Spatiotemporal and bifurcation characteristics of a nonlinear prey-predator model

Abstract

The fragile plateau pastoral ecosystem is vulnerable to human grazing activities. Studying the prey-predator relationship between species in plateau pastoral area helps predict the population change law of prey and predator, which is of significant meaning to protecting the stability of the ecosystem and promote the stable growth of economic animal husbandry on the plateau. Given this, this paper establishes a two time-delays prey-predator model of plateau pastoral ecosystem based on considering the time-delay of predators from larval to forage feeding stage and the time-delay of the age of the controlled predator and its delay-dependent coefficient. Firstly, to analyze the influence of human grazing on the spatial distribution of prey, the spatiotemporal characteristics of the system with time-delays of 0 are studied, and Turing instability and Hopf bifurcation conditions are obtained. The results show that the increase in predator migration rate will lead to a pattern. In order to analyze the influence of time-delays on the dynamical behavior of the model, the crossing curves composed of thresholds of dynamic behavior change on the double time-delays plane when the time-delays are greater than 0 and the type of threshold forming the crossing curves are obtained by using the crossing curves method. The results show that when the corresponding values on the crossing curves are taken, there are many depressions and bulges in the limit cycles of the system. Their appearance has certain rules. At this time, the system will move periodically. This shape of the limit cycles is found for the first time, which is worthy of further study.