Dynamic analysis of a size-structured model describing insect destruction in forests

Abstract

In this paper, we delve into the dynamics of a tree-insect model that considers size structure in trees, aiming to explore the effect of the maturity size at which the saplings are just coming into adulthood on the insect population in forests. By making full use of mathematical techniques and an ingenious variable transformation, we convert the model of nonlinear ordinary differential equation coupled to a partial differential equation into a model of delay differential equations with fixed delay. Combining the method of fluctuations and bifurcation theory, a relatively complete qualitative analysis of the model is conducted, including uniqueness, nonnegativity and boundedness of the solutions, global and local asymptotic stability of the equilibria, uniform persistence of the model, stability switches occurring at the positive equilibrium. In addition, numerical simulations indicate that the maturity size can lead to richer dynamics, such as regime shifts, bistability and long transients.