Periodic solution and control optimization of a prey-predator model with two types of harvesting

Abstract

In this work, a prey-predator model with both state-dependent impulsive harvesting and constant rate harvesting is investigated, where the replenishment rate of prey and the harvesting rate are linearly related with the selected threshold. By first using the successor function method and differential equation geometry theory, the existence, uniqueness and asymptotic stability of the order-1 periodic solution are discussed. And then numerical simulations with an example are given to illustrate the feasibility of the theorem-related results. Moreover, in order to increase the total profit, the optimization strategy is presented and the optimal threshold is obtained.

Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control

“…In the process of marine shery production, over shing often results in the exhaustion of shery resources. It is rewarding for humans to develop and utilize the ecological system of the population rationally, which also contributes to the sustainability of the system [7][8][9][10][11][12][13][14]. In 2003, Gopalsamy and Weng [15] studied the following population competition model with feedback control:…”

“…where u(t) is the feedback control variable, e and f denote the feedback control coefficients, a ii (i � 1, 2) denote the intraspecific competition rates, a ij (i ≠ j, i, j � 1, 2) stand for the capturing rates of the prey and predator populations, τ 1 is the time of catching prey, and τ 2 is maturation delay of predator. Shi et al [17] show that (i) e solution (x 1 (t), x 2 (t), u(t)) of system (3) is ultimately bounded (ii) When the conditions (r 1 /r 2 ) > (a 12 /(a 22 +(cf/e))), (a 11 /a 21 ) > (a 12 /a 22 ) are established, system (3) has a unique globally asymptotically stable positive equilibrium point (x * 1 , x * 2 , u * ), where x * 1 � (e(r 1 a 22 − r 2 a 12 ) + r 1 cf)/(e(a 11 a 22 + a 12 a 21 ) + cfa 11 ), x * 2 � e(r 2 a 11 + r 1 a 21 )/(e(a 11 a 22 + a 12 a 21 ) + cfa 11 ), and u * � (f/e)x * 2 In fact, in nature, ecosystems are inevitably affected by various environmental noises [18][19][20][21][22][23][24][25][26][27][28]. Mathematical models with environmental disturbances can usually be described by stochastic differential equations.…”

Dynamics in a diffusive phytoplankton–zooplankton system with time delay and harvesting

“…Hence, the stocks of these tiny zooplankton play a significant role in marine reserves and fishery management. For these reasons, many scholars have studied phytoplankton-zooplankton (or predator-prey) models with harvesting [16][17][18][19][20]. Generally, harvesting can be divided into three types [21]: (i) constant rate of harvesting, where a fixed number of individuals are harvested per unit time; (ii) proportionate harvesting, where the catch rate is proportional to the stock and effort; and (iii) nonlinear harvesting (Holling type II, Michaelis-Menten type).…”

Stability Analysis and Control Optimization of a Prey-Predator Model with Linear Feedback Control

“…In recent years, the theoretical studies on IDES have produced a lot of good research results [23][24][25][26][27][28][29][30][31][32][33][34]. Based on the theoretical research, some scholars have introduced impulsive differential equations in Lotka-Volterra system such as the regular release of predators [35][36][37]; the periodic release of infected pests [38][39][40]; the periodic release of predators together with regular spray of pesticides [41][42][43]; the periodic release of predators and infected pests together with regular spray of pesticides [39,44]. In the practical application, the two control measures can be adopted at two different levels of pest density concerning this case.…”