Narrow Results

Based on the classical Lotka–Volterra predator–prey system, an impulsive differential equation to model the process of periodically releasing natural enemies and spraying pesticides at different fixed times for pest control is proposed and investigated. It is proved that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Otherwise, the system can be permanent. We observe that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Numerical results show that the system we considered has more complex dynamics including period-doubling bifurcation, symmetry-breaking bifurcation, period-halving bifurcation, quasi-periodic oscillation, chaos and nonunique dynamics, meaning that several attractors coexist. Finally, a pest–predator stage-structured model for the pest concerning this kind of impulsive control strategy is proposed, and we also show that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some threshold.

Some weaknesses of 1D chaotic maps, such as lacking of ergodicity, multiple bifurcations, dense periodic windows, and short iteration period, limit their practical applications in cryptography. A higher-dimensional chaotic map with ergodicity can solve these problems. Based on 1D quadratic map, a 3D exponential hyperchaotic map (3D-EHCM) is constructed, and its dynamic behaviors, such as phase diagram, Lyapunov exponent spectrum, Kolmogorov entropy (KE), correlation dimension, approximate entropy and randomness, are analyzed and tested. The results demonstrate that the 3D-EHCM has ergodicity in a larger range of control parameter, and its state points have a longer period. To counteract dynamical degradation and make it suitable for a PRNG, the periodic point detection and random impulsive perturbation are applied to lengthen the aperiodic time sequence, and statistical results demonstrate that a full-period sequence can be obtained.

This paper deals with a periodic coefficients predator-prey system with impulsive harvesting for the predator, in which the prey has a history that takes them through two stages, juvenile and adult. We apply this system to model strategies for controlling apple snails which are serious pests of young rice seedlings in paddy fields. A sufficient and necessary condition to guarantee permanence of the system is provided. Finally, we give a brief discussion of our result.

We investigate a three-species food chain system with density-dependent birth rate and impulsive effect concerning biological and chemical control strategy — periodic releasing of natural enemies or spraying pesticide at different fixed times. Conditions for the extinction of the prey and top predator are given. By using the Floquet theory of impulsive differential equations and small amplitude perturbation skills, we consider the local stability of the prey and top predator eradication periodic solution. Further, we obtain the conditions of permanence of the system.

The effects of periodic forcing and impulsive perturbations on the predator–prey model with Beddington–DeAngelis functional response are investigated. We assume periodic variation in the intrinsic growth rate of the prey as well as periodic constant impulsive immigration of the predator. The dynamical behavior of the system is simulated and bifurcation diagrams are obtained for different parameters. The results show that periodic forcing and impulsive perturbation can very easily give rise to complex dynamics, including quasi-periodic oscillating, a period-doubling cascade, chaos, a period-halving cascade, non-unique dynamics, and period windows.

In this paper, we are concerned with the periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations with periodic inhomogenous terms in Banach spaces, where the operators in the linear part (possibly unbounded) depend on the time $t$ and generate an evolution family of linear operators. We first establish two new Gronwall–Bellman-type inequalities, and then prove a new and general existence theorem for periodic mild solutions to the nonautonomous impulsive delay evolution equations, which extends essentially some existing results even for the autonomous case as well as for the case when impulsive perturbations or delays are absent.

This paper deals with an impulsive predator-prey model with Beddington–DeAngelis functional response and time delay, in which the evolution of the predators takes them through two stages, juvenile and mature. It is assumed that only mature predators are able to hunt for prey and reproduce and the time delay is understood as being the time spent by the juvenile predators from birth to maturity. It is first seen that the dynamics of the model can be completely determined through the use of a reduced system consisting of the equations for prey and mature predators, respectively. Using the discrete dynamical system determined by the stroboscopic map, one first determines the mature predator-free periodic solution of the reduced system. By means of comparison techniques, one then deduces sufficient criteria for the global stability of the mature predator-free periodic solution and for the permanence of the reduced system, which yield similar properties for the initial system. As a result, it is observed that time delay and pulses have a crucial effect upon the dynamics of our model.

In this talk we present a brief summary of some recent results of the author on the subject of impulsive stochastic differential inclusions on Hilbert spaces. We also discuss briefly some problems of optimal control.