Abstract
In this paper, a predator‐prey system with pesticide dose‐responded nonlinear pulse of Beddington–DeAngelis functional response is established. First, we construct the Poincaré map of the impulsive semidynamic system and discuss its main properties including the monotonicity, differentiability, fixed point, and asymptote. Second, we address the existence and globally asymptotic stability of the order‐1 periodic solution and the sufficient conditions for the existence of the order‐k(k ≥ 2) periodic solution. Thirdly, we give the threshold conditions for the existence and stability of boundary periodic solutions and present the parameter analysis. The results show that the pesticide dosage increases with the extension of the control period and decreases with the increase of the threshold. Besides, the state pulse feedback control can manage the pest population at a certain level and avoid excessive application of pesticides.
Highlights
Where r, K, m, a, b, c, ε, and μ are positive constants. u(t) and v(t) represent the population density of prey and predator at time t, K is the environmental carrying capacity of the prey, and r is the intrinsic growth rate of prey
Function mu/(a + bv + cu) indicates the Beddington–DeAngelis functional response, and bv stands for mutual interference between the predators. e constants ε and μ represent the rate of conversion and death rate of predators, respectively
Feedback control of time pulse has certain defects, which may reduce crop yield and possibly increase management costs. erefore, we can choose to spray the pesticide when the quantity of pests reaches a certain threshold instead of spraying the pesticide at a fixed time. is measure avoids the possibility of the explosive growth of the number of pests and is more suitable for pest control. is paper studies the Beddington– DeAngelis system with pulse state feedback control strategies:
Summary
We give the definition of impulsive semidynamic system and order-k periodic solution. For trajectory Πf in (X, Π, M, I), if there are nonnegative integers m ≥ 0 and k ≥ 1, k is the minimum integer satisfying f+m f+m+k and Tk mi +mk− 1. E T-periodic solution (u, v) (ξ(t), η(t)) of thwe system. We know that the solution of system (3) is positive and bounded for all t. If d ≥ (1 + A)− 1, the equilibrium point (1, 0) is globally asymptotically stable [25]. If 0 < d < (1 + A)− 1, system (3) has three equilibrium points, (0, 0), (1, 0), and (u∗, v∗), respectively, where u∗ and v∗ are all positive and satisfy the following conditions:. We summarize global results for system (3) in Lemma 1 [25, 48, 49]
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