122 VOLUME 114 | SUPPLEMENT 1 | April 2006 • Environmental Health Perspectives
Monograph
There is a considerable body of observational
and experimental evidence of individual-level
effects produced by exposure to endocrine
disruptors in the laboratory (e.g., Panter et al.
1998, Jobling et al. 1998) and the field
(Jobling et al. 2002). However, observational
studies of population-level effects and model-
ing studies designed to predict or evaluate
such effects are still relatively rare.
Most modeling studies in the current litera-
ture have been carried out with a view to pre-
dicting population-level effects in a specific
natural system from information on individual-
level effects in the key species concerned. For
example, Brown et al. (2003) studied the likely
impact of exposure of fathead minnow and
brook trout to nonylphenol and methoxychlor.
Brown et al. (Brown AR, Riddle AM, Winfeld
IJ, Fletcher JM, James JB, unpublished data)
have also used the same approach to evaluate
the effects of nonylphenol and ethinylestradiol
on perch populations in Lake Windermere
(Northern England).
Their specific biological focus ensures that
models such as these contain intimidating
quantities of biological detail. In both studies
referred to above, a detailed demographic
model of a nonexposed population was para-
meterized using field data. Laboratory observa-
tions of the changes in growth and fecundity
induced by pollutant exposure were then used
to infer new values of key demographic para-
meters for exposed populations.
The target of quantitative risk prediction
generally predisposes investigators to primarily
computational methods of investigation, even
in cases where analytic results might be obtain-
able. Individual-based methods are also popular
in such studies (e.g., Van Winkle et al. 1993),
and these individual-based methods are almost
always entirely inaccessible to analytic methods.
In this article our purpose is rather different;
therefore, we adopt a very different modeling
philosophy. Our principal aim is to demon-
strate that relatively simple models can provide
a useful framework within which to consider
the expression of altered individual characteris-
tics at the population level. The objective of
such consideration is to identify those individ-
ual changes that are likely to be critical in a
population context, so that experimental effort
can be appropriately focused.
Because the investigations we seek to facil-
itate will almost certainly be carried out ahead
of the experimental studies they seek to
inform, we expect that a premium will be
placed on conclusions that are generic rather
than being sensitively dependent on specific
parameter combinations. In this context, the
elementary mathematical methods can be
used to gain a much more comprehensive
body of evidence from the simple models we
shall advocate than that which can be
obtained from their more complex cousins.
As an example of these methods, we con-
sider a simple delay-differential population
modeling framework, and we examine three
related process-based demographic models that
encompass both male and female individuals.
From this study we are able to distinguish
unambiguously those circumstances in which
changes in male reproductive performance will
have significant population-level effects from
those circumstances in which they will not.
In a similar spirit, we examine a simple
energetic modeling framework within which
changes in physiological rates can be related to
likely alterations in individual demographic
performance. We argue that a productive
model for the investigation we advocate will
almost certainly involve a combination of sim-
ple individual models and simple but demo-
graphically accurate population representations.
A Population Modeling
Framework
The irreducible minimum of biological detail
that our model must encompass is the follow-
ing: a) adults come in two sexes, one of which
produces eggs while the other fertilizes them;
b) a fertilized egg takes a finite time ( Ψ) to
develop into a reproductively functional adult;
c) a proportion, S
J
, of eggs will survive this
development process: and d) a proportion, Ω,
of these survivors become males, whereas the
rest become females. The minimal description
of the state of a population with these proper-
ties is the abundance of adult males and
females, which we denote by A
m
and A
f
,
respectively.
One of the more approachable for-
malisms to describe the dynamics of such a
population is that employed by Brown et al.
(2003) in modeling the demographic effects
of endocrine disruptors in populations of
trout, perch, and fathead minnow. Our vari-
ant of their description is formulated as a pair
of equations that specify the rate of change of
A
m
and A
f
. If we denote the rate of produc-
tion of fertilized eggs at time t by E(t), and
the per-capita mortality rate of male and
female adults by µ
m
and µ
f
, respectively, then
the rates of change of A
m
and A
f
are
[1]
and
[2]
dA
dt
SEt A
f
Jf
= ()() .1 Ω Ψ∝
dA
dt
SEt A
m
Jm
= Ω Ψ∝()
This article is part of the monograph “The
Ecological Relevance of Chemically Induced
Endocrine Disruption in Wildlife.”
Address correspondence to W.S.C. Gurney,
Department of Statistics and Modeling Science,
University of Strathclyde, Glasgow, G1 1XT, UK.
Telephone: 44 1415 483 385. Fax: 44 1415 522 079.
E-mail: bill@stams.strath.ac.uk
The authors declare they have no competing
financial interests.
Received 31 January 2005; accepted 29 June 2005.
Modeling the Demographic Effects of Endocrine Disruptors
William S.C. Gurney
Department of Statistics and Modelling Science, University of Strathclyde, Glasgow, Scotland, United Kingdom, and Fisheries Research
Services, Marine Laboratory, Aberdeen, Scotland, United Kingdom
In this article we describe a series of strategic models of populations and individuals subject to
challenge by endocrine disruptors. These models are not designed to be fitted to detailed data on
specific species but rather are intended to provide general insights on the relative importance of dif-
ferent demographic mechanisms in the population context. Therefore, the models contain the min-
imum necessary biological detail, but in recompense they are highly accessible to mathematical
analysis. We show that, over a range of models with contrasting biological detail, population viabil-
ity is controlled by the number of female offspring that result from the average female’s lifetime
reproductive activity. Thus, male fertility changes have little effect at the population level until they
become severe enough to reduce this average female output. We argue that in many circumstances
endocrine disruptors are likely to produce directly deleterious effects on female fecundity at levels
far below those required to reduce male fertility to dangerously low levels. Finally, we formulate a
simple model of individual energetics that we argue can form the basis of a strategic discussion of
the likely sensitivity of female demographic parameters to chemically induced changes in physiolog-
ical function. Key words: demographic model, endocrine disruption, fish demography, fish energetics.
Environ Health Perspect 114(suppl 1):122–126 (2006). doi:10.1289/ehp.8064 available via
http://dx.doi.org/ [Online 21 October 2005]

