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Mathematical Problems in Engineering
Volume 2014, Article ID 450935, 13 pages
http://dx.doi.org/10.1155/2014/450935
Research Article

The Effects of Resource Limitation on a Predator-Prey Model with Control Measures as Nonlinear Pulses

1College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China
2Natural Resources Institute, University of Greenwich at Medway, Kent ME4 4TB, UK

Received 16 October 2013; Accepted 26 January 2014; Published 11 March 2014

Academic Editor: Jui-Sheng Lin

Copyright © 2014 Wenjie Qin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.