Abstract
In this paper, a new stochastic four-species predator-prey model with disease in the first prey is proposed and studied. First, we present the stochastic model with some biological assumptions and establish the existence of globally positive solutions. Moreover, a condition for species to be permanent and extinction is provided. The above properties can help to save the dangered population in the ecosystem. Through Lyapunov functions, we discuss the asymptotic stability of a positive equilibrium solution for our model. Furthermore, it is also shown that the system has a stationary distribution and indicating the existence of a stable biotic community. Finally, our results of the proposed model have revealed the effect of random fluctuations on the four species ecosystem when adding the alternative food sources for the predator population. To illustrate our theoretical findings, some numerical simulations are presented.
1. Introduction
Mathematical modelling in ecology and epidemiology has become an important tool in analysing the spread and control of infectious diseases. Anderson and May [Citation3] were the first to merge the above two fields and formulate a predator-prey model where some disease infected the prey species. Most infected disease models are created from the classical SIR model of Kermack and McKendrick [Citation18]. Subsequently, many researchers have proposed and studied different predator-prey models in the presence of a disease. Studies in the field of eco-epidemiology have increased enormously in the last two decades, and we only mention a few of them [Citation16, Citation20, Citation21, Citation31, Citation32, Citation37]. Qin et al. performed Holling III predator-prey model with an infectious predator. A functional response plays a key role in the predator-prey model and describes the number of prey consumed per predator per unit of time for some given time. Here, we used the Beddington-DeAngelis functional response for the predator and the infected first prey interaction. Mathematically, we may consider both prey-dependent and ratio-dependent models as limiting cases of the general Beddington-DeAngelis type predator-prey model. Much progress [Citation10, Citation12, Citation13, Citation15, Citation35, Citation36, Citation39, Citation40] have been seen in the study of the predator-prey model with Beddington-DeAngelis functional response. Recently, Arora et al. [Citation1] proposed a predator-prey system highlighting the significance of additional food for predators with a Beddington-DeAngelis functional response.
Another important factor in ecological and eco-epidemiological species interaction is alternative food sources for the predator population. Generally, predators do not depend on a single prey species for food. Predators will change to alternative food when the density of their preferred prey is low. Fryxell and Lundberg [Citation11] has argued that alternative food plays an essential role in promoting the persistence of predator-prey systems. Many authors [Citation1, Citation9, Citation45] have considered one predator-two prey systems to show the importance of alternative food for the coexistence of species. Thus, a better understanding of the mechanisms, assumptions and biological hypotheses of predator-prey dynamic models involving alternative food is explicitly needed when the disease is present in the prey population. Here, we consider a model with one predator and two prey populations with the availability of alternative food sources for the predator population when a disease has affected only the first prey population. Srinivasu et al. [Citation38] showed that additional food serves as a tool for the biological conservation of predator-prey systems involving type III functional response.
In reality, a population model can be affected by environmental noise; hence, stochastic population systems play a major role in various branches of applied sciences, including population dynamics and biology. In real life, continuous fluctuations in the environment change the population to a lower or higher extent level. May [Citation30] discovered that all the parameters involved in the population system exhibit random fluctuation as the factors controlling them are not constant. Therefore most authors [Citation5,Citation14,Citation25,Citation41,Citation46] introduce stochastic perturbation into the deterministic models and make the model more realistic for examination; the resulting dynamics solely depend on the intensity of environmental fluctuations. Precisely, enormous efforts have been taken to analyse the dynamical behaviour with environmental fluctuations [Citation8, Citation21, Citation28, Citation34, Citation39, Citation40, Citation42, Citation44, Citation45]. Cao et al. [Citation5] transformed the stochastic system with a weak kernel case into an equivalent degenerate system and studied its behaviour. [Citation14] et al. proposed the predator-prey model with Holling II increasing function in the predator. Lu [Citation27] presented Crowley-Martin predator-prey model in a stochastic environment and analysed asymptotical stability in probability by combining the Khasminskii theory of stability with the Lyapunov method. Zhang et al. [Citation43] investigated the influence of anti-predator behaviour due to the fear induced by the predators for the stochastic eco-epidemiological predator-prey model. Recently, Liu et al. [Citation24] developed and analysed a Leslie-Gower stochastic predator-prey model with impulsive toxicant input in polluted environments. Liu [Citation26] carried out several two-species predator-prey models and showed the impact of randomness. A small amount of work has been carried out with the stochastic four-species predator-prey model and disease in the prey.
