A stochastic predator–prey model with Holling II increasing function in the predator

Abstract

This paper is concerned with a stochastic predator-prey model with Holling II increasing function in the predator. By applying the Lyapunov analysis method, we demonstrate the existence and uniqueness of the global positive solution. Then we show there is a stationary distribution which implies the stochastic persistence of the predator and prey in the model. Moreover, we obtain respectively sufficient conditions for weak persistence in the mean and extinction of the prey and extinction of the predator. Finally, some numerical simulations are given to illustrate our main results and the discussion and conclusion are presented.

A Dynamic Model of Multiple Time-Delay Interactions between the Virus-Infected Cells and Body’s Immune System with Autoimmune Diseases

“…As mentioned earlier, many researchers [7,17,22,28] have developed various preypredator models and recently developed mathematical models to investigate the interactions between the tumor cells and immune systems, and tumor-immune cells with consideration of an interaction between the tumor and immune cells with a time delay. In this section, we discuss a new virus-immune time-delay model of the body's immune system with considerations of the multiple interactions between the virus-infected cells and body's immune cells with an autoimmune disease.…”

“…Many models [9,10,[13][14][15][16] have been proposed using the ordinary differential equations and partial differential equations in the past several decades and using the delay partial differential equations in recent years for characterizing tumor-immune dynamic growth, but there is still no consensus on the modeling due to the complexity of virus-infected and tumor cancer growth in the body's immune system and the growth patterns of the tumors and virus-infected cells [16]. Many researchers [7,[17][18][19][20][21][22][23][24] have used the existing prey-predator modeling concept [25,26] to study and model the tumor-immune interactions [7,27,28] and the effects of tumor growth [17,29,30]. To simplify an understanding of the interaction between tumor and immune cells, several researchers used the concept of the prey-predator system [24,29].…”

“…The modeling studies of prey-predator systems and its related applications have beentremendously interesting in recent years to various disciplines including population disease [31,32], life expectancy [33,34], biomathematics [35,36], cancel growth [29,[37][38][39][40][41][42] and engineering science [23,35,43]. Many researchers have studied various dynamic prey-predator models including a two-dimension predator-prey model [44][45][46][47][48], multipredator models [20,23,49], multi-prey models [50][51][52] and time-delay prey-predator models [23,36,43,50] with various applications in biomathematics [53], population disease [22,23,29,30,54,[54][55][56][57][58][59] and recent COVID-19 disease analysis [1,12,[60]…”

“…Wang et al [43] studied a predator-prey model with distributed delays. Huang et al [22] recently studied a stochastic predator-prey model with a Holling II increasing function in the predator and discussed the analytic results of the dynamics of the stochastic predator-prey model. Jana and Kar [23] studied a three-dimensional epidemiological dynamic model incorporating time delay in the model for considering it as the time taken by a susceptible prey to become infected.…”

On the Dynamics of a Stochastic Ratio-Dependent Predator-Prey System with Infection for the Prey

“…In predator-prey system, functional response plays an important role in the population dynamics. Holling types functional response functions, namely Holling types I, II, III and IV, have been extensively used and investigated [8] [9]. In recent decades, Beddington-DeAngelis and Crowley-Martin type functional response are also widely chosen to model the predation [10] [11].…”