A Type Of Phytoplankton Growth Model With Crowding Effects

There are a large number of phytoplankton in lakes and oceans,which form the base of the aquatic food chain.Nutrition and light are important resources for the growth of phytoplankton.This paper deals with a reaction-diffusion-advection model with crowding effect of a single phytoplankton species in a water column:We study the following problems of this system,including the existence and uniqueness of the positive steady states,the long-time behavior of the solution and the effect of advection rate on the asymptotic profiles of positive steady states by means of the strong maximum principle,the comparison principle,the degree theory and the bifurcation theory.The main results are as follows:First of all,we investigate the existence and uniqueness of the positive steady states by the strong maximum principle,the degree theory and the bifurcation theory.We conclude that when the natural death rate is greater than or equal to the critical death rate,zero is the only nonnegative solution of the equilibrium system;when the natural death rate is less than the critical death rate,the equilibrium system has a unique positive solution.Futhermore,the positive solution will have a finite limit rather than go to infinity when the death rate disappears,which is due to the crowding effect.Secondly,we use the comparison principle and the method of upper and lower solutions to study the long-time behavior of the solution of the system.We obtain the following conclusions:when the natural death rate is greater than or equal to the critical death rate,the solution of the system converges uniformly to zero;when natural death rate is less than the critical death rate,the solution of the system converges uniformly to its unique positive equilibrium.Finally,we use scaling arguments and various analytical techniques to inves-tigate the asymptotic profiles of positive equilibria for large advection rates.The results indicate that for large sinking rate,the phytoplankton species concentrates at the bottom of the water column with a finite population density,and for large buoyant rate,the phytoplankton species concentrates at the surface of the water column with a finite population density.