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Zbl 0598.92017
Gard, Thomas C.
Stability for multispecies population models in random environments. (English)
[J] Nonlinear Anal., Theory Methods Appl. 10, 1411-1419 (1986). ISSN 0362-546X

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

MSC 2000:: *92D25 Population dynamics
60H10 Stochastic ordinary differential equations
92D40 Ecology
93E15 Stochastic stability

Keywords: stability; random environmental fluctuations; multispecies populations; deterministic multispecies model; nondegenerate case; stochastic equilibrium stability; global asymptotic stochastic stability; random noise fluctuations

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