Synchronism in a metapopulation model

QUASI-DETERMINISTIC POPULATION DYNAMICS IN STOCHASTIC COUPLED MAPS

We studied the stochastic dynamics of coupled map lattices for small local populations. Quasi-deterministic dynamics is lost when considering one isolated population or two populations linked by dispersal. In a one-dimensional ring linked by closest neighbors some intermittent synchronization is observed. Finally we show that in two-dimensional lattices long-term synchronization take place above a critical value of the dispersal probability and the system displays a quasi-deterministic dynamics.

Cluster formation in a heterogeneous metapopulation model

A spatially explicit heterogeneous metapopulation model with two different patch types is analyzed. Some network topologies support a partially synchronized dynamics, a state where two different clusters of patches are formed. Within each cluster the dynamics of all patches are synchronized. The linearized asymptotic stability of the partially synchronized attractor is studied. The transversal stability is analyzed and a simple expression for the transversal Lyapunov number of partially synchronized attractors is obtained.

How does intraspecific density regulation influence metapopulation synchrony and persistence?

Intraspecific density regulation influences the synchronization of local population dynamics through dispersal. Spatial synchrony in turn may jeopardize metapopulation persistence. Joining results from previous studies suggests that spatial synchrony is highest at moderate over-compensation and is low at compensating and at very strong over-compensating density regulation. We scrutinize this supposition of a unimodal relationship using a process-based metapopulation model with explicit local population dynamics. We extend the usually studied range of density regulation to under-compensation and analyse resulting metapopulation persistence. We find peaks of spatial synchrony not only at over-compensatory but also under-compensatory density regulation and show that effects of local density compensation on synchrony follow a bimodal rather than unimodal relationship. Persistence of metapopulations however, shows a unimodal relationship with a broad plateau of high persistence from compensatory to over-compensatory density regulation. This range of high persistence comprises both levels of low and high spatial synchrony. Thus, not synchrony alone jeopardizes metapopulation persistence, but only in interplay with high local extinction risk. The functional forms of the relations of density compensation with spatial synchrony and persistence are robust to increases in dispersal mortality, landscape dynamics, or density dependence of dispersal. However, with each of these increases the maxima of spatial synchrony and persistence shift to higher over-compensation and levels of synchrony are reduced. Overall, for over-compensation high landscape connectivity has negative effects while for under-compensation connectivity affects persistence positively. This emphasizes the importance of species life-history traits for management decisions with regard to landscape connectivity: while dispersal corridors are essential for species with under-compensatory density regulation, they may have detrimental effects for endangered species with over-compensation.

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.

Global asymptotic coherence in discrete dynamical systems

Spatial synchrony (coherence) in dynamical systems is of both theoretical and applied importance. We address this problem for a generalization of coupled map lattices (CMLs). In the systems we study, which we term "meta-CMLs," the map at each lattice point may be multidimensional (corresponding, for example, to multispecies ecological systems in which all species have the same dispersal pattern). Most previous work on coherence of CMLs has focused on local stability. Here, we prove a global theorem that provides a useful sufficient condition guaranteeing decay of incoherence in meta-CMLs regardless of initial conditions and regardless of the nature of the attractors of the system. This result facilitates useful analyses of a variety of applied problems, including conservation of endangered species and eradication of pests or infectious diseases.

Unexpected coherence and conservation

The effects of migration in a network of patch populations, or metapopulation, are extremely important for predicting the possibility of extinctions both at a local and a global scale. Migration between patches synchronizes local populations and bestows upon them identical dynamics (coherent or synchronous oscillations), a feature that is understood to enhance the risk of global extinctions. This is one of the central theoretical arguments in the literature associated with conservation ecology. Here, rather than restricting ourselves to the study of coherent oscillations, we examine other types of synchronization phenomena that we consider to be equally important. Intermittent and out-of-phase synchronization are but two examples that force us to reinterpret some classical results of the metapopulation theory. In addition, we discuss how asynchronous processes (for example, random timing of dispersal) can paradoxically generate metapopulation synchronization, another non-intuitive result that cannot easily be explained by the standard theory.

Coherence and conservation

A principal aim of current conservation policy is to reduce the impact of habitat fragmentation. Conservation corridors may achieve this goal by facilitating movement among isolated patches, but there is a risk that increased connectivity could synchronize local population fluctuations (causing coherent oscillations) and thereby increase the danger of global extinction. We identify general conditions under which populations can or cannot undergo coherent oscillations, and we relate these conditions to local and global extinction probabilities. We suggest a simple method to explore the potential success of conservation corridors and, more generally, any manipulations of dispersal patterns that aim to protect threatened species or control pests.