##
**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.

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.

Reviewer: Evelyn Buckwar (Berlin)

### 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 |

### Keywords:

stochastic delay differential equation; multiplicative noise; Brownian motion; stochastic Lotka-Volterra system; stochastic population dynamics; persistence; asymptotic stability; ultimate boundedness
PDFBibTeX
XMLCite

\textit{X. Mao} et al., J. Math. Anal. Appl. 304, No. 1, 296--320 (2005; Zbl 1062.92055)

Full Text:
DOI

### 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) · Zbl 0818.45004 |

[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: 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: 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: Kluwer Academic Dordrecht · Zbl 0917.34001 |

[11] | Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: 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: 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: Longman London · Zbl 0724.60059 |

[15] | Mao, X., Exponential Stability of Stochastic Differential Equations (1994), Dekker: Dekker New York · Zbl 0851.93074 |

[16] | Mao, X., Stochastic stabilisation and destabilisation, Systems Control Lett., 23, 279-290 (1994) · Zbl 0820.93071 |

[17] | Mao, X., Stochastic Differential Equations and Applications (1997), Horwood: 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 |

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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.