A discrete stochastic model of the COVID-19 outbreak: Forecast and control

1.   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:

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. $a_{ij}$ 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 $\mathcal{D}_{ji}(x_{i})(t) = a_{ji}x_{i}(t)+\int_{-\tau_{ji}}^{0}x_{i}(t+\theta)\mathrm{d}\mu_{ji}(\theta)$, $\int_{-\tau_{ji}}^{0}x_{i}(t+\theta)\mathrm{d}\mu_{ji}(\theta)$ are Lebesgue-Stieltjes integrals, $\tau_{ji} > 0$ are time delays, $\mu_{ji}(\theta)$ are nondecreasing bounded variation functions defined on $[-\tau, 0]$, $\tau = \max\left\{\tau_{ji}\right\}$.

On the other hand, the deterministic system has its limitation in mathematical modeling of ecosystems since the parameters involved in the system are unable to capture the influence of environmental noises ^[16,17]. Introducing Gaussian white noises into the corresponding deterministic model is one common way to characterize environmental noises ^{[18,19,20,21,22,23,24,25]}. As we all know, Gaussian white noise $\xi(t)$ is a stationary and ergodic stochastic process with $\langle \xi(t)\rangle = 0$ and $\langle \xi(t)\xi(s)\rangle = \sigma^{2}\delta(t-s)$, where $\sigma^{2}$ is the noise intensity ^[26]. The readers can refer to ^{[27,28,29,30,31,32,33,34]} for more related works. In this paper, we assume that $r_{i}$ are affected by Gaussian white noises, i.e., $r_{1} \hookrightarrow r_{1}+\sigma_{1}\dot{W}_{1}(t)$, $-r_{2} \hookrightarrow -r_{2}+\sigma_{2}\dot{W}_{2}(t)$, $-r_{3} \hookrightarrow -r_{3}+\sigma_{3}\dot{W}_{3}(t)$ and $-r_{4} \hookrightarrow -r_{4}+\sigma_{4}\dot{W}_{4}(t)$. Then, system (1.2) becomes

where $W_{i}(t)$ are mutually independent standard Wiener processes defined on a complete probability space $(\Omega, \mathcal{F}, P)$ satisfying the usual statistical properties, namely $\langle \mathrm{d}W_{i}(t)\rangle = 0$ and $\langle \mathrm{d}W_{i}(t)\mathrm{d}W_{j}(s)\rangle = \delta_{ij}\delta(t-s)\mathrm{d}t$ ^[35].

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:

where $\rho(t)$ is a continuous time Markov chain with finite state space $\mathbb{S} = \left\{1, 2, ..., S\right\}$, 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 $L(t)$ $(t \geq 0)$ 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 $\alpha \in (0, 2]$ is the stability parameter, $\sigma$ is the scale parameter, $\sigma^{\alpha}$ is the noise intensity, $\mu \in \mathbb{R}$ is the location parameter and $\beta \in [-1, 1]$ 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:

where $\mathcal{S}_{i}\left(t, \rho(t)\right) = \sigma_{i}(\rho(t))\mathrm{d}W_{i}(t)+\int_{\mathbb{Z}}\gamma_{i}(\mu, \rho(t))\widetilde{N}(\mathrm{d}t, \mathrm{d}\mu)$, $N$ is a Poisson counting measure with characteristic measure $\lambda$ on a measurable subset $\mathbb{Z}$ of $[0, +\infty)$, where $\lambda(\mathbb{Z}) < +\infty$ and $\widetilde{N}(\mathrm{d}t, \mathrm{d}\mu) = N(\mathrm{d}t, \mathrm{d}\mu)-\lambda(\mathrm{d}\mu)\mathrm{d}t$, $\gamma_{j}\left(\mu, \rho(t)\right) > -1$ $(\mu \in \mathbb{Z})$ 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:

where $\Delta x_{i}(t) = x_{i}\left(t^{+}\right)-x_{i}(t)$, $\Delta C_{i0}(t) = C_{i0}\left(t^{+}\right)-C_{i0}(t)$ and $\Delta C_{e}(t) = C_{e}\left(t^{+}\right)-C_{e}(t)$. 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.

2.   Preliminaries

We have four fundamental assumptions for system (1.6).