Model 1: A Well-Mixed
Population with No Density
Dependence
In this first application of the modeling
framework defined by Equations 1 and 2, we
make the simplifying assumption that males
and females have the same per capita mortal-
ity, which we denote by µ (µ
m
= µ
f
= µ). To
complete our population model, we must
specify the fertilized egg production rate E. If
adult females produce eggs at a per-capita rate
Γ, and the fraction of eggs fertilized at time t
is F(t), then
E(t) = ΓA
f
(t)F(t). [3]
Sperm are less expensive than eggs, so a single
male is likely to be capable of fertilizing the
reproductive output of many females. If there
are no males, then no eggs will be fertilized,
while many males will fertilize all released eggs.
To model the transition between these limiting
cases, we regard the population as well-mixed,
so the fraction of eggs fertilized depends on the
ratio of males to females, which we denote by
[4]
A convenient function relating R and F
changes smoothly from 0 to 1 as R increases,
and implies that 50% of eggs are fertilized
when R = R
h
; thus
[5]
We note that this formulation implies that
essentially all eggs will be fertilized when the
male-to-female ratio is large compared with
the characteristic value R
h.
Also, the propor-
tion fertilized will drop almost linearly with R
for values of R that are small compared with
R
h
. Suppose, for example, that R
h
= 0.01. In
this case, half the eggs will be fertilized if R =
0.01 (1 male to every 100 females), 90% of
eggs will be fertilized if R = 0.1 (1 male to
every 10 females), and 98% of eggs will be
fertilized if the sex ratio is 1:1.
To understand how the completed model
behaves, we differentiate Equation 4, substitute
from Equations 1 and 2, and, hence, show that
the rate of change R is
[6]
Equation 6 demonstrates that after a short
transient, the adult male-to-male ratio settles to
a steady-state value (which we denote by R
*
),
given by
[7]
and thus is equal to that of the fertilized eggs.
This scenario is illustrated by the solid line in
Figure 1A.
Once R reaches the steady state (R
*
), the
rate of change of the adult female population
will be given by
[8]
where
B = (1 – Ω)S
J
Γ, and F
*
= F(R
*
). [9]
Because B, F
*
, and µ are constants,
Equations 7 and 8 tell us that once F has
reached F
*
, the only possible behaviors this
model can exhibit are exponential population
growth or decay. Exponential growth is illus-
trated by the dashed line in Figure 1A.
To calculate the “long-run growth rate” Θ,
we seek a solution of Equation 8 of the form
A
f
= A
f 0
exp( Θt) [10]
and see that such a solution is possible only if Θ
is a solution of
Θ+ µ = BF
*
exp(– Θ Ψ). [11]
Although Equation 11 has no closed-form
solution, it is still useful. If we multiply both
sides by Ψ, we see that the relation of Θto 1/ Ψ
is determined by the two parameter groups
BF
*
Ψand µ Ψ. Equation 11 is also easy to solve
numerically, and in Figure 1B we show curves
relating Θ Ψto BF
*
Ψfor different values of µ Ψ.
A key observation from these curves, which
can be confirmed directly from Equation 11,
is that a viable population ( Θ> 0) requires
BF
*
> µ. [12]
Model 2: A Well-Mixed
Population with Density
Dependence
In the last section we saw that a model without
appropriate controlling density dependence
can predict only exponential growth or decline
in the population. To construct a variant of
our two-sex population model with a more
realistic behavioral repertoire, we assume that
survival between egg release and juvenile matu-
ration comprises the following two compo-
nents: survival from release to hatch, which we
denote by S
E
, and survival from hatch to matu-
rity, which we denote by S
J
. We continue to
assume that S
J
is a constant, but we imagine
that the egg deposition process gets more haz-
ardous as more females become involved. To
model this scenario, we assume that egg sur-
vival is a decreasing function of adult female
abundance (A
f
), which we model by writing
[13]
thus implying that all fertilized eggs survive to
hatch when A
f
is very small, 50% survive
when A
f
= A
h ,
and none survive when A
f
is
very large indeed.
The rest of this model variant is essentially
the same as before, except that we take the
opportunity to relax our previous assumption
that adult males and females have the same
per-capita mortality rate, by writing
[14]
and
[15]
where µ
m
and µ
f
represent the per-capita mor-
tality rates for adult males and females, and
the rate of production of fertilized eggs, E, is
as before, namely
E = ΓA
f
F. [16]
Unlike the formulation in the previous section,
this model has a stationary state in which the
ratio of adult male and female populations is
given by an expression similar to Equation 7
but reflecting the fact that males and females
have differing mortality rates, so that
dA
dt
SSEt A
f
EJ f f
= () () ,1 Ω Ψ∝
dA
dt
SSEt A
m
EJ mm
= Ω Ψ∝()
S
A
AA
E
h
hf
=
+
,
dA
dt
BFAt At
f
ff
=
*
() (), Ψ ∝
R
*
=

