Bifurcations of a predator-prey system with cooperative hunting and Holling III functional response

In this paper, we consider the dynamics of a predator-prey system of Gause type with cooperative hunting among predators and Holling III functional response. The known work numerically shows that the system exhibits saddle-node and Hopf bifurcations except homoclinic bifurcation for some special parameter values. Our results show that there are a weak focus of multiplicity three and a cusp of codimension two for general parameter conditions and the system can exhibit various bifurcations as perturbing the bifurcation parameters appropriately, such as the transcritical and the pitchfork bifurcations at the degenerate boundary equilibrium, the saddle-node and the Bogdanov-Takens bifurcations at the degenerate positive equilibrium and the Hopf bifurcation around the weak focus. The comparative study demonstrates that the dynamics are far richer and more complex than that of the system without cooperative hunting among predators. The analysis results reveal that appropriate intensity of cooperative hunting among predators is beneficial for the persistence of predators and the diversity of ecosystem.

Global Dynamics of a Delayed n + m-Species Competition Predator-Prey System on Time Scales

In this paper, we consider a delayed n + m-species competition predator-prey system on time scales with Holling III functional response and multiple exploited (or harvesting) terms. By using the continuation theorem based on Gaines and Mawhin’s coincidence degree theory, easily verifiable criteria are established for global existence of multiple positive periodic solutions for the above system.

Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response

A prey-predator system with the strong Allee effect and generalized Holling type III functional response is presented and discretized. It is shown that the combined influences of Allee effect and step size have an important effect on the dynamics of the system. The existences of Flip and Neimark-Sacker bifurcations and strange attractors and chaotic bands are investigated by using the center manifold theorem and bifurcation theory and some numerical methods.

Mathematical and Dynamic Analysis of a Gompertz Ecosystem with Impulsive Control Strategy and Stage Structure for Predator

In this paper, on the basis of the theories and methods of ecology and ordinary differential equations, an ecological Gompertz model with Holling III functional response and stage structure for predator is established. By use of the stroboscopic map, a predator extinction periodic solution is obtained, and the global attractivity of the predator extinction periodic solution is analyzed. By using comparison theorem of impulsive differential equation and small amplitude perturbation skills, we get the sufficient condition for permanence of the system under impulsive harvest strategy for the prey and maturation time delay of predator.

Analysis of an Ecoepidemiological Model with Prey Refuges

An ecoepidemiological system with prey refuges and disease in prey is proposed. Bilinear incidence and Holling III functional response are used to model the contact process and the predation process, respectively. We will study the stability behavior of the basic system from a local to a global perspective. Permanence of the considered system is also investigated.