Fractional Logistic Models With Variable Coefficients And Their Inverse Problems

The logistic equation is an important class of biological mathematical models that describe the population growth behavior under total resource constraints.The time fractional logistic diffusion model can describe the secondary diffusion process of multiple species under limited resource but uneven spatial distribution,it is of great scientific significance and ecological application prospect to study the inverse problem of fractional order variable coefficient logistic model simulation and parameter identification.This paper mainly studies the fractional order logistic model with spatially dependent environmental capacity and the numerical solution and differential order inversion of fractional order logistic diffusion equation with spatially dependent diffusion coefficients.The main research contents are as follows:In chapter 1,we provide an overview of the research significance and current research status at home and abroad,introduce the definition of Caputo fractional order and other preliminary knowledge,and provide the main work of this article.In chapter 2,we study the forward problem of a time fractional logistic model with spatially dependent environmental capacity.We use the Taylor expansion method to deal with nonlinear terms and construct a difference scheme by discretizing variable substitution and fractional derivatives.Under the condition that the environmental capacity is appropriately large,the stability and convergence of the difference scheme are proved by the estimation of the spectral radius of the coefficient matrix.The effectiveness of the difference scheme and t he influence of the change of differential order on the long-term diffusion behavior are verified by numerical examples.In chapter 3,we study the forward problem of a two species time fractional logistic diffusion model with spatially dependent diffusion coefficients.The difference scheme for solving the forward problem is established based on the discretization of fractional derivatives.The stability and convergence of the difference scheme are proved by applied mathematics induction,and the effectiveness of the difference scheme is verified by numerical examples.In chapter 4,we investigate the inverse problem of determining the capacity of a fractional order logistic model with spatially dependent environmental capacity and the parameter of determining the diffusion coefficient of a fractional order logistic system model with spatially dependent diffusion coefficient.Taking the additional data as the observation data of a given internal point,the homotopy regularization method is applied to carry out the inversion of accurate data and the inversion of data with random disturbance for the above two types of inverse problems.The numerical example results show that the inversion algorithm is numerically stable.In chapter 5,we summarize the main research findings of this article and provide future research prospects.