Ω
Ω1
dR
dt
SEt
A
R
J
f
=

←
→
…

()
().
Ψ
Ω Ω1
F
R
RR
h
=
+
.
R
A
A
m
f
= .
Demographic models with endocrine disruption
Environmental Health Perspectives • VOLUME 114 | SUPPLEMENT 1 | April 2006 123

Θ

Ψ
2
–2
10
0
0103
A
f

o
r

R
A B
A
f
R
t/ Ψ BF
*
Ψ
∝
Ψ
=
4
∝
Ψ
=
2
∝
Ψ
=
1
∝
Ψ
=
0
.5
Figure 1. A population with no density dependence. (A) The time history of adult female density and adult
male-to-female ratio for a population with a 1:1 sex ratio ( Ω= 0.5), half-saturation male-to-female ratio R
h
=
0.2, juvenile survival S
j
= 0.1, and normalized fecundity Γ Ψ= 10, initialized with 0.1 females and 0.005 males per
unit area. (B) Normalized long-run growth rate, Θ Ψ, as a function of BF
*
Ψ with µ Ψas a parameter. The dotted
line shows the boundary between growing and decaying populations ( Θ= 0).

[17]
The steady-state population of adult females
is given by
[18]
Inspection of Equation 18 shows that for the
steady-state population to be positive (i.e., for
a biologically meaningful steady state to exist)
we require that
BF
*
> µ. [19]
In Figure 2A we illustrate the temporal devel-
opment of a population with density-depen-
dent egg hatching, parameters that satisfy
Inequality 19, and an initial male-to-female
ratio of 5%. We see that R rapidly attains its
stationary value (R
*
= 1). While adult abun-
dance is small it exhibits (near) exponential
growth. As A
f
becomes comparable with A
h,
the
rate of population growth slows dramatically
and the female abundance eventually reaches an
asymptote at the value given by Equation 18.
In Figure 2B we illustrate the dependence
of the steady-state adult female abundance on
the controlling parameter groups BF
*
and µ.
For values of BF
*
< µ, a viable steady state can-
not be established. For BF
*
> µ the steady-state
adult female abundance increases with BF
*
at a
rate that depends on µ
–1
.
Model 3: Temporarily
Monogamous Pair Formation
In the last two sections we assumed that the
male and female populations are well-mixed,
so that any adult male has an equal chance of
fertilizing any released eggs. This scenario is
clearly inappropriate for many species, so we
now investigate an assumption at the opposite
end of the spectrum of possibilities, namely,
that fish form breeding pairs.
To keep the mathematics simple, and to
extend the range of possibilities that the model
encompasses, we shall assume that at any
instant, the eggs being released by a single
female will be fertilized by a single male. At
any given instant each male can pair with only
one female, so if there is a shortage of males,
some females will release unfertilized eggs.
Conversely, if there is an excess of males, all
females will be paired.
Provided that a single male can fertilize all
the eggs released by a single female, and eggs
released and not immediately fertilized are
permanently infertile, then the proportion of
eggs fertilized when the adult male-to-female
ratio is R, is
[20]
The rest of our model remains unchanged,
so the basic dynamics are specified by
Equations 14 and 15, and the controlling
density dependence is supplied by the adult
female dependent hatch probability described
by Equation 13. In addition, the fertilized egg
production rate is given by Equation 16, and
