Abstract
Whether the integrated control measures are applied or not depends not only on the current density of pest population, but also on its current growth rate, and this undoubtedly brings challenges and new ideas to the state control measures that only rely on the pest density. To address this, utilizing the tactics of IPM, we constructed a Lotka-Volterra predator-prey system with action threshold depending on the pest density and its changing rate and examined its dynamical behavior. We present new criteria guaranteeing the existence, uniqueness, and global stability of periodic solutions. With the help of Lambert W function, the Poincaré map is constructed for the phase set, which can help us to provide the satisfactory conditions for the existence and stability of the semitrivial periodic solution and interior order-1 periodic solutions. Furthermore, the existence of order-2 and nonexistence of order-k(k≥3) periodic solutions are discussed. The idea of action threshold depending on the pest density and its changing rate is more general and can generate new remarkable directions as well compared with those represented in earlier studies. The analytical techniques developed in this paper can play a significant role in analyzing the impulsive models with complex phase set or impulsive set.
Highlights
The predator-prey interaction systems attracted wide attention since the basic idea exhibited by Lotka and Volterra in the 1920s and widely studied for their rich dynamics [1–3]
In order to overcome those shortcomings, based on the classical Lotka-Volterra system, we propose the model with pest density and its change rate dependent feedback control; i.e., the action threshold depending on the pest density and its change rate determines whether the control strategy is implemented:
It is discussed in detail that the most important factor in integrated pest management (IPM) process is the economic threshold (ET) or action threshold
Summary
The predator-prey interaction systems attracted wide attention since the basic idea exhibited by Lotka and Volterra in the 1920s and widely studied for their rich dynamics [1–3]. A basic hypothesis in all existing models is that the IPM should be implemented once the density of the pest population reaches the given economic threshold; i.e., the state-dependent impulsive models have been used. Note that, in this way, the two crucial points related to analyzing the global dynamics of the proposed models, the impulsive set and phase set, are two straight lines. In order to overcome those shortcomings, based on the classical Lotka-Volterra system, we propose the model with pest density and its change rate dependent feedback control; i.e., the action threshold depending on the pest density and its change rate determines whether the control strategy is implemented:. To conclude the entire work, a brief discussion is given in the last section
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