Studies On Dynamics Of The Nonlinear Chemotaxis Models In Mathematical Biology

Chemotaxis,is the directional movement of cells or populations toward the concentration gradient of chemical signals.This is a widespread interaction mechanism,which is common in biological processes such as bacteria aggregation pattern,tumor-induced angiogenesis,population growth and competition.These biological phenomenons can be modeled by mathematical models which are formed by abstract PDE systems.This paper systematically investigates the dynamics of several nonlinear chemotaxis models in biological mathematics.We mainly discuss the following three types of models:a nonlinear chemotaxis-growth system with indirect attractant production,a chemotaxis model with singular sensitivities,a chemotaxis system with density-suppressed motilities.Moreover,this thesis is devoted to studying two kinds of problems:Global existence and uniform boundedness of solutions;Large time behavior of solutions,including large time convergence and convergence rate.The content of this paper is arranged as follows:In Chapter 1,introduction.We mainly introduce the source and development status of the research problem,and sketch the main research content of this paper.In Chapter 2,we mainly discuss the initial boundary value problem for a quasilinear chemotaxis system with indirect attractant production.Under the homogeneous Neumann boundary conditions,in light of some energy estimates,it is proved that this system admits a unique global classical bounded solution if the diffusion exponent is greater than 1 4/n(n 2).Moreover,suppose that the diffusion exponent is greater than zero and not equal to 1,then,by constructing the energy functional which decreases at an exponential rate when time increases,we also obtain the large time behavior of solutions.In Chapter 3,we mainly study the initial boundary value problem for a chemotaxis system with indirect attractant production and signal-dependent sensitivity under homogeneous Neumann boundary conditions.Firstly,based on weighted estimate techniques and heat semigroup theory,it is proved that this system possesses a unique global classical bounded solution in any spatial dimension.Next,when chemotactic sensitivity function and coefficient of logistic source satisfy some suitable conditions,we show that the solutions decay exponentially to the constant steady state and accurately calculate the convergence rates.In Chapter 4,we analysis a chemotaxis-competition model with singular sensitivities under homogeneous Neumann boundary conditions.Firstly,we invoke the classic transformationz:ln w₀ln wand derive the upper bound for z by semigroup theory,which results in the lower boundedness of w.With this,by using conventional processing skills,we further prove that this system admits a global classical solution if chemotactic coefficients of sensitivity functions and logistic source satisfy some suitable conditions when n 3.Next,when n 2,applying the transformed model and the classical energy method,it is shown that the system possesses a global bounded solution provided that the logistic damping effect is appropriately large.Finally,when n 1,by employing coupled estimate techniques and heat semigroup theory,we also eatablish the global existence and boundedness of solutions for the above system.In Chapter 5,we discuss the dynamics of a two-species chemotaxis-consumption system with density-suppressed motilities.Firstly,in view of energy estimate and the Moser-Alikakos iterative methods,it is shown that when n 2 or n 3 and the logistic damping effect is suitably strong,the initial-boundary value problem possesses a unique global bounded classical solution under appropriate assumptions on the motility functions.Moreover,as for the difference of the interspecific competition coefficients,by constructing the corresponding energy functional,we obtain that the solution decays exponentially or polynomially to the steady-state solution,also,the convergence rates are presented accurately.In Chapter 6,we summarize the main research content and the prospect of the future research plan.