Abstract
With a long history of theoretical development, biological model has focused on the interaction of a parasitoid and its host. In this paper, two Nicholson‐Bailey models with a nonlinear pulse control strategy are proposed and analyzed to examine how limited resource affects the pest control. For a fixed‐time discrete impulsive model, the existence and stability of the host‐free periodic solution are derived. Threshold analysis suggests that it is critical to release parasitoid in an optimal number in case of the happening of the intra‐specific competition, which will seriously affect the pest control. Bifurcation analysis reveals that the model exists complex dynamics including period doubling, chaotic solutions, coexistence of multiple attractors, and so on. For a state‐dependent discrete impulsive model, the numerical simulations for bifurcation analysis are studied, the results show that how the key parameters and the initial densities of both populations affect the pest outbreaks, and consequently the relative biological implications with respect to pest control are discussed.
Highlights
Mathematical models can assist in the design and understanding of basic problems in biology, medicine, and life sciences, which can take many forms including dynamical system, differential equation, difference equation, and so on [1,2,3]
In order to control pest more effective, at the same time, environmental influences and human interventions are taken into consideration, and the concept of Integrated Pest Management (IPM) strategy has been proposed [14,15,16,17], as a consequence it is challenging to evaluate the effectiveness of control measures such as biological, chemical, and physical control tactics
The results indicate that the parameters of model (3) are highly sensitive, and the dynamical behaviors of model (3) will have a complex change along with parameter variations, even if there is a small perturbation in some key parameters
Summary
Mathematical models can assist in the design and understanding of basic problems in biology, medicine, and life sciences, which can take many forms including dynamical system, differential equation, difference equation, and so on [1,2,3]. Some certain species, including many species of insect, have no overlap between successive generations and so their populations evolve in discrete time-steps. The discrete model governed by a difference equation is more appropriate than the continuous ones taking into account the fact that the population has a short life expectancy, non-overlapping generations in the real world [4,5,6], so it is reasonable to study biological models governed by difference equations [7,8,9,10,11,12,13]. Based on the concept of IPM strategy, some mathematical models to describe the interaction between the pest and its natural enemy have been developed [18,19,20,21]
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