Biological Pattern Formation And Parameter Inversion Based On Anomalous Diffusion System

Spatial pattern formations are core issues in ecology,biology,chemistry,dynamics and nonlinear sciences.It is of important scientific innovation and application values to study inverse problems arising in biological pattern formation based on anomalous diffusions.In this thesis,we focus on two fractional-order reaction-diffusion systems.One is the linear fractional-order activator-inhibitor system,we study the existence of solution to the forward problem,the numerical solution,the uniqueness of the parameter inversion and the numerical inversion.The other is the nonlinear fractional-order diffusion system,we study its positive boundedness of the solution of the forward problem,the numerical solution,the optimal solution of the inverse problem.The details are organized as follows.Chapter 1 gives the background,the significance of the research and the current state of research at home and abroad,and some preliminaries and the main work of this thesis are introduced.In Chapter 2,the forward problem for a linearized fractional-order activator-inhibitor system is studied.The unique existence of the solution to the forward problem is proved by using the Laplace inversion.A finite difference scheme is set forth to solve the forward problem,and numerical simulations are performed to show the convergence and stability of the scheme.Chapter 3 is devoted to an inverse problem of determining the fractional order and the degeneration coefficient using the observations at one interior point.The extremum principle is applied to prove the uniqueness of the inverse problem in the image space of Laplace transform,and numerical inversions are presented by the homotopy regularization algorithm.In Chapter 4,the forward problem for a nonlinear fractional-order diffusion system is studied.The positive boundedness of the solution of the forward problem is proved with the aid of the maximum principle,and the finite difference solution is established to give numerical solutions.Furthermore,we simulate the pattern formations with appropriate model parameters,and the streak-like spot formation becomes clear as the fractional orders become small.In Chapter 5,we study an inverse problem of determining the key parameters in the model by the final observations.From the view point of optimal control,the inverse problem is transformed to a minimal problem of an error functional,and the existence of an optimal solution to the inverse problem is proved by using the energy estimation and compactness method.Numerical inversions are performed by the homotopy regularization algorithm with noisy data to demonstrate the numerical stability of the inverse problem.Chapter 6 concludes with a summary of the work,and gives questions that can be further studied based on the research of this thesis.