Two-dimensional Predator-prey Model Research Articles

Bifurcation results and chaos in a two-dimensional predator-prey model incorporating Holling-type response function on the predator

Bifurcation results and chaos in a two-dimensional predator-prey model incorporating Holling-type response function on the predator

Sliding mode dynamics on a prey–predator system with intermittent harvesting policy

Unsustainable fishing has been identified as one of the most important direct drivers of the degradation of coral reef ecosystems. Herbivorous reef fish, which prevent excessive accumulation of coral-suffocating algae, are in a steady decline due to the presence of invasive predators together with overfishing in many coral reef systems. Here, a two-dimensional predator–prey model is proposed that enables exploration of intermittent harvesting policies, where a control function in the model is evoked that signals when harvesting should cease and commence to maintain a herbivorous fish density above a critical threshold necessary for population stability. We analyze the model by using Filippov’s regularization method, which is ideal for discontinuous dynamic systems, such as the system explored here with intermittent harvesting. We obtain the sliding segment of the Filippov’s system and its domains together with the conditions for the existence and stability of the regular, virtual and pseudo-equilibria. We show that regular equilibria and pseudo-equilibrium can coexist. Further, we show that intermittent harvesting, with time lags between fishery harvesting decisions and implementations considered, can lead to a bounded oscillation about pseudo-equilibria, or in other words, a sustainable herbivorous fishery. Finally, using a selective harvesting policy, where the invasive predator fish is targeted (here lionfish), we show that a longer herbivorous fish harvesting period can be achieved without changing the critical threshold for harvesting.

Stability and Hopf bifurcation of a modified predator-prey model with a time delay and square root response function

In this paper, we consider a two-dimensional predator-prey model with a time delay and square root response function. We analyze the stability of equilibria with the delay τ increasing and the critical value of τ when Hopf bifurcation occurs. Because the model has the term of square root, the zero point is a singularity. In order to clearly study the stability of the zero point, we rescale the variable x(t), say x(t)=X^{2}(t). The conclusion is that the zero point is not stable and the instability is not affected by the delay τ. We apply the normal form method and center manifold theorem to obtain the direction and stability of the Hopf bifurcation. Finally, we make several numerical simulations which is consistent with the conclusion of theoretical analysis.

RELAXATION OSCILLATIONS IN SINGULARLY PERTURBED GENERALIZED LIENARD SYSTEMS WITH NON-GENERIC TURNING POINTS

Based on the asymptotic analysis technique developed by Eckhaus [Lecture Notes in Math., vol. 985, pp 449-494. Springer, Berlin, 1983], this paper aims to study the existence and the asymptotic behaviors of relaxation oscillations of regular and canard types in a singularly perturbed generalized Lionard system with a non-generic turning point. The singularly perturbed Lionard system considered in this paper is very general and numerous real world models like some biological ones can be rewritten in the form of this system after a series of transformations. Under certain conditions, we rigorously prove the existence of regular relaxation oscillations and canard relaxation oscillations under the specific parameter conditions. As an application, two biological models, namely, a FitzHugh-Nagumo model and a twodimensional predator-prey model with Holling-II response are studied, in which, the existence of regular relaxation oscillations and canard relaxation oscillations as well as the bifurcation curves are obtained.