Global Dynamics of a Predator-Prey Model with Stage Structure for the Predator

Abstract

The global properties of a predator-prey model with nonlinear functional response and stage structure for the predator are studied using Lyapunov functions and LaSalle's invariance principle. It is found that, under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state, a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state. A condition which leads to the extinction of the predators is indicated. We also obtain results on the existence and stability of periodic solutions. In particular, when (4.2) fails to hold and the unique positive steady state E * becomes unstable, the coexistence of prey and predator populations is ensured for initial populations not on the one-dimensional stable manifold of E * , albeit with fluctuating population sizes.

Global dynmics of of a syrphid fly-aphid model with stage structure for predator popoluation

“…By Lemma 2.4, we obtain the following existence theorem of periodic solution, which is similar to the obtained result in [4]. Theorem 2.5.…”

“…By constructing a Lyapunov function (similarly to that of [4]), we obtain the following global stability property of the interior equilibrium.…”

“…The authors in [2][3][4] studied the global properties and the existence and stability of periodic solutions of a predatorprey model with nonlinear functional response and stage structure for the predator. W. Wang, et al [5] and R. Xu, et al [6] showed the combined effects of stage structure for predator and time delay on the global dynamics of predator-prey model.…”

“…Since W. G. Aiello and H. I. Freedman [1] built and studied a time delay model of single species growth with stage structure, model with stage structure consisting of immature and mature stages has been paid many attentions by numerous scholars due to it embodies the specific stages in different habits of its whole life history of biological population (particularly mammalian population) in the natural world: see [2][3][4][5][6][7][8][9][10] and references therein. The authors in [2][3][4] studied the global properties and the existence and stability of periodic solutions of a predatorprey model with nonlinear functional response and stage structure for the predator.…”

Global stability and Hopf bifurcation of a predator-prey model with stage structure and delayed predator response

“…In the natural world, there are many species whose individuals have a life history that takes them through two stages, immature and mature, where immature predators are raised by their parents, and the rate they attack at prey and the reproductive rate can be ignored. Stage-structured population models have received great attention in the last two decades (see, for example, [1,6,14,16,18,19]). In order to analyze the effect of a stage structure for the predator on the dynamics of a predator-prey system, in [16], Wang considered the following stage-structured predator-prey model:…”