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In this paper, we propose a Filippov switching model which is composed of the Lorenz and Chen systems. By employing the qualitative analysis techniques of nonsmooth dynamical systems, we show that the new Filippov system not only inherits the properties of the Lorenz and Chen systems but also presents new dynamics including new chaotic attractors such as four-wing butterfly attractor, Lorenz attractor with sliding segments, etc. In particular, we find that different new attractors can coexist such as the coexistence of two-point attractors and chaotic attractor, the coexistence of two-point attractors and quasi-periodic solution, the coexistence of transient transition chaos and quasi-periodic solution. Furthermore, nonsmooth bifurcations and numerical analyses reveal that the proposed Filippov system has a series of new sliding bifurcations including a symmetric pair of sliding mode bifurcations, a symmetric pair of sliding Hopf bifurcations, and a symmetric pair of Hopf-like boundary equilibrium bifurcations. [ABSTRACT FROM AUTHOR]

By assuming a periodic variation in the intrinsic growth rate of the prey, a nonlinear ecological system with periodic forcing and state-dependent feedback control is proposed. The main purpose of the present paper is to study the dynamical behavior generated by periodic forcing and nonlinear impulse perturbations and their effects on pest control. To do this, we first investigate the existence and stability of the boundary periodic solution, and then we employ the numerical bifurcation techniques, mainly including one-dimensional and two-dimensional parameter bifurcation analyses, to reveal that the system exhibits rich and complex dynamic behaviors. Especially, period-adding bifurcation with chaos is found in the two-parameter bifurcation plane. Moreover, we find the periodic structure similar to Arnold tongues, and they are arranged according to the sequence of a Farey tree. In addition, one-dimensional bifurcation diagrams reveal the existence of order- k periodic, and chaotic solutions, multiple coexisting attractors, period-doubling bifurcations, period-halving bifurcations, and so on. Finally, the effects of the initial population density of pests and natural enemies on the pulse frequency and the biological significance related to the numerical results are studied and discussed. [ABSTRACT FROM AUTHOR]

A planar predator–prey impacting system model with a nonmonotonic functional response function is proposed and analyzed. The existence and stability of a boundary order-1 periodic solution were investigated and the threshold conditions for a transcritical bifurcation and stable switching were obtained, and also the definition and properties of the Poincaré map are discussed. The main results indicate that multiple discontinuous points of the Poincaré map could induce the coexistence of multiple order-1 periodic solutions. Numerical analyses reveal the complex dynamics of the model including periodic adding and halving bifurcations, which could result in multiple active phases, among them rapid spiking and quiescence phases which can switch from one to another and consequently create complex bursting patterns. The main results reveal that it is beneficial to restore the stability and balance of a ecosystem for species with group defence by moderately reducing population densities and the group defence capacity. [ABSTRACT FROM AUTHOR]

Insects such as locusts and some moths can transform from a solitarious phase when they remain in loose populations and a gregarious phase, when they may swarm. Therefore, the key to effective management of outbreaks of species such as the desert locust Schistocercagregaria is early detection of when they are in the threshold state between the two phases, followed by timely control of their hopper stages before they fledge because the control of flying adult swarms is costly and often ineffective. Definitions of gregarization thresholds should assist preventive control measures and avoid treatment of areas that might not lead to gregarization. In order to better understand the effects of the threshold density which represents the gregarization threshold on the outbreak of a locust population, we developed a model of a discrete switching system. The proposed model allows us to address: (1) How frequently switching occurs from solitarious to gregarious phases and vice versa; (2) When do stable switching transients occur, the existence of which indicate that solutions with larger amplitudes can switch to a stable attractor with a value less than the switching threshold density?; and (3) How does random perturbation influence the switching pattern? Our results show that both subsystems have refuge equilibrium points, outbreak equilibrium points and bistable equilibria. Further, the outbreak equilibrium points and bistable equilibria can coexist for a wide range of parameters and can switch from one to another. This type of switching is sensitive to the intrinsic growth rate and the initial values of the locust population, and may result in locust population outbreaks and phase switching once a small perturbation occurs. Moreover, the simulation results indicate that the switching transient patterns become identical after some generations, suggesting that the evolving process of the perturbation system is not related to the initial value after some fixed number of generations for the same stochastic processes. However, the switching frequency and outbreak patterns can be significantly affected by the intensity of noise and the intrinsic growth rate of the locust population. [ABSTRACT FROM AUTHOR]

In the present paper, we perform a complete bifurcation analysis of a two-stage population model, in which the per capita birth rate and stage transition rate from juveniles to adults are density dependent and take the general Beverton-Holt functions. Our study reveals a rich bifurcation structure including codimension-one bifurcations such as saddle-node, Hopf, homoclinic bifurcations, and codimension-two bifurcations such as Bogdanov-Takens (BT), Bautin bifurcations, etc. Moreover, by employing the polynomial analysis and approximation techniques, the existences of equilibria, Hopf and BT bifurcations as well as the formulas for calculating their bifurcation sets have been provided. Finally, the complete bifurcation diagrams and associate phase portraits are obtained not only analytically but also confirmed and extended numerically. [ABSTRACT FROM AUTHOR]

