Complex Dynamics of a Leslie–Gower Predator–Prey Model with Crowley–Martin Functional Response

Predator–prey models, describing the predation interaction between predators and their prey, are fundamental frameworks in studying ecosystems. Their dynamics reflect the variation tendency and the asymptotic states of the populations, as well as the interaction mechanism between them. The complexity of dynamics further illustrates the diversity of their development. In this paper, by employing methods of qualitative and quantitative analyses, we investigate a Leslie–Gower predator–prey model with Crowley–Martin functional response. Qualitatively, first, instead of discussing the cubic equation satisfied by the component of the prey at a positive equilibrium, we reformulate the equation appropriately to obtain explicit conditions on the existence of positive equilibria. Then we study their local stability via linearization. When a positive equilibrium loses stability, we show that Hopf bifurcation can occur and calculate the corresponding Lyapunov number to determine the stability of bifurcated periodic orbits. Moreover, we elucidate the existence of periodic solutions by means of the Poincaré–Bendixon Theorem for annular regions under certain conditions. Based on the qualitative results, we conduct quantitative analysis by continuously changing the intrinsic growth rate of the predator while fixing the values of the other parameters in six sets. The rich numerical simulations demonstrate that the model can have two coexisting periodic solutions and experience a series of bifurcation phenomena including the saddle-node bifurcation of equilibria, supercritical and subcritical Hopf bifurcations, homoclinic bifurcation, and the saddle-node bifurcation of nonconstant periodic solutions. In particular, in one set, the model has three simple positive equilibria and six critical values determining its dynamics. There can be 11 different types of dynamical properties. These results illustrate the diversity of the asymptotic states of the model. Moreover, our qualitative and quantitative analyses suggest that the change in the intrinsic growth rate of the predator can lead to complexity in the predator–prey interaction process and asymptotic states in different situations.