The effect of habitat fragmentation on cyclic population dynamics: a reduction to ordinary differential equations

Habitat fragmentation is known to be a key factor affecting population dynamics. In a previous study by Strohm and Tyson (Bull Math Biol 71:1323?C1348, 2009), the effect of habitat fragmentation on cyclic population dynamics was studied using spatially explicit predator?Cprey models with four different sets of reaction terms. The difficulty with spatially explicit models is that often analytical tractability is lost and the mechanisms behind the behaviour of the models are difficult to analyse. In this study, we employ a simplification procedure based on a Fourier series first-term truncation of the spatially explicit models Strohm and Tyson (Bull Math Biol 71:1323?C1348, 2009) to obtain spatially implicit models. These simpler models capture the main features of the spatially explicit models and can be used to explain the dynamics observed by Strohm and Tyson. We find that the spatially implicit models and the spatially explicit models produce similar responses to habitat fragmentation for larger high-quality patch sizes. Additionally, we find that the critical patch size of the spatially implicit models provides an upper bound on the critical patch size of the spatially explicit models. Finally, we derive an approximation of the multi-patch habitat by a single-patch habitat with partial flux boundary conditions which allows for a lower bound on the critical patch size to be calculated.     

Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in M<Emphasis Type="SmallCaps">ATLAB</Emphasis>

We present two finite-difference algorithms for studying the dynamics of spatially extended predator–prey interactions with the Holling type II functional response and logistic growth of the prey. The algorithms are stable and convergent provided the time step is below a (non-restrictive) critical value. This is advantageous as it is well-known that the dynamics of approximations of differential equations (DEs) can differ significantly from that of the underlying DEs themselves. This is particularly important for the spatially extended systems that are studied in this paper as they display a wide spectrum of ecologically relevant behavior, including chaos. Furthermore, there are implementational advantages of the methods. For example, due to the structure of the resulting linear systems, standard direct, and iterative solvers are guaranteed to converge. We also present the results of numerical experiments in one and two space dimensions and illustrate the simplicity of the numerical methods with short programs MATLAB. Users can download, edit, and run the codes from http://www.uoguelph.ca/~mgarvie/, to investigate the key dynamical properties of spatially extended predator–prey interactions.     

Homogenization techniques for population dynamics in strongly heterogeneous landscapes

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features.     

Stabilization through spatial pattern formation in metapopulations with long-range dispersal

Many studies of metapopulation models assume that spatially extended populations occupy a network of identical habitat patches, each coupled to its nearest neighbouring patches by density-independent dispersal. Much previous work has focused on the temporal stability of spatially homogeneous equilibrium states of the metapopulation, and one of the main predictions of such models is that the stability of equilibrium states in the local patches in the absence of migration determines the stability of spatially homogeneous equilibrium states of the whole metapopulation when migration is added. Here, we present classes of examples in which deviations from the usual assumptions lead to different predictions. In particular, heterogeneity in local habitat quality in combination with long-range dispersal can induce a stable equilibrium for the metapopulation dynamics, even when within-patch processes would produce very complex behaviour in each patch in the absence of migration. Thus, when spatially homogeneous equilibria become unstable, the system can often shift to a different, spatially inhomogeneous steady state. This new global equilibrium is characterized by a standing spatial wave of population abundances. Such standing spatial waves can also be observed in metapopulations consisting of identical habitat patches, i.e. without heterogeneity in patch quality, provided that dispersal is density dependent. Spatial pattern formation after destabilization of spatially homogeneous equilibrium states is well known in reaction–diffusion systems and has been observed in various ecological models. However, these models typically require the presence of at least two species, e.g. a predator and a prey. Our results imply that stabilization through spatial pattern formation can also occur in single-species models. However, the opposite effect of destabilization can also occur: if dispersal is short range, and if there is heterogeneity in patch quality, then the metapopulation dynamics can be chaotic despite the patches having stable equilibrium dynamics when isolated. We conclude that more general metapopulation models than those commonly studied are necessary to fully understand how spatial structure can affect spatial and temporal variation in population abundance.     

