded most of the GYE regardless of possible 430
environmental barriers present at our 121 sites. The lack of correlation between heterogeneity in 431
blister rust infection rates and environmental heterogeneity might indicate that the disease has 432
pervaded all possible whitebark pine habitats in the GYE. The site-level variation in blister rust 433
progression in our analysis could be correlated to biotic environmental variables not accounted 434
for in our analysis, for example, genetic disease defenses. 435
The approach for our model could readily be applied to analyze infection dynamics in 436
other systems, where infected populations are relatively stationary (plants, amphibians, etc.) 437
relative to the infection agent. Our approach is unique in that it accounts for site-level and 438
ecosystem-wide infection parameters that are driving the infection of blister rust at specific sites. 439
Results of this study can be utilized by forest managers to track that rate of disease spread within 440
sites, as well as globally throughout the ecosystem. It might be used to identify sites with slow 441
rates of disease progression, which might indicate some genetic resistance within certain 442
populations. Additionally, the results might help to inform reforestation efforts in areas that 443
might be environmentally unsuitable for blister rust. The basic formulation of this model could 444
be applied to other multi-stage plant diseases, where managers have an interest in monitoring 445
disease progression at individual sites as well as large-scale ecosystem-wide levels of the 446
disease. 447
ACKNOWLEDGEMENTS 448
The NASA Biodiversity and Ecological Forecasting program supported this research 449
under award NNA07CN19A to Robert L. Crabtree. We also thank Gil Bohrer for helpful 450
well as the time to a global average of 90% infection across all sites. Although we were not able 428
to find any environmental explanations for variation in blister rust infection at the site level, our 429
analysis reinforces the fact that blister rust has pervaded most of the GYE regardless of possible 430
environmental barriers present at our 121 sites. The lack of correlation between heterogeneity in 431
blister rust infection rates and environmental heterogeneity might indicate that the disease has 432
pervaded all possible whitebark pine habitats in the GYE. The site-level variation in blister rust 433
progression in our analysis could be correlated to biotic environmental variables not accounted 434
for in our analysis, for example, genetic disease defenses. 435
The approach for our model could readily be applied to analyze infection dynamics in 436
other systems, where infected populations are relatively stationary (plants, amphibians, etc.) 437
relative to the infection agent. Our approach is unique in that it accounts for site-level and 438
ecosystem-wide infection parameters that are driving the infection of blister rust at specific sites. 439
Results of this study can be utilized by forest managers to track that rate of disease spread within 440
sites, as well as globally throughout the ecosystem. It might be used to identify sites with slow 441
rates of disease progression, which might indicate some genetic resistance within certain 442
populations. Additionally, the results might help to inform reforestation efforts in areas that 443
might be environmentally unsuitable for blister rust. The basic formulation of this model could 444
be applied to other multi-stage plant diseases, where managers have an interest in monitoring 445
