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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 $b\in \Bbb R\sp {n}$, $A\in \Bbb R\sp {n}\otimes \Bbb R\sp {n}$ be arbitrary. Let a matrix $\sigma \in \Bbb R\sp {n}\otimes \Bbb R\sp {n}$ be such that $\sigma \sb {ii}>0$ if $1\le i\le n$, $\sigma \sb {ij}\ge 0$ if $i\neq j$. The main theorem of the paper states that for any initial condition $x\sb 0\in \Bbb R\sp {n}\sb {+}$ there exists a unique nonnegative global solution to the system $$dx(t) = \text{diag} (x\sb 1(t),\dots ,x\sb {n}(t))[(b+Ax(t))\,dt + \sigma x(t) \,dW(t)].$$

##### MSC:
 60H10 Stochastic ordinary differential equations 92D25 Population dynamics (general)
##### Keywords:
stochastic differential equations; explosion; boundedness
Full Text:
##### References:
 [1] Arnold, L.: Stochastic differential equations: theory and applications. (1972) · Zbl 0216.45001 [2] H. (Ed.) Boucher, D.: The biology of mutualism. (1985) [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) · Zbl 0323.60057 [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) · 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) · Zbl 0466.60002 [12] Liptser, R.Sh., Shiryayev, A.N., 1989. Theory of Martingales. Kluwer Academic Publishers, Dordrecht (Translation of the Russian edition, Nauka, Moscow, 1986). [13] Mao, X.: Stability of stochastic differential equations with respect to semimartingales. (1991) · Zbl 0724.60059 [14] Mao, X.: Exponential stability of stochastic differential equations. (1994) · Zbl 0806.60044 [15] Mao, X.: Stochastic differential equations and applications. (1997) · Zbl 0892.60057 [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)