the fertilization fraction is described by
Equation 20.
Because it has controlling density depen-
dence, this model will exhibit a stationary state.
The analysis leading to Equation 17 carries
over unchanged to this model, so we know
that at the steady state the adult male-to-female
ratio will be as follows:
[21]
and the fertilization fraction will be F
*
= F(R
*
),
where F(R) is now specified by Equation 20.
The remaining analysis follows exactly on
from that of the last section, yielding the
result that
[22]
Examination of Equations 20, 21, and 22
shows that if there is an excess of males
(R
*
> 1), then all released eggs will be fertilized
and the steady state depends only on the bal-
ance of female fecundity and mortality. If there
is an excess of females, their steady-state abun-
dance depends also on the steady-state fertiliza-
tion fraction, which is just equal to the
stationary male-to-female ratio R
*
.
Effects of Endocrine Disruptors
The last three sections tell a remarkably con-
sistent story. A model with no limiting den-
sity dependence exhibits only exponential
growth or decay, whereas one with an appro-
priately limiting density dependence will
eventually approach a steady state. For
model 1, which has no limiting density
dependence, the ratio of the long-run growth
rate Θto the development delay Ψis deter-
mined by the two parameter groups BF
*
Ψand
µ Ψ. For models 2 and 3, the ratio of the
steady-state female abundance to the density
dependence parameter A
h
is determined by
the single parameter group BF
*
/µ
f
, which rep-
resents the number of adult female offspring
produced by a single adult female. Our task
in this section is to determine how exposure
to endocrine disruptors might be expected to
change these controlling parameter groups.
Changes in male fecundity and life span.
One of the most remarkable effects of expo-
sure to endocrine disruptors is the feminiza-
tion of male fish (Jobling et al. 1996, 1998).
Given a variety of exposure levels within the
population described by our model, this phe-
nomenon seems most likely to manifest itself
as a reduction in the population average male
fecundity, although it is also possible that
associated increases in metabolic costs could
produce a shortening of adult male life span.
Equations 9, 18, and 22 tell us that the
steady-state female abundance predicted by
both models with realistic density dependence
(models 2 and 3) is influenced only by prop-
erties of the adult male subpopulation
through the equilibrium proportion of eggs
that are fertilized (F
*
). Equations 9 and 11 tell
us that the same thing applies to the long-run
growth rate predicted by model 1.
In models 1 and 2, the equilibrium pro-
portion of eggs fertilized is given by
[23]
The effect of a reduction in average male
fecundity will be an increase in the adult sex
ratio at which 50% fertilization occurs (R
h
). A
decrease in male life span will be manifested as
an increase in the adult male per-capita mor-
tality µ
m
, which will result in a decrease in the
equilibrium adult sex ratio R
*
(Equation 21).
Inspection of Equation 23 shows us that the
effect of these changes will be very dependent on
F
R
RR
h
*
*
*
.=
+
A
A
BF
f
hf
*
*
.=
∝
1
R
A
A
m
f
f
m
*
*
*
==