Fragmented habitat patches between which plants and animals can disperse can be modeled as networks with varying degrees of connectivity. A predator-prey model with network structures is proposed for integrated pest management (IPM) with impulsive control actions. The model was analyzed using numerical methods to investigate how factors such as the impulsive period, the releasing constant of natural enemies and the mode of connections between the patches affect pest outbreak patterns and the success or failure of pest control. The concept of the cluster as defined by Holland and Hastings is used to describe variations in results ranging from global synchrony when all patches have identical fluctuations to n-cluster solutions with all patches having different dynamics. Heterogeneity in the initial densities of either pest or natural enemy generally resulted in a variety of cluster oscillations. Surprisingly, if n > 1, the clusters fall into two groups one with low amplitude fluctuations and the other with high amplitude fluctuations (i.e. duality in phase space), implying that control actions radically alter the system's characteristics by inducing duality and more complex dynamics. When the impulsive period is small enough, i.e. the control strategy is undertaken frequently, the pest can be eradicated. As the period increases, the pest's dynamics shift from a steady state to become chaotic with periodic windows and more multicluster oscillations arise for heterogenous initial density distributions. Period-doubling bifurcation and periodic halving cascades occur as the releasing constant of the natural enemy increases. For the same ecological system with five differently connected networks, as the randomness of the connectedness increases, the transient duration becomes smaller and the probability of multicluster oscillations appearing becomes higher. [ABSTRACT FROM AUTHOR]

A Filippov system is proposed to describe the stage structured nonsmooth pest growth with threshold policy control (TPC). The TPC measure is represented by the total density of both juveniles and adults being chosen as an index for decisions on when to implement chemical control strategies. The proposed Filippov system can have three pieces of sliding segments and three pseudo-equilibria, which result in 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 interesting 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 pseudo-saddle with a figure of eight shape, to a pseudo-saddle-node or to a standard saddle-node have been observed 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. One more sliding segment (or pseudo-equilibrium) is induced by the total density of a population guided switching policy, compared to only the juvenile density guided policy, implying that this control policy is more effective in terms of preventing multiple pest outbreaks or causing the density of pests to stabilize at a desired level such as an economic threshold. [ABSTRACT FROM AUTHOR]

We subject the classical Volterra predator--prey ecosystem model with age structure for predator to periodic forcing, which in its unforced state has a globally stable focus as its equilibrium. The periodic forcing is affected by assuming a periodic variation in the intrinsic growth rate of prey. By using the Poincaré surface-of-section technique and numerical experiments, an abundance of periodic solutions, quasiperiodic solutions and chaotic solutions are revealed. The route of transitions to chaos were found to be very complex which include quasiperiodicity and frequency locking. [ABSTRACT FROM AUTHOR]

In order to avoid high extinction risks of prey and keep the stability of the three-species food chain model, we introduce a Filippov food chain model (FFCM) with Holling type II under threshold policy control. The threshold policy is designed to play a pivotal strategy for controlling the three species in the FFCM. With this strategy, no control is applied if the density of the prey population is less than the threshold, thus the exploitation is forbidden. However, the exploitation is permitted if the density of the prey population increases and exceeds the threshold. The dynamic behaviors and the bifurcation sets of this model including the existence and stability of different types of equilibria are discussed analytically and numerically. Moreover, the regions of sliding and crossing segments are analyzed. The dynamic behaviors of sliding mode including the bifurcation sets of pseudo-equilibria are investigated. Numerically, the bifurcation diagram and maximum Lyapunov exponents are computed and plotted to show the complex dynamics of FFCM, for instance, it has stable periodic, double periodic and chaotic solutions as well as double periodic sliding bifurcation. It is demonstrated that the threshold policy control can be easily implemented and used for stabilizing the chaotic behavior of FFCM. [ABSTRACT FROM AUTHOR]

Surgery is the traditional method for treating cancers, but it often fails to cure patients for complex reasons so new therapeutic approaches that include both surgery and immunotherapy have recently been proposed. These have been shown to be effective, clinically, in inhibiting cancer cells while allowing retention of immunologic memory. This comprehensive strategy is guided by whether a population of tumour cells has or has not exceeded a threshold density. Conditions for successful control of tumours in an immune tumour system were modeled and the related dynamics were addressed. A mathematical model with state-dependent impulsive interventions is formulated to describe combinations of surgery with immunotherapy. By analyzing the properties of the Poincaré map, we examine the global dynamics of the immune tumour system with state-dependent feedback control, including the existence and stability of the semi-trivial order-1 periodic solution and the positive order- periodic solution. The main results showed that surgery alone can only control the tumour size below a certain level while there is no immunologic memory. If comprehensive therapy involving combining surgery with immunotherapy is considered, then not only can the cancers be controlled below a certain level, but the immune system can also retain its activity. The existence of positive order- periodic solutions implies that periodical therapy is needed to control the cancers. However, choosing the treatment frequency and the strength of the therapy remains challenging, and hence a strategy of individual-based therapy is suggested. [ABSTRACT FROM AUTHOR]

The switched discrete host-parasitoid model concerning integrated pest management (IPM) has been proposed in the present work, and the economic threshold (ET) is chosen to guide the switches. That is, if the density of host (pest) population increases and exceeds the ET, then the biological and chemical tactics are applied together. Those multiple control measures are suspended once the density of host falls below the ET. Firstly, the existence and stability of several types of equilibria of switched system have been discussed briefly, and two- or three-parameter bifurcation diagrams reveal the regions of different types of equilibria including regular and virtual equilibria. Secondly, numerical bifurcation analyses show that the switched discrete system may have very complex dynamics including the co-existence of multiple attractors and switched-like behavior among attractors. Finally, we address how the key parameters and initial values of both host and parasitoid populations affect the host outbreaks, switching frequencies or mean switching frequency, and consequently the relative biological implications with respect to pest control are discussed. [ABSTRACT FROM AUTHOR]