Butterfly Catastrophe Model On Population Dynamics Of Wheat Aphids

Catastrophe Theory (CT) was constructed by René Thom about40years ago. In Thom’sview, CT are all mathematical theories which describe how abrupt changes take place infamilies of objects depending smoothly on parameters. Because of Zeeman’s application workand advertisement, Elementary Catastrophe Theory (ECT), as a part of CT, attracted lots ofpublicity. After that, the theory was broadly applied in many catastrophic or uncontinuousphenomenons such as earthquake, vocano eruption, mudslide, heartbeat, embryonicdevelopment, cell division, neural osscilation and so on. But meanwhile, this kind ofapplications was suffered from harsh critisms. Most of these controversies originated not fromthe mathematical foundation of ECT, but from its application approach in biology and socialscience. This study will explore whether catastrophe can be applied in ecology field or notand if can, how to apply the theory. Since various ecological systems have multiple stableequilibrium points, ECT has the application potential for ecology. In this study, we adoptcatastrophe modeling approach to study an ecological system. In the last stage of the study,we notice and agree the viewpoint of Loehle (1989) that catastrophe analysis is morereasonable than catastrophe modeling. Studies with catastrophe analysis approach will be anew direction for this kind of researches.In agriculture, crops suffer heavily from aphid aggression. How to control aphidpopulation outbreak is a crucial question. Previous studies of aphid system using ECT modelsfocused on low dimentional models such as cusp. But in fact, the ecology system aphids staymay be affected by many control variables including whether, predators, pesticides and so on.In order to describe catastrophic behaviors of aphid numbers, we propose using a highdimentional model, butterfly catastrophe model, which has four control variables. Since aphidgrowth is influenced by meteorological conditions, natural enemies, environmental carryingcapacity and pesticide effect, we use these four factors as control factors of aphid populationdynamics and build population dynamic model, parameter estimation model and butterflycatastrophe model. We use stability analysis to analyze the bifurcation set, characterizedifferent catastrophe regions and predict catastrophe behaviors. We also use data to test ourmodels and evaluate these models have potential application for pest control. This study obtains results as follows.(1) We build the population dynamic model ofaphids incorporating environmental carrying capacity, whether conditions, predators andpesticide effects. This model is an improved version of Logistic model.(2) When thepopulation growth reaches balanced states, this study derivatives the butterfly catastrophemodel with four control variables correlated with the four factors. This work can offer areference for biological control of aphids.(3)This study performs parameter estimation ofthe butterfly catastrophe model. B-spline function approximation methods are employed toachieve the estimation.(4) Stability analysis is used to analyze the stability of differentcatastrophe regions in the bifurcation set and field data collected by Insect Control Laboratoryof Northwest A&F University is used to verify the theory analysis. Resulsts show the “jumps”in aphid data are predicted by the butterfly catastrophe model.