## Stochastic differential delay equations of population dynamics.(English)Zbl 1062.92055

The authors start with the deterministic $$n$$-dimensional delay Lotka-Volterra equation $dx(t)/dt= \text{diag}(x_1(t),\ldots,x_n(t))\big[b+ Ax(t)+ Bx(t-\tau)\big],\tag{1}$ where $$x$$ and $$b$$ are $$n$$-dimensional vectors and $$A$$ and $$B$$ are $$n\times n$$ matrices. Equation (1) can be seen as a basic model for the dynamical behaviour of a population of $$n$$ interacting species. The authors assume that the vector $$b$$, which represents the intrinsic growth rates of the $$n$$ species, is subject to noise. This gives rise to a stochastic delay Lotka-Volterra system with multiplicative noise. The drift and diffusion coefficients of this stochastic differential system are locally Lipschitz-continuous but do not satisfy a linear growth condition. In standard arguments the latter conditions ensures that a solution does not blow-up in finite time. Thus, the authors first consider several conditions that guarantee the global existence of a unique solution, which, in addition, stays positive almost surely.
Further, several asymptotic properties of the solutions are discussed. In particular, conditions for persistence with probability 1, asymptotic stability with probability 1 and stochastic ultimate boundedness are given. In the last section, an example of a $$3$$-dimensional stochastic Lotka-Volterra food chain is considered and, as an illustration, specific conditions for asymptotic stability with probability 1 are given.

### MSC:

 92D25 Population dynamics (general) 34K50 Stochastic functional-differential equations 60K99 Special processes 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K25 Asymptotic theory of functional-differential equations 60H20 Stochastic integral equations 93D99 Stability of control systems 92D40 Ecology
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### References:

 [1] Ahmad, A.; Rao, M.R.M., Asymptotically periodic solutions of n-competing species problem with time delay, J. math. anal. appl., 186, 557-571, (1994) [2] Bereketoglu, H.; Gyori, I., Global asymptotic stability in a nonautonomous lotka – volterra type system with infinite delay, J. math. anal. appl., 210, 279-291, (1997) · Zbl 0880.34072 [3] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064 [4] Gard, T.C., Persistence in stochastic food web models, Bull. math. biol., 46, 357-370, (1984) · Zbl 0533.92028 [5] Gard, T.C., Stability for multispecies population models in random environments, Nonlinear anal., 10, 1411-1419, (1986) · Zbl 0598.92017 [6] Gard, T.C., Introduction to stochastic differential equations, (1988), Dekker New York · Zbl 0682.92018 [7] Goh, B.S., Global stability in many species systems, Amer. nat., 111, 135-143, (1977) [8] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht · Zbl 0752.34039 [9] He, X.; Gopalsamy, K., Persistence, attractivity, and delay in facultative mutualism, J. math. anal. appl., 215, 154-173, (1997) · Zbl 0893.34036 [10] Kolmanovskii, V.; Myshkis, A., Applied theory of functional differential equations, (1992), Kluwer Academic Dordrecht · Zbl 0917.34001 [11] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002 [12] Kuang, Y.; Smith, H.L., Global stability for infinite delay lotka – volterra type systems, J. differential equations, 103, 221-246, (1993) · Zbl 0786.34077 [13] Liptser, R.Sh.; Shiryayev, A.N., Theory of martingales, (1989), Kluwer Academic Dordrecht, (translation of the Russian edition, Nauka, Moscow, 1986) · Zbl 0728.60048 [14] Mao, X., Stability of stochastic differential equations with respect to semimartingales, (1991), Longman London · Zbl 0724.60059 [15] Mao, X., Exponential stability of stochastic differential equations, (1994), Dekker New York · Zbl 0851.93074 [16] Mao, X., Stochastic stabilisation and destabilisation, Systems control lett., 23, 279-290, (1994) [17] Mao, X., Stochastic differential equations and applications, (1997), Horwood Chichester · Zbl 0874.60050 [18] Mao, X., A note on the Lasalle-type theorems for stochastic differential delay equations, J. math. anal. appl., 268, 125-142, (2002) · Zbl 0996.60064 [19] Mao, X.; Marion, G.; Renshaw, E., Environmental noise suppresses explosion in population dynamics, Stochastic process. appl., 97, 95-110, (2002) · Zbl 1058.60046 [20] Teng, Z.; Yu, Y., Some new results of nonautonomous lotka – volterra competitive systems with delays, J. math. anal. appl., 241, 254-275, (2000) · Zbl 0947.34066
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