Analysis of dynamics in an eco-epidemiological model with stage structure

This paper is devoted to the study of an eco-epidemiological model with stage structure in the predator and disease in the prey. To begin with, the positivity and boundedness of the solutions are obtained. This shows that the system possesses a bounded absorbing set. Then, by using the LaSalle-Lyapunov invariance principle, limit equation theory, and a geometrical criterion for analyzing the distribution of the eigenvalues, the stability of the boundary equilibria and interior equilibrium are established, respectively. Meanwhile, the existence of Hopf bifurcations is obtained when the delay τ varies in a limitary region. Furthermore, by employing center manifold theory and the normal form method, an algorithm for determining the direction and stability of the Hopf bifurcation is derived. At last, some numerical simulations are carried out for illustrating the analytic results.