Persistence and extinction for stochastic delay differential model of prey predator system with hunting cooperation in predators

Stochastic differential models provide an additional degree of realism compared to their corresponding deterministic counterparts because of the randomness and stochasticity of real life. In this work, we study the dynamics of a stochastic delay differential model for prey–predator system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Illustrative examples and numerical simulations, using Milstein’s scheme, are carried out to validate our analytical findings. It is observed that a small scale of white noise can promote the survival of both species; while large noises can lead to extinction of the predator population.


Introduction
Prey-predator (PP) interaction is one of the most extensively studied issues in ecological and mathematical literature; see [1][2][3]. The classic prey-predator models are mostly variations of the Lotka-Volterra model, which was proposed by Lotka [4] and Volterra [5]. Many studies have explored the effect of predator hunting cooperation on PP systems [6][7][8]. Most of these studies utilize deterministic models, which of course supported us with useful results for protecting species. However, the natural growth of populations is always affected by environmental stochastic perturbations which should be taken into account in the process of mathematical modeling. Ecological systems are often subject to environmental noise (e.g., temperature, precipitation), which is an important factor in ecosystems, to suppress a potential population explosion [9]. In reality, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time [10,11]. Therefore, it is useful to see how changes in environment affect the relationship between predator and prey populations. Many authors have investigated this phenomenon (see, e.g., [12][13][14][15]).
A key question in population biology is understanding the conditions under which populations coexist or go extinct. Extinction is one of the most important terms in population dynamics. A species is said to be extinct when the last existing member dies. Therefore, extinction becomes a certainty when there are no surviving individuals that can reproduce and create a new generation. In ecology, extinction is often used informally to refer to local extinction, in which a species ceases to exist in the chosen area of study, but may still exist elsewhere. There are a variety of causes that can contribute directly or indirectly to the extinction of species or group of species, such as lack of food and space or toxic pollution of the entire population habitat, competition for food to better adapted competitors, predation, etc. [16]. Due to the importance of this topic in population dynamics, our main goal in this paper is to investigate persistence and extinction in the considered model.
We should also mention here that time-delays (time-lags) have been extensively introduced into equations used in mathematical ecology to represent the time required for maturation period, reaction time, feeding time, etc. [17][18][19]. The presence of time-delay in a system greatly affects its stability. It can destabilize the equilibrium points and give rise to a stable limit cycle, oscillations grow, and enrich the dynamics of the model. Incorporating time-delays has been considered by many authors in prey-predator models and biological systems [20][21][22][23]. Hutchinson [23] first introduced the delay in a logistic differential equation. He proposed a delay differential model for a single species of the form Here, r (> 0) is the intrinsic growth rate and K (> 0) is the carrying capacity of the population, and time-delay τ was considered as hatching time. φ(θ ) is continuous on θ ∈ [-τ , 0]. (This equation is referred to as Hutchinson's equation or delayed logistic equation.) A simple general two-dimensional delayed model of interaction between prey x(t) and a generalist predator y(t) is represented by The function G 1 (x(tτ 1 ), K) is logistic per capita growth rate of prey, where K is the environmental carrying capacity, and G 2 (y) is the per capita growth rate of predator. F(x(t)) and μF (x(t)) are 0 extra responses of predator for a particular prey, and μ is the conversion efficiency (0 < μ < 1). Time-delay τ 1 represents the gestation period of the prey or reflects the impact of density-dependent feedback mechanism [24]. Time-delay τ 2 is incorporated in the functional response of predator equation to represent the reaction time with the prey. In reality, the reproduction of predators is not immediate to the consumption of prey, as there is some discrete time lag necessary for prey gestation [17]. There exist various and extensive studies of the dynamics of the delayed PP model; see, e.g., [25][26][27][28]. In [27], the authors investigated the complex dynamics of a delayed PP system with cooperation among the prey species, they have considered time delays in the growth components for each of the species. Berec [28] assumed a Holling type II functional response of the form F(x, y) = σ (y)x 1+c(y)σ (y)x , where σ is the consumption rate of prey where δ > 0 is the death rate of predator and a > 0 is an intra-specific competition rate for predators. The description of the model parameters is presented in Table 1.
It is known that deterministic models, such as (2), are stable with a cyclic behavior in the common period for the sizes of species populations. However, in practice, stochastic variations will occur in the values of x and y, which may produce a qualitatively different behavior. These variations may lead to an extinction of the predator as a result of a possible extinction of the prey. Deterministic models may be inadequate for capturing the exact variability in nature. Then, stochastic models are required for an accurate approximation of the dynamics of such interactions. The random fluctuations result in changing some degree of parameters in the deterministic environment. Many authors have studied stochastic population models and revealed the effects of environmental noises on the dynamics of population models (see [14,[29][30][31]). Hattaf et al. [14] studied the impact of random noise in the dynamics of delayed SIR epidemic model. They deduced a threshold parameter to determine the extinction and persistence of the disease. However, in [32], the authors studied the effect of environmental fluctuations of a delayed Harrison-type PP model, they analyzed the impact of the combination of delay and noise in the dynamical behavior of the model. In [33], the authors studied the effect of environmental fluctuations on a competitive model for two phytoplankton species where one species liberate toxic substances by considering a discrete time delay parameter in the growth equations of both species.
In this paper, we consider and investigate a stochastic version of PP system (2), so that B 1 (t), B 2 (t) are standard independent Wiener processes defined on a complete probability space (Ω, A, {A} t≥0 , P) with a filtration {A} t≥0 satisfying the usual conditions; and σ 1 , σ 2 are the positive intensities of white noises. The rest of this paper is organized as follows. In Sect. 2, we briefly investigate the qualitative behavior for deterministic model (2). In Sect. 3, we study stochastic delay differential equations (SDDEs) (3). Sufficient criteria for global existence, stochastically ultimate boundedness, persistence in mean, and extinction of the system are obtained. Some numerical simulations to validate our mathematical findings are given in Sect. 4, and finally the essential results and their ecological explanations are summarized in Sect. 5.

