How does adaptive consumer movement affect population dynamics in consumer���resource metacommunities with homogeneous patches? Peter A. Abrams n, Lasse Ruokolainen 1 Department of Ecology and Evolutionary Biology, University of Toronto, 25 Harbord St. Toronto, ON, Canada M5S 3G5 a r t i c l e i n f o Article history: Received 29 October 2010 Received in revised form 20 February 2011 Accepted 22 February 2011 Available online 1 March 2011 Keywords: Dispersal Metacommunity Predator���prey system Synchrony Stability a b s t r a c t This article uses simple models to explore the impact of adaptive movement by consumers on the population dynamics of a consumer���resource metacommunity consisting of two identical patches. Consumer���resource interactions within a patch are described by the Rosenzweig���MacArthur predator��� prey model, and these dynamics are assumed to be cyclic in the absence of movement. The per capita movement rate from one patch to the other is an increasing function of the difference between the per capita birth minus death rate in the destination patch and that in the currently occupied patch. Several variations on this model are considered. Results show that adaptive movement frequently creates anti- phase cycles in the two patches these suppress the predator���prey cycle and lead to low temporal variation of the total population sizes of both species. Paradoxically, even when movement is very sensitive to the fitness difference between patches, perfect synchrony of patches is often much less likely than in comparable systems with random movement. Under these circumstances adaptive movement of consumers often generates differences in the average properties of the two patches. In addition, mean global densities and responses to global perturbations often differ greatly from similar systems with no movement or random movement. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction The question posed by the title is important because most natural consumer���resource interactions occur in spatially subdivided ecolo- gical systems (metacommunities) connected by the dispersal of individuals. While movement behavior is likely to be adaptive (Bowler and Benton, 2005), relatively little is known about the effects of adaptive movement on the dynamics of metacommunities. Habitat selection models have usually assumed that individuals moving towards better habitat patches will achieve a stable ���ideal free��� distribution (Fretwell and Lucas, 1970), in which their fitness is equal across all occupied habitats (e.g., K�� rivan, 1997 Morris, 2003). If consumer���resource interactions within a patch are structurally identical across patches, equality of consumer fitness implies that resource densities are also equal across patches. This means that if the populations in the patches fluctuate, they must fluctuate in syn- chrony. However, some of the few studies that have explicitly modeled adaptive movement in sets of interacting species have found that ideal free distributions are often not necessarily attained, even when patches are equivalent in all parameter values (Schwinning and Rosenzweig, 1990 Bernstein et al., 1999 Abrams et al., 2007 Abrams, 2007). Other types of non-diffusive movement have been shown to affect stability differently than random movement in simple systems with interacting species (Huang and Diekmann, 2001 Amarasekare, 2008). The present article is a step towards a more systematic understanding of how adaptive movement affects dynamics and distributions of individuals in metacommunities. Most work on metacommunities (e.g., all but one chapter in Holyoak et al. (2005)) has assumed that movement consists of random dispersal. Although random movement is often thought to synchronize fluctuations across patches, this is not always true, even in very simple metacommunities consisting of two patches and two species (Jansen, 2001) or in single-species models with cycles driven by discrete density-dependent dynamics (Gyllenberg et al., 1993 Hastings, 1993). In metacommunity systems with three or more interacting species, small numbers of patches, and endogenously driven population cycles within a patch, it is even more likely that asynchronous dynamics will arise from random movement between patches (Koelle and Vandermeer, 2005 Wilson and Abrams, 2005). Nevertheless, for the simple case of two similar patches, each containing a consumer���resource (predator���prey) pair, most popula- tion growth parameters and movement rates imply that random movement synchronizes the fluctuations in different patches (Jansen and de Roos, 2000 Jansen, 2001 Goldwyn and Hastings, 2008 Vasseur and Fox, 2009). However, Jansen (2001) noted that anti- synchronized cycles are possible for some parameter values in this case. The fact that consumer movement towards a patch with more Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology 0022-5193/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2011.02.019 n Corresponding author. E-mail address: peter.abrams@utoronto.ca (P.A. Abrams). 1 Current address: University of Helsinki, Viikinkaari 1, P.O. Box 68, 00014 Helsinki, Finland. Journal of Theoretical Biology 277 (2011) 99���110

resource should reduce the difference in resource densities might suggest that synchronization would be more likely with adaptive than with random movement. This was found by Ruxton and Rohani (1998) in a study of fitness-dependent movement within a discrete time single species metapopulation model they con- cluded that such movement was more likely to stabilize and synchronize dynamics than was random movement. They also found that fitness-dependent movement generally simplified fluctuations when it did not produce point stability chaotic dynamics and complex cycles were usually converted to simple cycles. The work we describe here does not support either the intuition or the earlier results from discrete models. Defining ���fitness��� and ���adaptive��� can be problematic, particularly in the context of movement. However, for cost free movement of unstructured populations, whose dynamics are represented by ordin- ary differential equations, an individual that is always located in the patch that confers the largest per capita birth minus death rate will outcompete all others. ���Adaptive��� movement in the models considered here means that the per capita movement rate towards the patch currently characterized by the larger per capita birth minus death