Non-constant Positive Steady States Of A Prey-predator System

In this paper, a prey-predator model with cross-diffusions is considered. The following model will be considered:WhereThe organization of this paper is as follows:As the introduction of the whole paper, we state the background and make plans for the research of the problem.Next section of the paper presents some basic definitions and known theorems, which are the foundations and tools of the later work.The third section of this paper mainly investigates the stability of the ODE system.The forth section of this paper mainly investigates the stability of the above system.The fifth section of this paper will discuss the priori upper and lower bounds.The sixth section and the seventh section will, respectively, discuss the non-existence and the existence of non-constant positive solutions.At last, we summarize the work of the whole paper.