Analysis of Stochastic Predator-Prey Model with Disease in the Prey and Holling Type II Functional Response

A stochastic predator-prey model with disease in the prey and Holling type II functional response is proposed and its dynamics is analyzed. We discuss the boundedness of the dynamical system and find all feasible equilibrium solutions. For the stochastic systems, we obtain the conditions for the existence of the global unique solution, boundedness, and uniform continuity. We derive the conditions for extinction and permanence of species. Moreover, we construct appropriate Lyapunov functions and discuss the asymptotic stability of equilibria. To illustrate our theoretical findings, we have performed numerical simulations and presented the results.


Introduction
Mathematical models are used to study the interrelationship among species and their environment. The study of disease transmission has turned out to be a valuable field of research after the fundamental work of Kermac and McKendric [1] on susceptible-infected framework. Hadeler and Freedman [2] first proposed a disease spread model within interacting populations. Initially, epidemics are created if there are some people susceptible to the infection and some infected people in the population. It is especially essential to view the ecosystem with the influence of epidemiological factors to control the disease in the species. From the ecological point of view, the spread of disease can not be disregarded because its effects are serious. So, various authors have paid attention to the study of transmissible disease in ecology, see for example [3][4][5][6] and the references therein. Mondal [7] has examined the disease model with two species and analyzed the dynamical properties of the fractional order system. Haque and Venturino [8] investigated the stability behavior of the deterministic Holling-Tanner predator-prey model. In this paper, we propose the predator-prey model and consider the Holling type II response for predation.
In an ecological model, the interactions between two or more species and their dynamics are influenced by each other. So, the growth of one species depends on another and is described by the prey-predator system. Three primary kinds of interaction between the species are: predator-prey, mutualism, and competition. In all predator-prey interactions, Holling functions do not allow the growth of predators to very large extent even if the density of the prey is more. Specifically, Holling type II functional response is defined by a decelerating intake rate which follows from the assumption that the consumer is limited by its capacity to process food. In other words, Holling type II represents the fact that when prey density is small, the predator can take less time for handling prey and if the prey density increases, more prey are attacked so that the handling time also increases. In this article, we have used Holling type II response for both infected and susceptible prey interactions with the predator. This kind of functional response has been widely utilized as a part of biological systems, see few epidemic models [9][10][11] and chemostat model [12].
During the past decades, a study of dynamical behavior of the population species with stochastic impacts has been growing steadily. The interesting situation occurs at the global stability of all feasible equilibria. Pitchaimani and Rajaji [13] constructed the stochastic Nowak-May model and investigated the asymptotic stability. In addition to stability, for every population model, the problem of permanence and boundedness property is also important. Solutions of the population model are called ultimately bounded if they satisfy the following condition: if we find the existence of bounded region in the solution space of our system such that each solution enters the bounded region in limited time and remains within the region forever. The permanence gives a guarantee that if initially the density of all species is positive, then after a specific time the density of each species will be present in some sizeable amount. Ghosh et al. [14] illustrated a seasonally perturbed stochastic model and analyzed the persistence for three species. In the literature, many results that study stability, boundedness, and persistence have been presented for some ecological models with stochastic effect [6,10,[15][16][17][18].
