Application And Analysis Of Elliptic Ijmbilic Catastrophe On Enemy-Pest System

Catastrophe is a phenomenon which is widespread in the biological. Catastrophe theorywhich was created by Thom provides theoretic basis for the study of the catastrophephenomenon in the biology. However, the research so far mainly focus on fold and cuspbecause of the complexity of the ecological system. Catastrophe phenomenon plays animportant role in the analysis and study of ecological system stability theory. We can makepests-natural enemies system to achieve a stable state by adjusting the control variable andcan explain the causes of catastrophe by analyzing the properties of the catastrophic model.Because the elliptical catastrophe model involved two state variables and three controlvariables, therefore, the analysis of catastrophe area, catastrophe surface and umbilical pointsis very complex, especially balance surface and bifurcation.In the agricultural ecological system, the state variable of pests groups is controlled bymany control variables which influence the developmental direction of pest populations.However, the way of influence,the ratio of control variables and the application in theagricultural production need urgently to be solved in the next researches. The relationshipbetween predator and prey plays an important role in the stability of agroecosystem. In thisstudy, the isomorphic relation with Taylor after adding the concept of predatory strength inthe pest-enemy system is applied into constructing the expression of predatory strength. It isnot only having an important theoretical significance, but also having an important utilityvalue in agricultural production.To analyze the process of arthropod community succession and stability, the ellipticumbilic catastrophe model is developed based on diversity index, richness index, evennessindex and the ratio of pests and natural enemies of arthropod community in kiwifruit orchards,which include3years kiwifruit orchard,9years kiwifruit orchard,14years kiwifruit orchardand wild kiwifruit orchard. In this paper, we intend to offer some theoretical basis forecological regulation of pests according to analysing the curved surface and bifurcation set ofthe Cusp.The main conclusions obtained are as follows: 1. In this study, the isomorphic relation with Taylor after adding the concept of predatorystrength in the pest-enemy system is applied into constructing the expression of predatorystrength. Then obtain the elliptic umbilic catastrophe model which involved two statevariables and three control variables through mathematical deduction.2. We get the regions of the bifurcation and the stable according to the analysis of thebifurcation set and balance stability in each area of elliptic umbilic catastrophe. In addition,the conditions in which the catastrophe may occur can be gotten in this study.3. The phenomenon of alternative distribution can be discovered for the stable region andthe unsteady region according to analysing the characteristic of them.4. The results show that pest sub-populations of arthropod community are instable in3and9years kiwifruit orchard, and stable in14years kiwifruit orchard and wild kiwifruitorchard. It was illustrated that, community stability increased with the increasing of the ageof kiwifruit orchard, The control variable w is an important index of commuity stabilityaccording to the test in3and9years kiwifruit orchard. Based on the principal componentanalysis, we know that the stability level was a key factor for forecasting outbreak of pestpopulations, which is influenced by the balanced stable control variable.5. In this paper, we analyze five characters of catastrophe model by means of the curvedsurface and bifurcation set of the Cusp, and explain them based on ecology phenomena.In this paper, the elliptic umbilic catastrophe model was applied firstly to analyzecommunity evolution, structure stability and pest forecast. The model is more objective thanfold, cusp and swallow tail in describing the discipline of pest-enemy system. Therefore itmaybe provide theoretical basis for Integrated Pest Management.