An Impulsive Two-Prey One-Predator System with Seasonal Effects

In recent years, the impulsive population systems have been studied by many researchers. However, seasonal effects on prey are rarely discussed. Thus, in this paper, the dynamics of the Holling-type IV two-competitive-prey one-predator system with impulsive perturbations and seasonal effects are analyzed using the Floquet theory and comparison techniques. It is assumed that the impulsive perturbations act in a periodic fashion, the proportional impulses (the chemical controls) for all species and the constant impulse (the biological control) for the predator at different fixed time but, the same period. In addition, the intrinsic growth rates of prey population are regarded as a periodically varying function of time due to seasonal variations. Sufficient conditions for the local and global stabilities of the two-prey-free periodic solution are established. It is proven that the system is permanent under some conditions. Moreover, sufficient conditions, under which one of the two preys is extinct and the remaining two species are permanent, are also found. Finally, numerical examples and conclusion are given

Dynamic analysis of an impulsively controlled predator-prey system

In this paper, we study an impulsively controlled predator-prey model with Monod-Haldane functional response. By using the Floquet theory, we prove that there exists a stable prey-free solution when the impulsive period is less than some critical value, and give the condition for the permanence of the system. In addition, we show the existence and stability of a positive periodic solution by using bifurcation theory

The effects of resource limitation on a predator-prey model with control measures as nonlinear pulses

The dynamical behavior of a Holling II predator-prey model with control measures as nonlinear pulses is proposed and analyzed theoretically and numerically to understand how resource limitation affects pest population outbreaks. The threshold conditions for the stability of the pest-free periodic solution are given. Latin hypercube sampling/partial rank correlation coefficients are used to perform sensitivity analysis for the threshold concerning pest extinction to determine the significance of each parameter. Comparing this threshold value with that without resource limitation, our results indicate that it is essential to increase the pesticide’s efficacy against the pest and reduce its effectiveness against the natural enemy, while enhancing the efficiency of the natural enemies. Once the threshold value exceeds a critical level, both pest and its natural enemies populations can oscillate periodically. Furthermore,when the pulse period and constant stocking number as a bifurcation parameter, the predator-prey model reveals complex dynamics. In addition, numerical results are presented to illustrate the feasibility of our main results

A food chain system with Holling type IV functional response and impulsive perturbations

AbstractIn this paper, a three-trophic-level food chain system with Holling type IV functional response and impulsive perturbations is established. We show that this system is uniformly bounded. Using the Floquet theory of impulsive equations and small perturbation skills, we find conditions for the local and global stabilities of the prey and top predator-free periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent via the comparison theorem. We display some numerical examples to substantiate our theoretical results

The Dynamics of a Predator-Prey System with State-Dependent Feedback Control

A Lotka-Volterra-type predator-prey system with state-dependent feedback control is investigated in both theoretical and numerical ways. Using the Poincaré map and the analogue of the Poincaré criterion, the sufficient conditions for the existence and stability of semitrivial periodic solutions and positive periodic solutions are obtained. In addition, we show that there is no positive periodic solution with period greater than and equal to three under some conditions. The qualitative analysis shows that the positive period-one solution bifurcates from the semitrivial solution through a fold bifurcation. Numerical simulations to substantiate our theoretical results are provided. Also, the bifurcation diagrams of solutions are illustrated by using the Poincaré map, and it is shown that the chaotic solutions take place via a cascade of period-doubling bifurcations

Innovation, imitation and policy inaction

The paper deals with the controversial issue of intellectual property rights. We deal with an optimization problem to model the optimal government's behavior in presence of dynamic uncertainty and intervention costs. More specifically, we search for the optimal strategies to be implemented by a policy maker to optimally balance the number of innovators and imitators. The problem is first tackled from a purely theoretical perspective and then by implementing extensive numerical simulations on the basis of empirical data. By the theoretical perspective, we obtain a rigorous proof that optimal strategies depend on the initial value of the number of imitators and not on the initial ratio between innovators and imitators, whereas the simulations provide us with intuitive insights from an economic point of view, along with a validation of the theoretical results. The results support the evidence that governments choose the possible widest bandwidth and minimize the size of interventions so as to curb intervention costs

