The Qualitative Analysis Of Wheat Aphids Ecosystem Model Based On Catastrophe Theory

Wheat is the main grain crops which is second only to rice in our country. As an important factor of the wheat management, the pest controls make the farmer get higher production. There are more than 120 pests in the wheat fields, in which the aphids has a wide distribution in our country and has the following characteristics, such as strong reproductive, quicker diffusion etc. The effect of vegetation and weather etc on wheat aphids lead to its outbreak, which is described by catastrophe models.It is difficult for the catastrophe models to the choice of control variables, the estimation of parameters and the expression of potential functions. On the basis of the swallowtail catastrophe model and the butterfly catastrophe model, the normal forms of the corresponding model are given. According to the catastrophe theory and the qualitative theory of differential equations, the qualitative analysis of the models are studied. The conditions of catastrophe phenomenon are obtained and the principles of the outbreak of wheat aphids are proposed.The main contents are as follows:1. For the swallowtail catastrophe model, the bifurcation set of the models for wheat aphids ecosystem model is got, the corresponding geometric figures are drawn, and then the catastrophe regions with different properties are divided. Next, the stability of equilibria in different regions are investigated using the qualitative theory of differential equations. Finally, the simulations are carried to prove the results, and reveal the mechanism of the outbreak of wheat aphids.2. For the butterfly catastrophe model, the bifurcation set and corresponding geometric figures wheat aphids ecosystem model are got, and then the catastrophe regions with different properties are divided. Next, the stability of equilibria in different regions are investigated using the qualitative theory of differential equations. Finally, the simulations are carried to prove the results, which may provide the theoretical basis for wheat aphids control and management.3. Compared the swallowtail catastrophe model with the butterfly catastrophe model, we know that the former has more accurately explaination on the outbreak of wheat aphids than the latter, which results from the dependant of control variables. The further study is done in the future.