GLOBAL BEHAVIOR AND HOPF BIFURCATION OF A STAGE-STRUCTURED PREY–PREDATOR MODEL WITH MATURATION DELAY FOR PREY AND GESTATION DELAY FOR PREDATOR

Abstract

Age structured models are important for the dynamics and evolution of many insect populations. For such models the rates of survival, growth and reproduction depend on the age or developmental stage of individuals in the population. For example, the life cycles of many species are composed of at least two stages, immature individuals (immatures) and mature individuals (matures) which may exhibit fundamentally different developmental, morphological and behavioral characteristics. In this paper, we incorporate these biological properties into a predator–prey population model. Our model is stage-structured for the prey and contains maturation and gestation delays. Using comparison theorems, we derive sufficient conditions for dynamical system permanence. We found equilibrium points exhibit switching with boundary equilibrium points through a mechanism of decreasing maturation of the prey population. Furthermore, an analysis of the corresponding characteristic equations was performed to determine local stability of equilibria. Choosing the gestation delay as a bifurcation parameter, we were able to compute a critical value for the existence of small amplitude oscillation of population densities (i.e., a Hopf bifurcation). Sufficient conditions for the global stability of all non-negative equilibria were obtained using Kamke comparison theorems and a newly developed iteration technique. Numerical simulations were performed to illustrate the theoretical results.