Dynamical Behaviors in a Stage-Structured Model with a Birth Pulse

Abstract

This paper presents an exploitation model with a stage structure to analyze the dynamics of a fish population, where periodic birth pulse and pulse fishing occur at different fixed time. By utilizing the stroboscopic map, we can obtain an accurate cycle of the system and investigate the stability thresholds. Through the application of the center manifold theorem and bifurcation theory, our research has shown that the given model exhibits transcritical and flip bifurcation near its interior equilibrium point. The bifurcation diagrams, maximum Lyapunov exponents and phase portraits are presented to further substantiate the complexity. Finally, we present high-resolution stability diagrams that demonstrate the global structure of mode-locking oscillations. We also describe how these oscillations are interconnected and how their complexity unfolds as control parameters vary. The two parametric planes illustrate that the structure of Arnold’s tongues is based on the Stern–Brocot tree. This implies that the periodic occurrence of birth pulse and pulse fishing contributes to the development of more complex dynamical behaviors within the fish population.