Nonlinear state-dependent feedback control of a pest-natural enemy system

The numbers of pests and of natural enemies released to control them as part of integrated pest management strategies are density dependent. Therefore, the numbers of natural enemies to be released and the rate at which they kill pests should depend on their densities when the number of the pest population has reached the economic threshold. Bearing this in mind, a classic Lotka–Volterra system but with nonlinear state dependent feedback control tactics is proposed and analysed in this paper. Furthermore, the definition and properties of the Poincaré map which is defined in the phase set were investigated for various cases, allowing us to address the existence and global stability of an order-1 periodic solution of the model with nonlinear feedback control. Moreover, the existence and nonexistence of periodic solutions with an order larger than 2 or 3 are also discussed. The modelling methods and analytical techniques developed could be widely used and applied in other systems with threshold control such as the glucose insulin regulatory system

Optimum timing for integrated pest management: Modelling rates of pesticide application and natural enemy releases

Many factors including pest natural enemy ratios, starting densities, timings of natural enemy releases, dosages and timings of insecticide applications and instantaneous killing rates of pesticides on both pests and natural enemies can affect the success of IPM control programmes. To address how such factors influence successful pest control, hybrid impulsive pest–natural enemy models with different frequencies of pesticide sprays and natural enemy releases were proposed and analyzed. With releasing both more or less frequent than the sprays, a stability threshold condition for a pest eradication periodic solution is provided. Moreover, the effects of times of spraying pesticides (or releasing natural enemies) and control tactics on the threshold condition were investigated with regard to the extent of depression or resurgence resulting from pulses of pesticide applications. Multiple attractors from which the pest population oscillates with different amplitudes can coexist for a wide range of parameters and the switch-like transitions among these attractors showed that varying dosages and frequencies of insecticide applications and the numbers of natural enemies released are crucial. To see how the pesticide applications could be reduced, we developed a model involving periodic releases of natural enemies with chemical control applied only when the densities of the pest reached the given Economic Threshold. The results indicate that the pest outbreak period or frequency largely depends on the initial densities and the control tactics

Effects of additional food availability and pulse control on the dynamics of a Holling-(p+1) type pest-natural enemy model

In this paper, a novel pest-natural enemy model with additional food source and Holling-($p$+1) type functional response is put forward for plant pest management by considering multiple food sources for predators. The dynamical properties of the model are investigated, including existence and local asymptotic stability of equilibria, as well as the existence of limit cycles. The inhibition of natural enemy on pest dispersal and the impact of additional food sources on system dynamics are elucidated. In view of the fact that the inhibitory effect of the natural enemy on pest dispersal is slow and in general deviated from the expected target, an integrated pest management model is established by regularly releasing natural enemies and spraying insecticide to improve the control effect. The influence of the control period on the global stability and system persistence of the pest extinction periodic solution is discussed. It is shown that there exists a time threshold, and as long as the control period does not exceed that threshold, pests can be completely eliminated. When the control period exceeds that threshold, the system can bifurcate the supercritical coexistence periodic solution from the pest extinction one. To illustrate the main results and verify the effectiveness of the control method, numerical simulations are implemented in MATLAB programs. This study not only enriched the related content of population dynamics, but also provided certain reference for the management of plant pest

Dynamics of a Stage Structured Pest Control Model in a Polluted Environment with Pulse Pollution Input

By using pollution model and impulsive delay differential equation, we formulate a pest control model with stage structure for natural enemy in a polluted environment by introducing a constant periodic pollutant input and killing pest at different fixed moments and investigate the dynamics of such a system. We assume only that the natural enemies are affected by pollution, and we choose the method to kill the pest without harming natural enemies. Sufficient conditions for global attractivity of the natural enemy-extinction periodic solution and permanence of the system are obtained. Numerical simulations are presented to confirm our theoretical results