Dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment

: In this paper, a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment is investigated. We obtain the su ffi cient and necessary conditions for persistence in mean and extinction of each species. The results show that the stochastic dynamics of the system are closely correlated with both time delays and environmental noises. Some numerical examples are introduced to illustrate the main results.


Introduction
The predator-prey model is one of the hotspots in biomathematics.For example, Yavuz and Sene [1] considered a fractional predator-prey model with harvesting rate, Chatterjee and Pal [2] studied a predator-prey model for the optimal control of fish harvesting through the imposition of a tax and Ghosh et al. [3] presented a three-component model consisting of one prey and two predator species using imprecise biological parameters as interval numbers and applied a functional parametric form in the proposed prey-predator system.Because of its important role in the ecosystem, the food chain model has been extensively studied [4][5][6][7][8].Specifically, the classical four-species food chain model can be expressed as follows: dx 2 (t) = x 2 (t) [−r 2 + a 21 x 1 (t) − a 22 x 2 (t) − a 23 x 3 (t)] dt, dx 3 (t) = x 3 (t) [−r 3 + a 32 x 2 (t) − a 33 x 3 (t) − a 34 x 4 (t)] dt, dx 4 (t) = x 4 (t) [−r 4 + a 43 x 3 (t) − a 44 x 4 (t)] dt, (1.1) where x 1 (t), x 2 (t), x 3 (t) and x 4 (t) represent the densities of prey, primary predator, intermediate predator and top predator at time t, respectively.r 1 is the growth rate of prey, r 2 , r 3 and r 4 are the death rates of primary predator, intermediate predator and top predator, respectively.a i j and a ji (i < j) are the capture rates and food conversion rates, respectively.a ii are the intraspecific competition rates of species i.All parameters in system (1.1) are positive constants.
In ecology, biology, physics, engineering and other areas of applied sciences, continuous-time models, fractional-order models as well as discrete-time models have been widely adopted [9,10].However, "time delays occur so often that to ignore them is to ignore reality" [11,12], and in the models of population dynamics, the delay differential equations are much more realistic [13][14][15].We know that systems with discrete time delays and those with continuously distributed time delays do not contain each other but systems with S-type distributed time delays contain both.Introducing S-type distributed time delays into system (1.1) yields where D ji (x i )(t) = a ji x i (t) + 0 −τ ji x i (t + θ)dµ ji (θ), 0 −τ ji x i (t + θ)dµ ji (θ) are Lebesgue-Stieltjes integrals, τ ji > 0 are time delays, µ ji (θ) are nondecreasing bounded variation functions defined on [−τ, 0], τ = max τ ji .
Besides, population system may be affected by telephone noises which can cause the system to switch from one environmental regime to another [36][37][38].So, telephone noises should be taken into consideration in system (1.3), resulting the following model: (1.4) where ρ(t) is a continuous time Markov chain with finite state space S = {1, 2, ..., S }, which describes the telephone noises.
Moreover, the behaviour of real biological species, in different ecosystems, is affected by Lévy noises [39].Lévy processes are characterized by stationary independent increments [40].Assume that L(t) (t ≥ 0) is a Lévy process, using the decomposition [41] one can observe that the probability distribution of L(t) is infinitely divisible.The most general expression for the characteristic function of L(t) is where sgn(k) is the sign function with where α ∈ (0, 2] is the stability parameter, σ is the scale parameter, σ α is the noise intensity, µ ∈ R is the location parameter and β ∈ [−1, 1] is the skewness parameter [39].In addition, Lévy noises are statistically independent with zero mean.Now, let us further improve system (1.4) by considering Lévy noises.Some scholars pointed out that Lévy noises can be used to describe some sudden environmental perturbations, for instance, earthquakes and hurricanes [42][43][44][45][46][47].In the context of an epidemic situation, random jumps could refer to sudden and significant increases in the number of cases or spread of the disease that occur unpredictably [48].System (1.4) with Lévy noises can be expressed as follows: , N is a Poisson counting measure with characteristic measure λ on a measurable subset Z of [0, +∞), where λ(Z) < +∞ and N(dt, dµ) = N(dt, dµ) − λ(dµ)dt, γ j (µ, ρ(t)) > −1 (µ ∈ Z) are bounded functions ( j = 1, 2, 3, 4).
Finally, environmental pollution caused by agriculture, industries and other human activities has become a big challenge that is commonly concerned by international society.For example, with the rapid development of industrial and agricultural production, some chemical plants and other industries often periodically discharge sewage or other pollutants into rivers, soil and air [49].These pollutants can cause direct damage to ecosystems, such as species extinction, desertification and the greenhouse effect.Hence, we extend system (1.5) into the following form: and ∆C e (t) = C e (t + )−C e (t).For other parameters in system (1.6), see Table 1.
To the best of our knowledge to date, results about a stochastic hybrid delay four-species food chain model with jumps have not been reported.So, in this paper we investigate the dynamics of a stochastic hybrid delay four-species food chain model with jumps in an impulsive polluted environment.The organization of this paper is as follows: In Section 2, some basic preliminaries are presented.In Section 3, the sufficient and necessary conditions for stochastic persistence in mean and extinction of each species are obtained.In Section 4, some numerical examples are provided to illustrate our main results.Finally, we conclude the paper with a brief conclusion and discussion in Section 5.
Remark 1. Assumption 2 implies that the intensities of Lévy jumps are not too big to ensure that the solution will not explode in finite time.

Denote
The proof is rather standard and hence is omitted (see e.g., [52]).

The effects of Lévy jumps on the persistence in mean and extinction
Table 5. Changes of γ 4 (1) when γ    In view of Theorem 2, x 1 (t), x 2 (t) and x 3 (t) are persistent in mean, while x 4 (t) is extinctive and     (2.2510, 0, 0, 0)

Discussion and conclusions
This paper concerns the dynamics of a stochastic hybrid delay food chain model with jumps in an impulsive polluted environment.Theorem 2 establishes sufficient and necessary conditions for persistence in mean and extinction of each species.Our results reveal that the stochastic dynamics of the system is closely correlated with both time delays and environmental noises.Some interesting topics deserve further investigation, for instance, it is meaningful to consider the optimal harvesting problem of the stochastic hybrid delay food chain model with Lévy noises in an impulsive polluted environment.One may also propose some more realistic systems, such as considering the generalized functional response and the influences of impulsive perturbations.We will leave investigation of these problems to the future.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Table 1 .
Definition of some parameters in system (1.6).

Table 4 .
Source of some parameter values in system (1.6).