Reconstructing the dynamics of unobserved variables in spatially extended systems

Attractor reconstruction using embedding techniques is a widely used tool when analysing data from real systems. It allows reconstruction of the system dynamics from only one observable and is thus extremely powerful. We show here that this reconstruction is also possible from spatially coupled systems. We use a common host–parasitoid model as an example as ecological systems are virtually always spatially extended. Additionally, data from ecological systems has often only one observable, e.g. population density, from a potentially much higher-dimensional system. Singular value decomposition is used to show the existence of a functional relationship mapping the time delayed coordinates of one variable to the full spatially coupled system. We investigate the effects of noise and indicate two important spatial scales. Finally, we illustrate that a reconstruction can be obtained from a system that is only partially sampled.     

Area-dependent migration by ringlet butterflies generates a mixture of patchy population and metapopulation attributes

Interpretation of spatially structured population systems is critically dependent on levels of migration between habitat patches. If there is considerable movement, with each individual visiting several patches, there is one ”patchy population”; if there is intermediate movement, with most individuals staying within their natal patch, there is a metapopulation; and if (virtually) no movement occurs, then the populations are separate (Harrison 1991, 1994). These population types actually represent points along a continuum of much to no mobility in relation to patch structure. Therefore, interpretation of the effects of spatial structure on the dynamics of a population system must be accompanied by information on mobility. We use empirical data on movements by ringlet butterflies, Aphantopus hyperantus, to investigate two key issues that need to be resolved in spatially-structured population systems. First, do local habitat patches contain largely independent local populations (the unit of a metapopulation), or merely aggregations of adult butterflies (as in patchy populations)? Second, what are the effects of patch area on migration in and out of the patches, since patch area varies considerably within most real population systems, and because human landscape modification usually results in changes in habitat patch sizes? Mark-release-recapture (MRR) data from two spatially structured study systems showed that 63% and 79% of recaptures remained in the same patch, and thus it seems reasonable to call both systems metapopulations, with some capacity for separate local dynamics to take place in different local patches. Per capita immigration and emigration rates declined with increasing patch area, while the resident fraction increased. Actual numbers of emigrants either stayed the same or increased with area. The effect of patch area on movement of individuals in the system are exactly what we would have expected if A. hyperantus were responding to habitat geometry. Large patches acted as local populations (metapopulation units) and small patches simply as locations with aggregations (units of patchy populations), all within 0.5 km². Perhaps not unusually, our study system appears to contain a mixture of metapopulation and patchy-population attributes.     

Viability analyses with habitat-based metapopulation models

Population viability analysis (PVA) models incorporate spatial dynamics in different ways. At one extreme are the occupancy models that are based on the number of occupied populations. The simplest occupancy models ignore the location of populations. At the other extreme are individual-based models, which describe the spatial structure with the location of each individual in the population, or the location of territories or home ranges. In between these are spatially structured metapopulation models that describe the dynamics of each population with structured demographic models and incorporate spatial dynamics by modeling dispersal and temporal correlation among populations. Both dispersal and correlation between each pair of populations depend on the location of the populations, making these models spatially structured. In this article, I describe a method that expands spatially structured metapopulation models by incorporating information about habitat relationships of the species and the characteristics of the landscape in which the metapopulation exists. This method uses a habitat suitability map to determine the spatial structure of the metapopulation, including the number, size, and location of habitat patches in which subpopulations of the metapopulation live. The habitat suitability map can be calculated in a number of different ways, including statistical analyses (such as logistic regression) that find the relationship between the occurrence (or, density) of the species and independent variables which describe its habitat requirements. The habitat suitability map is then used to calculate the spatial structure of the metapopulation, based on species-specific characteristics such as the home range size, dispersal distance, and minimum habitat suitability for reproduction. Received: April 1, 1999 / Accepted: October 29, 1999     