Assumption 1. $W_{1}(t)$, $W_{2}(t)$, $W_{3}(t)$, $W_{4}(t)$, $\rho(t)$ and $\mathrm{N}$ are mutually independent. $\rho(t)$, taking values in $\mathbb{S} = \left\{1, 2, ..., S\right\}$, is irreducible with one unique stationary distribution $\pi = \left(\pi_{1}, \pi_{2}, ..., \pi_{S}\right)^{\mathrm{T}}$.

Assumption 2. $r_{j}(i) > 0$, $a_{jk} > 0$ and there exist $\gamma_{j}^{*}(i) \geq \gamma_{j*}(i) > -1$ such that $\gamma_{j*}(i) \leq \gamma_{j}(\mu, i) \leq \gamma_{j}^{*}(i)$ $(\mu \in \mathbb{Z})$, $\forall i \in \mathbb{S}$, $j, k = 1, 2, 3, 4$.

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.

Assumption 3. $0 < k_{i} \leq g_{i}+m_{i}$ $(i = 1, 2, 3, 4)$, $0 < b \leq 1-\mathrm{e}^{-h\gamma}$.

Remark 2. Assumption 3 means $0 \leq C_{i0}(t) < 1$ and $0 \leq C_{e}(t) < 1$, which must be satisfied to be realistic because $C_{i0}(t)$ and $C_{e}(t)$ are concentrations of the toxicant $(i = 1, 2, 3, 4)$.

Assumption 4. $A_{22}A_{33}A_{44}\left|{\bf A}\right|\left|{\bf \Xi}\right| > A_{12}A_{21}A_{23}A_{32}A_{44}\left|{\bf \Xi}\right|+A_{23}A_{32}A_{34}A_{43}\left|{\bf A}\right|^{2}$.

Lemma 1. ^[50,51] $C_{i0}(t)$ involved in system (1.6) satisfies

3.   Persistence in mean and extinction

Denote

Denote ${\bf \Sigma^{(2)}} = \left(\Sigma_{1}, \Sigma_{2}\right)^{\mathrm{T}}$, ${\bf \Sigma^{(3)}} = \left(\Sigma_{1}, \Sigma_{2}, \Sigma_{3}\right)^{\mathrm{T}}$, ${\bf \Sigma} = \left(\Sigma_{1}, \Sigma_{2}, \Sigma_{3}, \Sigma_{4}\right)^{\mathrm{T}}$. Denote ${\bf A_{j}}$ is ${\bf A}$ with column $j$ replaced by ${\bf \Sigma^{(2)}}$ $(j = 1, 2)$; ${\bf \Xi_{j}}$ is ${\bf \Xi}$ with column $j$ replaced by ${\bf \Sigma^{(3)}}$ $(j = 1, 2, 3)$; ${\bf \Delta_{j}}$ is ${\bf \Delta}$ with column $j$ replaced by ${\bf \Sigma}$ $(j = 1, 2, 3, 4)$.

Theorem 1. For any initial condition ${\bf \phi} \in \mathcal{C}\left([-\tau, 0], \mathbb{R}_{+}^{4}\right)$, system (1.6) has a unique global solution $\left(x_{1}(t), x_{2}(t), x_{3}(t), x_{4}(t)\right)^{\mathrm{T}} \in \mathbb{R}_{+}^{4}$ on $t \in \mathbb{R}_{+}$ a.s. Moreover, for any constant $p > 0$, there exists $K_{i}(p) > 0$ such that $\sup_{t \in \mathbb{R}_{+}}\mathbb{E}\left[x_{i}^{p}(t)\right] \leq K_{i}(p)$ $(i = 1, 2, 3, 4)$.

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

Lemma 2. ^[53] Suppose $Z(t)\in \mathcal{C}(\Omega\times [0, +\infty), \mathbb{R}_{+})$ and $\lim_{t\rightarrow +\infty}\frac{o(t)}{t} = 0$.

$\mathfrak{(i)}$ If there exists constant $\delta_{0} > 0$ such that for $t \gg 1$,

then

$\mathfrak{(ii)}$ If there exist constants $\delta > 0$ and $\delta_{0} > 0$ such that for $t \gg 1$,

then

Lemma 3. If $\left|{\bf \Delta_{4}}\right| > 0$, then $\left|{\bf \Delta_{j}}\right| > 0$ $(j = 1, 2, 3)$.

