A Delayed Prey–Predator System with Migration and Disease Infection

We propose a delayed prey–predator model involving the occurrence of infection in the interacting populations along with migration in prey population. In our model system, the interacting populations are categorized according to their health states as healthy prey, prey with infection, healthy predator and predators with infection. To study the dynamics of the model system, we analyze the boundedness of the solutions, existence of non negative equilibria and the stability of the proposed model. Our findings show that the time delay parameter is a leading bifurcation parameter to examine the Hopf-bifurcation existence around the positive equilibrium. We further investigate the direction of Hopf-bifurcation and stability of bifurcated periodic solutions employing the normal form theory, Riesz representation theorem and central manifold theorem. Finally, the numerical simulations for validating our theoretical findings are performed.