Stability and Hopf bifurcation analysis of a stage-structured delayed ecosystem with prey cannibalism
摘要
This paper develops a stage-structured predator-prey model with cannibalism and a Holling type II functional response, incorporating two key biological assumptions: (i) mature prey cannibalize eggs to reduce their own mortality, and (ii) predators experience a gestation delay in reproduction. To capture stage-specific cannibalism, the prey population is divided into egg and mature stages. The positive invariance of the model is established to ensure biological validity. We then analyze the existence of extinction, boundary, and positive equilibria for both the non-delayed and delayed systems, and derive local asymptotic stability criteria for the non-delayed case. By selecting the cannibalism rate (in the non-delayed system) and the predator gestation delay (in the delayed system) as bifurcation parameters, we investigate Hopf bifurcations and determine the critical thresholds. Applying the center manifold theorem and normal form theory, we further analyze the direction, stability, and periodicity of the bifurcating periodic solutions arising from the delayed system. Numerical simulations corroborate the theoretical findings and elucidate the dynamical effects of key biological parameters. The main results reveal that the non-delayed system undergoes a Hopf bifurcation when the cannibalism rate exceeds a critical value. For the delayed system, asymptotic stability holds for gestation delays below a threshold; stability is lost beyond this threshold, leading to periodic oscillations. Further increases in delay can induce period-doubling bifurcations and chaotic dynamics. This study elucidates the regulatory mechanisms of key parameters and time delay, providing a theoretical reference for the management of cannibalistic ecosystems and a mathematical framework for research on delayed ecological systems.