♥
↔
♠
≠
≈
≡
∝
∝
Ω
Ω1
F
if R
RifR
=
ϖ
<
↓
°
±
11
1
.
AA
BF
fh
f
=
♥
↔
♠
≠
≈
≡
*
.
∝
1
R
A
A
m
f
f
m
*
*
*
.==

♥
↔
♠
≠
≈
≡
∝
∝
Ω
Ω1
Gurney
124 VOLUME 114 | SUPPLEMENT 1 | April 2006 • Environmental Health Perspectives
R
A
f
A B
t/ Ψ
0 100
10
0
A
f

o
r

R
10
0
10
BF
*
∝

=

0
.
5
∝

=

1
∝

=
2
∝

=
4
A
f
/
A
h
Figure 2. A population with density-dependent egg hatching. (A) The time history of adult female density
and adult male-to-female ratio for a population with Ω= 0.5, R
h
= 0.2, µ
f
Ψ = µ
m
Ψ, S
j
= 0.1, Γ Ψ = 10, and R
h
= 5,
intitialized with 0.1 females and 0.005 males per unit area. (B) Normalized steady-state adult female abun-
dance, as a function of BF
*
with µ as a parameter.

the relative values of R
*
and R
h
. Provided that
R
*
>> R
h,
then F
*
will remain close to unity
regardless of the actual values of these two quan-
tities. Conversely, if R
*
<< R
h
, then F
*
will vary
in direct proportion to R
*
and in inverse propor-
tion to R
h
. We illustrate these two regimes, and
the transition between them in Figure 3.
These findings confirm the following con-
clusion, which is perhaps obvious in retro-
spect: If there is an excess of males beyond
that required to fertilize the entire egg output
of the female population, then changes in
their number and/or average fecundity will
have little effect until a change becomes severe
enough to reduce the proportion of fertilized
eggs significantly below 1. Because most nor-
mal populations will have an equilibrium sex
ratio of around 1 and a half-saturation sex
ratio between 0.01 and 0.1, R
h
or µ
m
would
need to change by almost an order of magni-
tude before a significant population-level
effect would be observed.
The generality of this conclusion is rein-
forced by model 3. This model variant
assumes that male–female interaction involves
individual pair formation, which in turn
changes the formula for the fraction of eggs
that are fertilized to Equation 20. Within the
limiting assumptions of this model (i.e., that a
single male is capable of fertilizing the entire
reproductive output of his pair), changing male
fecundity has no effect on the fraction of eggs
fertilized and, hence, on the steady-state popu-
lation abundance. Inspection of Equation 20
also shows that increasing adult male mortality,
and, hence, decreasing the steady–state
male/female ratio, does not affect steady-state
abundance while R
*
remains > 1, after which
F
*
decreases in proportion to R
*
.
Aside from fertilization effects, the proper-
ties of all three models we have examined
depend only on parameters that characterize
the demographic performance of females.
Moreover, the effects of changes in these para-
meters are neither magnified nor ameliorated
by the model dynamics. The severity of the
resultant demographic effects is therefore
dependent on the relative size of the parameter
changes induced by the chemical challenge
concerned.
A Framework for Modeling
Individuals
To provide a systematic way of thinking
about the linkage between chemical challenge
and changes in demographic performance,
we need a model of individual growth,
survival, and reproduction. To illustrate the
possibilities, we look briefly at a model of
female growth and reproduction closely
related to a model proposed by Kooijman
(1993, 2000).
The model characterizes an individual
female by her carbon weight W and asserts
that if she assimilates carbon at a rate A,
expends a fraction Πof that carbon assimilated
on reproduction, and has a basal metabolic
expenditure rate M, then her weight must
change at a rate given by
[24]
Although more detailed variants of this model
sometimes have the fractional expenditure on
reproduction varying with individual state, we
shall only consider its simplest incarnation, in
which Πis constant.
We now assume that the maintenance
rate M is proportional to the animal’s carbon
weight, while the assimilation rate scales with
W
Λ
, so we write
M = ]W, A = ΚW
Λ
, [25]
where the constant ]represents the weight-
specific basal maintenance rate and Κrepresents
the (environment dependent) assimilation
scale.
In a constant environment (requiring, at a
minimum, constant food availability and con-
stant temperature), the assimilation rate scale Κ
is independent of time. In this case, provided
that Λ< 1 (it is normally between 0.6 and 0.8),
this model predicts that the individual eventu-
ally reaches an asymptotic size (W
η
) that
depends on the assimilation rate scale Κ
(Figure 4A).
Examination of Equations 24 and 25
shows that in an environment that makes the
assimilation rate scale Κconstant, the mainte-
nance cost rate M rises linearly with W, while
(so long as Λ< 1) the assimilation rate rises
with W more slowly. Growth eventually stops
when the proportion of assimilation allocated
to growth and maintainance, (1 - Π)A, is equal
to the rate of expenditure on maintenance, M.
This state occurs when W = W
η,
as given by
[26]
It is also instructive to calculate the fecundity
of a female who has reached her asymptotic
weight, which we denote by Γ
η
. If the carbon
cost of an egg is e, then it is quite straightfor-
ward to see that Γ
η
= ( Π Κ/ ϑ)(W
η
)
Λ
. Reference
to Equation 26 then shows that
[27]
As Figure 4B shows, weight and fecundity at
age are sensitively dependent on both the
assimilation scale Κand the specific mainte-
nance cost rate ].
Γ
Π Κ
ϑ
Π Κ
]
Λ
Λ
η