Nonetheless, parameters associated with the system are not fully constant, and they always change with time around some average values. These fluctuations occur because of sudden changes in the environment [19] or often created by human interference and natural events in the ecosystem by disturbing the environment. Environmental changes are described as natural disasters, human intervention, or animal or bird contact or infestation of invasive species. So these environmental changes can be outlined as noise. These changes are extreme and produce more effect on the population size in a particular time. Bringing environmental fluctuations into the predator-prey model is the correct way to deal with this situation. May [20] has uncovered the reality that the birth rates, death rates, carrying capacities, and other parameters which describe the remaining factors involved in the ecological process carry the randomness to a large or low extent due to ecological fluctuations. Accordingly, as time tends to be large, every equilibrium solution does not achieve a steady-state value accurately but it fluctuates continuously around the steady state. Recently, Liu et al. [21] developed and analyzed a population model with Holling II response and random effect. To study the model with fluctuations, several authors have introduced ecological fluctuations into every population model to accentuate the reality [11,13,[16][17][18][22][23][24][25]. The predator-prey model with two species and ratio dependence is discussed to examine its stability of equilibrium solutions in [26]. Ji et al. [9] introduced two types of functional response and stochastic perturbation into the system. Zhang et al. [27] found the critical value for the stochastic predator-prey system which can be used to determine the extinction and persistence in the mean of the predator population. Zhang and Meng [6] developed the nonautonomous SIRI epidemic model with random disturbance. The above researchers used various noises and different types of functional response depending on the population model. By the above motivation, we consider the predatorprey model with environmental changes in this article.
The article is arranged as follows. In Section 2, we present few definitions, lemmas, and theorems which are utilized in further analysis. In Section 3, we discuss the detailed explanation about the formulation and the condition that solution of the deterministic model is bounded. For the stochastic system, we derive the existence of positive solution of the system and its uniqueness and also explore the conditions for stochastic boundedness in Section 4. In addition, we prove that the solution is uniformly continuous. In Section 5, stochastic permanence and extinction under certain parametric restriction are established. Using the corresponding Lyapunov function, we have examined the conditions on global asymptotic stability in Section 6. Next, we have obtained some figures to justify the results in Section 7. Finally, the conclusion based on our results is presented in Section 8.
Consider the stochastic model (SM) of d-dimension of the form with is an R m -valued Wiener process, and Z 0 is an R d -valued random variable. The differential operator L corresponding to the SM (1) is defined as Along with the existence and uniqueness assumptions, we make the assumption that g and h satisfy gðz * , tÞ = 0 and hðz * , tÞ = 0 for an equilibrium solution z * , for t ≥ t 0 . Definition 1. The equilibrium solution z * of the SM (1) is stochastically stable if it satisfies for every ε > 0 and s ≥ t 0 , where Z s,z 0 ðtÞ represents the solution of (1) with ZðsÞ = z 0 at time t ≥ s.
The equilibrium solution z * of the SM (1) is said to be stochastically asymptotically stable if it satisfies the stochastic stability condition and Definition 3. The equilibrium solution z * of the SM (1) is said to be globally stochastically asymptotically stable if it 2 Advances in Mathematical Physics satisfies the stochastic stability condition and for every z 0 and every s, Theorem 4 (see [28]). Let the functions g and h have continuous coefficients with respect to t and satisfy the existence and uniqueness properties.
(i) Suppose that a positive definite function V ∈ C 2,1 ðU k × ½t 0 , ∞ÞÞ exists, where U k = fz ∈ R d : ∥z − z * ∥ < kg, for k > 0, such that for all t ≥ t 0 , z ∈ U k : LVðz, tÞ ≤ 0, then, the solution z * of (1) is stochastically stable (ii) Additionally if V is decresing and a positive definite function V 1 exists such that then the equilibrium solution z * is stochastically asymptotically stable (iii) If the assumption (ii) holds for a radially unbounded function V ∈ C 2,1 ðR d × ½t 0 , ∞ÞÞ defined everywhere, then the equilibrium solution z * is globally stochastically asymptotically stable Lemma 5 (see [29,31]. Suppose that a stochastic process ZðtÞ on t ≥ 0 of n -dimension satisfies where α 1 , α 2 , and c are arbitrarily nonnegative constants and a continuous modificationZðtÞ of ZðtÞ exists having the property that, for every υ ∈ ð0, α 2 /α 1 Þ, there exists a random variable ψðωÞ > 0 such that that is, each sample path ofZðtÞ is locally but uniformly Hölder continuous with υ: Definition 6 (see [30]). The solution ZðtÞ of model (1) is said to be stochastically ultimately bounded, if, for any ε ∈ ð0, 1Þ, there is a constant δ = δðεÞ > 0, such that for any initial value Z 0 ∈ R 3 + , the solution ZðtÞ of (1) satisfies Definition 7 (see [30]). The solution ZðtÞ of (1) possesses stochastic permanent property, if there exists a pair of constants φ = φðνÞ > 0 and χ = χðνÞ > 0 for any ν ∈ ð0, 1Þ such that the solution ZðtÞ of (1) for any initial value Z 0 ∈ R 3 + satisfies the property