The present article is organized as follows. Section 2 presents some known definitions, theorems, and lemmas that will be used in the following analysis. In Section 3, we discuss the detailed explanation about the formulation of the stochastic model and analyse the existence and uniqueness of a global positive solution of the system. In Section 4, stochastic permanence and extinction are established under certain parametric restriction. We construct a suitable Lyapunov function to examine the global asymptotic stability for positive equilibrium in Section 5. In Section 6, we establish the conditions for unique stationary distribution. Additionally, we generate some figures to illustrate our main results in Section 7. Finally, we end the paper with a discussion in Section 8.
2. Preliminaries
Consider the d-dimensional stochastic model (SM) of the form (1) (1) with an initial value , where and are Borel measurable, is an -valued Wiener process and is an -valued random variable.
The differential operator associated with the SM (Equation1(1) (1) ) is given by In addition to the existence and uniqueness assumptions, we assume that f and g satisfy and for an equilibrium solution , for . Moreover we assume that is a non-random constant with probability 1.
Definition 2.1
see [Citation22]
The solution of (Equation1(1) (1) ) is said to possess stochastic permanent if for any there exists a pair of positive constants and such that for any initial value the solution to (Equation1(1) (1) ) has the properties
Theorem 2.1
see [Citation2]
Assume that f and g satisfy the existence and uniqueness assumptions and that they have continuous coefficients with respect to t.
Suppose that there exists a positive definite function where for h>0, such that Then, the equilibrium solution of (Equation1(1) (1) ) is stochastically stable.
If, in addition, V is decreasing and there exists a positive definite function such that then the equilibrium solution is stochastically asymptotically stable.
Let be a homogeneous Markov process defined in the l dimensional Euclidean space and described by the following system of stochastic differential equations: The diffusion matrix is defined by Hasminskii [Citation19], We assume there exists a bounded domain with regular boundary Γ, having the following properties:
In some neighbourhood of a domain U, the smallest eigenvalue of the diffusion matrix is bounded away from zero.
If , the mean time τ at which a path emerging from x reaches the set U is finite, and for every compact subset .
Lemma 2.1
If above assumptions hold, then the Markov process has a stationary distribution . Let be a function integrable with respect to the measure φ. Then for all .
Lemma 2.2
The following inequality holds
3. Problem formulation and assumptions
3.1. Mathematical model
In this section, we develop the predator-prey model with two prey and one predator population where the disease infects only the first prey population. Sharma and Samanta [Citation37] pay attention to a similar model and analyse the global stability of the positive equilibrium. We develop the model as follows: (2) (2) with initial densities (3) (3) Here, and denote the numbers of susceptible first prey, infected first prey, second prey and predator, respectively, and all parameters in the model are positive. We choose the Beverton–Holt functional response to represent the birth rate of the prey population. In the absence of disease and predator population, susceptible first prey population will follow the growth equation where Π is the maximum per-capita growth rate of prey and b is per-capita mortality rate; moreover, can be represented as , which approaches Π as goes to infinity.
The parameters used in the system modelling with their meaning are listed in the Table .
The given model requires the following assumptions:
Only the first prey population is infected by the disease and because of the presence of the disease, the first prey population is divided into two sub-classes, namely (i) susceptible first prey population and (ii) infected first prey population .
The infected first prey population dies before having the capability of reproducing and does not recover from the disease.
Disease does not spread to predators from the infected first prey.
Predators do not consume the susceptible first prey population. It has more possibilities to consume the infected first prey population as they are easily available to be preyed upon owing to weakening due to the infection, i.e. they lose their power to protect themselves from a predator. The predators also consume the second prey because of their significant availability in the total population. Thus, our assumption is reasonable in such cases. This explains why the infected first prey and second prey increase their availability to predation [Citation21].
The interaction between the infected first prey and predator is assumed to be governed by a Beddington-DeAngelis type functional response. This functional response was introduced by Beddington [Citation4] and DeAngelis et al. [Citation7]. The interaction between the second prey and predator is considered a Lotka-Volterra functional response. There is no interaction between the first prey and second prey population.
The growth of the second prey population is exponential, so the availability of second prey becomes very large when the predator becomes extinct. Because of this, there is no need to consider the search time for the second prey population.
