Abstract
The idea of action threshold depends on the pest density and its change rate is more general and furthermore can produce new modelling techniques related to integrated pest management (IPM) as compared with those that appeared in earlier studies, which definitely bring challenges to analytical analysis and generate new ideas to the state control measures. Keeping this in mind, using the strategies of IPM, we develop a prey-predator system with action threshold depending on the pest density and its change rate, and study its dynamical behavior. We develop new criteria guaranteeing the existence, uniqueness, local and global stability of order-1 periodic solutions. Applying the properties of Lambert W function, the Poincaré map is portrayed for the exact phase set, which is helpful to provide the sufficient conditions for the existence and stability of the interior order-1 periodic solutions and boundary order-1 periodic solution, also confirmed by numerical simulations. It is studied in detail that how and under what conditions the fixed point of Poincaré map and its stability are affected by the newly introduced action threshold. The analytical methods developed in this paper will be very beneficial to study other generalized models with state-dependent feedback control.
Highlights
Most of the researchers considered systems with impulses at fixed moments [21,22,23,24,25,26,27,28,29]. e shortcomings of this kind of systems are that they did not pay enough attention to the Discrete Dynamics in Nature and Society y (t) y(t)
Prey-predator models have received a high concentration of scholars due to their prosperous dynamic behavior
Prey and predator can impact each other’s development, and such pairs exist throughout nature. It represents one of the primary models in mathematical ecology. Another fundamental concept of integrated pest management (IPM) process is that of using sound ET
Summary
In view of the reasons specified above, we consider the commonly used prey-predator system with pest density and its change rate dependent feedback control, i.e., the action threshold depends on pest density and on its change rate, which can be modeled by. 푥(푡) and 푦(t) respectively, represent the quantities of prey and predator. E quantities 1 − 푝 푥(푡) and 푦(푡) + 휏 are known as the controlling quantities; whenever the pest population touches the action threshold, the management activities are adapted and the quantities of prey and predator are adjusted according to the controlling actions 1 − 푝 푥(푡) and 푦(푡) + 휏 respectively. If the weighted parameter vanishes, the ratio-dependent AT transformed into 푦 = (푐푥 − 퐴푇)/ 푞푥. Tang et al [46] presented the following prey-predator model with state-dependent feedback control which is the special case of model (1) for 훼 = 1 and 훽 = 0. We will see that the results associated to model (8) can be obtained based on the results for model (2)
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