Influence of Intrapredatory Interferences on Impulsive Biological Control Efficiency

International audienceIn this paper, a model is proposed for the biological control of a pest by its natural predator. The model incorporates a qualitative description of intrapredatory interference whereby predator density decreases the per capita predation efficiency and generalises the classical Beddington-DeAngelis formulation. A pair of coupled ordinary differential equations are used, augmented by a discrete component to depict the periodic release of a fixed number of predators into the system. This number is defined in terms of the rate of predator release and the frequency at which the releases are to be carried out. This formulation allows us to compare different biological control strategies in terms of release size and frequency that involve the same overall number of predators. The stability properties of the zero-pest solution are analysed. We obtain an upper bound on the interference strength (the biological condition) and a minimal bound on the predator release rate (the managerial condition) required to eradicate a pest population. We demonstrate that increasing the frequency of releases reduces this minimal rate and also increases the rate of convergence of the system to the zero-pest solution for a given release rate. Additionally, we show that other conclusions are to be expected if the interferences between predators have weaker or stronger effects than the generalised Beddington-DeAngelis formulation proposed in this paper

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

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

Modelling and Analysis of a Pest-Control Pollution Model with Integrated Control Tactics

A hybrid impulsive pest control model with stage structure for pest and Holling II functional response is proposed and investigated, in which the effects of impulsive pesticide input in the environment and in the organism are considered. Sufficient conditions for global attractiveness of the pest-extinction periodic solution and permanence of the system are obtained, which show that there exists a globally asymptotically stable pest-extinction periodic solution when the number of natural enemies released is more than some critical value, whereas the system can be permanent when the number of natural enemies released is less than another critical value. Furthermore, numerical simulations are carried out to illustrate our theoretical results and facilitate their interpretation

Two models of interfering predators in impulsive biological control

International audienceIn this paper, we study the effects of Beddington-DeAngelis interference and squabbling, respectively, on the minimal rate of predator release required to drive a pest population to zero. A two-dimensional system of coupled ordinary differential equations is considered, augmented by an impulsive component depicting the periodic release of predators into the system. This periodic release takes place independently of the detection of the pests in the field. We establish the existence of a pest-free solution driven by the periodic releases, and express the global stability conditions for this solution in terms of the minimal predator rate required to bring an outbreak of pests to nil. In particular, we show that with the interference effects, the minimal rate will only guarantee eradication if the releases are carried out frequently enough. When Beddington-DeAngelis behaviour is considered, an additional constraint for the existence itself of a successful release rate is that the pest growth rate should be less than the predation pressure, the latter explicitly formulated in terms of the predation function and the interference parameters

An Impulsive Three-Species Model with Square Root Functional Response and Mutual Interference of Predator

An impulsive two-prey and one-predator model with square root functional responses, mutual interference, and integrated pest management is constructed. By using techniques of impulsive perturbations, comparison theorem, and Floquet theory, the existence and global asymptotic stability of prey-eradication periodic solution are investigated. We use some methods and sufficient conditions to prove the permanence of the system which involve multiple Lyapunov functions and differential comparison theorem. Numerical simulations are given to portray the complex behaviors of this system. Finally, we analyze the biological meanings of these results and give some suggestions for feasible control strategies

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

Dynamics of a Stage Structure Pest Control Model with Impulsive Effects at Different Fixed Time

Many existing pest control models, which control pests by releasing natural enemies, neglect the effect that natural enemies may get killed. From this point of view, we formulate a pest control model with stage structure for the pest with constant maturation time delay (through-stage time delay) and periodic releasing natural enemies and natural enemies killed at different fixed time and perform a systematic mathematical and ecological study. By using the comparison theorem and analysis method, we obtain the conditions for the global attractivity of the pest-eradication periodic solution and permanence of the system. We also present a pest management strategy in which the pest population is kept under the economic threshold level (ETL) when the pest population is uniformly permanent. We show that maturation time delay, impulsive releasing, and killing natural enemies can bring great effects on the dynamics of the system. Numerical simulations confirm our theoretical results

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