Environmental noise impact on regularity and extinction of population systems with infinite delay

Abstract

a b s t r a c tPopulations of biological species are often subject to different types of environmental noises. These noises play crucial roles and have significant impact on the evolution and biodiversity. To understand the effects of different types of noises on the asymptotic properties of the populations, this paper reveals the influence of inherent net birth noise and the interaction noise among stochastically perturbed population systems. By considering systems with infinite delays, (1) this paper discovers that (a) the interaction noise may suppress the potential explosion and guarantee the existence of the global solution, which yields regularity of the stochastic system, and (b) the inherent net birth noise may lead to extinction of the populations; (2) this paper provides sufficient conditions that ensure the stochastic persistence, stochastic boundedness, and asymptotic polynomial growth of the population system; (3) this paper demonstrates that these asymptotic properties are completely determined by stochastic noises and are independent of other parameters.

Asymptotic Behaviour and Extinction of Delay Lotka-Volterra Model with Jump-Diffusion

“…From condition (82) in Theorem 12, we can observe that if intensities , of white noises, the random downward jump magnitude ( , ), or intensity ] (average rate of jump events arrival) is sufficiently large, the species will be extinct. Compared with the former results [17], this result gives an interesting and important condition under which the frequent nature disasters can force the population to become extinct.…”

“…where 1 ( ) and 2 ( ) are mutually independent Brownian motions with (0) = 0, = 1, 2, defined on a complete probability space (Ω, F, P) with a filtration {F } ≥0 satisfying the usual conditions (i.e., it is right continuous and increasing while F 0 contains all P-null sets). In recent years, several authors introduced white noises into deterministic systems with delays to reveal the effect of environmental variability on the population dynamics (see, e.g., [14][15][16][17][18]). Particularly, it has also been revealed by Bahar and Mao [14] and Wu and Yin [17] that the environmental noise can suppress a potential population explosion of the Lotka-Volterra model with delays.…”

“…In recent years, several authors introduced white noises into deterministic systems with delays to reveal the effect of environmental variability on the population dynamics (see, e.g., [14][15][16][17][18]). Particularly, it has also been revealed by Bahar and Mao [14] and Wu and Yin [17] that the environmental noise can suppress a potential population explosion of the Lotka-Volterra model with delays. These indicate clearly that environmental noise may change the properties of population systems significantly.…”

Kolmogorov-type systems with regime-switching jump diffusion perturbations

“…When the environmental noise intensity depends weakly on the population size (for example, the net birth rate noise), the stochastic system behaves similarly to the deterministic one and asymptotic properties are also independent of the noise. [49] further showed that although the interactions between the species plays the crucial roles for the existence of the global positive solution of the stochastic Lotka-Voltera systems and some asymptotic properties, the most important factor of the extinction of the species is still from the noise of the birth rate.…”

Stochastic periodic solutions of stochastic periodic differential equations

“…[11]). Many researchers have studied qualitative properties such as existence, uniqueness, boundedness and stability for various stochastic differential systems, for instance [12][13][14][15][16][17][18][19][20][21]. However, as far as we know, there are few papers on periodic stochastic differential equation.…”