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Stability for multispecies population models in random environments. (English) Zbl 0598.92017

In this paper, the stability consequences of taking into account random environmental fluctuations by stochastic differential equation models for multispecies populations are discussed. The starting point is a deterministic multispecies model with a globally stable feasible equilibrium. When noise terms are added, the stochastic model that arises exhibits a degree of stability directly related to how these noise terms respect the equilibrium.

In particular, section 3 reviews the situation when the noise terms do not vanish at the equilibrium (the nondegenerate case); although stochastic equilibrium stability is impossible in this case, weaker types of stability may hold. In particular, there may exist a stable invariant distribution with positive density.

It is mentioned in section 4 that global asymptotic stochastic stability may hold if noise intensities vanish at the equilibrium (the degenerate case). However, there is some question concerning the biological relevance of this assumption.

Finally, the discussion and results in sections 3 and 4 indicate that random noise fluctuations need not have a destabilizing effect in multispecies models.

92D25Population dynamics (general)
60H10Stochastic ordinary differential equations
93E15Stochastic stability