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.
Mathematical models can assist to design and understand control strategies for limited resources in Integrated Pest Management (IPM). This paper studies the dynamical behavior of a Filippov predator–prey model with periodic forcing. Firstly, bifurcation analyses are carried out to show that the Filippov predator–prey ecosystem may have very complex dynamics, i.e. the system may have periodic, quasi-periodic, chaotic solutions, as well as period doubling bifurcations. Meanwhile, the model is analyzed theoretically and numerically to understand how resource limitation and periodic forcing affect pest population outbreaks, the intersection between the initial densities (pest and natural enemy populations) and pest control has been discussed. Furthermore, the sliding surface, sliding mode dynamics, the existence and stability of sliding periodic solution of the proposed model and its application in IPM strategy are investigated. Our results show that several hidden factors can adversely affect our control strategy in limited resource and fluctuating environment. Thus, choosing a proper threshold value ET may play a decisive role in pest control, which confirms that IPM is the optimal control strategy.
In this work, we propose a family of six new quasi-quintic trigonometric blending functions with two shape parameters. Based on these blending functions, a class of quasi-quintic trigonometric Bézier curve is proposed, which has some properties analogous to the classical quintic Bézier curves. For the same control points, the resulting quasi-quintic trigonometric Bézier curves can be closer to the control polygon than the classical quintic Bézier curves. The shape of the quasi-quintic trigonometric Bézier curves can be flexibly adjusted by altering the values of the two shape parameters without changing their control points. Under the C 2 smooth connection conditions, the resulting composite quasi-quintic trigonometric Bézier curves can automatically reach C 2 ∩ FC 3 continuity.
This paper studies a Filippov predator-prey system, where chemical control strategies are proposed and analyzed. Initially, the exact sliding segment and its domains are addressed. Then the existence and stability of the regular, virtual, pseudo-equilibria and tangent points are discussed. It shows that two regular equilibria and a pseudo-equilibrium can coexist. By employing theoretical and numerical techniques several kinds of bifurcations are investigated, such as sliding bifurcations related to the boundary node (focus) bifurcations, touching bifurcations, sliding crossing bifurcation and buckling bifurcations (or sliding switching). Furthermore, it makes comparison of the obtained results with previous studies for the Filippov predator-prey system without control strategies. Some biological implications of our results with respect to pest control are also given.
This paper investigates the impact of the threshold control strategy and environmental randomness on pest control. Firstly, a fixed-time impulsive stochastic ecosystem with IPM strategy is proposed, where the local and global existence of positive solution and the boundedness of expectation are discussed in detail. Moreover a sufficient condition for the extinction of the pest population with probability-1 is given. Then, a state-dependent stochastic ecosystem with IPM strategy is proposed. By employing the numerical simulations, the effects of ambient noise intensity on pest-outbreak are discussed. The result shows that there is a close relationship among the frequency of pest-outbreak, ET, the environmental perturbation intensity, and control measures. This study helps us to understand the impact of random factors on pest-outbreak frequency by theoretical derivations and numerical simulations; the results have directive significance in the design of an optimal control strategy for the department of ecological agriculture.
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