=
←
→
↑
…


()
.
1
1
W
η

=
←
→
↑
…


()
.
1
1
1
Π Κ
]
Λ
dW
dt
AM= () .1 Π
Demographic models with endocrine disruption
Environmental Health Perspectives • VOLUME 114 | SUPPLEMENT 1 | April 2006 125
A B
4
3
2
1
0
4
3
2
1
0
10
–2
10
–1
10
0
10
1
10
2
∝
m
/∝
f
10
–3
10
–2
10
–1
10
0
10
1
10
2
R
h
A
*
f
/
A
h
A
*
f
/
A
h
Time
C
a
r
b
o
n

w
e
i
g
h
t
1,000
0
0 200
Κ = 1.0
Κ = 0.84
Κ = 0.67
] = 0.06
] = 0.09
] = 0.12
Assimilation scale
800
0
0.5 1.5

ϑ

Γ
i
n
f
i
n
i
t
y
/

Π
A B
Figure 3. Steady-state variation with male parameters under model 2. (A) Normalized steady-state abun-
dance of adult females as a function of half-saturation sex ratio (R
h
) for a population with R
*
= 1, and B/µ
f
=
4. (B) Normalized steady-state adult female abundance, as a function of µ
m
/µ
f
for a population with B/µ
f
= 4
and R
h
= 0.1.
Figure 4. Simple individual growth model. (A) Carbon weight against time for an individual with ]= 0.09,
Π= 0.1 growing in a constant environment, implies a value of Κas marked. (B) Asymptotic fecundity against
assimilation scale ( Κ) with ] as a parameter for an individual with Π= 0.1.

