Permanence and Stability of a Stage-Structured Predator–Prey Model

Abstract

A predator᎐prey model with a stage structure for the predator which improves the assumption that each individual predator has the same ability to capture prey is proposed. It is assumed that immature individuals and mature individuals of the predator are divided by a fixed age and that immature predators do not have the ability to attack prey. We obtain conditions that determine the permanence of the populations and the extinction of the populations. Furthermore, we establish necessary and sufficient conditions for the local stability of the positive equilibrium of the model. By exploiting the monotonicity of one equation of the model, we obtain conditions for the global attractivity of the positive equilibrium, which allow for long delay as long as the predator birth rate is large or the death rate of immature predators is small. By constructing Liapunov functionals, we also obtain conditions under which the positive equilibrium is globally stable when the delay is small. ᮊ

Periodicity in a ratio‐dependent predator‐prey system with stage structure for predator

“…Already, there were many scholars considering the stage-structure ecosystem. In [23], a predator-prey system with stage structure was established under the assumptions that the predator is divided into two groups, one immature and the other mature, and that the mature predator attacks the prey and admits reproductive ability, while the immature predator does not attack the prey and does not have reproductive ability. It is found that an orbitally asymptotically stable periodic orbit exists in the model.…”

“…In [22], they further incorporated Holling-II-type functional response into the model and investigated extinction and permanence of the model. Stimulated by the works of [22,23], in [26], Xiao considered the ratio-dependent predator-prey system with stage structure, that is…”

Permanence and partial extinction in a delayed three-species food chain model with stage structure and time-varying coefficients

“…Just because of the biological meaning, the permanence and extinction analysis for dynamic systems have attracted considerable attention, and many interesting results were presented (for example, see [12,22,27,31,35]). Different from the two dimensional predator-prey model, partial extinction instead of extinction is the key characteristic property for the food chain model.…”

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

“…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.…”

“…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. The literatures [7][8][9][10] proposed and investigated the dynamics of predator-prey models with harvesting and stage structure.…”