In reality, environmental noises significantly affect population dynamics, which are essential components in an ecosystem. Taking into account the effect of a randomly fluctuating environment, we included white noise in each equation of the system (Equation2(2) (2) ). Let us assume that the fluctuations in the environment incorporate themselves mainly as fluctuations in the death rate of the susceptible first prey population, the death rate of the infected first prey population, the growth rate of the second prey population, and the death rate of the predator population, as follows: where represent the independent Brownian motions and represent the intensities of the white noises. Therefore the corresponding stochastic system can be described by the Itô equations (4) (4) In the past several years, no work has been done on the given stochastic predator-prey system (Equation4(4) (4) ). Our aim is to analyse the dynamical properties of the stochastic model and show the impact of environmental noise in the population system.
Theorem 3.1
All solutions of the system (Equation2(2) (2) ) with initial conditions (Equation3(3) (3) ) exist in the interval and for all .
Proof.
The proof of this theorem is similar to the proof of Theorem 1 in [Citation37].
3.2. Existence of global positive solution
Theorem 3.2
For , the system (Equation4(4) (4) ) possesses a unique positive local solution for almost surely, where is the explosion time.
Proof.
Consider the following transformation of variables Let us use Itô's formula Using this, we obtain Similarly, from system (Equation4(4) (4) ), we obtain (5) (5) with initial condition . Now, the functions associated with the above system have an initial growth and satisfy a local Lipchitz condition. Therefore there exists a unique local solution defined in some interval . Consequently, is the unique positive local solution of (Equation4(4) (4) ) starting from an interior point of the first quadrant.
Theorem 3.3
For any initial condition , the system (Equation4(4) (4) ) possesses a unique solution for and the solution remains in with probability one; therefore, for all almost surely.
Proof.
To prove that this solution is global, it is enough to show that almost surely. Let be a sufficiently large nonnegative integer such that the closed ball contains . We choose for any and define the stop-time by using the ideas in [Citation17] as where (∅ being the empty set). Clearly is increasing as .
Let ; then almost surely. If almost surely is true, then almost surely. If this statement is not true, that is, , then there exist two constants T>0 and such that (6) (6) Thus, by denoting there is an integer such that, for all , (7) (7) Define a function by where the function is positive definite for all . Using Itô's formula, we obtain Taking the differential of along the solution trajectories of the system (Equation4(4) (4) ) using above equation, we obtain Positivity of and implies that Define three positive constants We utilize Lemma 2.2 (the proof is given in [Citation6]) in the form , we get that . Therefore where . For where . Using the properties of Itô's integral and with the above inequality we obtain that Taking the expectations of both sides of the above inequality and applying Fubini's theorem, we obtain By Grownwall's inequality given in [Citation29] and the above inequality, we obtain (8) (8) where From (Equation7(7) (7) ), we have . It is noteworthy that there is at least one of and belongs to the set , for and hence is no less than the smallest of and . Consequently, Hence from (Equation6(6) (6) ) and (Equation8(8) (8) ), it follows that where is the Indicator function of .
Let lead to a contradiction: Therefore almost surely. Thus and almost surely. The proof is complete.
Theorem 3.3 tells us that the solution of system (Equation4(4) (4) ) will remain in . With this property, we continue to discuss how the solution varies in in more detail.
Theorem 3.4
The solutions of the system (Equation4(4) (4) ) are stochastically ultimately bounded for any initial value .
Proof.
This proof is similar to the proof of Theorem 3 in [Citation44].
4. Long time behaviour of system
Here, we look at the long-time behaviour of the system (Equation4(4) (4) ), such as permanence and extinction to determine how the solution varies in in more detail. First, we impose the following hypotheses, which will be useful later on.
In population dynamics, stochastic permanence is one of the most essential properties for species survival. We discuss this property with the fruitful ideas of [Citation28] as follows:
Theorem 4.1
Suppose that assumption holds; then system (Equation4(4) (4) ) will be stochastically permanent.
Proof.
