Feedback control effect on the Lotka-Volterra prey-predator system with discrete delays

Abstract

In this paper, we study a Lotka-Volterra prey-predator system with feedback control. We establish sufficient conditions under which a unique positive equilibrium is globally stable. Further, we show that a suitable feedback control on predator species can make prey species that is on the brink of extinction become globally stable, but under the conditions of small feedback control on predator, the prey species still extinct, whereas the predator species is stable at certain values. Several examples are presented to show the feasibility of the main results.

Stability and bifurcation in a Holling type II predator–prey model with Allee effect and time delay

“…All authors read and approved the final manuscript. 1 College of Mathematics and Computer Science, Fuzhou University, Fuzhou, China.…”

“…The predator-prey model is one of the basic models between different species in nature which has been widely researched [1][2][3][4][5][6]. In the natural world, the population generally has a saturation effect.…”

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

“…If the two-species competition model is extinct, by choosing the suitable values of feedback control variables, they can make extinct species become globally stable, or still keep the property of extinction. In 2017, Shi et al [17] discussed a Lotka-Volterra predator-prey model with discrete delays and feedback control as follows:…”

“…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.…”

Asymptotic behavior of age-structured and delayed Lotka-Volterra models

“…Some of the papers concern the global stability of equilibria (see e.g. [2], [3], [14], [37], [36], [31], [39]). However, in the papers mentioned above, a carrying capacity is present in the prey equation, meaning that preys grow logistically instead of exponentially.…”