Stability in a Metapopulation Model with Density-dependent Dispersal

Rethinking the Prevalence and Relevance of Chaos in Ecology

Chaos was proposed in the 1970s as an alternative explanation for apparently noisy fluctuations in population size. Although readily demonstrated in models, the search for chaos in nature proved challenging and led many to conclude that chaos is either rare or nigh impossible to detect. However, in the intervening half-century, it has become clear that ecosystems are replete with the enabling conditions for chaos. Chaos has been repeatedly demonstrated under laboratory conditions and has been found in field data using updated detection methods. Together, these developments indicate that the apparent rarity of chaos was an artifact of data limitations and overreliance on low-dimensional population models. We invite readers to reevaluate the relevance of chaos in ecology, and we suggest that chaos is not as rare or undetectable as previously believed.

Impact of dispersal on the stability of metapopulations

Dispersal is a key ecological process that enables local populations to form spatially extended systems called metapopulations. In the present study, we investigate how dispersal affects the linear stability of a general single-species metapopulation model. We discuss both the influence of local within-patch dynamics and the effects of various dispersal behaviours on stability. We find that positive density-dependent dispersal and positive density-dependent settlement are destabilizing dispersal behaviours while negative density-dependent dispersal and negative density-dependent settlement are stabilizing. It is also shown that dispersal has a stabilizing impact on heterogeneous metapopulations that correlates positively with the number of patches and the connectance of metapopulation networks.

The integration of climate change, spatial dynamics, and habitat fragmentation: A conceptual overview

A growing number of studies have looked at how climate change alters the effects of habitat fragmentation and degradation on both single and multiple species; some raise concern that biodiversity loss and its effects will be exacerbated. The published literature on spatial dynamics (such as dispersal and metapopulation dynamics), habitat fragmentation and climate change requires synthesis and a conceptual framework to simplify thinking. We propose a framework that integrates how climate change affects spatial population dynamics and the effects of habitat fragmentation in terms of: (i) habitat quality, quantity and distribution; (ii) habitat connectivity; and (iii) the dynamics of habitat itself. We use the framework to categorize existing autecological studies and investigate how each is affected by anthropogenic climate change. It is clear that a changing climate produces changes in the geographic distribution of climatic conditions, and the amount and quality of habitat. The most thorough published studies show how such changes impact metapopulation persistence, source-sink dynamics, changes in species' geographic range and community composition. Climate-related changes in movement behavior and quantity, quality and distribution of habitat have also produced empirical changes in habitat connectivity for some species. An underexplored area is how habitat dynamics that are driven by climatic processes will affect species that live in dynamic habitats. We end our discussion by suggesting ways to improve current attempts to integrate climate change, spatial population dynamics and habitat fragmentation effects, and suggest distinct areas of study that might provide opportunities for more fully integrative work.

Population distribution and synchronized dynamics in a metapopulation model in two geographic scales

In this paper, a metapopulation model composed of patches distributed in two spatial scales is proposed in order to study the stability of synchronous dynamics. Clusters of patches connected by short-range dispersal are assumed to be formed. Long distance dispersal is responsible to link patches that are in different clusters. During each time step, we assume that there are three processes involved in the population dynamics: (a) the local dynamics, which consists of reproduction and survival; (b) short-range dispersal of individuals between the patches of each cluster; and (c) the movement between the clusters. First we present an analytic criterion for regional synchronization, where the clusters evolve with the same dynamics. We then discuss the possibility of a full synchronism, where all patches in the network follow the same time evolution. The existence of such a state is not always ensured, even considering that all patches have the same local dynamics. It depends on how the individuals are distributed among the local patches that compose a cluster after long-range dispersal takes place in the regional scale. An analytic criterion for the stability of synchronized trajectories in this case is obtained.

Modeling density dependence in heterogeneous landscapes: dispersal as a case study

Population models often pose density-dependent rates as relations between current population size on a habitat patch, n, and some threshold size defined by limiting resources, r. In fourteen recent modeling studies incorporating density-dependent dispersal, formulations of the density-dependent rate (or probability) fall into two distinct groups, expressing the rate as a function of n-r or n/r. These two depictions of the same process differ fundamentally: they can cause strikingly different dynamics in otherwise identical systems and they have different scaling properties in heterogeneous landscapes. Here I consider the implications of the two formulations under two broad ecological scenarios: scramble competition for an equally divided resource (e.g. food) and contest competition for an unequally divided resource (e.g. nest sites). In both cases, simple arguments show that the n/r form is preferable when density dependence is driven by individual access to resources. Other circumstances may require different formulations, but modelers must ensure that these have appropriate scaling and non-equilibrium behavior.

Sudden shifts in ecological systems: intermittency and transients in the coupled Ricker population model

Many real ecological systems show sudden changes in behavior, phenomena sometimes categorized as regime shifts in the literature. The relative importance of exogenous versus endogenous forces producing regime shifts is an important question. These forces' role in generating variability over time in ecological systems has been explored using tools from dynamical systems. We use similar ideas to look at transients in simple ecological models as a way of understanding regime shifts. Based in part on the theory of crises, we carefully analyze a simple two patch spatial model and begin to understand from a mathematical point of view what produces transient behavior in ecological systems. In particular, since the tools are essentially qualitative, we are able to suggest that transient behavior should be ubiquitous in systems with overcompensatory local dynamics, and thus should be typical of many ecological systems.

Effects of density-dependent migrations on stability of a two-patch predator–prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.

Density-dependent migration and synchronism in metapopulations

A spatially explicit metapopulation model with density-dependent dispersal is proposed in order to study the stability of synchronous dynamics. A stability criterion is obtained based on the computation of the transversal Liapunov number of attractors on the synchronous invariant manifold. We examine in detail a special case of density-dependent dispersal rule where migration does not occur if the patch density is below a certain critical density, while the fraction of individuals that migrate to other patches is kept constant if the patch density is above the threshold level. Comparisons with density-independent migration models indicate that this simple density-dependent dispersal mechanism reduces the stability of synchronous dynamics. We were able to quantify exactly this loss of stability through the frequency that synchronous trajectories are above the critical density.

Stability in an age-structured metapopulation model

We present a discrete model for a metapopulation of a single species with overlapping generations. Based on the dynamical behavior of the system in absence of dispersal, we have shown that a migration mechanism which depends only on age can not stabilize a previously unstable homogeneous equilibrium, but can drive a stable uncoupled system to instability if the migration rules are strongly related to age structure.