Abstract
The success of integrated pest management (IPM) depends on spraying the correct amount of pesticides at an appropriate time and releases of natural enemies or pathogens of the pest in appropriate proportions at critical times, with little cost and minimal effects on the environment. Therefore, control decisions require information on instantaneous killing rates of pesticides and numbers of natural enemies to be released, variables that should depend on the densities of both pest and natural enemy population densities in the field. To describe such a control strategy we have proposed a mathematical model of IPM involving releases of natural enemies in relation to a regulatory factor. The threshold condition for the existence and stability of the pest free periodic solution is provided using a cobweb model, the comparison principle and Floquet theory, which reveals the effects of nonlinear control actions on pest outbreaks. Bifurcation analyses show that the dynamics of the proposed model can be very complex, including multiple attractors and switch-like transition patterns following small random perturbations. Moreover, the random perturbations and nonlinear impulsive control measures could generate complex switching patterns, which show that the pest population could have outbreaks in complex ways due to environmental noise.
Highlights
It is well known that spraying pesticides can kill beneficial organisms as well as the target pests, and can result in outbreaks of secondary pests or rapid resurgence of pests that were initially suppressed [25]
The stability of the pest free solution switches on and off as T increases, and we emphasize here that the pest population goes to extinction very slowly if the threshold condition is satisfied once T lies in the oscillation region and close to the threshold value, while the dynamics could be much more complex if the period T is chosen such that the pest free periodic solution becomes unstable
We have extended a model with linear impulsive control tactics to a model with nonlinear impulsive control measures, which revealed more realistic situations for pest control
Summary
It is well known that spraying pesticides can kill beneficial organisms as well as the target pests, and can result in outbreaks of secondary pests or rapid resurgence of pests that were initially suppressed [25]. The releasing of natural enemies such as predators, parasites and pathogens to control pests is a type of biological control known as augmentation This approach uses commercially available species that are applied in a timely manner to prevent population increases, or to suppress a pest population [3, 11, 13]. Mathematical models can be used to evaluate the effectiveness of multiple biological control factors including the timing of releases, release ratios and density dependent regulatory factors affecting the natural enemies to be augmented [7,14,15,20,21]. Taking more factors into account, to address the effects of the density regulatory factor for the natural enemies (predators and not parasitoids or pathogens in this case) on the releasing ratios and pest control, we propose the following model dx(t) x(t).
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