Abstract
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.
Highlights
Ecological models that reveal the amounts of prey and predator have long been and still will be investigated for their universal existence and importance
The existence and properties of a Hopf bifurcation are investigated in Sections . and . , respectively
In the following we study the stability of the nonnegative equilibria
Summary
Ecological models that reveal the amounts of prey and predator have long been and still will be investigated for their universal existence and importance. An eco-epidemiological model of SI type contains three variables: the susceptible prey S(t), infected prey I(t), and predator P(t). The predation on infected prey follows a Holling type II response function. ). The rest of this paper is organized as follows: In Section , the properties of the solutions such as positivity and boundedness are obtained.
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