Deterministic Model
In this section, we propose a predator-prey model with disease among the prey population. Chattopadhyay and Bairagi [32] framed the ecoepidemiological model with two species dividing into three compartments in the Salton sea and analyzed the stability of the positive equilibrium. Because of the disease, susceptible prey and infected prey are there as two groups in the prey population. The predator mostly eats infected prey because they are easy to catch. So these infected preys become more attractive to the predator. We have assumed that both the preys are subject to predation by the predator. In our article, we considered the population model as in [32] with the inclusion of the susceptible prey and predator interaction and functional response as Holling type II for interaction in the following form: Here, SðtÞ, IðtÞ, and PðtÞ denote the population densities of susceptible prey, infected prey, and predator at any time t with Sð0Þ = S 0 ≥ 0, Ið0Þ = I 0 ≥ 0, and Pð0Þ = P 0 ≥ 0. r, K, and λ represent the growth rate of S, carrying capacity of susceptible prey, and disease transmission coefficient. α is the search rate of the predator towards susceptible prey and β is the search rate of predator towards infected prey, μ and d are the natural death rates of infected prey and predator. Parameters m and a are half saturation constants. System (11) can have at most five equilibrium solutions: Let S * be a nonnegative root of the following equation The roots of the above quadratic equation are When any one of the following cases is satisfied, the equilibrium solution S * can have one or two positive values.
(i) C < 0 and D < 0 The following relations must hold for the positiveness of I * and P * : The positive equilibrium solution plays a major role in changing the dynamical behavior. It is the only solution where all the species exist. All other equilibria are the subcases of the coexisting equilibrium solution. Therefore, it is essential to analyze the dynamical properties of positive equilibrium and also it gives the behavior of each species exactly. Now, we provide certain conditions to bound the solutions of the system through the boundedness of the model equation (11). Theorem 8. All the solutions of system (11) in R 3 + with positive initial conditions are uniformly bounded.
Proof. To get the boundedness of solutions of given system (11), we consider the function Differentiate the above equation with respect to time t to obtain For each 0 ≤ η ≤ min ðμ, dÞ, the following inequality holds The maximum value of the quadratic function ax 2 + bx + c is c − ðb 2 /4aÞ when a < 0. In this way, we get the max R + ðrð1 − ðS/KÞÞ + ηÞS as ðK/4rÞðr + ηÞ 2 (refer [33]). Assume that L = ðK/4rÞðr + ηÞ 2 > 0, this implies By the theory of differential inequalities, we get and letting t tend to infinity, the above solution is of the form From the above discussion, we conclude that the solution space of system (11) lies within Hence, the theorem is proved.

Stochastic Model
In the natural world, each population in an ecosystem is greatly affected by environmental noises which play a major role in population dynamics. By considering the effect of random environment fluctuations, we have included environmental noise in every equation of our deterministic system (11). In our system, the randomness in the environment will directly affect themselves as fluctuations in the growth rate of the susceptible prey, death rate of the infected prey population, and predator population like where B i ðtÞ, i = 1, 2, 3 are independent Brownian motions and σ 2 i ði = 1, 2, 3Þ denote the intensities of the environmental fluctuations and σ i ði = 1, 2, 3Þ represent the standard 4 Advances in Mathematical Physics deviation. With this fact, we have framed the stochastic system by using the Itô equations as follows: During the past several years, no work has been reported on the above stochastic model (23). Our aim is to find the dynamics of the stochastic system (23) and show how each population varies with respect to environmental fluctuations. Now, we discuss some important properties like positiveness, boundedness, and continuity of solution of the stochastic model (23). (23) has a unique positive local solution ðSðtÞ, IðtÞ, PðtÞÞ for t ∈ ½0, τ e Þ almost surely, where τ e is the explosion time.
Proof. Consider the transformation of variables Using the Itô formula, we get Similarly, we obtain, from system (23), with xð0Þ = log Sð0Þ,yð0Þ = log Ið0Þ, and zð0Þ = log Pð0Þ. Now, the functions corresponding to system (28) have initial growth and they satisfy the local Lipchitz property. Hence, a unique local solution ðxðtÞ, yðtÞ, zðtÞÞ exists and it is defined in ½0, τ e Þ. Consequently, there exists a unique positive local solution of (23) as SðtÞ = e xðtÞ , IðtÞ = e yðtÞ , and PðtÞ = e zðtÞ .
With the existence of solution, next, we analyze how the solution changes in R 3 + .
Proof. By Theorem 10, the solution WðtÞ remains in R 3 + for all t ≥ 0. Consider the function V 1 ðt, SÞ = e t S θ for θ > 0. Using the Itô formula, we compute By considering the integral and expectation on two sides of the above equation, we get e t EðS θ ðtÞÞ − EðS θ 0 Þ ≤ M 1 ðθÞe t : So, we have lim sup t⟶∞ ES θ ðtÞ ≤ M 1 ðθÞ < +∞.

