Some Studies For Lesile Model With A Protection Zone And The Ratio-dependent Functional Response

Ecological security and the protection of the endangered species are the main issues with the great concern.In view of the human activities and the deterioration of the environment,a large number of species are on the verge of extinction.If these endangered species are not protected,they will eventually become extinct,which cause some irreparable damage to local ecosystems and food chains.How to protect effectively the endangered species and whether such protection will bring unpredictable changes to the species involved,are the issues.In this paper,by investigating an appropriate mathematical model with a protection zone,we will find its mathematical mechanism.In this paper,we study the Lesile model with a protection zone and the ratio-dependent functional response,where the prey is the endangered species,and a protection zone is established for the prey.We mainly study the following problems:(1)what are the size or the volume of the protection zone ?0 to protect effectively the endangered species?(2)what does the protection zone ?0 impact on the coexistence of the prey and predator?By using the estimates theory of partial differential equations,the bifurcation theorems,the Lyapunov-Schmidt reduction method and the perturbation theory,we obtain a critical value ?1N(b?(x),Q)which is different from the critical value ?1D(?0)in other models with the protection zone,and prove that if ?>?1N(b?(x),?),then the prey species persists regardless of the growth rates,u of the predator.If ?<?1N(b?(x),?),Q),then its results is similar to that of the corresponding model without the protection zone,namely,there exists a positive number ?*,such that the prey is ultimately extinct when ?>?*;while if ?<?1N(b?(x),?)and ?<?*,the prey and predator can coexist.Meantime,we also obtain the uniqueness and the stability of the positive solution of the corresponding model under various conditions.