#### Abstract

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.

#### 1. 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 [3–6] 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 [9–11] 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, 15–18].

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, 16–18, 22–25]. 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 predator-prey 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.

#### 2. Preliminaries

Here, we give certain notations, definitions, theorems, and lemmas which are used in the following analysis. For more details, see [28–31].

Consider the stochastic model (SM) of -dimension of the form with . The functions and are Borel measurable, is an -valued Wiener process, and is an -valued random variable.

The differential operator corresponding to the SM (1) is defined as

Along with the existence and uniqueness assumptions, we make the assumption that and satisfy and for an equilibrium solution , for .

*Definition 1. *The equilibrium solution of the SM (1) is stochastically stable if it satisfies for every and ,
where represents the solution of (1) with at time .

*Definition 2. *The equilibrium solution of the SM (1) is said to be stochastically asymptotically stable if it satisfies the stochastic stability condition and

*Definition 3. *The equilibrium solution of the SM (1) is said to be globally stochastically asymptotically stable if it satisfies the stochastic stability condition and for every and every ,

Theorem 4 (see [28]). *Let the functions and have continuous coefficients with respect to and satisfy the existence and uniqueness properties.
*(i)*Suppose that a positive definite function exists, where , for , such that for all, then, the solution of (1) is stochastically stable*(ii)*Additionally if is decresing and a positive definite function exists such that**then the equilibrium solution is stochastically asymptotically stable
*(iii)*If the assumption (ii) holds for a radially unbounded function defined everywhere, then the equilibrium solution is globally stochastically asymptotically stable*

Lemma 5 (see [29, 31]. *Suppose that a stochastic process on of -dimension satisfies
where , , and are arbitrarily nonnegative constants and a continuous modification of exists having the property that, for every , there exists a random variable such that
that is, each sample path of is locally but uniformly Hölder continuous with *

*Definition 6 (see [30]). *The solution of model (1) is said to be stochastically ultimately bounded, if, for any there is a constant , such that for any initial value the solution of (1) satisfies

*Definition 7 (see [30]). *The solution of (1) possesses stochastic permanent property, if there exists a pair of constants and for any such that the solution of (1) for any initial value satisfies the property

#### 3. 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, , , and denote the population densities of susceptible prey, infected prey, and predator at any time with , , and . , , and represent the growth rate of , carrying capacity of susceptible prey, and disease transmission coefficient. is the search rate of the predator towards susceptible prey and is the search rate of predator towards infected prey, and are the natural death rates of infected prey and predator. Parameters and are half saturation constants. System (11) can have at most five equilibrium solutions: (i)The trivial equilibrium solution (ii)The equilibrium solution lying on the boundary(iii)The planar equilibrium solution on the plane where and (iv)Another planar equilibrium solution on the plane where and (v)The positive equilibrium solution which is obtained as follows:

Let be a nonnegative root of the following equation where and .

The roots of the above quadratic equation are

When any one of the following cases is satisfied, the equilibrium solution can have one or two positive values. (i) and (ii) and (iii)where

The following relations must hold for the positiveness of and :

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 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 to obtain
For each the following inequality holds
The maximum value of the quadratic function is when . In this way, we get the max as (refer [33]). Assume that , this implies
By the theory of differential inequalities, we get
and letting 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.

#### 4. 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 are independent Brownian motions and denote the intensities of the environmental fluctuations and represent the standard 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).

Theorem 9. *For , system (23) has a unique positive local solution for almost surely, where is the explosion time.*

*Proof. *Consider the transformation of variables
Using the Itô formula,
we get
Similarly, we obtain, from system (23),
with , and . Now, the functions corresponding to system (28) have initial growth and they satisfy the local Lipchitz property. Hence, a unique local solution exists and it is defined in . Consequently, there exists a unique positive local solution of (23) as , , and .

Theorem 10. *System (23) has a unique solution for and for any initial condition and the solution remains in with probability one. Therefore, for all , almost surely.*

*Proof. *To show that the global solution exists, it is enough to prove that almost surely. Assume that is a large nonnegative integer such that the closed ball contains . We choose for any and define the stop-time as
Here, ( is the empty set). Therefore, is increasing as .

Let ; then, almost surely. If almost surely is true, then almost surely. If this statement fails, that is, if , then the two constants and exist with
Thus, by denoting then is an integer such that, for all ,
Define by
where the function for all . Using Itô’s formula, we get
Taking the differential of , one gets
where
From [29], it is easy to show that is bounded above, say by , in (that is to say , for all ). From equation (35), we have
where . Taking the expectations of the above inequality, one gets
Note that no less than one of , , and belongs to the set , for every ; therefore, we get
Hence, from (32),
in which denotes the indicator function of . It follows from (39) that
leads to a contradiction: Therefore, almost surely. Thus, then almost surely.

With the existence of solution, next, we analyze how the solution changes in .

Theorem 11. *For any initial value , the solutions of system (23) are stochastically ultimately bounded.*

*Proof. *By Theorem 10, the solution remains in for all . Consider the function for . Using the *Itô* formula, we compute
By considering the integral and expectation on two sides of the above equation, we get So, we have .

Define the function for ; using the *Itô* formula, we get
Then, So, we have . Similarly, defining the function for and applying the *Itô* formula, we get
Then, So, we have .

For , we may get
Consequently,
where . Applying Chebyshev inequality, we get that all solutions are stochastically bounded.

Using fundamental properties and suitable Lyapunov functions, we continue to show that the positive solution is uniformly Hölder continuous.

Theorem 12. *Every sample path of is uniformly continuous, where is a solution of system (23) on with .*

*Proof. *The modified form of the first equation of system (23) is
Assume that and

From Theorem (28), we deduce that
For stochastic integrals, we observe the moment inequality and apply for and , to get
Then, for , we have
where By Lemma 5, for each exponent , we get that each sample path of is uniformly and locally *Hölder* continuous and which shows the uniform continuity of each sample path of on . Similarly, the uniform continuity of and is proved on . Therefore, we get the uniform continuity of each sample path of to system (23) on .

#### 5. 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.

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 holds, then system (23) is stochastically permanent.*

*Proof. *For any initial value , we show that there exists a solution such that
where arbitrary nonnegative constant satisfies
By (53), there is a positive constant such that
Define for and ; from the Itô formula, we have
Under the hypothesis , we introduce as a positive constant such that condition (53) holds. By Itô’s formula, we obtain
Next, we choose to be small such that condition (54) holds. Then,
where
The upper bound of the function is defined in the following way:
Hence,
From (53) and (54), we get a nonnegative constant satisfying . This implies that
Therefore,
Note that,
where . Accordingly,
Assume that for any ; then, by Chebyshev’s inequality, we get
that is,
Similarly, we can get for any such that