zbMATH — the first resource for mathematics

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 \[ dx(t) = \text{diag} (x_ 1(t),\dots ,x_ {n}(t))[(b+Ax(t))\,dt + \sigma x(t) \,dW(t)]. \]

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
92D25 Population dynamics (general)
Full Text: DOI
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.