Analysis Of The Nature Of The Solutions Of Two Types Of Biological Models

Partial differential equations occurrenced in 18 century as a branch of math-ematics. With building mathematical model, theoretical analysis, explaining ob-jective phenomenon and solving practical problems, partial differential equations became a branch mathematics. By study and exploration of 200 years, partial dif-ferential equations have gained many important results in theories and applying.Reaction diffusion equations as an important branch of partial differential equa-tions are concerned widely by many scholars. Nowdays, reaction diffusion equations have owned an important domain in mathematics, and are one of the important bridges between mathematics and other science. Furthermore, reaction diffusion equations are also one of basic sources in development of fundamental mathemat-ics. At present, the dynamics of biological models have received intensive studies makeing use of reaction diffusion equations, and have been an important aspect in the field of non-linear reaction diffusion equations, and have gained many important and useful results.In this article, on the base of above research, using the theories of nonlinear analysis and nonlinear partial differential equations, especially those of parabolic equations and corresponding elliptic equations, we will respectively discuss the dy-namic behavior of following two specific biological models: stability, coexistence, non-existence, bifurcation of positive steady states have stud-ied. The tools used here include super-sub solutions method, comparsion principle, global bifurcation theory, linear stability theory and fixed-point theory of topology. The main contents and results in this article are as follows: In the first part, we state the global bifurcation and stability of (0.1) system. Some prior-estimates of the positive steady-states are proved by using the maximum principle and lower-upper solutions. Applying the local bifurcation theory and the global bifurcation theory, sufficient conditions for the existence of local bifurcating solutions of the steady-states of (0.1) and its stability are obtained, the existence of global bifurcating solution are also proved. Finally, making use of numerial value theories, numerial number of this system is produced.In the second part, main analytical results on equilibria of (0.2) system are dis-cussed. First, stability of the positive steady-states are given, some prior-estimates of the positive steady-states are proved by using the maximum principle. The non-existence and existence of non-constant positive solution are obtained. Bifurcation solutions of non-constant positive steady-state are given. Simultaneously, numerial number of this system is produced.