Bifurcation Analysis of a Predator–Prey Model with Age Structure

Abstract

In this paper, a predator–prey model with age structure in predator is studied. Using maturation period as the varying parameter, we prove the existence of Hopf bifurcation for the model and calculate the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The method we employed includes Hopf bifurcation theorem, center manifolds and normal form theory for the abstract Cauchy problems with nondense domain. Under a certain set of parameter values, it turns out that subcritical Hopf bifurcation may occur, indicating that the increment of maturation period could stabilize the steady state, which is initially unstable and enclosed by a stable periodic solution. In addition, stability switches will also take place. Numerical simulations are finally carried out to show the theoretical results.

Hopf bifurcation in an age-structured predator–prey system with Beddington–DeAngelis functional response and constant harvesting

“…Yang and Wang (2020b) proposed an age-dependent predator-prey system with strong Allee effect, discussed the existence and uniqueness of a nonnegative steady state by transforming the system into a non-densely defined abstract Cauchy problem, and studied Hopf-Zero bifurcation by applying the center manifold theorem (Magal and Ruan 2009a) and the normal form theory (Liu et al 2014;Chu et al 2016). We refer to Li (1990), Zhang and Liu (2019, 2021, Yang (2019), Yang andWang (2019, 2020a), Cai et al (2020) and Yuan and Fu (2022) on further studies of predator-prey models with age structure and to Webb (1985) and Magal and Ruan (2008) for fundamental theories in classical age-structured models.…”

Bifurcation analysis of an age‐structured prey–predator model with infection developed in prey

“…The age‐structured models are the subject of interest of the recent research activities; we cite for instance previous researches 11–13 . This case of models also been applied to understand the ecological interactions; we cite for instance other studies 14–20 . The main purpose of this research is to prove the existence of Hopf bifurcation, where the age of maturation plays an important role in achieving this result.…”

“…The authors proved the existence of the Hopf bifurcation for the unique positive equilibrium state. Numerous similar research is the subject of the recent research activities; we cite for instance the papers of Liu and Li, 14 Cai et al, 15 and Tang and Liu 18 …”