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

This paper studies the effect of jump-diffusion random environmental perturbations on the asymptotic behaviour and extinction of Lotka-Volterra population dynamics with delays. The contributions of this paper lie in the following: (a) to consider delay stochastic differential equation with jumps, we introduce a proper initial data space, in which the initial data may be discontinuous function with downward jumps; (b) we show that the delay stochastic differential equation with jumps associated with our model has a unique global positive solution and give sufficient conditions that ensure stochastically ultimate boundedness, moment average boundedness in time, and asymptotic polynomial growth of our model; (c) the sufficient conditions for the extinction of the system are obtained, which generalized the former results and showed that the sufficiently large random jump magnitudes and intensity (average rate of jump events arrival) may lead to extinction of the population. 1. Introduction Populations of biological species in some regions are often subject to sudden environmental shocks, for example, earthquakes, floods, tsunami, hurricanes, and so forth. As it is well known, occurrence of these disasters has the properties of random unpredictability and great destruction. To illustrate our mathematical results in terms of their ecological implications, we consider the protection of wildlife rare species, in the southwest region Sichuan in China, best-preserved panda habitat on earth, which belongs to Longmen Shan active fault zone. Both the 2008？M8.0 Wenchuan and the 2013？M7.0 Ya’an earthquakes occurred in this region. Earthquakes and secondary disasters caused by the earthquake such as mudslides, landslides, barrier lake, and other geological disasters may destroy natural habitat of wildlife and even may lead to extinction of endangered wildlife. Thus, it is very interesting to reveal how these sudden environmental shocks have an effect on the populations through stochastic analysis of the underlying dynamic systems. The classical deterministic Lotka-Volterra model with delays is generally described by the integrodifferential equation which is used to describe the population dynamics of -species with interactions, where represents the population size of the th species; , , , and are constant parameters; is the inherent net birth rate of the th species; , , and represent the interaction rates; and is a probability measure on that may be any function defined on of bounded variations. There is an extensive literature concerned with the dynamics of model (1)