disease progression at individual sites as well as large-scale ecosystem-wide levels of the 446
disease. 447
ACKNOWLEDGEMENTS 448
The NASA Biodiversity and Ecological Forecasting program supported this research 449
under award NNA07CN19A to Robert L. Crabtree. We also thank Gil Bohrer for helpful 450
Page 21
21
comments and suggestions in the modeling design, and the Yellowstone Ecological Research 451
Center for assistance with data aggregation. 452
453
LITERATURE CITED 454
Arthur, J. C. 1934. Manual of the rusts in United States and Canada. Purdue Research 455
Foundation, Layfayette, IN. 456
457
Benedict, W. V. 1966. Experience with antibiotics to control white pine blister rust. Journal of 458
Forestry 64(6): 382-388. 459
460
Bingham, R. T. 1972. Taxonomy, crossability, and relative blister rust resistance of 5-needled 461
white pines. in Biology of Rust Resistance in Forest Trees. Ed. R. T. Bingham, R. J. 462
Hoff, and G. I. McDonald, p. 271-278. 463
464
Brown, J. K. M. and M. S. Hovmoller. 2002. Epidemiology: Aerial dispersal of pathogens on the 465
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Campbell, E. and J. Antos. 2000. Distribution and severity of white pine blister rust and 468
mountain pine beetle on whitebark pine in British Columbia. Canadian Journal of Forest 469
Research 30:1051-1059. 470
471
Castello, J. D., D. J. Leopold, and S. P. J. 1995. Pathogens, patterns, and processes in forest 472
ecosystems. Bioscience 45:16-24. 473
comments and suggestions in the modeling design, and the Yellowstone Ecological Research 451
Center for assistance with data aggregation. 452
453
LITERATURE CITED 454
Arthur, J. C. 1934. Manual of the rusts in United States and Canada. Purdue Research 455
Foundation, Layfayette, IN. 456
457
Benedict, W. V. 1966. Experience with antibiotics to control white pine blister rust. Journal of 458
Forestry 64(6): 382-388. 459
460
Bingham, R. T. 1972. Taxonomy, crossability, and relative blister rust resistance of 5-needled 461
white pines. in Biology of Rust Resistance in Forest Trees. Ed. R. T. Bingham, R. J. 462
Hoff, and G. I. McDonald, p. 271-278. 463
464
Brown, J. K. M. and M. S. Hovmoller. 2002. Epidemiology: Aerial dispersal of pathogens on the 465
global and continental scales and its impact on plant disease. Science 297:537-541. 466
467
Campbell, E. and J. Antos. 2000. Distribution and severity of white pine blister rust and 468
mountain pine beetle on whitebark pine in British Columbia. Canadian Journal of Forest 469
Research 30:1051-1059. 470
471
Castello, J. D., D. J. Leopold, and S. P. J. 1995. Pathogens, patterns, and processes in forest 472
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V. Sobczak, K. A. Stinson, J. K. Stone, C. M. Swan, J. Thompson, B. Von Holle, and J. 481
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dynamics of forested ecosystems. Frontiers in Ecology and the Environment 3:479-486. 483
484
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ecosystems.in Proceedings of a symposium on whitebark pine ecosystems, Ed. W. C. 486
Schmidt and K. J. McDonald. General Technical Report INT-270, USDA Forest Service 487
Intermountain Research Center. 488
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Frelich, L. E. and P. B. Reich. 1999. Neighborhood effects, disturbance severity, and community 494
stability in forests. Ecosystems 2:151-166. 495
496
474
Clinton, G. P. and F. A. McCormick. 1919. Infection experiments of Pinus strobus with 475