Local variation in plant quality influences large‐scale population dynamics

Spatial variation in ecological systems can arise both as a consequence of variation in the quality and availability of resources and as an emergent property of spatially structured interactions. We used a spatially explicit model to simulate populations of herbivore hosts and their parasitoids in landscapes with different levels of variance in plant patch quality and different spatial arrangements of high‐ and low‐quality plant patches. We found that even small variation in patch quality at a fine spatial scale decreased overall herbivore populations, as parasitoid populations on low‐quality plant patches were subsidized by those from high‐quality neighbors. On landscapes with large, homogeneous regions of high‐ and low‐quality plant patches, herbivore populations increased with variation in patch quality. Overall, our results demonstrate that local variation in resource quality profoundly influences global population dynamics. In particular, fine‐scale variation in plant patch quality enhanced biological control of herbivores by parasitoids, suggesting that adding back plant genetic variation into perennial production systems may enhance the biological control of herbivores by their natural enemies.     

Mapping the Information Landscape: Discerning Peaks and Valleys for Ecological Monitoring

We investigate previously unreported phenomena that have a potentially significant impact on the design of surveillance monitoring programs for ecological systems. Ecological monitoring practitioners have long recognized that different species are differentially informative of a system’s dynamics, as codified in the well-known concepts of indicator or keystone species. Using a novel combination of analysis techniques from nonlinear dynamics, we describe marked variation among spatial sites in information content with respect to system dynamics in the entire region. We first observed these phenomena in a spatially extended predator–prey model, but we observed strikingly similar features in verified water-level data from a NOAA/NOS Great Lakes monitoring program. We suggest that these features may be widespread and the design of surveillance monitoring programs should reflect knowledge of their existence.     

Using connectivity metrics in conservation planning – when does habitat quality matter?

Aim The objective of conservation planning is often to prioritize patches based on their estimated contribution to metapopulation or metacommunity viability. The contribution that an individual patch makes will depend on its intrinsic characteristics, such as habitat quality, as well as its location relative to other patches, its connectivity. Here we systematically evaluate five patch value metrics to determine the importance of including an estimate of habitat quality into the metrics. Location We tested the metrics in landscapes designed to represent different degrees of variability in patch quality and different levels of patch aggregation. Methods In each landscape, we simulated population dynamics using a spatially explicit, continuous time metapopulation model linked to within patch logistic growth models. We tested five metrics that are used to estimate the contribution that a patch makes to metapopulation viability: two versions of the probability of connectivity index, two versions of patch centrality (a graph theory metric) and the metapopulation capacity metric. Results All metrics performed best in environments where patch quality was very variable and high quality patches were aggregated. Metrics that incorporated some measure of patch quality did better in all environments, but did particularly well in environments with high variance of patch quality and spatial aggregation of good quality patches. Main conclusions Including an estimate of patch quality significantly increased the ability of a given connectivity metric to rank correctly habitat patches according to their contribution to metapopulation viability. Incorporating patch quality is particularly important in landscapes where habitat quality is highly variable and good quality patches are spatially aggregated. However, caution should be used when applying patch metrics to homogeneous landscapes, even if good estimates of patch quality are available. Our results demonstrate that landscape structure and the degree of variability in patch quality need to be assessed prior to selecting a suitable method for estimating patch value.     

The effect of migration on metapopulation stability is qualitatively unaffected by demographic and spatial heterogeneity

