Equilibrium Stability of Single-species Metapopulations

Abstract

We investigate the effect of migration between local populations of a single discrete-generation species living in a ring or an array of habitats. The commonly used symmetric dispersal assumption is relaxed to include the biologically more reasonable asymmetric dispersion. It is demonstrated analytically that density independent migration has no effect on the equilibrium stability of individual populations. However, the positive equilibrium may be destabilizing if the migration is density dependent in such a way that it increases with increasing population density at the source patch.

Costly dispersal can destabilize the homogeneous equilibrium of a metapopulation

“…I prove that the homogeneous equilibrium ˆ( ,..., ) N N of a metapopulation is stable if and only if N is a stable equilibrium of an isolated population, which is subject to extra mortality that equals the average dispersal-related mortality of the metapopulation. This direct link between the dynamics of the metapopulation and the dynamics of an isolated population is an intuitive and straightforward extension of the result obtained by Rohani et al (1996), Jang and Mitra (2000) and Jansen and Lloyd (2000), but it does not guarantee that the stability of the metapopulation is independent of dispersal. It is easy to construct examples of the single-population dynamics such that adding extra mortality destabilizes the equilibrium.…”

“…Cost-free dispersal has no effect (cf. Rohani et al 1996;Jang and Mitra 2000;Jansen and Lloyd 2000). Because the effect on stability is independent of any aspect of dispersal but the average cost, costly dispersal can destabilize the homogeneous equilibrium even in the simplest metapopulation model with a global dispersal pool.…”

“…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). …”

“…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).…”

“…One such general result was given by Rohani et al (1996) and, under more relaxed conditions, by Jang and Mitra (2000); both are special cases of the multi-species model of Jansen and Lloyd (2000). These authors proved that in a homogeneous metapopulation of a single species, where local populations are unstructured and interact solely via passive and cost-free dispersal, the metapopulation equilibrium where all local populations are characterised by the same equilibrium population density N is locally stable if and only if N is a stable equilibrium of a single population in absence of dispersal.…”