Bifurcation analysis of a single population reaction-diffusion model with discrete memory delay and distributed competition delay

In this paper, a reaction-diffusion model that incorporates both discrete memory delay and distributed competition delay is introduced, we primary focus on the stability and Hopf bifurcation of its internal equilibrium point. A general methodology for analyzing the characteristic equations of this kind of model is established. Theoretical findings reveal that the stability of the system is contingent upon the relationship between the memory diffusion coefficient d2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_2$$\end{document} and the free diffusion coefficient d1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_1$$\end{document}. In addition, we also find a sudden change in the number of pure imaginary eigenvalues from 0 to infinity. In order to study the properties of the bifurcations, we transformed the model into an equivalent system without any integral term and provided a new formula for calculating the normal form of Hopf bifurcation. In the numerical simulation section, we delineate the crossing curves within the dual-delay sigma-tau\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma -\tau $$\end{document} plane that signify the occurrence of Hopf bifurcations in the system. These curves reveal the presence of periodic solutions with varying amplitudes, along with the intriguing phenomenon of stability switching. Furthermore, the model demonstrated a distinct dynamic behavior, diverging from that observed in models with discrete competitive delays, these results suggest that the disruptive impact of distributed delays on the stability of system is significantly less pronounced than that of discrete delays.