Dynamical analysis of a delayed pine wilt disease model under temperature conditions

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Abstract

Pine wilt disease represents a devastating forest pest with severe ecological risks. Biological control based on the natural enemy Dastarcus helophoroides has become a core management strategy. Most existing dynamical models mainly focus on a single time delay or the effects of natural enemies, and studies comprehensively incorporating the coupled interplay of temperature, interspecific relationships, and time delay are still limited. Motivated by this situation, we establish a three-dimensional model of delayed differential equations, which captures the tripartite interaction among pine wood nematodes,Monochamus alternatus, and Dastarcus helophoroides. The model combines logistic population growth, Holling type II functional response, and the Briére type II nonlinear temperature-dependent developmental function, and introduces parasitic time delay to reflect temperature-driven regulation of development and reproduction across all three species. Analytically, we prove the existence of positive equilibria and derive criteria for the local asymptotic stability of the model. Taking the parasitic time delay as the bifurcation parameter, we obtain the critical time delay threshold and the transversality condition for Hopf bifurcation. The multiple time scales method is applied to derive the normal form of Hopf bifurcation, which further determines the stability and bifurcation direction of the periodic orbits. Numerically, we perform systematic simulations to examine how key parameters shape the model’s dynamical behavior. Simulation results indicate that as the time delay increases, attracting periodic solutions emerge under the tested parameter values and initial histories. The ecological insights and biological control implications derived from this work are model-based hypotheses that may serve as a reference for further studies on pine wilt disease.

Keywords

pine wilt disease / delay differential equation / Hopf bifurcation / stability / temperature effect

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