Coupled map lattices (CMLs), using two coupled logistic equations, have been extensively used to model the dynamics of two-patch ecological systems. Such studies have revealed that migration rate plays an important role in determining the dynamics of the system, particularly when the two maps differ in their intrinsic growth rate parameter, r. However, under more realistic assumptions, a metapopulation can be expected to consist of more than two subpopulations, each with its own demographic parameters, which will in part be a function of the environment of that patch. The role of the spatial arrangement of heterogeneous (i.e. with different r values) subpopulations in shaping the dynamics of such a metapopulation has rarely been investigated. Here, we study the effect of demographic and spatial heterogeneity on the stability of one- and two-dimensional systems of 64 coupled Ricker maps with different r values, under periodic and absorbing boundary conditions. We show that the effects of migration rate on metapopulation stability do not depend upon either the precise spatial arrangement of the subpopulations in the lattice, or on the presence of a moderate proportion of vacant (uninhabitable) patches in the lattice. The results, thus, suggest that metapopulation models are robust to variation in spatial arrangement of patch quality and, hence, of demographic parameters. We also show that for any given arrangement of the patches, maximum stability of the metapopulation occurs when the migration levels are intermediate, a result that agrees well with previous studies on two-map CML systems.     

Spatial heterogeneity enhance robustness of large multi-species ecosystems

Understanding ecosystem stability and functioning is a long-standing goal in theoretical ecology, with one of the main tools being dynamical modelling of species abundances. With the help of spatially unresolved (well-mixed) population models and equilibrium dynamics, limits to stability and regions of various ecosystem robustness have been extensively mapped in terms of diversity (number of species), types of interactions, interaction strengths, varying interaction networks (for example plant-pollinator, food-web) and varying structures of these networks. Although many insights have been gained, the impact of spatial extension is not included in this body of knowledge. Recent studies of spatially explicit modelling on the other hand have shown that stability limits can be crossed and diversity increased for systems with spatial heterogeneity in species interactions and/or chaotic dynamics. Here we show that such crossing and diversity increase can appear under less strict conditions. We find that the mere possibility of varying species abundances at different spatial locations make possible the preservation or increase in diversity across previous boundaries thought to mark catastrophic transitions. In addition, we introduce and make explicit a multitude of different dynamics a spatially extended complex system can use to stabilise. This expanded stabilising repertoire of dynamics is largest at intermediate levels of dispersal. Thus we find that spatially extended systems with intermediate dispersal are more robust, in general have higher diversity and can stabilise beyond previous stability boundaries, in contrast to well-mixed systems.     

Global existence for semilinear reaction–diffusion systems on evolving domains

We present global existence results for solutions of reaction–diffusion systems on evolving domains. Global existence results for a class of reaction–diffusion systems on fixed domains are extended to the same systems posed on spatially linear isotropically evolving domains. The results hold without any assumptions on the sign of the growth rate. The analysis is valid for many systems that commonly arise in the theory of pattern formation. We present numerical results illustrating our theoretical findings.     

Spatial mechanisms for coexistence of species sharing a common natural enemy

We examine the conditions under which spatial structure can mediate coexistence of apparent competitors. We use a spatially explicit, host-parasitoid metapopulation model incorporating local dynamics of Nicholson-Bailey type and global dispersal. Depending on the model parameters, the resulting system displays a plethora of asynchronous dynamical behaviors for which permanent or transient coexistence is observed. We identify a number of spatially mediated tradeoffs which apparent competitors can utilize and demonstrate that the dynamics of spatial coexistence can typically be understood from consideration of two and three patch systems. The phase relationships of species abundances are different for our model than for some other mechanisms of spatial coexistence. We discuss the implications of our findings relative to issues of community organization and biological conservation.     

Effective dispersal rate is a function of habitat size and corridor shape: mechanistic formulation of a two‐patch compartment model for spatially continuous systems

Two‐patch compartment models have been explored to understand the spatial processes that promote species coexistence. However, a phenomenological definition of the inter‐patch ‘dispersal rate’ has limited the quantitative predictability of these models to community dynamics in spatially continuous habitats. Here, we mechanistically rederived a two‐patch Lotka–Volterra competition model for a spatially continuous reaction‐diffusion system where a narrow corridor connects two large habitats. We provide a mathematical formula of the dispersal rate appearing in the two‐patch compartment model as a function of habitat size, corridor shape (ratio of its width to its length), and organism diffusion coefficients. For most reasonable settings, the two‐patch compartment model successfully approximated not only the steady states, but also the transient dynamics of the reaction–diffusion model. Further numerical simulations indicated the general applicability of our formula to other types of community dynamics, e.g. driven by resource‐competition, in spatially homogeneous and heterogeneous environments. Our results suggest that the spatial configuration of habitats plays a central role in community dynamics in space. Furthermore, our new framework will help to improve experimental designs for quantitative test of metacommunity theories and reduce the gaps among modeling, empirical studies, and their application to landscape management.     

