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.

International audienceIn this paper, the effects of periodic partial harvesting of a continuously grown crop on augmentative biological control are analyzed. Partial harvesting can remove a proportion of both pests and biological control agents, so its influence on the control efficiency cannot be a priori neglected. An impulsive model consisting of a general predator-prey model in ode, augmented by a discrete component to depict releases of biological control agents and the periodic partial harvesting is used. The periods are taken as integer multiples of each other. A stability condition for pest eradication is expressed as the minimal value of the budget per unit time to spend on predators. We consider the partial harvesting period to be fixed by both the plant's physiology and market forces so that the only manipulated variable is the release period. It is shown that varying the release period with respect to the harvest period influences the minimal budget value when the former is carried out more often than the latter and has no effect when releases take place as often as or less frequently than the partial harvests

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

In this paper, the effects of periodic partial harvesting of a continuously grown crop on augmentative biological control are analysed. Partial harvesting can remove a proportion of both pests and biological control agents, so its influence on the control efficiency cannot be a priori neglected. An impulsive model consisting of a general predator-prey model in ode is used. It is augmented by a discrete component to depict releases of biological control agents and the periodic partial harvesting. A stability condition for pest eradication is expressed as the minimal value of the budget per unit time to spend on predators. We consider the partial harvesting period to be fixed so that the only manipulated variable is the release period. One period is taken as the integer multiple of the other. We show that when the releases are carried out more often than the harvests, the release period influences the minimal budget. Conversely, there is no effect on this budget when releases take place as often as or less frequently than harvests.