The Impulsive Model with Pest Density and Its Change Rate Dependent Feedback Control

Abstract

e 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 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. e analytical methods developed in this paper will be very beneficial to study other generalized models with state-dependent feedback control.Hindawi is a trajectory tangent to curve Γ at point 푆 = 푥 , 푦 , Γ 2 is a trajectory tangent to curve Γ at point 푇 = 푥 , 푦 , Γ 3 is homoclinic trajectory which touches curve Γ at two points 푥 1 , 푦 1 and 푥 2 , 푦 2 , and its upper and lower right branches touch curve Γ at points 푄 1 = 푥 푄 1 , 푦 푄 1 and 푄 2 = 푥 푄 2 , 푦 푄 2 , respectively.3 Discrete Dynamics in Nature and Society weighted parameters on the fixed point of Poincaré map. In the same section, the effect of weighted parameters on the different cases is also discussed. To sum up the whole work, a detailed conclusion is given in Section 7.

Complex Dynamics of Beddington–DeAngelis-Type Predator-Prey Model with Nonlinear Impulsive Control

“…Let Φ be the solution of the pulse-free system associated with system (30). Also, we get X(t) � Φ(t, X 0 ) with X 0 � X(0).…”

“…Furthermore, to establish the nontrivial periodic solution of system (30), we employ the techniques in [33]. en, we have…”

“…erefore, the release ratios of numbers of natural enemies according to current density in the field could significantly affect the effect of pest control strategy. In order to take the resource limitation into the IPM strategy, several predator-prey models with nonlinear impulse have been proposed [29][30][31][32] and mainly focused on establishing the global stability conditions. However, the nonlinear impulsive function mentioned above is only related to the density of the natural enemy population in the field.…”

Border-collision bifurcation in the Logistic map with state-dependent impulsive forces

“…Therefore, the stability analysis and the controller synthesis problems of these systems have drawn considerable attention from researchers (see, e.g., [33]). In particular, the dynamical behaviors of continuous systems with periodic impulsive forces [26] or state-dependent impulsive forces [16] have been studied extensively. On the other hand, considerable attention has been devoted to studying the behavior of discrete systems that experience periodic impulsive forces due to their wide range of applications, including biology and internet congestion control, e.g., [10,14,15,19].…”