Proof. Compute

Noting that $\Sigma_{j} < 0$ $(j = 2, 3, 4)$, we obtain the desired assertion. □

First, let us consider the following auxiliary system:

Lemma 4. System (3.1) satisfies Table 2, where

Proof. Consider the following stochastic hybrid delay logistic model with Lévy jump in an impulsive polluted environment:

Thanks to Lemma 1 and Lemma 2.3 in ^[54], system (3.2) satisfies

By Itô's formula, we compute

where

$\mathfrak{Case} \, \, (1):$ $B_{1} < 0$. Based on Eq (3.3), $\lim_{t \rightarrow +\infty} X_{1}(t) = 0$ a.s. Therefore, for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Since $\Sigma_{2} < 0$, then $\lim_{t \rightarrow +\infty} X_{2}(t) = 0$ a.s. Similarly, $\lim_{t \rightarrow +\infty} X_{j}(t) = 0$ a.s. $(j = 3, 4)$.

$\mathfrak{Case} \, \, (2):$ $B_{1} \geq 0$. Then,

Consider the following auxiliary function:

Then $X_{2}(t) \leq \widetilde{X_{2}(t)}$ a.s. By Itô's formula, we get

In view of Lemma 2, we obtain:

If $B_{1} \geq 0$, $B_{2} < 0$, then $\lim_{t \rightarrow +\infty} \widetilde{X_{2}(t)} = 0$ a.s.

If $B_{1} \geq 0$, $B_{2} \geq 0$, then

Therefore, for arbitrary $\zeta > 0$, we have

According to system (3.4) and Eq (3.6), we obtain

Thanks to Lemma 2, we deduce:

If $B_{1} \geq 0$, $B_{2} < 0$, then $\lim_{t \rightarrow +\infty} X_{j}(t) = 0$ a.s. $(j = 2, 3, 4)$.

If $B_{1} \geq 0$, $B_{2} \geq 0$, then

$\mathfrak{Case} \, \, (3):$ $B_{1} \geq 0$, $B_{2} \geq 0$. Consider the following SDE:

Then $X_{3}(t) \leq \widetilde{X_{3}(t)}$ a.s. By Itô's formula, we get

In the light of Lemma 2, we obtain:

If $B_{1} \geq 0$, $B_{2} \geq 0$, $B_{3} < 0$, then $\lim_{t \rightarrow +\infty} \widetilde{X_{3}(t)} = 0$ a.s.

If $B_{1} \geq 0$, $B_{2} \geq 0$, $B_{3} \geq 0$, then

Hence, for arbitrary $\zeta > 0$,

Thanks to system (3.4) and Eq (3.7), we obtan

Based on Lemma 2, we obtain:

If $B_{1} \geq 0$, $B_{2} \geq 0$, $B_{3} < 0$, then $\lim_{t \rightarrow +\infty} X_{j}(t) = 0$ a.s. $(j = 3, 4)$.

If $B_{1} \geq 0$, $B_{2} \geq 0$, $B_{3} \geq 0$, then

$\mathfrak{Case} \, \, (4):$ $B_{1} \geq 0$, $B_{2} \geq 0$, $B_{3} \geq 0$. Consider the following SDE:

Then $X_{4}(t) \leq \widetilde{X_{4}(t)}$ a.s. By Itô's formula, we get

In view of Lemma 2, we obtain:

If $B_{1} \geq 0$, $B_{2} \geq 0$, $B_{3} \geq 0$, $B_{4} < 0$, then $\lim_{t \rightarrow +\infty} \widetilde{X_{4}(t)} = 0$ a.s.

If $B_{1} \geq 0$, $B_{2} \geq 0$, $B_{3} \geq 0$, $B_{4} \geq 0$, then

Hence, for arbitrary $\zeta > 0$,

Thanks to systems (3.4) and (3.8), we deduce

Based on Lemma 2 and the arbitrariness of $\epsilon$, we obtain:

If $B_{1} \geq 0, \, \, B_{2} \geq 0, \, \, B_{3} < 0, \, \, B_{4} < 0$, then $\lim_{t \rightarrow +\infty} X_{4}(t) = 0 \, \, a.s.$

If $B_{1} \geq 0, \, \, B_{2} \geq 0, \, \, B_{3} \geq 0, \, \, B_{4} \geq 0$, then

The proof is complete. □

Lemma 5. For system (1.6):

$(i)$ $\limsup_{t \rightarrow +\infty}t^{-1}\ln x_{i}(t) \leq 0$ a.s. $(i = 1, 2, 3, 4)$.

