HOPF BIFURCATION OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH NONLOCAL INTRASPECIFIC COMPETITION
Abstract
The study of predator-prey models and reaction-diffusion equations helps us to more comprehensively and accurately explain the changes in population density in the natural world, and is an important aspect of biological and mathematical research. The study of Hopf bifurcations is a significant topic of research on reaction-diffusion equations, and it is of great importance for our understanding of population behavior. Firstly, we modify a predator-prey system with local effects studied by Geng et al. [1] and conduct further research based on this system. Secondly, we investigate the existence of the positive equilibrium in the system. We find that the positive equilibrium exists only under certain conditions, and we provide criteria through the study of the properties of a cubic function. Thirdly, we present the characteristic equations for two different systems under the scenarios of $n = 0$ and $n \neq 0$. Since the model includes two integral terms, we categorize into two different Hopf bifurcation scenarios based on the magnitude of the value of certain parameters. We provide conditions under which the system undergoes Hopf bifurcation for each case.
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