First, we show that for any initial value , the solution satisfies where κ is an arbitrary positive constant satisfying (9) (9) By (Equation9(9) (9) ), there exists an arbitrary positive constant satisfying (10) (10) Define for and ; by Itô's formula, we have and Under Assumption , we can certainly choose a positive constant κ satisfying (Equation9(9) (9) ). Again, applying Itô's formula, we obtain Then, we can choose sufficiently small such that it satisfies (Equation10(10) (10) ); consequently where Let us now discuss the upper boundedness of the function . It is easy to see that and Hence From (Equation9(9) (9) ) and (Equation10(10) (10) ), we know that there exists a positive constant Q such that which implies that Therefore Note that where . Accordingly Thus, for any , let By Chebyshev's inequality, we obtain that is, Similarly, we obtain that for any , there exists such that . Hence the theorem is proved.
In ecology, the worse occurs when a species disappears completely. Here we prove that if the noise is sufficiently large, the solution will become extinct with probability one.
Theorem 4.2
Assume that holds, then for any given initial value , the solution of system (Equation4(4) (4) ) will be extinct with probability one.
Proof.
Define the Lyapunov function . Then, by Itô's formula, we have Integrating it from 0 to t yields which implies that Dividing by t on both sides and letting , we obtain Furthermore, define the Lyapunov function ; by Itô's formula, we have Integrating it from 0 to t, we obtain Consequently Dividing by t and letting , we obtain Define the Lyapunov function ; by Itô's formula, we arrive at Integrating it, we get Accordingly, Then, dividing it by t and letting , we obtain Similarly, we define the Lyapunov function and by Itô's formula, we get Integrating it, we have Then Dividing it by t and letting , we have The desired assertion is thus proved.
5. Stochastic asymptotic stability
Using the ideas of Pitchaimani and Rajaji [Citation33], we shall analyse the stochastic stability, which will lead to certain conditions. Here, the discussion is entirely about the mathematical theory on stochastic asymptotic stability of the positive equilibrium solution of the system (Equation4(4) (4) ).
The positive equilibrium point where and others are obtained as follows:
Let be a positive root of the following quadratic equation where and The roots of the above quadratic equation are When any one of the following cases is satisfied, then the equilibrium points can have one or two positive values.
C<0 and D<0,
C<0, D>0 and ,
,
Theorem 5.1
The equilibrium solution of the system (Equation4(4) (4) ) is stochastically asymptotically stable, if (11) (11)
Proof.
Consider the Lyapunov function The infinitesimal generator acting on the Lyapunov function can be written as follows One can observe that , if . By assumption, becomes negative definite; therefore the equilibrium solution of the system (Equation4(4) (4) ) is stochastically asymptotically stable.
6. Existence of stationary distribution
Here, we shall give the existence of the stationary distribution of prey and predator populations. For this purpose, we must find the stationary distribution for solutions of the system (Equation4(4) (4) ), which implies stability in the stochastic sense. We use lemma 2.1, (that is given in [Citation19]) to prove the main theorem on the stationary distribution.
Theorem 6.1
Assume where is the equilibrium of system (Equation2(2) (2) ), simultaneously. Then there exists a stationary distribution for the stochastic system (Equation4(4) (4) ).
Proof.
The proof of this theorem is based on the following positive definite function , where , Applying Itô's formula, we obtain and Therefore where Note that Now if υ satisfies the following condition, Here, the ellipsoid lies entirely in Int. Let U to be a neighbourhood of the ellipsoid with Ū , where Ū is the closure of U. Then, for every , we have , which implies condition in Lemma 2.1 is satisfied.
In addition, there exist Ū such that for all Ū, , which implies that condition in above Lemma 2.1 is satisfied. Therefore we can conclude from the above discussion that the stochastic system (Equation4(4) (4) ) has a stationary distribution .
7. Numerical simulation
In this section, let us consider some examples to visualize our theoretical findings by simulating the stochastic system (Equation4(4) (4) ). To find the strong solution of system (Equation4(4) (4) ) with a given initial value, we use the Milstein method. Consider the following discretization equations of model (Equation4(4) (4) ) where time increment and , are independent random Gaussian random variables that can be generated numerically by pseudo random number generators.
We can construct suitable simulations for the stochastic system (Equation4(4) (4) ) and the corresponding deterministic system (Equation2(2) (2) ) with the help of MATLAB software and possible parameters. Choose the initial value as for all figures that are going to be discussed in this platform. (12) (12) The only difference between Figure (a,b) is the intensity of white noise. From the figures, we see that the solution curve of the stochastic model (Equation4(4) (4) ) always fluctuates around the solution curves of the deterministic system (Equation2(2) (2) ). We observe that with the decreasing of values, the dynamics of the stochastic system become increasingly similar to the deterministic system.