Cronartium ribicola. Connecticut Agricultural Experiemental Station Bulletin 214:428-476
459. 477
478
Ellison, A. M., M. S. Bank, B. D. Clinton, E. A. Colburn, K. Elliott, C. R. Ford, D. R. Foster, B. 479
D. Kloeppel, J. D. Knoepp, G. M. Lovett, J. Mohan, D. A. Orwig, N. L. Rodenhouse, W. 480
V. Sobczak, K. A. Stinson, J. K. Stone, C. M. Swan, J. Thompson, B. Von Holle, and J. 481
R. Webster. 2005. Loss of foundation species: consequences for the structure and 482
dynamics of forested ecosystems. Frontiers in Ecology and the Environment 3:479-486. 483
484
Farnes, P. E. 1990. SNOTEL and snow course data describing the hydrology of whitebark pine 485
ecosystems.in Proceedings of a symposium on whitebark pine ecosystems, Ed. W. C. 486
Schmidt and K. J. McDonald. General Technical Report INT-270, USDA Forest Service 487
Intermountain Research Center. 488
489
ForestPests.org. 18 May 2006. The University of Georgia Warnell School of Forestry and 490
Natural Resources and College of Agricultural and Environmental Sciences. 15 Jan 2009. 491
http://www.forestpests.org/images/768x512/0355052.jpg. 492
493
Frelich, L. E. and P. B. Reich. 1999. Neighborhood effects, disturbance severity, and community 494
stability in forests. Ecosystems 2:151-166. 495
496
Page 23
23
Gelfand, A. E. and S. K. Ghosh. 1998. Model choice: a minimum posterior predictive loss 497
approach. Biometrika 85(1): 1-11. 498
499
Greater Yellowstone Interagency Whitebark Pine Monitoring Group (GYIWPMG). 2007. 500
Interagency whitebark pine monitoring protocol for the Greater Yellowstone Ecosystem. 501
Greater Yellowstone Whitebark Pine Monitoring Working Group. 502
503
Hagle, S. K., G. I. McDonald, and E. A. Norby. 1989. White pine blister rust in Northern Idaho 504
and Western Montana: alternatives for integrated management. General Technical Report 505
INT-261, USDA Forest Service Intermountain Research Station. 506
507
Halligan, K. Q., R. L. Crabtree, and M. O. Jones. 2003. Hyperspectral data analysis of whitebark 508
pine (Pinus albicaulis) and white pine blister rust (Cronartium ribicola) at four sites in the 509
Greater Yellowstone Ecosystem. Yellowstone Ecological Research Center, 55pp. 510
511
Hepting, G. H. and G. M. Jemison. 1955. Forest protection against destructive agencies. Timber 512
Resource Review 4:1-69. 513
514
Hoff, R., R. T. Bingham, and G. I. McDonald. 1980. Relative blister rust resistance of white 515
pines. European Journal of Forest Pathology 10:307-316. 516
517
Holdenrieder, O., M. Pautasso, W. P. J., and L. D. 2004. Tree diseases and landscape processes: 518
the challenge of landscape pathology. Trends in Ecology and Evolution 19:446-452. 519
Gelfand, A. E. and S. K. Ghosh. 1998. Model choice: a minimum posterior predictive loss 497
approach. Biometrika 85(1): 1-11. 498
499
Greater Yellowstone Interagency Whitebark Pine Monitoring Group (GYIWPMG). 2007. 500
Interagency whitebark pine monitoring protocol for the Greater Yellowstone Ecosystem. 501
Greater Yellowstone Whitebark Pine Monitoring Working Group. 502
503
Hagle, S. K., G. I. McDonald, and E. A. Norby. 1989. White pine blister rust in Northern Idaho 504
and Western Montana: alternatives for integrated management. General Technical Report 505
INT-261, USDA Forest Service Intermountain Research Station. 506
507
Halligan, K. Q., R. L. Crabtree, and M. O. Jones. 2003. Hyperspectral data analysis of whitebark 508
pine (Pinus albicaulis) and white pine blister rust (Cronartium ribicola) at four sites in the 509
Greater Yellowstone Ecosystem. Yellowstone Ecological Research Center, 55pp. 510
