Two Types Of Competition-competition-reciprocity Model Dynamics Research

This paper is concerned with two types of competitor-competitor-mutualist models.Part one,we investigate coexistence of the homogeneous Dirichlet boundary condition and Neumann bundary condition problem to a elliptic system of a competitor-competitor-mutualist model which the reaction rate is a smooth positive function of x.The existence of the positive solution is established by means of the method of upper and lower solutions and comparison principle.Then,we study a reaction-diffusion system of a competitor-competitor-mutualist model with Dirichlet boundary condition and analysis the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities.The main method used in studying of the stabilities is the spectral analysis to the linearized operators.In part two,we investigate a general class of strongly coupled elliptic systems(competitor-competitor-mutualist three-species Lotka-Volterra model).Firstly,by Schauder fixed point theory,the coexistence state of the strongly coupled system is given,the existence and method of construction of quasi-solutions for a strongly coupled elliptic system by the method of upper and lower solutions and its associated monotone iterations.Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak.