Two Groups Of Mutual Solutions Of The Model Cyclical And Blasting,

In nature, in addition to biological species between predator and prey relations, mutually beneficial relationship is a relationship prevailing. Two-species cooperating models have been extensively studied. Most models studied before are based on the assumptions that each individual admits a stable birth-rate, death-rate and diffusivity. However, because of the birth-rate, death-rate and environmental capital of the cyclical and other reasons, therefore, it is clearly unrealistic for many animals. Therefore, it is necessary to introduce the coefficient in the model of a function of time"t".For most models, there are a number of papers before just considering the periodicity of the solutions or just considering the blowup of the solutions. For example, Gan and Lin considered a competitor-competitor-mutualist model with Dirichlet boundary condition, and studied the existence, uniqueness and the global asymptotic behavior of the system mainly by Schauder fixed point theory, the method of upper and lower solutions and its associated monotone iterations. Lin considered a three-species cooperating model, and gave sufficient conditions for the solution to exist globally and to blow up in a finite time, and gave the estimation of the rate of blowup, further. In this paper, our main idea is with the same combination of these two parts of the same model, giving the response conditions for the periodicity and blowup of the solutions.Based on the two above factors, in this paper we major on periodicity and blowup in a two-species cooperating model. To begin with, the background and history of related work are introduced. In the second part, we will research the existence of the two-species cooperating model using the methods of upper and lower solutions and its associated monotone iterations and give the sufficient conditions of the stability and attractivity of the maximum and minimum T -periodic solutions. This condition shows that periodic solutions exist if the intra-specific competitions are strong. In the third part, the sufficient conditions of the blowup of the solutions are proven after several lemmas and definition of the blowup. This condition shows that blowup solutions exist under certain conditions if the inter-specific competitions are strong. The last part, numerical illustrations are carried out. From the tables and graphs are given, we can see that the values of parameters meet the conditions of the theorems, the simulated images also meet the conditions of the theorems.