511
Hepting, G. H. and G. M. Jemison. 1955. Forest protection against destructive agencies. Timber 512
Resource Review 4:1-69. 513
514
Hoff, R., R. T. Bingham, and G. I. McDonald. 1980. Relative blister rust resistance of white 515
pines. European Journal of Forest Pathology 10:307-316. 516
517
Holdenrieder, O., M. Pautasso, W. P. J., and L. D. 2004. Tree diseases and landscape processes: 518
the challenge of landscape pathology. Trends in Ecology and Evolution 19:446-452. 519
Page 24
24
520
Kendall, K. C. 1995. Whitebark pine sampling instructions for Ecodata and Survey Plots. U.S. 521
Geologic Survey. 522
523
Kendall, K. C. and S. F. Arno. 1990. Whitebark pine: an important but endangered wildlife 524
resource. in Proceedings of a symposium on whitebark pine ecosystems. Ed. W. Schmidt 525
and K. McDonald. General Technical Report INT-270, USDA Forest Service 526
Intermountain Research Center. 527
528
Logan, J. A., J. Regniere, and J. A. Powell. 2003. Assessing the impacts of global warming on 529
forest pest dynamics. Frontiers in Ecology and Environment 1(3): 130-137. 530
531
Maloy, O. C. 1997. White pine blister rust control in North America: A case history. Annual 532
Review of Pytopathology 35:87-109. 533
534
Mattson, D. J., B. M. Blanchard, and R. R. Knight. 1992. Yellowstone grizzly bear mortality, 535
human habituation, and whitebark pine seed crops. Journal of Wildlife Management 536
56:432-442. 537
538
Moorcroft, P. R., S. W. Pacala, and M. A. Lewis. 2006. Potential role of natural enemies during 539
tree range expansions following climate change. Journal of Theoretical Biology 241:601-540
616. 541
542
520
Kendall, K. C. 1995. Whitebark pine sampling instructions for Ecodata and Survey Plots. U.S. 521
Geologic Survey. 522
523
Kendall, K. C. and S. F. Arno. 1990. Whitebark pine: an important but endangered wildlife 524
resource. in Proceedings of a symposium on whitebark pine ecosystems. Ed. W. Schmidt 525
and K. McDonald. General Technical Report INT-270, USDA Forest Service 526
Intermountain Research Center. 527
528
Logan, J. A., J. Regniere, and J. A. Powell. 2003. Assessing the impacts of global warming on 529
forest pest dynamics. Frontiers in Ecology and Environment 1(3): 130-137. 530
531
Maloy, O. C. 1997. White pine blister rust control in North America: A case history. Annual 532
Review of Pytopathology 35:87-109. 533
534
Mattson, D. J., B. M. Blanchard, and R. R. Knight. 1992. Yellowstone grizzly bear mortality, 535
human habituation, and whitebark pine seed crops. Journal of Wildlife Management 536
56:432-442. 537
538
Moorcroft, P. R., S. W. Pacala, and M. A. Lewis. 2006. Potential role of natural enemies during 539
tree range expansions following climate change. Journal of Theoretical Biology 241:601-540
616. 541
542
Page 25
25
Newcomb, M. 2003. White pine blister rust, whitebark pine, and Ribes species in the Greater 543
Yellowstone Area. Ph.D. Dissertation, University of Montana. 544
545
O'Neill, R. V., R. H. Gardner, T. M. G., and R. W. H. 1992. Epidemiology theory and 546
disturbance spread on landscapes. Landscape Ecology 7:19-26. 547
548
Park, A. W., S. Gubbins, and G. C. A. 2001. Invasion and persistence of plant parasites in a 549
spatially structured host population. Oikos 94:162-174. 550
551
Park, A. W., S. Gubbins, and G. C. A. 2002. Extinction times for closed epidemics: the effects of 552
host spatial structure. Ecology Letters 5:747-755. 553
554
Reinhart, D. P., M. A. Haroldson, D. J. Mattson, and K. A. Gunther. 2001. Effects of exotic 555
