Spatially discrete metapopulation models with directional dispersal

We use an age-structured discrete-time metapopulation model linking two sub-populations through larval transport and directed movements of adults to study the implications of linkages among subpopulations for the stability and resilience of exploited species. Our two-habitat model, a generalization of Fogarty's inshore-offshore lobster population model, includes isolated habitats under compensatory (monotone) or overcompensatory (oscillatory) dynamics [M.J. Fogarty, Implications of migration and larval interchange in American lobster (Homarus americanus) stocks: spatial structure and resilience, in: G.S. Jamieson, A. Campbell (Eds.), Proc. of North Pacific Symposium on Invertebrate Stock Assessment and Management, Can. Spec. Publ. Fish. Aquat. Sci. 125 (1998) 273]. Pre-migration local dynamics are selected from general classes of functions that capture the effects of competition for resources via contest (compensatory) and scramble (overcompensatory) intraspecific competitions. We explore the implications of these mechanisms on the long-term survival of exploited species. In particular, we use threshold parameters R(d)1 for Habitat 1 and R(d)2 for Habitat 2 together with precise mathematical definitions to prove that species persistence is possible at high levels of fishing in one habitat and low to moderate levels of fishing in the other. Our results support Fogarty's conclusion that conservative management of larval source populations could contribute to the resilience of exploited species.     

Interplay between local dynamics and dispersal in discrete-time metapopulation models

The effects of synchronous dispersal on discrete-time metapopulation dynamics with local (patch) dynamics of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are explored. Single-species metapopulation models behave as single-species single-patch models, whenever all local patches are governed by compensatory dynamics. Dispersal gives rise to multiple attractors with complex basin structures, whenever some local patches are under overcompensatory dynamics. In mixed systems, dispersal is capable of altering the local dynamics from compensatory to overcompensatory dynamics and vice versa. Examples are provided of metapopulation models supporting multiple attractors with intermingled basins of attraction.     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

The effects of small dispersal rates on extinction times in structured metapopulation models

Habitat destruction is a critical factor that affects persistence in several taxa, including Pacific salmon. Salmon are noted for their ability to home to their natal streams for reproduction. Since straying (i.e., spawners reproducing in nonnatal streams) is typically low in salmon, its effects have not been appreciated. In this article, we develop both a general analytical model and a simple simulation model describing structured metapopulations to study how weak connections between subpopulations affect the ability of a species to tolerate habitat destruction and/or declines in habitat quality. Our goals are to develop general principles and to relate these principles to salmon population dynamics. The analytical model describes the dynamics of two density-dependent subpopulations, connected by dispersal, whose growth rates fluctuate in response to environmental and demographic stochasticity. We find that, for moderate levels of environmental variability, small dispersal rates can significantly increase mean extinction times. This effect declines with increasing habitat quality, increasing temporal correlation, and increasing spatial correlation, but it is still significant for realistic parameter values. The simulation model shows there is a threshold rate of dispersal that minimizes extinction probabilities. These results cannot be seen in classical metapopulation models and provide new insights into the rescue effect.     

Spatially coupled larval supply of marine predators and their prey alters the predictions of metapopulation models

Oceanographic forces can strongly affect the movement of planktonic marine larvae, often producing predictable spatial patterns of larval delivery. In particular, recent empirical evidence suggests that in some coastal systems, certain locations consistently receive higher (or lower) larval supplies of both predators and their prey. As a consequence, rates of settlement and predation may be coupled spatially, a phenomenon I term the "coupled settlement effect." To investigate the metapopulation consequences of this phenomenon, I created discrete-time, patch-based analytical and simulation models with a common larval pool and uneven larval supply among patches. Using two complementary measures of subpopulation value as a basis of comparison, I found that models with and without the coupled settlement effect yielded strikingly different predictions. When prey and predator larval supplies were not coupled, patches supplied with a larger proportion of the larval pool made a greater contribution to the metapopulation. When settlement of prey and predator was strongly coupled, however, the opposite was true: subpopulations with lower rates of larval supply (above some minimum) were more essential to metapopulation persistence. These considerations could facilitate more effective selection of sites for protection in marine reserves.     

Evolution of density-dependent dispersal in a structured metapopulation

We study the evolution of density-dependent dispersal in a structured metapopulation subject to local catastrophes that eradicate local populations. To this end we use the theory of structured metapopulation dynamics and the theory of adaptive dynamics.The set of evolutionarily possible dispersal functions (i.e., emigration rates as a function of the local population density) is derived mechanistically from an underlying resource-consumer model. The local resource dynamics is of a flow-culture type and consumers leave a local population with a constant probability per unit of time κ when searching for resources but not when handling resources (i.e., eating and digesting). The time an individual spends searching (as opposed to handling) depends on the local resource density, which in turn depends on the local consumer density, and so the average per capita emigration rate depends on the local consumer density as well.The derived emigration rates are sigmoid functions of local consumer population density. The parameters of the local resource-consumer dynamics are subject to evolution. In particular, we find that there exists a unique evolutionarily stable and attracting dispersal rate κ^∗ for searching consumers. The κ^∗ increases with local resource productivity and decreases with resource decay rate. The κ^∗ also increases with the survival probability during dispersal, but as a function of the catastrophe rate it reaches a maximum before dropping off to zero again.     