For the parametric and intensity values condition in Theorem 4.2 is verified by the observer; hence, Figure (a) shows that all the four species of the stochastic system (Equation4(4) (4) ), as well as the corresponding deterministic system (Equation2(2) (2) ), will die out. Furthermore, in Figure (b), the parameters chosen are the same as those used in (Equation12(12) (12) ); moreover, the intensities satisfy condition in Theorem 4.1, and simulations also confirm that all three species of both systems are permanent.
Figure illustrates Theorem 5.1. We choose the parameters values as in (Equation12(12) (12) ) with different initial values and . Our model admits a stochastically asymptotically stable solution around the positive equilibrium for the above parameters. From this, we can conclude that the positive equilibrium agrees with Theorem 5.1 under the assumed conditions. The stochastic stability shown for each of the four species in the phase-portrait plane is presented in Figure .
The parameter values were chosen (Equation12(12) (12) ), and the choice of and was based on the conditions needed for the existence of a stationary distribution (see Theorem 6.1). We kept all parameter values fixed and repeated the simulation 10, 000 times but never observed any extinction scenario appear at any future time. Figure shows that all species are distributed normally around the mean values . After that, we increased the strengths of environmental forcing to and, again, we focussed on the species distribution fluctuate around the deterministic steady-state value; however, the amplitude of fluctuation is more comparable to that of the earlier case given in Figure . In this case, the susceptible first prey population is distributed within ; the infected first prey population occupies within , the second prey lies within , and the predator population remains within the range . In the earlier case, the distribution of each species was within and . Figures and show that in stochastic stability, all the solution curves will converge to one particular region. In deterministic stability, all the solution curves will converge to a single point. In Figures and , the left-hand side shows that the mean solution of the stochastic system coincided with the solution curves of the deterministic system. Figure (a,b) present a clear representation of the stationary distribution for all species in a single graph, similar to Figures and .
8. Discussion
From an applications viewpoint, the population model has become more interesting and studied extensively by several researchers. Our work is the first attempt to consider the stochastic four species predator-prey model with disease in the first prey (Equation4(4) (4) ). The main objective of the present paper is to study the model regarding how the intensities of environmental noises affect each population in the system (Equation4(4) (4) ) where an alternative food source is considered for predator population and disease has affected only the first prey population. Our analytical findings have pointed out that the intensity of noises can play a significant role in the survival of all species. These result can be verified with numerical support.
First, we explain the rigour of our model (Equation2(2) (2) ) under several assumptions. We introduce the stochastic term to the system (Equation2(2) (2) ) and show that the stochastic system (Equation4(4) (4) ) has a local and global positive solution for any positive initial value. The property of permanence and extinction is more desirable since it means the long-time and short time survival in a population dynamics. Under assumption , our stochastic system (Equation4(4) (4) ) attains the permanence. The stochastic system goes to the extinction state if assumption holds. In real life, this may happen when a serious disease or severe weather occurs. Furthermore, we discuss the stochastic asymptotic stability of positive equilibrium with the help of the invariance principle and Lyapunov's second method that are given in Theorem 5.1. Thereafter, we have shown the existence of a stationary distribution for all species under certain parameter restriction discussed in Theorem 6.1. This parametric restriction follows the idea that a large intensity of noise can destabilize the system, and in this particular situation we cannot find a stationary distribution.
In a stochastic system, we see that larger intensities of the white noise lead to increasingly larger fluctuations of the solutions. The numerical result shows that strong noise may make the species extinct, which can occur in reality. In permanence, if we reduce the intensity of noise, solutions become stable. From this aspect, one can see the major difference when comparing the deterministic with stochastic models: that is, in the deterministic system the parameters are assumed to be constant, which is not true biologically in real life. Thus, to make our model more fruitful there arises a need to include the environmental variations while modelling the real world scenario mathematically. The use of additional food has been widely recognized by experimental scientists as one of the important tools for biological balance. The present study gives the importance of additional food for predator and the impact of randomness into the system. With the proper choice of parameters satisfying conditions and , remedial measures could be taken to maintain the balance of the ecosystem.
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No potential conflict of interest was reported by the author(s).
Data availability statement
Our paper contains numerical experimental results, and values for these experiments are included in the paper. The data is freely available.
