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.     

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.     

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.     

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.     

A reconciliation of local and global models for bone remodeling through optimization theory

Remodeling rules with either a global or a local mathematical form have been proposed for load-bearing bones in the literature. In the local models, the bone architecture (shape, density) is related to the strains/energies sensed at any point in the bone, while in the global models, a criterion believed to be applicable to the whole bone is used. In the present paper, a local remodeling rule with a strain error form is derived as the necessary condition for the optimum of a global remodeling criterion, suggesting that many of the local error-driven remodeling rules may have corresponding global optimization-based criteria. The global criterion proposed in the present study is a trade-off between the cost of metabolic growth and use, mathematically represented by the mass, and the cost of failure, mathematically represented by the total strain energy. The proposed global criterion is shown to be related to the optimality criteria methods of structural optimization by the equivalence of the model solution and the fully stressed solution for statically determinate structures. In related work, the global criterion is applied to simulate the strength recovery in bones with screw holes left behind after removal of fracture fixation plates. The results predicted by the model are shown to be in good agreement with experimental results, leading to the conclusion that load-bearing bones are structures with optimal shape and property for their function.     

Fitting parameters of stochastic birth-death models to metapopulation data

Populations that are structured into small local patches are a common feature of ecological and epidemiological systems. Models describing this structure are often referred to as metapopulation models in ecology or household models in epidemiology. Small local populations are subject to demographic stochasticity. Theoretical studies of household disease models without resistant stages (SIS models) have shown that local stochasticity can be ignored for between patch disease transmission if the number of connected patches is large. In that case the distribution of the number of infected individuals per household reaches a stationary distribution described by a birth-death process with a constant immigration term. Here we show how this result, in conjunction with the balancing condition for birth-death processes, provides a framework to estimate demographic parameters from a frequency distribution of local population sizes. The parameter estimation framework is applicable to estimate parameters of disease transmission models as well as metapopulation models.     

Spatially implicit plankton population models: Transient spatial variability

Ocean plankton models are useful tools for understanding and predicting the behaviour of planktonic ecosystems. However, when the regions represented by the model grid cells are not well mixed, the population dynamics of grid cell averages may differ from those of smaller scales (such as the laboratory scale). Here, the ‘mean field approximation’ fails due to ‘biological Reynolds fluxes’ arising from nonlinearity in the fine-scale biological interactions and unresolved spatial variability. We investigate the domain-scale behaviour of two-component, 2D reaction-diffusion plankton models producing transient dynamics, with spatial variability resulting only from the initial conditions. Failure of the mean field approximation can be quite significant for sub grid-scale mixing rates applicable to practical ocean models. To improve the approximation of domain-scale dynamics, we investigate implicit spatial resolution methods such as spatial moment closure. For weak and moderate strengths of biological nonlinearity, spatial moment closure models generally yield significant improvements on the mean field approximation, especially at low mixing rates. However, they are less accurate given weaker transience and stronger nonlinearity. In the latter case, an alternative ‘two-spike’ approximation is accurate at low mixing rates. We argue that, after suitable extension, these methods may be useful for understanding and skillfully predicting the large-scale behaviour of marine ecosystems.     

Daphnia revisited: local stability and bifurcation theory for physiologically structured population models explained by way of an example

We consider the interaction between a general size-structured consumer population and an unstructured resource. We show that stability properties and bifurcation phenomena can be understood in terms of solutions of a system of two delay equations (a renewal equation for the consumer population birth rate coupled to a delay differential equation for the resource concentration). As many results for such systems are available (Diekmann et al. in SIAM J Math Anal 39:1023–1069, 2007), we can draw rigorous conclusions concerning dynamical behaviour from an analysis of a characteristic equation. We derive the characteristic equation for a fairly general class of population models, including those based on the Kooijman–Metz Daphnia model (Kooijman and Metz in Ecotox Env Saf 8:254–274, 1984; de Roos et al. in J Math Biol 28:609–643, 1990) and a model introduced by Gurney–Nisbet (Theor Popul Biol 28:150–180, 1985) and Jones et al. (J Math Anal Appl 135:354–368, 1988), and next obtain various ecological insights by analytical or numerical studies of special cases.     

Spatial scale of local adaptation in a plant-pathogen metapopulation

The rate and scale of gene flow can strongly affect patterns of local adaptation in host-parasite interactions. I used data on regional pathogen occurrence to infer the scale of pathogen dispersal and to identify pathogen metapopulations in the interaction between Plantago lanceolata and its specialist phytopathogen, Podosphaera plantaginis. Frequent extinctions and colonizations were recorded in the metapopulations, suggesting substantial gene flow at this spatial scale. The level of pathogen local adaptation was assessed in a laboratory inoculation experiment at three different scales: in sympatric host populations, in sympatric host metapopulations and in allopatric host metapopulations. I found evidence for adaptation to sympatric host populations, as well as evidence indicating that local adaptation may extend to the scale of the sympatric host metapopulation. There was also variation among the metapopulations in the degree of pathogen local adaptation. This may be explained by regional differences in the rate of migration.