Structured models of metapopulation dynamics

I develop models of metapopulation dynamics that describe changes in the numbers of individuals within patches. These models are analogous to structured population models, with patches playing the role of individuals. Single species models which do not include the effect of immigration on local population dynamics of occupied patches typically lead to a unique equilibrium. The models can be used to study the distributions of numbers of individuals among patches, showing that both metapopulations with local outbreaks and metapopulations without outbreaks can occur in systems with no underlying environmental variability. Distributions of local population sizes (in occupied patches) can vary independently of the total population size, so both patterns of distributions of local population sizes are compatible with either rare or common species. Models which include the effect of immigration on local population dynamics can lead to two positive equilibria, one stable and one unstable, the latter representing a threshold between regional extinction and persistence.     

From individual behavior to metapopulation dynamics: unifying the patchy population and classic metapopulation models

Spatially structured populations in patchy habitats show much variation in migration rate, from patchy populations in which individuals move repeatedly among habitat patches to classic metapopulations with infrequent migration among discrete populations. To establish a common framework for population dynamics in patchy habitats, we describe an individual-based model (IBM) involving a diffusion approximation of correlated random walk of individual movements. As an example, we apply the model to the Glanville fritillary butterfly (Melitaea cinxia) inhabiting a highly fragmented landscape. We derive stochastic patch occupancy model (SPOM) approximations for the IBMs assuming pure demographic stochasticity, uncorrelated environmental stochasticity, or completely correlated environmental stochasticity in local dynamics. Using realistic parameter values for the Glanville fritillary, we show that the SPOMs mimic the behavior of the IBMs well. The SPOMs derived from IBMs have parameters that relate directly to the life history and behavior of individuals, which is an advantage for model interpretation and parameter estimation. The modeling approach that we describe here provides a unified framework for patchy populations with much movements among habitat patches and classic metapopulations with infrequent movements.     

Single-species metapopulation dynamics: concepts, models and observations

This paper outlines a conceptual and theoretical framework for single-species metapopulation dynamics based on the Levins model and its variants. The significance of the following factors to metapopulation dynamics are explored: evolutionary changes in colonization ability; habitat patch size and isolation; compensatory effects between colonization and extinction rates; the effect of immigration on local dynamics (the rescue effect); and heterogeneity among habitat patches. The rescue effect may lead to alternative stable equilibria in metapopulation dynamics. Heterogeneity among habitat patches may give rise to a bimodal equilibrium distribution of the fraction of patches occupied in an assemblage of species (the core-satellite distribution). A new model of incidence functions is described, which allows one to estimate species' colonization and extinction rates on islands colonized from mainland. Four distinct kinds of stochasticity affecting metapopulation dynamics are discussed with examples. The concluding section describes four possible scenarios of metapopulation extinction.     

Allee effect and self-fertilization in hermaphrodites: reproductive assurance in a structured metapopulation

Reproductive assurance through selfing during colonization events or when population densities are low has often been put forward as a mechanism selecting for the evolution of self-fertilization. Such arguments emphasize on the role of both local demography and metapopulation processes. We developed a model for the evolution of self-fertilization in a structured metapopulation in which local densities are not steady because of population growth. Reproduction by selfing is density-independent (reproductive assurance) but selfed seeds endure inbreeding depression, whereas reproduction by outcrossing is density-dependent (Allee effect). First, we derived an analytical criterion for metapopulation viability as a function of the selfing rate and metapopulation parameters. We show that outcrossers can develop a viable metapopulation when they produce a high amount of dispersal seeds that counterbalances their incapacity to found new populations from low densities. Second, the model shows there is a positive feedback between demography and outcrossing rates, leading to either complete outcrossing or selfing. Specifically, we illustrate that inbreeding depression can paradoxically favor the evolution of selfing because of its negative effect on density. Also, complete outcrossing can be selected despite pollen limitation, although it does not provide a full seed set. This model underlines the influence of the mating system both on demography and gene dynamics in a metapopulation context.     

Spatially structured superinfection and the evolution of disease virulence

When pathogen strains differing in virulence compete for hosts, spatial structuring of disease transmission can govern both evolved levels of virulence and patterns in strain coexistence. We develop a spatially detailed model of superinfection, a form of contest competition between pathogen strains; the probability of superinfection depends explicitly on the difference in levels of virulence. We apply methods of adaptive dynamics to address the interplay of spatial dynamics and evolution. The mean-field approximation predicts evolution to criticality; any small increase in virulence capable of dynamical persistence is favored. Both pair approximation and simulation of the detailed model indicate that spatial structure constrains disease virulence. Increased spatial clustering reduces the maximal virulence capable of single-strain persistence and, more importantly, reduces the convergent-stable virulence level under strain competition. The spatially detailed model predicts that increasing the probability of superinfection, for given difference in virulence, increases the likelihood of between-strain coexistence. When strains differing in virulence can coexist ecologically, our results may suggest policies for managing diseases with localized transmission. Comparing equilibrium densities from the pair approximation, we find that introducing a more virulent strain into a host population infected by a less virulent strain can sometimes reduce total host mortality and increase global host density.     

