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Environmental Brownian noise suppresses explosions in population dynamics. (English) Zbl 1058.60046
It is well known that by adding a noise term to the right hand side of an ordinary differential equation it is often possible to change completely stability properties of solutions. In particular, solutions to a stochastic differential equation may exist globally while solutions to the corresponding deterministic problem blow up in finite time. Such phenomena are studied for stochastic Lotka-Volterra systems. Let \(W\) be an \(n\)-dimensional Wiener process, and let \(b\in \mathbb R^ {n}\), \(A\in \mathbb R^ {n}\otimes \mathbb R^ {n}\) be arbitrary. Let a matrix \(\sigma \in \mathbb R^ {n}\otimes \mathbb R^ {n}\) be such that \(\sigma _ {ii}>0\) if \(1\leq i\leq n\), \(\sigma _ {ij}\geq 0\) if \(i\neq j\). The main theorem of the paper states that for any initial condition \(x_ 0\in \mathbb R^ {n}_ {+}\) there exists a unique nonnegative global solution to the system \[ dx(t) = \text{diag} (x_ 1(t),\dots ,x_ {n}(t))[(b+Ax(t))\,dt + \sigma x(t) \,dW(t)]. \]

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
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[1] Arnold, L., Stochastic differential equations: theory and applications, (1972), Wiley New York
[2] Boucher, D.H.(Ed.), The biology of mutualism, (1985), Groom Helm London
[3] Butler, G.; Freedman, H.I.; Waltman, P., Uniformly persistence systems, Proc. amer. math. soc., 96, 425-430, (1986) · Zbl 0603.34043
[4] Friedman, A., Stochastic differential equations and their applications, (1976), Academic Press New York
[5] He, X.; Gopalsamy, K., Persistence, attractivity, and delay in facultative mutualism, J. math. anal. appl., 215, 154-173, (1997) · Zbl 0893.34036
[6] Hutson, V.; Schmitt, K., Permanence and the dynamics of biological systems, Math. biosci., 111, 1-71, (1992) · Zbl 0783.92002
[7] Jansen, W., A permanence theorem for replicator and lotka – volterra system, J. math. biol., 25, 411-422, (1987) · Zbl 0647.92021
[8] Khasminskii, R.Z., Stochastic stability of differential equations, (1981), Sijthoff and Noordhoff Alphen a/d Rijn · Zbl 1259.60058
[9] Kifer, Y., Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel J. math., 70, 1-47, (1990) · Zbl 0732.58037
[10] Kirlinger, G., Permanence of some ecological systems with several predator and one prey species, J. math. biol., 26, 217-232, (1988) · Zbl 0713.92025
[11] Ladde, G.S.; Lakshmikantham, V., Random differential inequalities, (1980), Academic Press New York
[12] Liptser, R.Sh., Shiryayev, A.N., 1989. Theory of Martingales. Kluwer Academic Publishers, Dordrecht (Translation of the Russian edition, Nauka, Moscow, 1986). · Zbl 0728.60048
[13] Mao, X., Stability of stochastic differential equations with respect to semimartingales, (1991), Longman Scientific and Technical New York · Zbl 0724.60059
[14] Mao, X., Exponential stability of stochastic differential equations, (1994), Marcel Dekker New York
[15] Mao, X., Stochastic differential equations and applications, (1997), Horwood New York · Zbl 0874.60050
[16] Marion, G., Mao, X., Renshaw, E., 2001. Convergence of the Euler scheme for a class of stochastic differential equation. Internat. Math. J., in press. · Zbl 0987.60068
[17] Ramanan, K.; Zeitouni, O., The quasi-stationary distribution for small random perturbations of certain one-dimensional maps, Stochastic process. appl., 86, 25-51, (1999) · Zbl 0997.60074
[18] Wolin, C.L.; Lawlor, L.R., Models of facultative mutualism: density effects, Amer. natural., 124, 843-862, (1984)
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