rate is higher than the movement in the reverse direction. Although a few previous models of metacommunities have assumed this type of movement, most have considered scenarios more complicated than just consumer movement in a two-patch consumer resource system. In some, both consumers and resources move adaptively (Abrams, 2007) others involve systems with very many patches (Armsworth and Roughgarden, 2008), or systems in which the biological commu- nity has more than two species (Abrams, 2007 Amarasekare, 2007, 2008, 2009 K�� rivan et al., 2008). These studies have shown that in specific cases, adaptive movement in a set of interacting species can produce dynamics that differ greatly from the dynamics arising via random movement and may not always equalize fitness of the moving species across patches. Understanding the generality of these findings requires further studies to determine how adaptive move- ment alters the dynamics of the simplest ecological communities. The present study will analyze several differential equation models of population dynamics in two identical patches, each containing the same consumer and the resource species. The resource is assumed to be immobile, while the consumer moves adaptively between patches. We concentrate on cases in which the consumer��� resource system within an isolated patch would undergo limit cycle dynamics. This assumption of cyclic dynamics is appropriate because movement in general is an adaptation to changing conditions. The other population dynamical assumptions mirror those of much of the previous work on random movement in coupled 2-patch consumer��� resource systems (Jansen, 1995, 2001 Goldwyn and Hastings, 2008, 2009). The purpose of the analyses is to determine how consumer movement affects the variation of the populations of both species, their synchrony across patches, and how closely the consumers approach an ideal free distribution. Our results show that moving towards the patch with more resource often produces anti-syn- chrony, in which patch dynamics are 1801 out of phase, and the variability in total density of each species is greatly reduced. A variety of other outcomes are possible, and many parameter sets can have two or more alternative attractors. We compare these results to those for similar systems with random movement, and examine how adaptive movement changes the response of the entire system to global environmental variables that affect both patches in the same way. Similar questions for systems with non-equivalent patches are considered elsewhere (Ruokolainen et al., submitted for publication). 2. The primary model The following consumer���resource (predator���prey) model is the familiar Rosenzweig-MacArthur (1963) model with the addition of fitness-dependent consumer movement as in Abrams (2000, 2007): dRi dt �� Ri��r kRi�� CRiNi 1��ChRi dNi dt �� Ni bCRi 1��ChRi d mNi exp l��Wi Wj�� ��mNj exp l��Wj Wi�� i, j �� 1,2, where Wi �� bCRi 1��ChRi d ��1a b�� These equations describe resource (1a) and consumer (1b) population growth rate in patch i, where Ri and Ni are resource and consumer densities, respectively, in patch i, and Wi is consumer fitness (instantaneous per capita birth minus death) in patch i. In the absence of the consumer, resources have logistic growth. The parameterization of the logistic used here separates the per capita resource growth rate into a maximum per capita growth rate r and a per capita reduction in that rate of k (Rueffler et al., 2006). Under this parameterization, the equilibrium resource density in the absence of consumption is r/k. Consump- tion of resources is given by a Holling type II functional response with an attack rate C and handling time h. Consumed resources are converted to additional consumers (numbers or biomass) with an efficiency of b. The parameter d defines the (constant) per capita mortality of the consumer. The form of the movement function assumes that a consumer individual has the knowledge of resource densities in both patches. Information about condi- tions in the other patch may be obtained by cues that may be detected at a distance (e.g., diffusing chemicals) or by short- duration non-foraging visits. Eq. (1b) implies that a baseline per capita movement rate m applies when consumer fitness (i.e., per capita growth rate W) is equal in the two patches. This is modified by an exponential function of the difference in fitness between the patches. A larger value of the fitness sensitivity (l) makes the movement rate increase more rapidly with the fitness difference between patches. A larger l means that the consumer moves into a more-rewarding patch at a greater rate and moves into a less- rewarding patch at a lower rate l��0 implies random movement. Alternative forms for both the movement function and population dynamics are considered in Appendix B, which is summarized in Section 3.6. Here we consider movements that occur on a behavioral time scale, which is assumed to be more rapid than the population dynamical time scale. As a result, in choosing parameter values, we require that the maximum per capita movement rate (which occurs when Ri ��r/k and Rj ��0 i.e., m exp[bC(r/k)/(1��Ch(r/k))]) exceeds the maximum per capita population growth rate within a patch (i.e., bC(r/k)/(1��Ch(r/k)) d). The ratio of these two quan- tities is referred to as the ���movement ratio��� (MR). We assume a low value of the baseline rate m, reflecting the idea that ���adaptive��� implies low rates of movement to poorer quality patches. The analysis examines a wide range of the fitness sensitivity para- meter l. The minimum l considered in each case is one that implies MR��1, although most behavioral movement is likely to have MRb1. The maximum l considered in all cases represents a MR of at least 106. Many of these maximum movement rates may be too large to be realistic in most biological systems. However, the large differences between resource densities required to produce near-maximal movement do not occur except during transient dynamics for some initial conditions. The realized movement rates and their effects are discussed for the example in Fig. 1C and D. Appendix B discusses several alternative move- ment functions, including one with an upper limit to the movement rate. The population dynamic parameters are constrained by the assumption that consumer and resource populations in a single P.A. Abrams, L. Ruokolainen / Journal of Theoretical Biology 277 (2011) 99���110 100