| In recent years,more and more attention has been paid to the study of reaction-diffusion equations,which involve a large number of problems from many mathemati-cal models of physics,chemistry and biology.Thus,they have a strong practical back-ground.It is not only of great theoretical significance,but also of great practical value to study the nutrient-phytoplankton model in biology.Chakraborty et al.study a compli-cated nutrient-phytoplankton model,it can be shown that the local asymptotic stability and Turing instability of the system depending on the toxic effect ?[9].As they increase?,they find Turing instability.This thesis mainly studies a simple nutrient-phytoplankton model with toxic ef-fects,by using the comparison principle,energy estimates in partial differential theory,implicit function theorem and other methods to help us to study the qualitative proper-ties of the solution.Main studies are as follows:Chapter 1 introduces the background and significance of a nutrient-phytoplankton model,and the main work of this thesis simply.Chapter 2 studies a priori estimates and long-time behavior of solutions to a reaction-diffusion model,and discusses the existence of a compact attractor of the parabolic system.Chapter 3 concerns the stability of the constant solution to the corresponding el-liptic equation,especially Turing instability.Chapter 4 is devoted to a priori estimates of solutions to elliptic system,which will become fundamental in discussing the nonexistence and existence of nonconstant steady states.In Chapter 5 and Chapter 6,we mainly analyze the existence and nonexistence of nonconstant steady states,when the diffusion coefficients change within a certain range.By using the energy estimate,the implicit function theorem and Leray Schauder topological degree to carry on the discussion,respectively.Chapter 7 summarizes the research results of this thesis,compared to the model with toxic effect of the Holling-type ? functional proposed in[9]. |
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