Stability in a Metapopulation Model with Density-dependent Dispersal

Abstract

A spatially explicit metapopulation model with positive density-dependent migration is analysed. We obtained conditions under which a previously stable system can be driven to instability caused by a density-dependent migration mechanism. The stability boundary depends on the rate of increase of the number of migrants on each site at local equilibrium, on the intrinsic rate of increase at local level, on the number of patches, and on topological aspects regarding the connectivity between patches. A concrete example is presented illustrating the dynamics on the dispersal-induced unstable regime.

Costly dispersal can destabilize the homogeneous equilibrium of a metapopulation

“…The homogeneous equilibrium can be destabilized if dispersal is cost-free but depends on local population density (Ruxton 1996;Jang and Mitra 2000;Silva et al 2001). …”

“…by age or because there are several interacting species (such as predator-prey or host-parasitoid), provided that different age/stage classes or different species differ in dispersal (Hastings 1992;Rohani and Ruxton 1999a,b;Jansen and Lloyd 2000;White and White, 2005;de Castro et al 2006). Concerning the reverse case, cost-free dispersal cannot stabilize the homogeneous equilibrium if it is unstable in absence of dispersal: this holds for density-dependent dispersal (Silva et al 2001) and for multi-species systems (Jansen and Lloyd 2000) as well. A simple continuity argument shows that adding dispersal cost can change this conclusion.…”

“…Since the leading eigenvalue of a stochastic matrix is 1, the leading eigenvalue of B must be ; and the eigenvalue of the Jacobian matrix This proof is a straightforward extension of Jang and Mitra (2000) to the case of costly dispersal (but as Silva et al (2001) also noticed, matrix ( ) ij ( ) ij need not be irreducible as Jang and Mitra (2000) required for the proof to hold; indeed, two unconnected networks of populations with identical within-population dynamics and identical value of yield a reducible matrix for ( ) ij ( ) ij , yet the two networks will independently equilibrate to the same homogeneous equilibrium upon a small perturbation). The same result can also be obtained from the framework of Jansen and Lloyd (2000) by introducing some cost to dispersal (but their technical assumption that ( ) ij ( ) ij is diagonalizable is not necessary for the proof above).…”

Double-Jump migration and diffusive instability

“…In order to study the effects of the (cross-) density-dependence at the end of a migration event on diffusive instability we formulated the continuous-time model with double-jump migration. Clearly one can also study the issue by discrete-time models which may be formulated by extending the standard models (e.g., Ruxton, 1996;Rohani and Ruxton, 1999;Silva et al, 2001).…”

Sincronização em Metapopulações com Migração Dependente da Densidade

“…Por exemplo, Rohani et al [17], Jang e Mitra [13], Silva et al [19] concluíram que a estabilidade do estado homogêneo nãoé afetada quando a migraçãoé independente da densidade. Mas o estado homogêneo pode perder a estabilidade quando a migraçãoé dependente da idade (Hastings [11]), em sistemas do tipo hospedeiro-parasitóide (Rohani e Ruxton [18]) e no caso da migração dependente da densidade (ver Silva et al [19] …”