Long Time Behavior of System
Here, we look at the solution behaviour of system (23) as time becomes very large. For that, we define the hypotheses which are useful in further analysis.

Advances in Mathematical Physics
First, we will prove stochastic permanence which plays an essential part in population dynamics. We discuss this property as follows: Theorem 13. If the assumption ðH1Þ holds, then system (23) is stochastically permanent.
In population dynamics, there is a chance to lose the species population fully. So, the study of disappearance of species is also much important in the ecosystem.

Theorem 14.
If the assumption ðH2Þ holds, then the solution WðtÞ = ðSðtÞ, IðtÞ, PðtÞÞ of system (23) will be extinct with probability one for any given initial value Wð0Þ = ðSð0Þ, Ið0Þ, Proof. Define V 4 = ln S. By Itô's formula, we obtain Integrating it from 0 to t gives Then, Divide the above inequality on two sides by t and taking t ⟶ ∞, we get almost surely. Also, define the Lyapunov function V 5 = ln I; use Itô's formula to obtain Integrating this from 0 to t, we have Consequently, Dividing above inequality by t and taking t ⟶ ∞, we obtain lim sup almost surely. Similarly, we define the Lyapunov function V 6 = ln P and by Itô's formula, we get Integrating it, we get Advances in Mathematical Physics Accordingly, Then, dividing this by t and taking t ⟶ ∞, we get almost surely. Hence, the desired assertion is proved.

Stochastic Asymptotic Stability
In this section, we prove that both planar and coexistence equilibrium of system (23) are stochastically asymptotically stable under certain assumptions.

Concluding Remarks
In reality, predator-prey system reacts to more consequences in the environmental changes in the ecosystem. So, the study of dynamical behavior of the predator-prey model has been receiving more attention, and many efforts have been taken in the field of population dynamics by several authors. To overcome the effect of random fluctuations and control the 12 Advances in Mathematical Physics disease in an ecosystem, we have proposed the stochastic Holling type II predator-prey model (23) with disease in the prey. We are interested to know the changes of dynamical behavior of model (23) with respect to the intensities of environmental fluctuations. There is also need to reveal the relationship between the intensities of the environmental fluctuations and parameters associated with the system since the predator-prey model is disturbed by environmental noises. First, we explain the basic assumptions of our model (11) with boundedness of solutions. We have introduced the stochastic term to the deterministic model (11) and proved that     (23). For a stochastic predator-prey model, the boundedness of a solution is also verified because it gives a guarantee of system validity. In Theorem 12, we have checked the property of uniform conti-nuity of the positive solution of system (23). Permanence and extinction property are provided for system (23) since as time tends to be large it validates the long time and short time survival in a ecosystem. Under assumption ðH1Þ, our stochastic system (23) attains the permanence. The stochastic system     shows that if the strong environmental changes happen in an ecosystem making each species to disappear it can occur in reality while the stochastic system (23) can maintain permanence under sufficiently small environmental noise. In permanence, if we decrease the level of environmental fluctuations, solutions become stable. By the above fact, the stochastic results give more accurate results compared to the deterministic results.

Data Availability
Our paper contains numerical experimental results, and values for these experiments are included in the paper. The data is freely available.

Conflicts of Interest
The authors declare that they have no conflicts of interest.  Advances in Mathematical Physics