Bifurcation Analysis Of Filippov Nonsmooth Ecosystem

In nature, the life cycle of each individual pest is composed of different age structures, such as, egg stage, larval stage and the adult stage. Pests in different age stages may have different morphology and sizes. The outbreak of pest popula-tions is closely associated with periodic variation of the environment where they live. The individual differences and seasonal periodic variation are very important factors that affect not only the growth of the pest population, but also bring challenges to the implementation of pest control strategies. How to describe and assess impact of those differences and periodic disturbance on pest control remain challenges both in agriculture and applied mathematics. In recent years impulsive differential equa-tion models, designed of integrated pest management strategies in view of the above factors, have been sufficiently developed and systemically examined, and have been employed for optimal control pest population. However, a common assumption of the work is the comprehensive control strategy including chemical control, biologi-cal control is instantaneously realized, the role of strategies on insect pests is also instantaneous. So, when pest population reach the economic threshold the existent control strategy does not instantaneously stopped, but only when they return back below the economic threshold the strategy can be stopped. A typical feature of the threshold strategy is whether the control measures are implemented or not depends on the given threshold. Research shows that the non-smooth Filippov system can provide a very natural depiction and description of this switching phenomenon, and this kind of system has been used in many fields of science, engineering and is also widely studied.Based on the juvenile or adult population density or total density as an in-dex to determine whether the control strategies are implemented, the interesting questions may include:are there any difference in outbreak frequency, amplitude of pest population, and frequency of control strategies implemented? What does the stage structure of pest population affect on the successful implementation of pest control strategies? To this end, we propose a non-smooth Filippov stage-structured pest growth model to describe three different threshold control strategies. Using the Filippov convex combination and Utkin equivalent control method, the bifur-cation theory of non-smooth systems, the definition and properties of the Lambert W function, we carefully analyze all kinds of equilibria of the proposed system, the existence of the sliding mode, sliding dynamics, and then obtain that system can have rich local sliding bifurcation and global sliding bifurcation. The main results show that for the Filippov system with total number of adult and larval popula-tion as an index, there may exist three pieces of sliding segments, while for the Filippov system with either the adult or larve population as an index, at most two pieces of sliding segments are possible. The increase of sliding segments results in that the proposed system exists the rich sliding mode bifurcations and local sliding bifurcations including boundary node (boundary focus, or boundary saddle) and tangency bifurcations. As the threshold density varies the model exhibits the in-teresting global sliding bifurcations sequentially:touching→buckling→crossing →sliding homoclinic orbit to a pseudo-saddle→crossing→touching bifurcations. In particular, bifurcation of a homoclinic orbit to a pseudosaddle with a figure of eight shape, to a pseudo-saddle-node or to a standard saddle-node have been ob-served for some parameter sets. This implies that control outcomes are sensitive to the threshold level, and hence it is crucial to choose the threshold level to initiate control strategy.In order to study impact of the periodic disturbance on pest control strategy and the dynamics of Filippov non-smooth system, we formulate a non-smooth Fil-ippov prey-predator model with periodic forcing and threshold policy. Using the Utkin equivalent control method, we systemically investigate the existence of the sliding mode, sliding dynamics, obtain the sufficiently conditions under which the sliding periodic solution is feasible and stable. Further, we numerically investigated the complex dynamics including coexistence of multiple attractors, period adding sequences and chaotic solutions with respect to bifurcation parameters of forcing amplitude and economic threshold (ET). Moreover, the stability of the sliding pe-riodic solution, the switching transients associated with pest outbreaks and their biological implications have been discussed. The results indicate that the sliding periodic solution could be globally stable, and consequently the pest population can be controlled below the given threshold. The magnitude and frequency of switching transients depend on the initial values of both pest and natural enemy population, forcing amplitude and ET.This paper establishes non-smooth population ecology models with multi-segments sliding modes, develops the branch analysis and numerical studies of the complex non-smooth ecosystem. The corresponding biological conclusions can provide cer-tain guidance for the agricultural pest control, the development of the modeling techniques and analytical methods can offer a reference for the Filippov systems applied in other fields.