species on Yellowstone's grizzly bears. Western North American Naturalist 61:277-288. 556
557
Six, D. L. and M. Newcomb. 2005. A rapid system for rating white pine blister rust incidence, 558
severity, and within-tree distribution in whitebark pine. Northwest Science 79:189-195. 559
560
Smith, J. P. and J. T. Hoffman. 2000. Status of white pine blister rust in the Intermountain West. 561
Western North American Naturalist 60:165-179. 562
563
Snell, W. H. 1942. The production of sporidia of Cronartium ribicola on cultivated red currants 564
in relation to infection of white pine. American Journal of Botany 29:506-513. 565
Newcomb, M. 2003. White pine blister rust, whitebark pine, and Ribes species in the Greater 543
Yellowstone Area. Ph.D. Dissertation, University of Montana. 544
545
O'Neill, R. V., R. H. Gardner, T. M. G., and R. W. H. 1992. Epidemiology theory and 546
disturbance spread on landscapes. Landscape Ecology 7:19-26. 547
548
Park, A. W., S. Gubbins, and G. C. A. 2001. Invasion and persistence of plant parasites in a 549
spatially structured host population. Oikos 94:162-174. 550
551
Park, A. W., S. Gubbins, and G. C. A. 2002. Extinction times for closed epidemics: the effects of 552
host spatial structure. Ecology Letters 5:747-755. 553
554
Reinhart, D. P., M. A. Haroldson, D. J. Mattson, and K. A. Gunther. 2001. Effects of exotic 555
species on Yellowstone's grizzly bears. Western North American Naturalist 61:277-288. 556
557
Six, D. L. and M. Newcomb. 2005. A rapid system for rating white pine blister rust incidence, 558
severity, and within-tree distribution in whitebark pine. Northwest Science 79:189-195. 559
560
Smith, J. P. and J. T. Hoffman. 2000. Status of white pine blister rust in the Intermountain West. 561
Western North American Naturalist 60:165-179. 562
563
Snell, W. H. 1942. The production of sporidia of Cronartium ribicola on cultivated red currants 564
in relation to infection of white pine. American Journal of Botany 29:506-513. 565
Page 26
26
566
Spaulding, P. C. 1916. Foresters have a vital interest in the white pine blister rust. Proceedings of 567
the Society for American Forestry 11:40-47. 568
569
Spaulding, P. C. 1922. Investigations of the white pine blister rust, USDA Bulletin 957. 570
571
Toko, E. V., D. A. Graham, C. E. Carlson, and D. E. Ketcham. 1967. Effects of past Ribes 572
eradications on controlling white pine blister rust in Northern Idaho. Phytopathology 573
57:1010. 574
575
VanArsdel, E. P., A. J. Riker, and R. F. Patton. 1956. The effects of temperature and moisture on 576
the spread of white pine blister rust. Phytopathology 46:307-318. 577
578
Yellowstone National Park. 1968-71. Blister rust incidence survey, Reference FSH 5277 R1. 579
580
566
Spaulding, P. C. 1916. Foresters have a vital interest in the white pine blister rust. Proceedings of 567
the Society for American Forestry 11:40-47. 568
569
Spaulding, P. C. 1922. Investigations of the white pine blister rust, USDA Bulletin 957. 570
571
Toko, E. V., D. A. Graham, C. E. Carlson, and D. E. Ketcham. 1967. Effects of past Ribes 572
eradications on controlling white pine blister rust in Northern Idaho. Phytopathology 573
57:1010. 574
575
VanArsdel, E. P., A. J. Riker, and R. F. Patton. 1956. The effects of temperature and moisture on 576
the spread of white pine blister rust. Phytopathology 46:307-318. 577
578
Yellowstone National Park. 1968-71. Blister rust incidence survey, Reference FSH 5277 R1. 579
580
Page 27
27
TABLE LEGENDS 581
