Dynamical Analysis of Two‐Microorganism and Single Nutrient Stochastic Chemostat Model with Monod‐Haldane Response Function

Abstract

In this paper, we formulate and investigate a two-microorganism and single nutrient chemostat model with Monod-Haldane response function and random perturbation. First, for the corresponding deterministic system, we introduce the conditions of the stability of the equilibrium points. Then, using Lyapunov function and Itô’s formula, we investigate the existence and uniqueness of the global positive solution of the stochastic chemostat model. Furthermore, we explore and obtain the criterions of the extinction and the permanence for the stochastic model. Finally, numerical simulations are carried out to illustrate our main results.

Dynamics Analysis of a Stochastic Leslie–Gower Predator‐Prey Model with Feedback Controls

“…where u 1 (t) and u 2 (t) are the feedback control variables and c 1 , c 2 , f 1 , f 2 , g 1 , and g 2 are all positive constants from the realistic biological significance. e rest of this paper is structured as follows: in Section 2, we prove the existence and uniqueness of global positive solutions to system (4). In Section 3, we obtain the conditions for the existence of positive recurrence of the solutions to system (4).…”

“…+ , the model (4) has a unique positive solution (N 1 (t), N 2 (t), u 1 (t), u 2 (t)) on t ≥ 0 and the solution will remain in R 4 + with probability one.…”

“…In the above section, we obtained the existence conditions for the positive recurrence of system (4). In this section, we discuss the persistence and extinction of species of the stochastic model (4) with feedback controls.…”

“…e relationship between biological populations is usually expressed and analyzed by differential equations [1][2][3][4][5]. Among numerous studies of biological populations, Leslie [1] presented a classical predator-prey model as follows:…”

Dynamic Analysis of Stochastic Lotka–Volterra Predator‐Prey Model with Discrete Delays and 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.…”