| This paper deals with a Lotka-Volterra competition model of allelopathy The existence and stability of coexistence solutions of the corresponding equilibrium system are studied. By means of the upper and lower solution method, the degree theory and the bifurcation theory, we study the existence and stability of coexistence solutions, and analyze the effect of the allelopathy on coexistence solutions. The main contents of this paper are as follows:In chapter 1, we introduce the background and the recent works on the competi-tion model in details. And some important lemmas are given. Finally, we investigate the existence and uniqueness of the semi-trivial solutions, and discuss the related properties of the semi-trivial solutions.In chapter 2, we mainly study the existence of coexistence solutions of steady state system. By applying the upper and lower solution method, the variational principle, we obtain some estimates of the equilibrium problem. Finally, we prove the existence of coexistence solutions by using the degree theory.In chapter 3, we mainly study the bifurcation structure of coexistence solutions, and analyze the effect of the allelopathy on coexistence solutions. Firstly, by taking c as the bifurcation parameter, the local bifurcation branch of positive solutions from the semi-trivial solution (?a,0) is constructed by using the bifurcation theory. At the same time, we investigate the global bifurcation solution by using the global bifurcation theory. It turns out that the bifurcation curve connects two semi-trivial solutions{(a,?a,0):a E R+} and{(c,0, (?)c):c ? R+}. Secondly, the existence and stability of the bifurcation solutions from a double eigenvalue are proved. We also discuss the stability of the bifurcation solutions. It shows that coexistence solutions are stable as long as bd< 1. Finally, we analyze the allelopathic effects on coexistence solutions. The results show that when the allelopathy within a certain range, it is helpful for the species coexistence. |
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