Stable Metapopulations Persisting among Locally Unstable Habitats

Stable metapopulations have long been recognized as an example where a stabilizing balance between migrations and local extinctions provides for persistence in a metapopulation, that is, a population that inhabits a spatial system of patchy habitats connected by migrations. This is also an example where spatial heterogeneity provides for persistence; hence, more complex spatial organization does beget stability. The issue can also be considered in terms of the community matrix as follows.

Let the environment consist of h > 2 habitats, each harboring a community of n > 2 species whose local dynamics N(t) obey a system of ODEs [1]. If the habitats were isolated, they would have equilibria N*, whose stability/instability would depend on the eigenvalues of community matrix J(N*) [4]. The state of the 'metacommunity' is now described by nh vector M(t) composed of h versions of the n vector:

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