Existence and uniqueness of periodic orbits in a discrete model on Wolbachia infection frequency

In this paper, we study a discrete model on Wolbachia infection frequency. Assume that a periodic and impulsive release strategy is implemented, where infected males are released during the first N generations with the release ratio α, and the release is terminated from (N + 1)-th generation to T-th generation. We find a release ratio threshold denoted by α *(N, T), and prove the existence of a T-periodic solution for the model when α ∈ (0, α *(N, T)). For the special case when N = 1 and T = 2, we prove that the model has a unique T-periodic solution which is unstable when α ∈ (0, α *(N, T)). While α ≥ α *(N, T), no periodic phenomenon occurs and the Wolbachia fixation equilibrium is globally asymptotically stable. Numerical simulations are also provided to illustrate our theoretical results. One main contribution of this work is to offer a new method to determine the exact number of periodic orbits to discrete models.

Dynamics of a Seasonally Forced Phytoplankton-Zooplankton Model with Impulsive Biological Control

This paper investigates the dynamics of a seasonally forced phytoplankton-zooplankton model with impulsive biological control. It shows that the periodic eradicated solution is unstable. Further, the condition for permanence of the system is established by relations between the model parameters and the intensity of the impulses. The numerical analysis is performed to study the effect of seasonality and impulsive perturbations on plankton dynamics. The numerical results imply that the seasonal forcing can trigger more periodic mode and the impulsive period for control of the size of phytoplankton is more practicable to the system than the impulsive release of zooplankton. These conclusions provide a better understanding of controlling harmful algae blooms.

Two Generalized Predator-Prey Models for Integrated Pest Management with Stage Structure and Disease in the Prey Population

Stage-structured predator-prey models with disease in the prey are constructed. For the purpose of integrated pest management, two types of impulsive control strategies (impulsive release of infective prey and impulsive release of predator) are used. For Case 1, infective prey applications are more frequent than releases of predator (natural enemies). For Case 2, predator (natural enemies) releases are more frequent than infective prey applications. In both cases, we get the sufficient conditions for the global attractivity of the susceptible prey-eradication periodic solution. In addition, the persistence of the systems is also discussed. At last, the results are discussed and some possible future work is put forward.

Impulsive Biological Pest Control Strategies of the Sugarcane Borer

We propose an impulsive biological pest control of the sugarcane borer (Diatraea saccharalis) by its egg parasitoidTrichogramma galloibased on a mathematical model in which the sugarcane borer is represented by the egg and larval stages, and the parasitoid is considered in terms of the parasitized eggs. By using the Floquet theory and the small amplitude perturbation method, we show that there exists a globally asymptotically stable pest-eradication periodic solution when some conditions hold. The numerical simulations show that the impulsive release of parasitoids provides reliable strategies of the biological pest control of the sugarcane borer.

A Higher Casting and Jump Motions Realized by Robots Using Magnetic Brake Cylinder

A casting motion or a jumping motion can enhance the traverse ability and agility simultaneously of a mobile robot. This paper describes the development of a novel actuator, based on a pneumatic driving unit, which enables the generation of high-speed motion necessary to realize the motions mentioned above. The proposed actuator, named Magnetic Brake (MB) Cylinder, is composed of a pneumatic cylinder, a permanent magnet, a portable tank, and small valves. The speed of conventional pneumatic cylinders highly depends on the size of the valve which drives it. Since the magnet plays a role to enhance the impulsive release function of pneumatic energy instead of using a big and heavy valve, the pressure inside the cylinder can be kept in high condition, enabling the generation of high velocity with light structure. The height control method of casted objects with the MB Cylinder and its design method are also described in this paper. The analysis of the performance of the MB Cylinder and its simulation method are described for when using the MB Cylinder for both casting motion and jumping motion. After the developed unit is installed on both the casting device and the jumping robot, the validity of the proposed methods is experimentally verified in addition to discussion on its application to rescue operation.

DYNAMIC BEHAVIORS OF A KIND OF PREDATOR–PREY SYSTEM WITH IVLEV'S AND BEDDINGTON–DEANGELIS' FUNCTIONAL RESPONSE AND IMPULSIVE RELEASE

A kind of one-prey two-predator system with Ivlev's and Beddington–DeAngelis' functional response and impulsive release at fixed moments is presented. It is shown that the system has a prey-free periodic solution. By using the Floquet theory and small amplitude perturbation skills, it is proved that the prey-free periodic solution of the system is locally asymptotically stable when the period of impulsive release is less than a critical value. Furthermore, permanence of the system is investigated and the condition of permanence is obtained. Finally, numerical simulations show that the system has complex properties which include periodic solution, period-doubling bifurcation, chaos, chaotic windows, half-period bifurcation. A brief discussion on our results and their relation between the continuous system and the impulsive system are given. The results obtained in this paper are confirmed by numerical simulations.