A Positive Solution Branch Of A Two-species Competition Model In An Open Convection Environment

The effect of competition on population dynamics is an important topic in the study of application of biomathematical differential equations.Generally,we apply reaction-diffusion model to depict the dynamic behaviors of species.This paper deals with a two-species competition system in open advective and heterogeneous environments:The characteristics of this model are heterogeneity of resources distribution and different competition coefficients.The loss rate of the species at the downstream end is a hundred percent due to the effect of water flow.That is,the species will be completely washed out when they are washed downstream,and the downstream end presents a free flow state.Firstly,based on the related linear eigenvalue problems,the critical values on the interspecific competition coefficients are established,which determine the local stability of two semi-trivial steady states.The results show that the corresponding semi-trivial steady states are locally asymptotically stable when the competition coefficients are larger than the corresponding critical values and they are unstable when the competition coefficients are less than the corresponding critical values.In particular,the theory of monotonic dynamical systems indicates that two species coexist when the competition coefficients of both species are less than or greater than their corresponding critical values.Secondly,for the global dynamics of the system,it is very difficult to study the nonexistence of the positive steady-state solutions.We study the the nonexistence of coexistence steady-state solutions by means of the limit techniques when one of the competition coefficients is sufficiently large,and the other competition coefficient is less than the corresponding critical competition coefficient.Combining with stability analysis and monotone dynamical system theory,we further find that the competition exclusion principle holds.That is,the species whose the competition coefficient does not exceed the corresponding critical value will be eliminated in the competition.Finally,in terms of these critical values,the structure and direction of bifurcation branches of positive equilibria arising from two semi-trivial steady states are given by means of the bifurcation theory and stability analysis.Consequently,we derive the existence and multiplicity of positive steady states.In addition,we show that multiple positive steady-state solutions exist under certain regimes.