Allee threshold and extinction threshold for spatially explicit metapopulation dynamics with Allee effects

In this paper, we investigate a spatially explicit metapopulation model with Allee effects. We refer to the patch occupancy model introduced by Levins (Bull Entomol Soc Am 15:237–240, 1969) as a spatially implicit metapopulation model, i.e., each local patch is either occupied or vacant and a vacant patch can be recolonized by a randomly chosen occupied patch from anywhere in the metapopulation. When we transform the model into a spatially explicit one by using a lattice model, the obtained model becomes theoretically equivalent to a “lattice logistic model” or a “basic contact process”. One of the most popular or standard metapopulation models with Allee effects, developed by Amarasekare (Am Nat 152:298–302, 1998), supposes that those effects are introduced formally by means of a logistic equation. However, it is easier to understand the ecological meaning of associating Allee effects with this model if we suppose that only the logistic colonization term directly suffers from Allee effects. The resulting model is also well defined, and therefore we can naturally examine it by Monte Carlo simulation and by doublet and triplet decoupling approximation. We then obtain the following specific features of one-dimensional lattice space: (1) the metapopulation as a whole does not have an Allee threshold for initial population size even when each local population follows the Allee effects; and (2) a metapopulation goes extinct when the extinction rate of a local population is lower than that in the spatially implicit model. The real ecological metapopulation lies between two extremes: completely mixing interactions between patches on the one hand and, on the other, nearest neighboring interactions with only two nearest neighbors. Thus, it is important to identify the metapopulation structure when we consider the problems of invasion species such as establishment or the speed of expansion.     

Small Heterogeneity Has Large Effects on Synchronization of Ecological Oscillators

Heterogeneity in habitat plays a crucial role in the dynamics of spatially extended populations and is often ignored by both empiricists and theoreticians. A common assumption made is that spatially homogeneous systems and those with slight heterogeneity will behave similarly and, therefore, the results and data from studies of the former can be applied to the latter. Here, we test this assumption by deriving a phase model from two weakly coupled predator-prey oscillators and analyze the effect of spatial heterogeneity on the phase dynamics of this system. We find that even small heterogeneity between the two patches causes substantial changes in the phase dynamics of the system which can have dramatic effects on both population dynamics and persistence. Additionally, if the prey and predator time scales are similar, the effect of heterogeneity is much greater.     

Pattern does not equal process: what does patch occupancy really tell us about metapopulation dynamics?

Patch occupancy surveys are commonly used to parameterize metapopulation models. If isolation predicts patch occupancy, this is generally attributed to a balance between distance-dependent recolonization and spatially independent extinctions. We investigated whether similar patterns could also be generated by a process of spatially correlated extinctions following a unique colonization event (analogous to nonequilibrium processes in island biogeography). We simulated effects of spatially correlated extinctions on patterns of patch occupancy among pikas (Ochotona princeps) at Bodie, California, using randomly located extinction disks to represent the likely effects of predation. Our simulations produced similar patterns to those cited as evidence of balanced metapopulation dynamics. Simulations using a variety of disk sizes and patch configurations confirmed that our results are potentially applicable to a broad range of species and sites. Analyses of the observed patterns of patch occupancy at Bodie revealed little evidence of rescue effects and strong evidence that most recolonizations are ephemeral in nature. Persistence will be overestimated if static or declining patterns of patch occupancy are mistakenly attributed to dynamically stable metapopulation processes. Consequently, simple patch occupancy surveys should not be considered as substitutes for detailed experimental tests of hypothesized population processes, particularly when conservation concerns are involved.