Turing instability of the periodic solutions in the homogeneous diffusive host-parasite-hyperparasite interaction model

In this paper, we examine a homogeneous diffusive model depicting the interactions among hosts, parasites and hyperparasites. Our primary focus is on the asymptotic dynamics of the solutions in this model. For the corresponding ordinary differential equations, we have investigated both the existence and stability of equilibrium points. Furthermore, we have explored the potential for Hopf bifurcations, delving into their associated characteristics, such as the direction of bifurcation and the stability of the resulting periodic solutions. For the reaction-diffusion equations, our emphasis is on the Turing instability of the periodic solutions that arise from Hopf bifurcations. The results we obtained provide critical insights into the dynamics of the model and help in understanding the complex interactions within this biological system.