Table 1. This table outlines the field protocol for the datasets that were assimilated into the 582
metapopulation study. 583
584
Table 2. The predictive loss model scores for the four models, used for model selection. 585
A lower score indicates a better-fit model. 586
587
Table 3. This model shows the transition parameters computed for the four models 588
through the MCMC routine. 589
590
TABLE LEGENDS 581
Table 1. This table outlines the field protocol for the datasets that were assimilated into the 582
metapopulation study. 583
584
Table 2. The predictive loss model scores for the four models, used for model selection. 585
A lower score indicates a better-fit model. 586
587
Table 3. This model shows the transition parameters computed for the four models 588
through the MCMC routine. 589
590
Page 28
28
TABLES 591
Table 1: 592
Dataset
collector
Years Number
of sites
Survey
method
Site establishment method Citation
IWPMGa 2004, 2007 22 10m by
50m
transects
Randomly located within the
GYE grizzly bear Primary
Conservation Area, within
whitebark pine stands greater
than 2.5 hectares
(IWPMG
2007)
NPSb 1968/1970,
2008*
19 Variable
length and
width 28
tree plots
Placed within Yellowstone
National Park in areas of
probable blister rust
infestation
(Yellowstone
National
Park, 1968-
71)
USGSc 1995/1996,
2008*
136 300ft long
variable-
width
transects
and 0.1
acre plots
Dispersed throughout the
GYE to encompass the
widest variety of whitebark
pine habitat possible
(Kendall
1995)
UMTd 2001/2002,
2007*
20 10m wide
variable-
length
transects
Randomly located within
known whitebark pine
habitat
(Newcomb
2003, Six and
Newcomb
2005)
YERCe 2000, 2007 8 Ten 5.2m
radius
plots at
each site
Sites selected to fall within
hyperspectral imagery
flightline
(Halligan et
al. 2003)
a Interagency Whitebark Pine Monitoring Working Group, Bozeman, MT 593
b National Park Service, Yellowstone National Park, Mammoth, WY 594
c U.S. Geological Survey, Northern Rocky Mountain Science Center, Bozeman, MT 595
d University of Montana, School of Forestry, Missoula, MT 596
e Yellowstone Ecological Research Center, Bozeman, MT 597
* plots were recensused in 2007/2008 by the Yellowstone Ecological Research Center, Bozeman, MT, and the data 598
is previously unpublished 599
600
601
602
603
604
605
TABLES 591
Table 1: 592
Dataset
collector
Years Number
of sites
Survey
method
Site establishment method Citation
IWPMGa 2004, 2007 22 10m by
50m
transects
Randomly located within the
GYE grizzly bear Primary
Conservation Area, within
whitebark pine stands greater
than 2.5 hectares
(IWPMG
2007)
NPSb 1968/1970,
2008*
19 Variable
length and
width 28
tree plots
Placed within Yellowstone
National Park in areas of
probable blister rust
infestation
(Yellowstone
National
Park, 1968-
71)
USGSc 1995/1996,
2008*
136 300ft long
variable-
width
transects
and 0.1
acre plots
Dispersed throughout the
GYE to encompass the
widest variety of whitebark
pine habitat possible
(Kendall
1995)
UMTd 2001/2002,
2007*
20 10m wide
variable-
length
transects
Randomly located within
known whitebark pine
habitat
(Newcomb
2003, Six and
Newcomb
2005)
YERCe 2000, 2007 8 Ten 5.2m
radius
plots at
each site
Sites selected to fall within
hyperspectral imagery
flightline
(Halligan et
al. 2003)
a Interagency Whitebark Pine Monitoring Working Group, Bozeman, MT 593
b National Park Service, Yellowstone National Park, Mammoth, WY 594
c U.S. Geological Survey, Northern Rocky Mountain Science Center, Bozeman, MT 595
d University of Montana, School of Forestry, Missoula, MT 596
