A Class H (?) Lling Ii Type Functional Response Predator Model The Asymptotic Nature,

Originating from the Population Statistics, Applied Entomology and Aquatic Science, population ecology is a science focusing on the relations between the dynamic amount of population and the environment.With the appearance of modeling theory by Lotka-Volterra in 1925 and 1926, well-known as a milestone of theoretical ecology, its development, afterwards, entered a golden age, and has grown gradually into a testing, quantitative and academic discipline from the ones aiming mainly to description since the last half century.However, population ecology became the mainstream of ecology marking from the Leng Quan Harbor international conference on the theory of population control in 1957.Since then, population ecology has stepped into the most developmental and active field both on the theories and methods.The presentation focuses on a H?lling II predator-prey model with the non-local delay and stage structure.After giving the existence and uniqueness of the system by upper and lower solutions, we consider the globally asymptotical behavior of steady-state solutions under self-diffusion by means of Lyapunov Functionals, Comparison Principles and Soblev Imbedding theorems, respectively.The results yield three cases.Case (1): Both of the prey and predator are permanent if the intra-specific competitions of the predator and the prey are sufficiently large.Case (2): The prey will be permanent, while the predator will go to extinction, if the death rate of predator is large enough while the death rate of the mature prey is relevantly small together with a large birth rate of the immature.Case (3): If the death rate of the mature prey are large enough and the birth rate as well as the transformation rate from immature individuals to mature ones is sufficiently low.Also, the influence on the system of stage-structure, non-local delay and cross-diffusion have been analyzed showing that the two populations maybe persistent if each individual admits an unchanged density-dependent rate and an equal change to compete with others; on the other hand, they may be extinct since the stage-structure makes negative effects on the prey.Meanwhile, the incorporating of non-local delay brings with no effect on the global stability of the system.Finally, it is know that the diffusion makes a smooth effect on the system, which means the solution becomes smooth and eventually converges to a homogenous equilibrium stage for any initial data, it is evident in our paper that the introduction of cross-diffusion in the system can drive the instability (called"Turing patterns"). The arrangement of the presentation is as follows: to begin with, the background and history of population ecology are introduced, followed by the description of Turing patterns, and then we consider the reaction-diffusion equations with H?lling II predator-prey model. In section two, the existence and uniqueness of the global solution to the system are given by means of upper and lower solutions, and then the global stabilities of the corresponding equilibriums are proved through the way of Lyapunov functions. In the fourth section some analysis has been made to search for the conditions under which the Turing patterns occur.The corresponding numerical illustrations are carried out with the aid of Matlab in the last two sections.