The Asymptotic Analysis For Twoclasses Of Single Species Population Model With Singular Perturbation Problems

Logistic model is one of core theory in population ecology, it plays a leading role to describe population growth. This paper considers a class of Logistic models with slowly varying parameters, including the harvested Logistic model with slowly varying parameters and generalized Logistic model with slow varying coefficients. The models have the form as below respectively andThe preface introduces the knowledge of the model which related to biomathematics background and singular perturbation method. Chapter two is the necessary basic knowledge. In chapter three, the harvested Logistic model with small varying parameter has been studied, and then the approximate solution through two kinds of singular perturbation method is calculated. On the one hand, firstly we rewrite the harvested Logistic model with slowly varying parameters in the form of singularly perturbed systems and separate their fast and slow limits. Then apply matching method to obtain the approximate solution, finally prove the uniform validity of the approximate solution, and give the error estimate between the approximate solution and the exact solution via the methods of upper and lower solution. On the other hand, the solution is calculated through the nonlinear method of multiple scales. Firstly, choose two proper time scales, regard x(t,ε) as a function of these two scales, note that the ordinary different equation is now transformed to the partial differential equation, then turn the partial differential equation into the two linear ordinary differential equations by the expanding theory of power series, obtain the solution of the two respectively which formed the asymptotic solution. The fourth chapter, a class of generalized Logistic model with small varying parameter is discussed. Firstly, we construct their outer solution, and applying stretched variables to obtain the corresponding inner solution, then the approximate solution of theoriginal singularly perturbed problems is given by matching method, finally, theuniform validity of the approximate solution is proved, meanwhile the error estimateis given under some suitable conditions.