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References
- C. Arora and V. Kumar, Dynamics of one-prey and two-predator system highlighting the significance of additional food for predators with Beddington-DeAngelis functional response, Differ. Equat. Dyn. Syst. 30 (2018), pp. 411–431.
- L. Arnold, Stochastic Differential Equations: Theory and Applications, Wiley, New York, 1974.
- R.M. Anderson and R.M. May, The invasion, persistence and spread of infectious diseases within animal and plant communities, Philos. Trans. R. Soc. London B 314(1167) (1986), pp. 533–570.
- J.R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol. 44 (1975), pp. 331–340.
- Z. Cao, W. Feng, X. Wen, and L. Zu, Stationary distribution of a stochastic predator-prey model with distributed delay and higher order perturbations, Phys. A: Statis. Mech. Its Appl. 521 (2019), pp. 467–475.
- N. Dalal, D. Greenhalgh, and X. Mao, A stochastic model for internal HIV dynamics, J. Math. Anal. Appl. 341(2) (2008), pp. 1084–1101.
- D.L. DeAngelis, R.A. Goldstein, and R.V. O'Neill, A model for trophic interaction, Ecology 56 (1975), pp. 881–892.
- B. Du, M. Hu, and X. Lian, Dynamical behavior for a stochastic predator-prey model with HV type functional response, Bull. Malaysian Math. Sci. Soc. 40(1) (2017), pp. 487–503.
- B. Dubey and R.K. Upadhyay, Persistence and extinction of one-prey and two-predators system, Nonlinear Anal.: Modell. Cont. 9(4) (2004), pp. 307–329.
- M. Fan and Y. Kuang, Dynamics of a nonautonomous predator-prey system with the Beddington–DeAngelis functional response, J. Math. Anal. Appl. 295(1) (2004), pp. 15–39.
- J.M. Fryxell and P. Lundberg, Diet choice and predator-prey dynamics, Ecol. Evol. 8 (1994), pp. 407–421.
- C. Gokila and M. Sambath, Analysis on stochastic predator-prey model with distributed delay, Rand. Oper. Stochast. Equat. 29(2) (2021), pp. 97–110.
- C. Gokila, M. Sambath, K. Balachandran, and Y.-K. Ma, Analysis of stochastic predator-prey model with disease in the prey and Holling type II functional response, Adv. Math. Phys. 2020 (2020), pp. 1–17.
- Y. Huang, W. Shi, C. Wei, and S. Zhang, A stochastic predator-prey model with Holling II increasing function in the predator, J. Biol. Dyn. 15(1) (2021), pp. 1–18.
- T.W. Hwang, Global analysis of the predator-prey system with Beddington–DeAngelis functional response, J. Math. Anal. Appl. 281(1) (2003), pp. 395–401.
- C. Ji and D. Jiang, Analysis of a predator-prey model with disease in the prey, Int. J. Biomath. 6(3) (2013), p. 1350012.
- C. Ji, D. Jiang, and N. Shi, Analysis of a predator-prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation, J. Math. Anal. Appl. 359(2) (2009), pp. 482–498.
- W. Kermack and A. McKendrick, Contributions to the mathematical theory of epidemics, I II, III, Bullet. Math. Biol. 53 (1991), pp. 33–55;57–87;89–118.
- R. Khasminskii, Stochastic Stability of Differential Equations, Springer, Berlin, 2011.
- B.W. Kooi and E. Venturino, Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey, Math. Biosci. 274 (2016), pp. 58–72.
- J. Li and W. Gao, Analysis of predator-prey model with disease in prey, Appl. Math. Comput. 217 (2010), pp. 4024–4035.
- X. Li and X. Mao, Population dynamical behavior of non-autonomous Lotka–Volterra competitive system with random perturbation, Discrete Continuous Dyn. Syst.-Ser. A 24(2) (2009), pp. 523–593.
- S. Li and X. Wang, Analysis of a stochastic predator-prey model with disease in the predator and Beddington-DeAngelis functional response, Adv. Differ. Equat. 2015(1) (2015), pp. 1–21.
- M. Liu, C. Du, and M. Deng, Persistence and extinction of a modified Leslie–Gower Holling-type II stochastic predator-prey model with impulsive toxicant input in polluted environments, Nonlinear Anal.: Hybrid Syst. 27 (2018), pp. 177–190.
