Two models of interfering predators in impulsive biological control

Abstract

In 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.

Influence of Intrapredatory Interferences on Impulsive Biological Control Efficiency

“…In fact, it is also apparent when the interference is modelled through a density dependent increase of the predator mortality rate (Nundloll et al, 2010) rather than by modifying the trophic function and, although we do not show it here, similar properties can be derived in the case of weak and strong interference as presented in Sections 4.2 and 4.3. Conversely, no such dependence in T is obtained when no interference effects are modelled (Mailleret and Grognard, 2009).…”

“…Both conditions are only sufficient. The dependence of the above mentioned threshold in T is presented explicitly, and generalises the work of Zhang and Chen (2006), Nundloll et al (2010) on explicit Beddington-DeAngelis trophic functions in that we ensure that the conclusions we draw are generic and not specifically related to a model form; additionally, we provide global stability conditions that guarantee the efficiency of the biological control for any initial size of infestation, as opposed to local stability conditions that only ensure pest eradication in cases of small outbreaks. Our study of the release rate condition's dependence on T also reveals that small frequent releases lower the minimal release rate required for pest eradication and maximise the speed of convergence to the pest-free solution.…”

“…were of Holling II type). As such, we note that this formulation generalises the results by Zhang and Chen (2006) and Nundloll et al (2010). The stability conditions are then described by the following theorem.…”

“…Other studies, involving semi-discrete predator-prey dynamics that also take into account intra-predatory competition of the type presented here (that is, involving a modification of the trophic response) consider explicit Beddington-DeAngelis responses (Negi and Gakkhar, 2007;Zhang and Chen, 2006;Nundloll et al, 2010;Zhang et al, 2008). As will be shown subsequently, the qualitative formulation in this paper is able to provide us with general insights on the influence of intra-predatory interference on an impulsive system.…”

A population model with birth pulses, age structure, and non-overlapping generations

“…Whilst our model is new, as far as we are aware, we note that a number of other models of populations subject to birth pulses and age structure have been studied in recent years [7,[22][23][24]. Furthermore, hybrid systems have been investigated in many other contexts, including (but not limited to) vaccination strategies [2,20,28], chemotherapeutic treatment of diseases [12,18], and pest control [16,17,26,27,30,33,36].…”