Augmentative biocontrol when natural enemies are subject to Allee effects

Abstract

Intraspecific interactions such as Allee effects are key properties that can guide population management. This contribution considers component Allee effects that are elementary mechanisms leading to declines of fitness at the population scale, i.e. demographic Allee effects. It especially focuses on the consequences of such properties in predator populations, and investigates their repercussions in a biological control context. A modelling framework able to account for reproductive and/or foraging component Allee effects is proposed. From this, four models of augmentative biological control, corresponding to the periodic introduction of natural enemies, have been investigated. This is done using semi-discrete models: ordinary differential equations are used to depict predator-prey dynamics and a discrete equation describes the abrupt augmentation of predators at periodic intervals. In that context, stability of a prey-free solution corresponding to pest eradication has been analyzed. It has been found that rare but large introductions should be preferred over frequent and small ones, when Allee effects influence predator populations. In particular, the occurrence of foraging, rather than reproducing, Allee effects significantly hinders pest eradication. Cases where the pest-free solution is locally, but not globally, stable were also observed and were shown to be favored by the occurrence of reproductive Allee effects among predators.

On the existence and uniqueness of ( N , λ ) $(N,\lambda )$ -periodic solutions to a class of Volterra difference equations

“…Moreover, for a matrix L such that e LT = Φ(T), there is a periodic matrix function t → Z(t) such that x(t) = Z(t)e LT x 0 where L, Z(t) ∈ C n×n and Z(t) = z(t + T). An application of this result to the model of predator-prey interactions can be seen in [7]. For additional related applications, see [9,11] and the references therein.…”

Multiple dynamics in a delayed predator‐prey model with asymmetric functional and numerical responses

“…In this paper, we considered a predator-prey system with Holling type I functional response and intra-specific competition among the predators, but the numerical response is not proportional to the functional response. The role of such a unbalanced functional and numerical responses has been studied by a limited number of researchers to explore population dynamics in the contexts of Allee induced dynamics 1,34 biological pest control, 35 and hydra effect. 36 A special feature of our model is that the number of coexisting equilibria (if exist) is always two.…”

“…In Verdy, 1 the Allee effect was found to modulate the predation rate. Bajeux et al 35 have discussed about biological control tactics in a predator‐prey model in which two component Allee effects are present, viz., foraging efficiency and reproductive Allee effects. Costa and dos Anjos 36 have incorporated reproductive Allee effect in the Rosenzweig‐MacArthur predator‐prey model with intra‐specific competition among predators.…”

Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

“…The main purpose of this paper is to construct a generalised predator-prey model with nonlinear impulsive control to investigate the effect of limited predator releases on the outbreak of pest populations. Other generalised predator-prey models with linear or constant pulse actions have already provided some analytical techniques to deal with the global stability of periodic solutions [27][28][29]. In Sect.…”