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

Gard, Thomas C.

Nonlinear Anal., Theory Methods Appl. 10, 1411-1419 (1986).

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.

MSC:

92D25	Population dynamics (general)
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
92D40	Ecology
93E15	Stochastic stability in control theory

Keywords:

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

References:

[1]	Arnold, L., Stochastic Differential Equations, Theory and Applications (1974), Wiley: Wiley New York
[2]	Barra, M.; Del Grosso, G.; Gerardi, A.; Koch, G.; Marchetti, F., Some basic properties of stochastic population models, systems theory in immunology, (Lecture Notes in Biomathematics, 32 (1979), Springer: Springer Berlin), 165-174 · Zbl 0433.92018
[3]	Brauman, C. A., Population growth in random environments, Bull. math. Biol., 45, 635-641 (1983) · Zbl 0511.92017
[4]	Chesson, P., Prey-predator theory and variability, Ann. Rev. Ecol. Syst., 9, 323-347 (1978)
[5]	Chesson, P., The stabilizing effect of a random environment, J. math. Biol., 15, 1-36 (1982) · Zbl 0505.92021
[6]	Brauer, F.; Nohel, J. A., Qualitative Theory of Ordinary Differential Equations (1969), Benjamin: Benjamin New York · Zbl 0179.13202
[7]	Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Dekker: Dekker New York · Zbl 0448.92023
[8]	Friedman, A., Stochastic Differential Equations and Applications, Vols. 1, 2 (1975), Academic Press: Academic Press New York
[9]	Gard, T. C., Persistence in stochastic food web models, Bull. math. Biol., 46, 357-370 (1984) · Zbl 0533.92028
[10]	Gard, T. C.; Kannan, D., On a stochastic differential equation modelling of prey-predator evolution, J. appl. Prob., 13, 429-443 (1976) · Zbl 0352.92013
[11]	Gihman, I. I.; Skorohod, A. V., Introduction to the Theory of Random Processes (1969), Sounders: Sounders Philadelphia · Zbl 0305.60027
[12]	Goel, N. S.; Maitra, S. C.; Montroll, E. W., On the Volterra and other nonlinear models of interacting populations, Rev. Mod. Phys., 43, 231-276 (1971)
[13]	Goel, N. S.; Richter-Dyn, N., Stochastic Models in Biology (1974), Academic Press: Academic Press New York
[14]	Goh, B. S., Global stability in many species systems, Am. Natur., 111, 135-143 (1977)
[15]	Has’minskii, R. Z., Stochastic Stability of Differential Equations (1980), Sijthoff & Noordhoff, Alphen aan den Rijn: Sijthoff & Noordhoff, Alphen aan den Rijn Netherlands · Zbl 0276.60059
[16]	Kendall, D. G., Stochastic processes and population growth, J.R. Statist. Soc., B11, 230-264 (1949) · Zbl 0038.08803
[17]	Kesten, H.; Ogura, Y., Reccrrence properties of Lotka-Volterra models with random fluctuations, J. Math. Soc. Japan, 32, 335-366 (1981) · Zbl 0449.92016
[18]	Kliemann, W., Qualitative theory of stochastic dynamical systems—applications to life sceinces, Bull. math. Biol., 45, 483-506 (1983) · Zbl 0521.92004
[19]	Kushner, H. J., On the weak convergence of interpolated Markov chains to a diffusion, Ann. Prob., 2, 40-50 (1974) · Zbl 0285.60064
[20]	Ludwig, D., Persistence of dynamical systems under random perturbations, SIAM Rev., 17, 605-640 (1975) · Zbl 0312.60040
[21]	May, R. M.; MacArthur, R. H., Niche overlap as a function of environmental variability, Proc. Natn. Acad. Sci. U.S.A., 69, 1109-1113 (1972)
[22]	Nisbet, R. M.; Gurney, W. S.C., Modelling Fluctuating Populations (1982), Wiley: Wiley New York · Zbl 0593.92013
[23]	Polansky, P., Invariant distributions for multipopulation models in random environments, Theor. Pop. Biol., 16, 25-34 (1979) · Zbl 0417.92019
[24]	Ricciardi, L. M., Diffusion processes and related topics in biology, (Lecture Notes in Biomathematics, 14 (1974), Springer: Springer Berlin) · Zbl 0575.92009
[25]	Schuss, Z., Theory and Applications of Stochastic Differential Equations (1980), Wiley: Wiley New York · Zbl 0439.60002
[26]	Stephanopoulos, G.; Fredrickson, A. G., Extinction probabilities in microbial predation: a birth-death approach, Bull. math. Biol., 43, 165-181 (1981) · Zbl 0446.92019
[27]	Turelli, M., Random environments and stochastic calculus, Theor. Pop. Biol., 12, 140-178 (1977) · Zbl 0444.92013
[28]	Turelli, M., A reexamination of stability in randomly varying versus deterministic environments with comments on the stochastic theory of limiting similarity, Theor. Pop. Biol., 13, 244-267 (1978) · Zbl 0407.92019
[29]	Turelli, M., Stochastic community theory: a partially guided tour, (Hallam, T. G.; Levin, S. A., A Course in Mathematical Ecology (1986), Springer: Springer New York)