Forest Evolution Equations With Age Structure And Integrated Pest Management Models

In this paper, we applied the mathematical theory to the forest evolution system and the integrated pest management, and mainly discussed the forest evolution system with age structure and the forest diseases and pests dynamical systems. First of all, we discussed some intrinsic properties of the forest system with age structure and growth function, secondly we study the existence of continuous solutions, the stability of the equilibrium and the continuity and controllability of pest disease by establishing a continuous pests dynamical systems with age structure and growth function. This paper is mainly divided into two large parts:The first part, basing on the traditional forest evolution system model, and considering the influence of the dissipation rate caused by the hyperplasia of the forest itself and the different age of trees, we established the forest evolution model with age structure and growth function described by the partial differential equation. Then we get the existence and uniqueness of nonnegative solution by using the operator semigroup theory in functional analysis and the integral equation theory, and then we get the conditions of stability and asymptotical stability of the system by establishing the Lyapunov function.The second part, the human SIR epidemic dynamics model with age structure is introduced into the forest pest model, and then we study the area of plant diseases and insect pests by considering the influence of growth function on forest system and forest diseases and insect pests, and we also assume the area that we study is a closed system. Firstly we divided the trees into three classes:the susceptible class of plant diseases and insect pests, the infective class, the immune class, and considering the integrated pest management, we established the pest model with age structure and growth function. Secondly using the operator semigroup theory and fixed point principle to discuss the existence and uniqueness of solution, then solving the equilibrium state of plant diseases and insect pests and disease-free equilibrium corresponding to the system, and giving the expression of reproductive number and characteristic equation, giving the condition to control the spread of diseases and insect pests by analyzing the impact of the distribution of characteristic equation root on the stability of the equilibrium system, which provides an important theoretical reference for the forestry pest prevention management department, on this account we can achieve the goal of integrated pest management.