Research In Nonlinear Age-structured Insect Pests’ System Model

This paper introduced the mathematic theory into forest-diseasedynamic system，mainly discussed the forest insect pests system. Throughestablish the non-linear，continuous，age-structure forest-insect dynamicmodel，study the existence of positive solution，the stability of equilibrium，the controllability of the system. The article mainly consists of two parts:In the first part，basing on the classical forest development systemsmodel and considering the age’s impact on growth function and cut rate，we establish a new forest epidemic model which contains growth functionby dividing the forest into susceptible and infective. The model is anage-structure development system that described by partial differentialequations with initial conditions and boundary conditions (including theharvest and plant factors). By using the operatorC0-semigroup theory andrelated properties，prove the existence and uniqueness of the system’s localand global non-negative solution. At last，analysis the equilibrium stabilityby defining the threshold of the system，get the relationship between thestability and the threshold: case when the threshold is less or equal to1，then disease-free equilibrium is locally asymptotically stability，case when the threshold is more than1，then disease-free equilibrium is not stability.And give the conditions of system endemic equilibrium is locallyasymptotical stability. Case when the condition holds，disease will belocally continuing epidemic.In the second part，we introduce the human age-structured SIRepidemic dynamics model into the forest insect pest’ s system，study animportant class of diseases—Bursaphelenchus Xylophilus in some specialareas (e.g. scenic area). And put the forest into three parts of susceptible，infective and immune. Assuming the area is a closed area，we establishedthe forest insect pest’s model with immune measures. Get thecorresponding equilibrium solution and the nontrivial equilibrium solutionof the system，calculate the basic reproductive number，analysis the impactof each indicator on the stability of system equilibrium，and give theconditions of controlled the pests spread. Provide a theoretical basis toprevent forest insect pests for administrative department. Finally，selectrealistic numerical to simulate，verify the conclusion of the theorem and thededuction.