$(ii)$ $\lim_{t \rightarrow +\infty}x_{i}(t) = 0\Rightarrow \lim_{t \rightarrow +\infty}x_{j}(t) = 0$ a.s. $(1 \leq i < j \leq 4)$.

Proof. From Lemma 4, system (3.1) satisfies $\lim_{t \rightarrow +\infty}t^{-1}\ln X_{i}(t) = 0$ a.s. $(i = 1, 2, 3, 4)$. By the stochastic comparison theorem, we obtain the desired assertion $(i)$. The proof of $(ii)$ is similar to that of Lemma 4 and here is omitted. □

Theorem 2. Under Assumption 4 system (1.6) satisfies Table 3, where

Proof. Compute $\left|{\bf \Delta_{4}}\right| < A_{43}\left|{\bf \Xi_{3}}\right| < A_{32}A_{43}\left|{\bf A_{2}}\right| < A_{21}A_{32}A_{43}B_{1}$. By Eq (3.8), for any $\zeta > 0$,

By Itô's formula, we compute

$\mathfrak{Case \, \, (i):}$ $\left|{\bf \Delta_{4}}\right| > 0$. According to system (3.9), we compute

In view of Lemma 5 $(i)$ and Lemma 2, we get

Based on system (3.9), we compute

By Lemma 5 $(i)$ and Eq (3.12), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

In view of Eq (3.11), we deduce

where $\left|{\bf A}\right| > 0$ and $\left|{\bf \Delta}\right| > 0$. From Lemma 3, we have $\left|{\bf \Delta_{1}}\right| > 0$. Based on Lemma 2 and the arbitrariness of $\epsilon$, we obtain

According to system (3.9), we compute

Thanks to Lemma 5 $(i)$ and Eq (3.15), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

If

then by Lemma 2 and the arbitrariness of $\epsilon$, we obtain

If

since $\liminf_{t \rightarrow +\infty}t^{-1}\int_{0}^{t}x_{3}(s)\mathrm{d}s\geq 0$, we also obtain Eq (3.16).

According to system (3.9), Eq (3.14) and Eq (3.16), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

On the basis of Eq (3.13), Eq (3.14) and Eq (3.16), we have

From Lemma 3, we have $\left|{\bf \Delta_{2}}\right| > 0$. By Lemma 2 and the arbitrariness of $\epsilon$, we obtain

By system (3.9), Eq (3.11) and Eq (3.18), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

In view of Eq (3.17) and Eq (3.18), we obtain

In the light of Lemma 3, we have $\left|{\bf \Delta_{3}}\right| > 0$. Thanks to Lemma 2 and the arbitrariness of $\epsilon$, we obtain

By system (3.9) and Eq (3.20), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Thanks to Eq (3.19), we obtain

In the light of Lemma 2 and the arbitrariness of $\epsilon$, we obtain

In other words, we have

According to Eq (3.21) and Assumption 4, we obtain

Combining Eq (3.11) and Eq (3.22) yields

Substituting Eq (3.22) into Eq (3.14) yields

Substituting Eq (3.22) into Eq (3.16) yields

Substituting Eq (3.24) and Eq (3.25) into Eq (3.18) yields

Substituting Eq (3.26) into Eq (3.20) yields

Combining Eq (3.25) and Eq (3.27) yields

In view of system (3.9), we have

By Eq (3.26) and Eq (3.29), for $\forall \epsilon \in (0, 1)$ and $t\gg 1$,

where $B_{1}-A_{12}\frac{\left|{\bf \Delta_{2}}\right|}{\left|{\bf \Delta}\right|} = A_{11}\frac{\left|{\bf \Delta_{1}}\right|}{\left|{\bf \Delta}\right|}$. From Lemma 3, we have $\left|{\bf \Delta_{1}}\right| > 0$. According to Lemma 2 and the arbitrariness of $\epsilon$, we have

Combining Eq (3.24) with Eq (3.30) yields

Substituting Eq (3.31) into system (3.9) yields

From Lemma 3, we have $\left|{\bf \Delta_{2}}\right| > 0$. By Lemma 2, we obtain

$\mathfrak{Case \, \, (ii)}:$ $\left|{\bf \Xi_{3}}\right| > 0 > \left|{\bf \Delta_{4}}\right|$. Thanks to Eq (3.10), we deduce

which implies

From Lemma 5 $(ii)$ and Eq (3.33), we obtain

In other words,

According to system (3.9), we compute

Combining Lemma 5 $(i)$ with Lemma 2 yields

Based on system (3.9), we compute

By Lemma 5 $(i)$ and Eq (3.37), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