Conservation of a domestic metapopulation structured into related and partly admixed strains

Preservation of genetic diversity is one of the most pressing challenges in the planetary boundaries concept. Within this context, we focused on genetic diversity in a native, unselected and highly admixed domesticated metapopulation. A set of 1,828 individuals from 60 different cattle breeds was analysed using a medium density SNP chip. Among these breeds, 14 Bu?a strains formed a metapopulation represented by 350 individuals, while the remaining 46 breeds represented the global cattle population. Genetic analyses showed that the scarcely selected and less differentiated Bu?a metapopulation contributed a substantial proportion (52.6%) of the neutral allelic diversity to this global taurine population. Consequently, there is an urgent need for synchronized maintenance of this highly fragmented domestic metapopulation, which is distributed over several countries without sophisticated infrastructure and highly endangered by continuous replacement crossing as part of the global genetic homogenization process. This study collected and evaluated samples, data and genomewide information and developed genome‐assisted cross‐border conservation concepts. To detect and maintain genetic integrity of the metapopulation strains, we designed and applied a composite test that combines six metrics based on additive genetic relationships, a nearest neighbour graph and the distribution of semiprivate alleles. Each metric provides distinct information components about past admixture events and offers an objective and powerful tool for the detection of admixed outliers. The here developed conservation methods and presented experiences could easily be adapted to comparable conservation programmes of domesticated or other metapopulations bred and kept in captivity or under some other sort of human control.     

Attractivity of coherent manifolds in metapopulation models

The likelihood that coupled dynamical systems will completely synchronize, or become “coherent”, is often of great applied interest. Previous work has established conditions for local stability of coherent solutions and global attractivity of coherent manifolds in a variety of spatially explicit models. We consider models of communities coupled by dispersal and explore intermediate regimes in which it can be shown that states in phase space regions of positive measure are attracted to coherent solutions. Our methods yield rigorous and practically useful coherence criteria that facilitate useful analyses of ecological and epidemiological problems.     

Simple models for complex systems: exploiting the relationship between local and global densities

Simple temporal models that ignore the spatial nature of interactions and track only changes in mean quantities, such as global densities, are typically used under the unrealistic assumption that individuals are well mixed. These so-called mean-field models are often considered overly simplified, given the ample evidence for distributed interactions and spatial heterogeneity over broad ranges of scales. Here, we present one reason why such simple population models may work even when mass-action assumptions do not hold: spatial structure is present but it relates to global densities in a special way. With an individual-based predator–prey model that is spatial and stochastic, and whose mean-field counterpart is the classic Lotka–Volterra model, we show that the global densities and densities of pairs (or spatial covariances) establish a bi-power law at the stationary state and also in their transient approach to this state. This relationship implies that the dynamics of global densities can be written simply as a function of those densities alone without invoking pairs (or higher order moments). The exponents of the bi-power law for the predation rate exhibit a remarkable robustness to changes in model parameters. Evidence is presented for a connection of our findings to the existence of a critical phase transition in the dynamics of the spatial system. We discuss the application of similar modified mean-field equations to other ecological systems for which similar transitions have been described, both in models and empirical data.     

Simple discrete-time metapopulation models of patch occupancy

Simple models in theoretical ecology have a long-standing history of being used to understand how specific processes influence population dynamics as well as providing a foundation for future endeavors. The Levins model is the seminal example of this for continuous-time metapopulation dynamics. However, many natural populations have a distinct separation between processes and data is not collected continuously leading to the need for using a discrete-time model. Our goal is to develop a simple discrete-time metapopulation model of patch occupancy using difference equations. In our formulation, we consider the two fundamental processes of colonization and extinction that will be treated as sequential events and will only consider patch occupancy. To achieve this, we use a composition of two functions where one will reflect the extinction process and the other for the colonization process. Under some mild assumptions, we are able determine the dynamic behavior of the metapopulation. In addition, we provide numerous examples for the functions used to emulate the colonization and extinction processes. Our results illustrate that the dynamics of the model are tied to properties such as convexity and monotonicity of the colonization and extinction functions. In particular, if the model is non-monotone, then complex dynamics can arise such as cyclic and even chaotic behavior. Overall, our approach shows how certain properties of the colonization and extinction functions can influence metapopulation dynamics.     

局域种群的Allee效应和集合种群的同步性

从包含Allee效应的局域种群出发,建立了耦合映像格子模型,即集合种群模型.通过分析和计算机模拟表明:(1)当局域种群受到Allee效应强度较大时,集合种群同步灭绝;(2)而当Allee效应强度相对较弱时,通过稳定局域种群动态(减少混沌)使得集合种群发生同步波动,而这种同步波动能够增加集合种群的灭绝风险;(3)斑块间的连接程度对集合种群同步波动的发生有很大的影响,适当的破碎化有利于集合种群的续存.全局迁移和Allee效应结合起来增加了集合种群同步波动的可能,从而增加集合种群的灭绝风险.这些结果对理解同步性的机理、利用同步机理来制定物种保护策略和害虫防治都有重要的意义.