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 thatthen 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 . Hence, by Definition 7, 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 holds, then the solution of system (23) will be extinct with probability one for any given initial value .
Proof. Define . By Itô’s formula, we obtain
Integrating it from 0 to gives
Then,
Divide the above inequality on two sides by and taking , we get
almost surely.
Also, define the Lyapunov function ; use Itô’s formula to obtain
Integrating this from 0 to , we have
Consequently,
Dividing above inequality by and taking , we obtain
almost surely.
Similarly, we define the Lyapunov function and by Itô’s formula, we get
Integrating it, we get
Accordingly,
Then, dividing this by and taking , we get
almost surely.
Hence, the desired assertion is proved.
6. Stochastic Asymptotic Stability
In this section, we prove that both planar and coexistence equilibrium of system (23) are stochastically asymptotically stable under certain assumptions.
Theorem 15. The equilibrium solution of system (23) is stochastically asymptotically stable on , if where
Proof. We use Theorem (1) and construct the suitable Lyapunov function as follows: We define the infinitesimal generator on the Lyapunov function and get If , it follows that . By assumptions, on . Therefore, is stochastically asymptotically stable.
Theorem 16. The equilibrium solution of system (23) is stochastically asymptotically stable on , if where and .
Proof. Consider the Lyapunov function for the equilibrium solution Define the infinitesimal generator on the above function as follows: If where , we obtain . By assumptions, on . Therefore, is stochastically asymptotically stable.
Theorem 17. The equilibrium solution of system (23) is stochastically asymptotically stable on , if where, , and .
Proof. The Lyapunov function for the equilibrium solution can be written as Define the infinitesimal generator on as One can see that , if where and . By assumptions, on . Therefore, of system (23) is stochastically asymptotically stable.
Theorem 18. The equilibrium solution of system (23) is stochastically asymptotically stable on , if
where , , and
for and .
Proof. Construct the suitable Lyapunov function for the coexistence equilibrium as follows where are suitable constants to be determined in further analysis. Define the infinitesimal generator on and we get One can observe that , if where and . By assumptions, on . Therefore, of system (23) is stochastically asymptotically stable.
7. Numerical Simulation
From the practical point of view, numerical results play a major role in population dynamics. Here, we have provided graphical solution curves to carry out and check our theoretical results of system (23). To find the strong solution of system (23) with given initial value, Higham [34] introduced Milstein method. By using this method, we have derived the following discretization equations from our given system (23): where is a positive time increment parameter and represent independent random Gaussian variables , it is given with the help of pseudorandom number generators. Using MATLAB software and possible parameters, we have generated the solution curves of the deterministic system (11) and corresponding stochastic system (23).
In Figures 1–3, we take the initial conditions , , and and parameters , , , , , , , , and . The only difference between Figures 1–3 is the intensities of environmental changes. Specially, we choose in Figure 1; in Figure 2, we have chosen and in Figure 3. From Figures 1–3, we have observed that the coexistence equilibrium solution curves of stochastic model (23) always oscillates with respect to the curves of the deterministic model (11). From Figures 1–3, we have arrived at the conclusion that with the decrease in the values of the fluctuations of the solution curves of the stochastic system are reduced and coincide with that of the solution curves of the deterministic system.
In Figure 4, the parametric values are , , , , , , , , and with initial values . By Theorem 14, we observe that all the species in the stochastic case (23) disappear completely under the hypothesis and we have seen this graphically via Figure 4. Further, we have simulated the corresponding deterministic case in Figure 4 which satisfies condition (14) and leading to the permanence of each species. Under parametric values , , , , , , , , and and initial parametric condition , by Theorem 14, all the species of the stochastic system (23) as well as corresponding deterministic system (11) disappear fully under the hypothesis and the solution curves are shown in Figure 5. If the deterministic system goes to extinction, the stochastic system also goes to extinction, in which case solution is not dependent on the intensity of noise value (Compare Figure 4 with Figure 5).
Furthermore, in Figure 6, parameters chosen are , , , , , , , , and with initial values and simulations also confirm the result that all three species of system (11) and (23) are permanent by Theorem 13.
Finally, Figure 7 illustrates the Theorem 18. We choose the parameters , , , , , , , , and with different initial values satisfying condition (88). Theorem 18 says that of (23) is stochastically asymptotically stable.
8. 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 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 for any positive initial value there exists a positive global solution of the stochastic system (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 continuity 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 , our stochastic system (23) attains the permanence. The stochastic system goes to extinction state if the assumption holds. Furthermore, with the help ofthe Lyapunov function theory we have examined the stochastic asymptotic stability of all feasible equilibrium solutions of (23). The parametric restriction for stochastic stability is established. (see the inequalities in Theorem 15, 16, 17, and 18).
In a stochastic system, we have observed that if the intensities of the environmental changes increase, the fluctuations of the solutions increase. The numerical result 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.
Acknowledgments
The research work of the first author is supported by the DST-INSPIRE Fellowship (No. DST/INSPIRE Fellowship/2017/IF170244), Department of Science and Technology, New Delhi. The second author is thankful to UGC(BSR)-Start Up Grant (Grant No. F.30-361/2017(BSR)), University Grants Commission, New Delhi and the third author is thankful to UGC BSR Faculty Fellowship, University Grants Commission, New Delhi. This work was supported by the research grant of the Kongju National University in 2020 and this research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2018R1D1A1B07049623) to the last author.