A Three Species Strong Coupling System Constant Positive Equilibrium

In chemistry, physics and biology, there are so many questions waiting to be solved. In order to find the answer, we need to build mathematical models, among which reaction-diffusion equation is widely used. Through study of reaction-diffusion equation, reasonable explanations and predictions of practical questions and natural phenomena can be concluded. At present, more and more biologists adopt it in the field. For example, when a certain population quantity is large, in order to consider the interaction and growth laws between species and in the species, biologists will build competitive models, cooperative models and prey-predator models based on the actual situation. And through study of these models by mathematical theory, certain conclusions can be made, for example, what condition a population needs to co-exist or extinguish. Then, these conclusions can be applied to solve practical problems.In 1920s. A. J. Lotka and V. Volterra proposed the famous Lotka-Volterra Model. Since then, biomathematicians have never stopped research and study on it. In order to solve practical questions better, the model has been improved to be more practical through deep research. In this paper, on the basis of above research, we mainly concern with a strongly coupled reaction-diffusion system arising from a three-species prey-predator model, which is introduced cross-diffusion terms in all the three equations. The model is written as followsFor the above model, we consider the effect of cross-diffusion coefficientsα21,α23. andα32 on the existence of non-constant positive steady-state solutions under cer- tain hypotheses, respectively. First, some prior upper and lower bounds for positive steady states are given by the Maximum Principle and Harnack Inequality. Sec-ond, the non-existence of non-constant positive steady states is established whenαij= 0.i≠j. Then, based on Leray-Schauder degree theory, we investigate the dependence of the existence of non-constant positive steady-state solutions on cross-diffusion coefficients, respectively. We show that the model has at least one non-constant positive solution for suitable values ofα21,α23 andα30, respectively. Fi-nally, we discuss the bifurcation of non-constant positive solutions of the model with respect to d2, we prove that the bifurcation can be extended to infinity.