On the basis of Eq (3.36), we deduce

In view of Lemma 2 and the arbitrariness of $\epsilon$, we obtain

According to system (3.9), Eq (3.36) and Eq (3.38), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Combining Eq (3.38) with system (3.39) yields

In the light of Lemma 2 and the arbitrariness of $\epsilon$, we obtain

From system (3.9) and Eq (3.41), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Thanks to Eq (3.40), we obtain

In the light of Lemma 2 and the arbitrariness of $\epsilon$, we obtain

In other words, we have

In view of Eq (3.42) and Assumption 4, we obtain

Combining Eq (3.36) and Eq (3.43) yields

Substituting Eq (3.44) into Eq (3.38) yields

Substituting Eq (3.45) into Eq (3.41) yields

In view of system (3.9), we compute

By Lemma 5 $(i)$ and Eq (3.47), for $\forall \epsilon \in (0, 1)$ and $t\gg 1$,

From Eq (3.40), we have $\left|{\bf \Xi_{2}}\right| > 0$ and $\left|{\bf \Xi}\right| > 0$. Based on system (3.48) and Lemma 2,

Combining Eq (3.46) with Eq (3.49) yields

Substituting Eq (3.50) into system (3.9) yields

From Eq (3.38), we have $\left|{\bf \Xi_{1}}\right| > 0$ and $\left|{\bf \Xi}\right| > 0$. Therefore, by Lemma 2, we obtain

$\mathfrak{Case \, \, (iii)}:$ $\left|{\bf A_{2}}\right| > 0 > \left|{\bf \Xi_{3}}\right|$. Then $\lim_{t \rightarrow +\infty} x_{4}(t) = 0$ a.s. Thanks to Eq (3.35), we deduce

which implies

According to Lemma 5 $(ii)$ and Eq (3.52), we obtain

In other words, we derive

In view of Eq (3.47) and Eq (3.54), we get

Based on Lemma 5 $(i)$ and Lemma 2, we have

In the light of Eq (3.37) and Eq (3.54), we obtain

By Lemma 5 $(i)$ and Lemma 2, we obtain

Substituting Eq (3.54) and Eq (3.57) into system (3.9) yields

On the basis of Lemma 2 and the arbitrariness of $\epsilon$, we have

Combining Eq (3.56) with Eq (3.58) yields

By system (3.9) and Eq (3.59), we compute

In the light of Lemma 2, we obtain

$\mathfrak{Case \, \, (iv)}:$ $B_{1} > 0 > \left|{\bf A_{2}}\right|$. Then

According to Eq (3.55), we gain

Hence, $\limsup_{t \rightarrow +\infty}x_{1}^{A_{21}}(t)x_{2}^{A_{11}}(t) = 0$. By Lemma 5 $(ii)$ and Eq (3.62),

In other words, we derive

Substituting Eq (3.64) into system (3.9) yields

On the basis of Lemma 2 and the arbitrariness of $\epsilon$, we obtain

$\mathfrak{Case \, \, (v)}:$ $B_{1} < 0$. Compute

By Eq (3.66), we have

In view of Lemma 2, we obtain $\lim_{t \rightarrow +\infty}x_{1}(t) = 0$ a.s. According to Lemma 5 $(ii)$, we get

The proof is complete. □

4.   Numerical examples

In this section we introduce some numerical examples to illustrate our main results. For simplicity, we suppose that $\mathbb{S} = \left\{1, 2\right\}$. Then system (1.6) is a hybrid system of the following two subsystems:

and

with initial conditions $x_{1}(\theta) = 2 \mathrm{e}^{\theta}$, $x_{2}(\theta) = 1.5 \mathrm{e}^{\theta}$, $x_{3}(\theta) = 0.8 \mathrm{e}^{\theta}$ $x_{4}(\theta) = 0.5 \mathrm{e}^{\theta}$ and $\theta \in [-\ln 2, 0]$.