- Q. Liu and D. Jiang, Stationary distribution and extinction of a stochastic predator-prey model with distributed delay, Appl. Math. Lett. 78 (2018), pp. 79–87.
- Q. Liu, D. Jiang, T. Hayat, and A. Alsaedi, Stationary distribution and extinction of a stochastic predator-prey model with herd behavior, J. Franklin Inst. 355(16) (2018), pp. 8177–8193.
- C. Lu, Dynamical analysis and numerical simulations on a Crowley–Martin predator-prey model in stochastic environment, Appl. Math. Comput. 413 (2022), p. 126641.
- P.S. Mandal and M. Banerjee, Stochastic persistence and stationary distribution in a Holling–Tanner type prey-predator model, Phys. A: Statis. Mech. Its Appl. 391(4) (2012), pp. 1216–1233.
- X. Mao, Stochastic versions of the Lassalle theorem, J. Differ. Equat. 153(1) (1999), pp. 175–195.
- R.M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, Princeton, 2001.
- A. Mondal, A.K. Pal, and G.P. Samanta, On the dynamics of evolutionary Leslie–Gower predator-prey eco-epidemiological model with disease in predator, Ecol. Genet. Genom. 10 (2019), p. 100034.
- D. Mukherjee, Persistence aspect of a predator-prey model with disease in the prey, Differ. Equat. Dyn. Syst. 24(2) (2016), pp. 173–188.
- M. Pitchaimani and R. Rajaji, Stochastic asymptotic stability of Nowak-May model with variable diffusion rates, Methodol. Comput. Appl. Probab. 18(3) (2016), pp. 901–910.
- R. Rudnicki, Long-time behaviour of a stochastic prey-predator model, Stochast. Process. Their Appl.108(1) (2003), pp. 93–107.
- M. Sambath and K. Balachandran, Bifurcation analysis of a diffusive predator-prey model with Beddington–DeAngelis response, Nonlinear Stud. 22(2) (2015), pp. 263–280.
- M. Sambath, K. Balachandran, and M. Suvinthra, Stability and Hopf bifurcation of a diffusive predator prey model with hyperbolic mortality, Complexity 21(1) (2016), pp. 34–43.
- S. Sharma and G.P. Samanta, Analysis of a two prey one predator system with disease in the first prey population, Int. J. Dyn. Cont. 3(3) (2015), pp. 210–224.
- P.D.N. Srinivasu, D.K.K. Vamsi, and V.S. Ananth, Additional food supplements as a tool for biological conservation of predator-prey systems involving type III functional response: A qualitative and quantitative investigation, J. Theor. Biol. 455 (2018), pp. 303–318.
- T.V. Ton and A. Yagi, Dynamics of a stochastic predator-prey model with the Beddington–DeAngelis functional response, Commun. Stochast. Anal. 5(2) (2011), pp. 371–386.
- J.P. Tripathi, S. Abbas, and M. Thakur, Dynamical analysis of a prey-predator model with Beddington–DeAngelis type function response incorporating a prey refuge, Nonlinear Dyn. 80(1-2) (2015), pp. 177–196.
- C. Qin, J. Du, and Y. Hui, Dynamical behavior of a stochastic predator-prey model with Holling-type III functional response and infectious predator, AIMS Math. 7(5) (2022), pp. 7403–7418.
- R. Wu, X. Zou, and K. Wang, Asymptotic behavior of a stochastic non-autonomous predator-prey model with impulsive perturbations, Commun. Nonlinear Sci. Numerical Simul. 20(3) (2015), pp. 965–974.
- Y. Zhang, S. Gao, and S. Chen, A stochastic predator-prey eco-epidemiological model with the fear effect, Appl. Math. Lett. 134 (2022), p. 108300.
- Y. Zhang, S. Gao, K. Fan, and Y. Dai, On the dynamics of a stochastic ratio-dependent predator-prey model with a specific functional response, J. Appl. Math. Comput. 48(1-2) (2015), pp. 441–460.
- Q. Zhang, D. Jiang, Z. Liu, and D. O'Regan, The long-time behavior of a predator-prey model with disease in the prey by stochastic perturbation, Appl. Math. Comput. 245 (2014), pp. 305–320.
- D. Zhou, M. Liu, and Z. Liu, Persistence and extinction of a stochastic predator-prey model with modified Leslie–Gower and Holling-type II schemes, Adv. Differ. Equat. 2020(1) (2020), pp. 1–15.