Gurney
126 VOLUME 114 | SUPPLEMENT 1 | April 2006 • Environmental Health Perspectives
Individual Effects of Endocrine
Disruptors
Except for the more specific and dramatic
effects on male sexual function and fertility,
endocrine disruptors that are systematically
present in modest concentrations seem likely
to present a range of effects that have much in
common with many other chemical pollu-
tants. Although specific pollutants produce
their individual effects through myriad physio-
logical pathways, the most common manifes-
tations of their action at one level of detail
below the individual’s demographic perfor-
mance are increased specific basal metabolic
cost ( ]in our model) and decreased assimila-
tion in a given environment (decreased Κin
our model).
Before examining the dependence of either
growth rate or asymptotic size and fecundity
on such changes, it is worth considering the
relationship between environment and assimi-
lation scale. The normal form of this depen-
dence is something like
[28]
where F represents food availability, F
H
is the
half-saturation food availability (which is
related to search volume), and Κ
max
is the
maximum assimilation rate scale (which is
related to ingestate handling time).
The effects of an environmental chemical
challenge on assimilation can thus manifest
themselves either as changes in external activ-
ity (search volume) or as changes in internal
physiology (handling time). Although changes
in Κ
max
will result in proportional changes in
Κ, changes in external activity (and, hence, in
F
H
) will produce significant changes in Κonly
when food is not in abundance (i.e., when F is
not large compared with F
H
).
Broadly speaking, chemical challenges seem
likely to increase ]and decrease Κin any given
environment. For a species whose individual
allometry is such that Λ= 0.75, we see from
Equation 26 that asymptotic size will change
with the ratio ( Κ/ ]) to the power of 4 and from
Equation 27 that asymptotic fecundity will
change with 1/ ]to the power of 3 and with Κ
to the power of 4.
We thus conclude that increases in ]and
decreases in Κ
max
are unequivocally very bad
news indeed, but that reductions in F
H
are
serious only if food is limiting, or if the
decrease is sufficiently large to make it so.
Discussion
In this article we have formulated and analyzed a
series of strategic models of populations and indi-
viduals subject to challenge by endocrine disrup-
tors. The models are concerned principally with
egg-bearing species and neglect special biological
features such as parental care. The payoff from
this reductionist approach is the simplicity of the
resulting models, which has made it possible to
obtain useful analytic results by the application
of very elementary mathematical methods. The
advantage of such methods, where they can be
deployed, is that general findings can be obtained
much more quickly and definitively than is pos-
sible with computational approaches.
Our first investigation examined the
demographic importance of alterations to
male fertility. Because we sought general
results, we took special care to establish not
only the generality of our results for a particu-
lar model but also their robustness to changes
in model structure, representing either uncer-
tainty about basic biological process or vari-
ability in process between systems of interest.
Even where the model structure lacks a
controlling dependence on density (so that the
resulting model can predict only exponential
population growth or decay), we were able to
show a remarkable consistency of outcome. In
all the structural variants we examined, the pre-
