Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations.

*(English)* Zbl 06157314
Summary: This paper is concerned with two $n$-species stochastic cooperative systems. One is autonomous, the other is non-autonomous. For the first system, we prove that for each species, there is a constant which can be represented by the coefficients of the system. If the constant is negative, then the corresponding species will go to extinction with probability 1; If the constant is positive, then the corresponding species will be persistent with probability 1. For the second system, sufficient conditions for stochastic permanence and global attractivity are established. In addition, the upper- and lower-growth rates of the positive solution are investigated. Our results reveal that, firstly, the stochastic noise of one population is unfavorable for the persistence of all species; secondly, a population could be persistent even the growth rate of this population is less than the half of the intensity of the white noise.

##### MSC:

34F05 | ODE with randomness |

60H10 | Stochastic ordinary differential equations |

92B05 | General biology and biomathematics |

60J27 | Continuous-time Markov processes on discrete state spaces |