On the impulsive controllability and bifurcation of a predator–pest model of IPM

Abstract

From a practical point of view, the most efficient strategy for pest control is to combine an array of techniques to control the wide variety of potential pests that may threaten crops in an approach known as integrated pest management (IPM). In this paper, we propose a predator-prey (pest) model of IPM in which pests are impulsively controlled by means of spraying pesticides (the chemical control) and releasing natural predators (the biological control). It is assumed that the biological and chemical control are used with the same periodicity, but not simultaneously. The functional response of the predator is allowed to be predator-dependent, in the form of a Beddington-DeAngelis functional response, rather than to have a perhaps more classical prey-only dependence. The local and global stability of the pest-eradication periodic solution, as well as the permanence of the system, are obtained under integral conditions which are shown to have biological significance. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations.

Modeling and Analysis of Integrated Pest Control Strategies via Impulsive Differential Equations

“…As was already mentioned in Section 3.2, Zhang et al [44] studied the stability of pest-free (also referred to as pest-eradication) periodic solutions of system (14), as the impulse period is varied. They determined a threshold 0 , depending on some of the remaining system parameters, so that for < 0 the pest-free solution is asymptotically stable, while for > 0 the system response is dominated by periodic solutions for which the pest and its natural enemies coexist.…”

“…As was pointed out, several authors have contributed to the theoretical analysis of those systems, with particular emphasis on the existence and stability of pest-free solutions, as well as finding explicit thresholds for control parameters at which stability is lost (bifurcations). While this is generally a straightforward task for models belonging to the class of smooth ordinary differential equations, the situation can be more involved for other types of systems, for example, impulsive differential equations; see [43,44,48,55,56]. Although impulsive systems describing pest control methods have received a good deal of attention in the past, numerical investigations of such models are rather scarce in the literature and are mostly carried out at the simulation level.…”

“…The theoretical foundations for a numerical study of the impulsive system (14) have been established by Zhang et al [44]. They addressed the question of existence and uniqueness of nontrivial periodic solutions in terms of a fixed point problem of a suitably defined operator.…”

“…In the study presented in [44], the authors established a critical value of the impulse period that separates the system behavior into two cases. In the first one, model (4) admits a stable periodic solution of the form (0,̃( )), > 0; that is, the pest population can be completely eradicated by means of the control measures.…”

“…Zhang and coworkers have proposed several pest control models based on the scheme described above, for example [44],…”