Let $r_{ii} = 0.3$, $\tau_{ji} = \ln2$, $\mu_{ji}(\theta) = \mu_{ji} \mathrm{e}^{\theta}$, $\gamma_{j}(\mu, i) = \gamma_{j}(i)$ and $\lambda(\mathbb{Z}) = 1$, see Table 4. Denote

Then system (1.6) may be regarded as the result of regime switching between subsystems (4.1) and (4.2) with the following estimated parameters, respectively,

Compute $\left|{\bf \Delta}\right| = 0.066525$, $\left|{\bf \Xi}\right| = 0.1005$ and $\left|{\bf A}\right| = 0.195$. Denote

4.1.   The effects of Markovian switching on the persistence in mean and extinction

Let $\overrightarrow{{\bf r}}(1) = \left(0.9, 0.5, 0.3, 0.2\right)$. Compute

Based on Theorem 2, all species in subsystem (4.1) are persistent in mean and

Let $\overrightarrow{{\bf r}}(2) = \left(0.6, 0.3, 0.2, 0.1\right)$. Then $B_{1} = -0.1527 < 0$. From Theorem 2, all species in subsystem (4.2) are extinctive.

${\bf Case \, \, 1:}$ $\left(\pi_{1}, \pi_{2}\right) = \left(0.8, 0.2\right)$. Compute

By Theorem 2, all species in system (1.6) are persistent in mean and

${\bf Case \, \, 2:}$ $\left(\pi_{1}, \pi_{2}\right) = \left(0.6, 0.4\right)$. Compute

From Theorem 2, $x_{1}(t)$, $x_{2}(t)$ and $x_{3}(t)$ are persistent in mean, while $x_{4}(t)$ is extinctive and

${\bf Case \, \, 3:}$ $\left(\pi_{1}, \pi_{2}\right) = \left(0.5, 0.5\right)$. Compute

Thanks to Theorem 2, $x_{1}(t)$ and $x_{2}(t)$ are persistent in mean, while $x_{3}(t)$ and $x_{4}(t)$ are extinctive and

${\bf Case \, \, 4:}$ $\left(\pi_{1}, \pi_{2}\right) = \left(0.3, 0.7\right)$. Compute

Based on Theorem 2, $x_{1}(t)$ is persistent in mean, while $x_{2}(t)$, $x_{3}(t)$ and $x_{4}(t)$ are extinctive and

${\bf Case \, \, 5:}$ $\left(\pi_{1}, \pi_{2}\right) = \left(0.1, 0.9\right)$. Compute $B_{1} = -0.0499 < 0$. On the basis of Theorem 2, all species in system (1.6) are extinctive.

4.2.   The effects of Lévy jumps on the persistence in mean and extinction

Let $\overrightarrow{{\bf r}}(1) = \left(0.7, 0.5, 0.3, 0.2\right)$. We study the effects of Lévy jumps on the persistence in mean and extinction of system (4.1) by changing the values of $\gamma_{j}(1)$ and setting the remaining parameters of the examples to be the same as those in system (4.1). Denote $I_{4} = \left\{-0.3, 0.4\right\}$, $\alpha_{4} \in I_{4}$; $I_{3} = \left\{-0.6, 1.1\right\}$, $\alpha_{3} \in I_{3}$; $I_{2} = \left\{-0.9, 1.9\right\}$, $\alpha_{2} \in I_{2}$; $I_{1} = \left\{-0.8, 1.7\right\}$, $\alpha_{1} \in I_{1}$.

4.2.1.   The effects of $\gamma_{j}(1)$ on the persistence in mean and extinction of system (4.1)

${\bf Case \, \, 1:}$ Let $\overrightarrow{{\bf \gamma}}(1) = \left(0.1, 0.1, 0.1, {\bf \alpha_{4}}\right)$. Then $\left|{\bf \Delta_{4}}\right| < 0$, $\left|{\bf \Xi_{3}}\right| = 0.0606 > 0$. According to Theorem 2, $x_{1}(t)$, $x_{2}(t)$ and $x_{3}(t)$ are persistent in mean, while $x_{4}(t)$ is extinctive.

Let $\overrightarrow{{\bf \gamma}}(1) = \left(0.1, 0.1, 0.1, {\bf 0.1}\right)$. Then $\left|{\bf \Delta_{4}}\right| = 0.0047 > 0$. By Theorem 2, all species in system (4.1) are persistent in mean. See Table 5.

