Abstract
The dynamical behavior of a Holling II predator-prey model with control measures as nonlinear pulses is proposed and analyzed theoretically and numerically to understand how resource limitation affects pest population outbreaks. The threshold conditions for the stability of the pest-free periodic solution are given. Latin hypercube sampling/partial rank correlation coefficients are used to perform sensitivity analysis for the threshold concerning pest extinction to determine the significance of each parameter. Comparing this threshold value with that without resource limitation, our results indicate that it is essential to increase the pesticide’s efficacy against the pest and reduce its effectiveness against the natural enemy, while enhancing the efficiency of the natural enemies. Once the threshold value exceeds a critical level, both pest and its natural enemies populations can oscillate periodically. Further-more, when the pulse period and constant stocking number as a bifurcation parameter, the predator-prey model reveals complex dynamics. In addition, numerical results are presented to illustrate the feasibility of our main results.
Highlights
It is well known that pest outbreaks often cause serious ecological and economic problems, requiring complex control measures to reduce harm due to insect pests of agriculture and insect vectors of important plant, animal, and human diseases
We explored the parameter space by performing an uncertainty analysis and sensitivity analysis using the Latin hypercube sampling (LHS) method and evaluating partial rank correlation coefficients (PRCCs) for various input parameters against threshold conditions and the key factors which are most significantly related to the threshold conditions were determined
Note that the threshold values R0i depend on all parameters of model (3), the most interesting parameters here are the pulse period T and some parameters related to resource limitation such as the maximum fatality rates pimax (i = 1, 2) and the constant θ2, so it makes sense to know how T, half saturation pimax, θ2 affect the threshold conditions (R0i < 1; i = 1, 2) which guarantee the global stability of the pest-free periodic solution (12)
Summary
It is well known that pest outbreaks often cause serious ecological and economic problems, requiring complex control measures to reduce harm due to insect pests of agriculture and insect vectors of important plant, animal, and human diseases. The above traditional predator-prey model with IPM has assumed that the fatality rate of pesticide applications with respect to the pest is constant, which implies that the agricultural resources such as pesticides, labor forces, equipments, and costs are very effective and sufficient for controlling pests. The main purposes of this paper are to construct a simple mathematical model including the features of periodic biological and chemical control for pest control to understand how limited resources affect pest outbreaks since the limited resources and the fatality rate of pesticide applications on the pest could depend on its population’s density. In order to investigate the effect of the limited capacity of pesticides, a nonlinear continually differentiable function to characterize the saturation phenomenon of the limited resources is introduced. The paper ends with some interesting biological conclusions and numerical bifurcation analyses, which complement the theoretical findings
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