e Yellowstone Ecological Research Center, Bozeman, MT 597
* plots were recensused in 2007/2008 by the Yellowstone Ecological Research Center, Bozeman, MT, and the data 598
is previously unpublished 599
600
601
602
603
604
605
Page 29
29
Table 2: 606
Model Predictive Loss Score
1: Static local infection, no
global infection
48800.54
2: Dynamic local infection, no
global infection
29834.29
3: Dynamic local infection,
static global infection
33497.91
4: Dynamic local infection,
dynamic global infection
28463.08
607
Table 3: 608
Model alpha_12 alpha_23 alpha_34 sigma_1 sigma_2 sigma_3 sigma_4 tau_12 tau_23 tau_34 g
1.lowerCI 0.0315 0.0718 0.0842 0.0431 0.0419 0.0310 0.0374 0.8345 0.3965 0.2175 N/A
1.median 0.0288 0.0792 0.1042 0.0627 0.0538 0.0376 0.0438 1.1760 0.5252 0.3952 N/A
1.upperCI 0.0237 0.0923 0.1123 0.0743 0.0597 0.0418 0.0497 1.4870 0.6459 0.5218 N/A
2.lowerCI 0.1187 0.0679 0.0786 0.0172 0.0229 0.0282 0.0249 1.0890 0.6687 0.4349 N/A
2.median 0.1314 0.0835 0.1043 0.0182 0.0244 0.0301 0.0264 1.3120 0.8452 0.6208 N/A
2.upperCI 0.1526 0.0983 0.1202 0.0194 0.0263 0.0343 0.0288 1.6210 1.0230 0.8852 N/A
3.lowerCI 0.0785 0.0721 0.1186 0.0172 0.0219 0.0277 0.0254 1.2930 0.4786 0.3734 1.34E-003
3.median 0.0978 0.0824 0.1051 0.0181 0.0235 0.0301 0.0279 1.5760 0.6330 0.4928 1.59E-003
3.upperCI 0.1372 0.0938 0.0841 0.0192 0.0252 0.0338 0.0302 2.0580 0.8872 0.6892 1.77E-003
4.lowerCI 0.1247 0.0832 0.0897 0.0175 0.0228 0.0287 0.0242 0.9472 0.7302 0.6587 1.99E-005
4.median 0.1493 0.0916 0.1068 0.0185 0.0243 0.0317 0.0264 1.1690 0.9182 0.8027 4.76E-005
4.upperCI 0.1812 0.1043 0.1352 0.0203 0.0279 0.0362 0.0299 1.4010 1.1103 1.1000 6.09E-005
609
610
Table 2: 606
Model Predictive Loss Score
1: Static local infection, no
global infection
48800.54
2: Dynamic local infection, no
global infection
29834.29
3: Dynamic local infection,
static global infection
33497.91
4: Dynamic local infection,
dynamic global infection
28463.08
607
Table 3: 608
Model alpha_12 alpha_23 alpha_34 sigma_1 sigma_2 sigma_3 sigma_4 tau_12 tau_23 tau_34 g
1.lowerCI 0.0315 0.0718 0.0842 0.0431 0.0419 0.0310 0.0374 0.8345 0.3965 0.2175 N/A
1.median 0.0288 0.0792 0.1042 0.0627 0.0538 0.0376 0.0438 1.1760 0.5252 0.3952 N/A
1.upperCI 0.0237 0.0923 0.1123 0.0743 0.0597 0.0418 0.0497 1.4870 0.6459 0.5218 N/A
2.lowerCI 0.1187 0.0679 0.0786 0.0172 0.0229 0.0282 0.0249 1.0890 0.6687 0.4349 N/A
2.median 0.1314 0.0835 0.1043 0.0182 0.0244 0.0301 0.0264 1.3120 0.8452 0.6208 N/A
2.upperCI 0.1526 0.0983 0.1202 0.0194 0.0263 0.0343 0.0288 1.6210 1.0230 0.8852 N/A
3.lowerCI 0.0785 0.0721 0.1186 0.0172 0.0219 0.0277 0.0254 1.2930 0.4786 0.3734 1.34E-003
3.median 0.0978 0.0824 0.1051 0.0181 0.0235 0.0301 0.0279 1.5760 0.6330 0.4928 1.59E-003
3.upperCI 0.1372 0.0938 0.0841 0.0192 0.0252 0.0338 0.0302 2.0580 0.8872 0.6892 1.77E-003
4.lowerCI 0.1247 0.0832 0.0897 0.0175 0.0228 0.0287 0.0242 0.9472 0.7302 0.6587 1.99E-005
4.median 0.1493 0.0916 0.1068 0.0185 0.0243 0.0317 0.0264 1.1690 0.9182 0.8027 4.76E-005
4.upperCI 0.1812 0.1043 0.1352 0.0203 0.0279 0.0362 0.0299 1.4010 1.1103 1.1000 6.09E-005
609
610
Page 30
30
FIGURE LEGENDS 611
Figure 1. Blister rust spreads through two alternate hosts, members of the genus Ribes and 612
whitebark pine. The fungus can only over-winter in whitebark pine, so the spread from whitebark 613
pine to Ribes to another whitebark pine must occur within a single growing season. Note that the 614
fungus cannot spread from pine to pine, but must pass through an intermediate Ribes species. 615
616