${\bf Case \, \, 2:}$ Let $\overrightarrow{{\bf \gamma}}(1) = \left(0.1, 0.1, {\bf \alpha_{3}}, \alpha_{4}\right)$. Then $\left|{\bf \Xi_{3}}\right| < 0$, $\left|{\bf A_{2}}\right| = 0.2478 > 0$. Based on Theorem 2, $x_{1}(t)$ and $x_{2}(t)$ are persistent in mean, while $x_{3}(t)$ and $x_{4}(t)$ are extinctive. See Table 6.

${\bf Case \, \, 3:}$ Let $\overrightarrow{{\bf \gamma}}(1) = \left(0.1, {\bf \alpha_{2}}, \alpha_{3}, \alpha_{4}\right)$. Then $\left|{\bf A_{2}}\right| < 0$, $B_{1} = 0.6753 > 0$. From Theorem 2, $x_{1}(t)$ is persistent in mean, while $x_{2}(t)$, $x_{3}(t)$ and $x_{4}(t)$ are extinctive. See Table 7.

${\bf Case \, \, 4:}$ Let $\overrightarrow{{\bf \gamma}}(1) = \left({\bf \alpha_{1}}, \alpha_{2}, \alpha_{3}, \alpha_{4}\right)$. Then $B_{1} < 0$. Thanks to Theorem 2, all species are extinctive. See Table 8.

4.2.2.   The effects of $\gamma_{1}(1)$ on the persistence in mean and extinction of system (4.1)

${\bf Case \, \, 1:}$ Let $\gamma_{1}(1) = -0.8$. Then $B_{1} = -0.1294 < 0$. According to Theorem 2, all species in system (4.1) are extinctive.

Let $\gamma_{1}(1) = -0.7$. Then $\left|{\bf A_{2}}\right| = -0.0518 < 0$, $B_{1} = 0.1760 > 0$. By Theorem 2, $x_{1}(t)$ is persistent in mean, while $x_{2}(t)$, $x_{3}(t)$ and $x_{4}(t)$ are extinctive and

Let $\gamma_{1}(1) = -0.6$. Then $\left|{\bf \Xi_{3}}\right| = -0.0329 < 0$, $\left|{\bf A_{2}}\right| = 0.0608 > 0$. Based on Theorem 2, $x_{1}(t)$ and $x_{2}(t)$ are persistent in mean, while $x_{3}(t)$ and $x_{4}(t)$ are extinctive and

Let $\gamma_{1}(1) = -0.3$. Then $\left|{\bf \Delta_{4}}\right| = -0.0023 < 0$, $\left|{\bf \Xi_{3}}\right| = 0.0450 > 0$. 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

Let $\gamma_{1}(1) = -0.1$. Then $\left|{\bf \Delta_{4}}\right| = 0.0046 > 0$. From Theorem 2, all species are persistent in mean and

${\bf Case \, \, 2:}$ Let $\gamma_{1}(1) = 0.2$. Then $\left|{\bf \Delta_{4}}\right| = 0.0029 > 0$. Thanks to Theorem 2, all species are persistent in mean and

Let $\gamma_{1}(1) = 0.6$. Then $\left|{\bf \Delta_{4}}\right| = -0.0122 < 0$, $\left|{\bf \Xi_{3}}\right| = 0.0230 > 0$. On the basis of Theorem 2, $x_{1}(t)$, $x_{2}(t)$ and $x_{3}(t)$ are persistent in mean, while $x_{4}(t)$ is extinctive and

Let $\gamma_{1}(1) = 0.9$. Then $\left|{\bf \Xi_{3}}\right| = -0.0155 < 0$, $\left|{\bf A_{2}}\right| = 0.0957 > 0$. By Theorem 2, $x_{1}(t)$ and $x_{2}(t)$ are persistent in mean, while $x_{3}(t)$ and $x_{4}(t)$ are extinctive and

Let $\gamma_{1}(1) = 1.3$. Then $\left|{\bf A_{2}}\right| = -0.0297 < 0$, $B_{1} = 0.2129 > 0$. From Theorem 2, $x_{1}(t)$ is persistent in mean, while $x_{2}(t)$, $x_{3}(t)$ and $x_{4}(t)$ are extinctive and

Let $\gamma_{1}(1) = 1.7$. Then $B_{1} = -0.0267 < 0$. In view of Theorem 2, all species are extinctive.

5.   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.

Acknowledgments

This work is supported by National Natural Science Foundation of China (No. 11901166).

Conflict of interest

The authors declare that there are no conflicts of interest.