System Of Cooperation With The Smith Conjecture,

In this paper, we mainly study some relevant problems of cooperative system and this paper includes two parts. In the first part the global behavior of solution of n dimensional cooperative Lotka-Volterra system is studied. Simple conditions are established in terms of the parameters of the system for the asymptotic behavior forinitial conditions in the open orthant and as for the system x =, we obtain three theorems:Theorem then equation 1 0.(2) If then the above system has a positive steady state xsuch that and in which , and Moreover, for every x>0.These conditions are simple.In second part we prove a important assumed proposition that gives a relatively simple graph-theoretic test for determining and solving if a given system is cooperative or competitive in a domain D with respect to some Km. Some results aredrawn about relations between type of the system and existence of Km .The paper divides into two chapters. In the first chapter we mainly establish some simple conditions in terms of the parameters of the system for the asymptotic behavior. Firstly it gives the origin of the problem, secondly some definitions and some preliminary lemmas, finally the proof of three theorems. The second chapter mainly gives an assumed proposition a proof showed in Smith[2] on determining the existence of Km with respect to a given system.