Deterministic analysis
Herein, we study the qualitative behavior of the deterministic model. Under some restrictions on the parameters of system (2), there exist three equilibrium points E 0 , E 1 , and E * (see Appendix). We linearize the system around E * = (x * , y * ), so that x(t) = x * +x(t), y(t) = y * +ỹ(t), then we have where the coefficients are given by The characteristic equation of linearization model (4) is given by λ 2 -(a 1 + a 4 )λ + a 1 a 4 + (a 3 a 4a 3 λ)e -λτ 1a 2 a 5 e -λτ 2 = 0.
Thus all the roots of (6) have negative real parts if (H 1 ) a 3 + a 4 < -a 1 and a 1 a 4 + a 3 a 4 > a 2 a 5 hold.
If τ 1 is fixed in its stable interval and τ 2 varies, we arrive at the following remark.

Stochastic analysis
In this section, we extend the analysis to the stochastic model, where we incorporate white noise into the growth equations of both prey and predator. Recall model (3) We The initial value of system (17) becomes Now, we investigate the existence and uniqueness of positive solutions.

Existence and uniqueness of positive solution
In order to prove that the model of SDDEs (17) has a unique global solution (i.e., no explosion in a finite-time) for any given initial condition, the coefficients of system (17) are generally required to satisfy the linear growth condition and local Lipschitz condition [34,35]. To show that model (17) has a global positive solution, let us firstly prove that the model has a positive local solution by making the change of variables. Then we prove that this solution will also not explode to infinity at any finite time by using a suitable stochastic Lyapunov functional. (17) be locally Lipschitz continuous, then for any given initial data (18) there is a unique positive solution (x(t), y(t)) of system (17) on t ≥ -τ , and the solution will remain in R 2 + with probability one.

Stochastically ultimate boundedness
After discussion on the existence and uniqueness of positive solution of SDDEs (17), we show that the positive solution does not explode to infinity in a finite time. (17) is said to be stochastically ultimately bounded a.s. if, for any ∈ (0, 1), there is a positive constant ϕ = ϕ( ) such that lim t→∞ sup P{|(x(t), y(t))| > ϕ} < .

Persistence
Under certain restrictions on the parameter values with small intensities of white noise, we investigate the conditions under which the persistence of system SDDEs (17) occurs. Let us first define persistence in the mean of a dynamical system.