dicted outcome is controlled by a quantity that
can be interpreted as the number of mature
female offspring that result from the average
female’s lifetime reproductive activity.
This suggests, and our more detailed
investigations confirm, that changes in male
fertility have little effect at the population level
unless the reduction is so general and severe
that the chance of released eggs being fertilized
is significantly reduced. An adult sex ratio of
anything approaching 1:1 guarantees that this
reduction in fertilization will happen only
when almost all the male population is suffer-
ing serious fertility reduction. Hence, we argue
that endocrine disruptor exposure will cer-
tainly produce deleterious changes in female
demographic characteristics at levels much
lower than those required to produce popula-
tion-level effects by altering male fertility.
Our investigation emphasized the impor-
tance of identifying those parameters to
which population level effects are insensitive,
but it also showed that changes in many indi-
vidual demographic parameters produced
population-level changes more or less in
proportion to change in the parameter or
parameter group concerned.
Even where it is practical to measure indi-
vidual summary parameters such as fecundity
and mortality accurately under realistic expo-
sure conditions, such measurements inevitably
occupy a time-span comparable with the life-
time of an individual, which may be many
years. By contrast, exposure-induced changes
in low-level physiological parameters such as
respiration or ingestion rates can be measured
on a time scale of days or hours.
Our final demonstration of the utility of
simple models therefore examined a strategic
model that describes how an individual allo-
cates resources among the three fundamental
demographic tasks: growth, maintenance, and
reproduction. We showed that the techniques
that yield good biological gain for low mathe-
matical pain in the context of population
dynamic models also work when applied to
models that describe individual energy bud-
gets. Thus, we have provided a viable route for
investigating the sensitivity of an individual’s
demographic summary parameters to changes
in low-level physiological rates.
REFERENCES
Brown AR, Riddle AM, Cunningham NL, Kedwards TJ, Shillabeer
N, Hutchinson TH. 2003. Predicting the effects of endocrine-
disrupting chemicals on fish populations. Hum Ecol Risk
Assess 9(3):761–788.
Jobling S, Beresford N, Nolan M, Rodgers-Gray R, Tyler CR,
Sumpter JP. 2002. Altered sexual maturation and gamete
production in wild roach (Rutilus rutilus) living in rivers that
receive treated sewage effluents. Biol Reprod 66:272–281.
Jobling S, Nolan M, Tyler CR, Brighty G, Sumpter J. 1998.
Widespread sexual disruption in wild fish. Environ Sci Tech
32:2498–2506.
Jobling S, Sheahan D, Osborne J, Mattheisen P, Sumpter J. 1996.
Inhibition of testicular growth in rainbow trout
(Oncorhynchus mykiss) exposed to estrogenic alkylphenolic
chemicals. EnvironToxicol Chem 15:194–202.
Kooijman SALM. 1993. Dynamic Energy Budgets in Biological
Systems. Cambridge, UK:Cambridge University Press.
Kooijman SALM. 2000. Dynamic Energy and Mass Budgets in
Biological Systems. Cambridge, UK:Cambridge University
Press.
Panter GH, Thomson RS, Sumpter JP. 1998. Adverse reproductive
effects in male fathead minnows (Pimephales promelas)
exposed to environmentally relevant concentrations of the
natural oestrogens oestradiol and oestrone. Aqua Toxicol
42:243–253.
Van Winkle W, Rose KA, Chambers RC. 1993. Individual-based
approach to fish population dynamics—a review. Trans Am
Fish Soc 380:397–403.
Κ Κ=
+
max
,
F
FF
H