Figure 2. For our study, data is aggregated from four different sources across the Greater 617
Yellowstone Ecosystem from the years 1968-2008. 618
619
Figure 3. The outline of the three hierarchical levels of model parameters that operate through 620
time and space within our model. 621
622
Figure 4. This figure demonstrates the effects of the global transition rates , the variance on 623
the global transition rates , and the site effects within the model. The inclusion of all three 624
parameters clearly improves overall model performance. 625
626
Figure 5. These panels show the output of the model at six different field plots. The outputs at 627
these different field sites demonstrate that overall, model performance from the dynamic local 628
and global density dependence performs best throughout the time period 1968-2008. 629
630
Figure 6. The averaged ecosystem-wide level of blister rust infestation for the time period 1968-631
2008, averaged from the Markov-chain Monte-carlo output for all sites. 632
633
FIGURE LEGENDS 611
Figure 1. Blister rust spreads through two alternate hosts, members of the genus Ribes and 612
whitebark pine. The fungus can only over-winter in whitebark pine, so the spread from whitebark 613
pine to Ribes to another whitebark pine must occur within a single growing season. Note that the 614
fungus cannot spread from pine to pine, but must pass through an intermediate Ribes species. 615
616
Figure 2. For our study, data is aggregated from four different sources across the Greater 617
Yellowstone Ecosystem from the years 1968-2008. 618
619
Figure 3. The outline of the three hierarchical levels of model parameters that operate through 620
time and space within our model. 621
622
Figure 4. This figure demonstrates the effects of the global transition rates , the variance on 623
the global transition rates , and the site effects within the model. The inclusion of all three 624
parameters clearly improves overall model performance. 625
626
Figure 5. These panels show the output of the model at six different field plots. The outputs at 627
these different field sites demonstrate that overall, model performance from the dynamic local 628
and global density dependence performs best throughout the time period 1968-2008. 629
630
Figure 6. The averaged ecosystem-wide level of blister rust infestation for the time period 1968-631
2008, averaged from the Markov-chain Monte-carlo output for all sites. 632
633
Page 31
31
Figure 7. The globally average projections for the future level of blister rust in the Greater 634
Yellowstone Ecosystem differ in the rates of increase for each dynamic model formulation. 635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
Figure 7. The globally average projections for the future level of blister rust in the Greater 634
Yellowstone Ecosystem differ in the rates of increase for each dynamic model formulation. 635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
Page 32
32
FIGURE 1 657
658
659
660
661
FIGURE 1 657
658
659
660
661
Page 33
33
FIGURE 2 662
663
FIGURE 2 662
663
Page 34
34
FIGURE 3 664
665
666
667
668
669
670
671
672
673
674
FIGURE 3 664
665
666
667
668
669
670
671
672
673
674
Page 35
35
FIGURE 4 675
676
677
678
FIGURE 4 675
676
677
678
Page 36
36
FIGURE 5.AB 679
680
FIGURE 5.AB 679
680
Page 37
37
FIGURE 5.CD 681
682
FIGURE 5.CD 681
682
Page 38
38
Figure 5.EF 683
684
Figure 5.EF 683
684
Page 39
39
FIGURE 6 685
686
687
688
689
FIGURE 6 685
686
687
688
689
Page 40
40
FIGURE 7 690
691
FIGURE 7 690
691
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