Definition 2
The species y(t) is said to be persistent (see [16]) in the mean if Let us define a threshold parameter R s 0 as follows: (17) with initial conditions (18). Assume that 2r > σ 2 1 , then system (17) will be persistent if R s 0 > 1, so that lim inf t→∞  (17) yields

Extinction
Extinction is one of the most important terms in population dynamics. A species is said to be extinct if there is no existing member in the habitat. Although, under some conditions, the solution to the original deterministic DDEs (2) may be persistent, the solution to the associated SDDEs will become extinct with probability one. This reveals the important fact that the environmental noise may make the population extinct. Now, we establish the conditions under which extinction of predator population occurs.

Definition 3
The species y(t) is said to go to extinction with probability one if lim t→∞ y(t) = 0 a.s.

Theorem 5
Let a > μα. If R s 0 < 1, then the solution (x(t), y(t)) of model (17), for any given initial value (18), satisfies which means lim t→∞ y(t) = 0 exponentially almost surely. In other words, the predators die out with probability one. In addition, 2r .
Proof According to (32), we have since (a > μα), then Thus, we have two cases.
• When x(t) ≤ 1, according to (42), we can get therefore, we have So, where ds.
In view of the strong law of large numbers of Brownian motion, we can easily obtain that lim t→∞ χ 1 (t) = 0 a.s.
Thus, it follows from (45) and since R s • When x(t) > 1, by (42), we have then Therefore, where In view of the strong law of large numbers of Brownian motion, we can easily obtain that lim t→∞ χ 2 (t) = 0 a.s.
which implies that y(t) tends to zero exponentially with probability one, lim t→∞ y(t) = 0 a.s.
By taking the limit on both sides of (28) and (30) at the same time, we can get 2r .
This completes the proof.

Numerical simulations
In this section, we attempt to validate the mathematical results obtained in the previous sections. We utilize Milstein's scheme with a strong order of convergence one discussed in [40]. The corresponding discretization system to SDDEs (17) is y n+1 = y n + hy n μ(1 + αy n )x n-m 2 1 + c(1 + αy n )x n-m 2 δay n + σ 2 y n ξ 2,n + σ 2 2 2 y n ξ 2,n (h) Here, ξ 1,n and ξ 2,n are mutually independent N(0, 1) random variables, m 1 , m 2 are integers such that the time delays can be expressed in terms of the step-size as τ 1 = m 1 h & τ 2 = m 2 h. We provide some numerical simulations of the stochastic model (17) and its corresponding deterministic model (2).
Remark 3 Extinction of the predator population possibly occurs when the intensity of white noise is large, such that R s 0 < 1. This would not happen in deterministic system (2) without noises (see Fig. 4). If the predator death rate is large, extinction of the predators can also occur in deterministic model (2); while with a small noise in the stochastic model, extinction of the predator population occurs faster than in the deterministic model (see Fig. 3).

Discussion and conclusion
In this paper, we studied the dynamics of SDDEs for a prey-predator system with hunting cooperation in predators. We investigated the effect of environmental fluctuations on the model. We first studied the qualitative behaviors of the deterministic model through local stability analysis of the interior equilibrium points and obtained critical values of delays, where the Hopf bifurcation occurs for the deterministic model. We then considered how environmental fluctuations affect extinction of predator and prey populations. We established the existence and uniqueness of global positive solution and the stochastically ultimate boundedness of the system. We have shown the effect of environmental noises on the persistence and possible extinction of prey and predator populations. We verified the obtained analytical results with supportive numerical simulations using Milstein's scheme. Conditions under which persistence of the system occurs have been established when R s 0 > 1; while with R s 0 < 1, extinction of the predator occurs. It can also be observed that the extinction of the predator population occurs more rapidly for stochastic system (17) when the intensity of white noise increases, see Figs. 3-4. It has also been shown numerically that the predator population dominates the prey population as cooperative hunting parameter increases (see Fig. 1).
Our main findings, theoretically and numerically, are all represented in terms of the system parameters and the intensity of randomly fluctuating driving forces. This indicates that time-delay and white noise have a considerable impact on the dynamics and presence of prey and predator populations. For future investigation, one may extend our results with other kinds of environmental noise such as color noise [41] or telephone noise [42].
Using (60) and from the first equation of (2), we get i.e., for > 0, there exists T 1 > 0 such that x(t) ≤ Ke rτ 1 + for all t > T 1 . Similarly, from the second equation of (2), we get Thus, y(t) ≤ ξ + for all t > T 2 , conclusion of this theorem can be achieved by letting → 0.