A Kind Of Plankton Food Chain Model With Internal Storage

Plankton plays a key role in the energy flow of marine ecosystems.Phyto-plankton is the basis of the aquatic food chain,and nutrients are the main resources for the growth of phytoplankton.By using the reaction diffusion equation theory,we study a food chain model of plankton with internal storage under homogeneous Robin boundary conditions.We mainly use comparison principle,upper lower solu-tion method,fixed point method and theory of monotone dynamical system in the cones to study the dynamic behavior of the system.In the first chapter,the background and related works on food chain model of plankton are introduced.Then we give some known results as preliminaries.In the second chapter,we investigate coexistence and long-term behavior of the model.The main results are as follows.Firstly,the eigenvalue problem is constructed to study the long-term behavior of single species model.For the growth model of phytoplankton,we obtain a critical diffusion coefficient.There is only one positive steady-state solution of the system when the diffusion coefficient is smaller than the critical diffusion coefficient,and the species can survive.Conversely,the system has only zero steady-state solution,and the species goes to extinct.For the growth model of zooplankton,the species can survive when the principal eigenvalue of the associated eigenvalue problem is greater than zero.Conversely,the species goes to extinct.Secondly,the well-posedness of the model is discussed by means of constructing special functions,upper lower solution method and comparison principle.Then the global existence of classical solutions is established.The stability of trivial and semi-trivial solutions are determined by the signs of the principal eigenvalues of associated eigenvalue problems.Then we use the theory of monotone dynamical system to prove the uniform persistence of the system.Finally,we establish the sharp a priori estimates of the steady state system.We know that all nonnegative solutions are contained in a special cone.According to the selection of this cone,the existence of positive solutions of the model is obtained by applying the topological fixed point method.The results show that the existence of positive steady-state solutions depends on the principal eigenvalues of the associated eigenvalue problems.The positive steady-state solutions exist when the associated principal eigenvalues have the same signs.