Spatial decay estimates for a coupled system of wave-plate type

1.   Introduction

Optimal harvesting problem is an important and interesting topic from both biological and mathematical point of view. The classical two-species deterministic Lotka-Volterra system with harvesting under catch-per-unit-effort hypothesis ^[1] can be expressed as follows:

where $x_{i}(t)$ represent the population densities of species $i$ at time $t$. $\overline{r_{i}}$ and $\overline{a_{ij}}$ are constants. $h_{i} \geq 0$ represents the harvesting effort of $x_{i}(t)$ $(i, j = 1, 2)$.

However, 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 ^[2,3]. Hence, it is of enormous importance to study the effects of environmental noises on the dynamics of population systems. Introducing white Gaussian noises into the deterministic system is the most common way to characterize environmental noises ^[4,5]. Assume that $\overline{r_{i}}$ are affected by white Gaussian noises, i.e., $\overline{r_{i}} \hookrightarrow \overline{r_{i}}+\sigma_{i}\dot{W}_{i}(t)$, where $W_{i}(t)$ are standard Wiener processes defined on a complete probability space $(\Omega, \mathcal{F}, P)$ with a filtration $\{\mathcal{F}_{t}\}_{t\geq 0}$ satisfying the usual conditions. Then, system (1.1) becomes

On the other hand, many academics argue that parameters in ecosystems often switch because of environmental changes, for example, some species have different growth rates at different temperatures, and these changes can be well described by a continuous-time Markov chain $\rho(t)$ with finite-state space, instead of white Gaussian noises ^{[6,7,8,9,10,11,12,13]}. System (1.2) under regime switching can be expressed as follows:

where $\rho(t)$ is a right-continuous Markov chain with finite values $\mathbb{S} = \left\{1, 2, ..., S\right\}$, $\overline{r_{i}(\rho(t))}$ and $\overline{a_{ij}(\rho(t))}$ are functions with finite values. Furthermore, population systems may suffer sudden environmental perturbations, such as earthquake, torrential flood, typhoon and infectious disease. Some scholars claimed that Lévy noise can be used to describe these sudden environmental perturbations ^{[14,15,16,17,18,19]}. Introducing Lévy noise into system (1.2) yields

where $N$ is a Poisson counting measure with characteristic measure $\lambda$ on a measurable subset $\mathbb{Z} \subseteq [0, +\infty)$ with $\lambda(\mathbb{Z}) < +\infty$ and $\widetilde{N}(\mathrm{d}t, \mathrm{d}\mu) = N(\mathrm{d}t, \mathrm{d}\mu)-\lambda(\mathrm{d}\mu)\mathrm{d}t$, $\gamma_{j}(\mu, \rho(t))$ are bounded functions. For the sake of simplicity, we define

Hence, system (1.4) can be rewritten into

Given the growing importance of environmental noises in the dynamics of complex physical and biological systems, interdisciplinary stochastic systems driven by two different types of environment noises have attracted great attention in the last few decades ^{[20,21,22,23,24,25,26,27,28,29,30,31,32]}. Particularly, Giorgio Parisi's Nobel Prize in Physics (2021) expounded on the importance of fluctuations on physics and systems from microscopic to macroscopic physics.

In the natural environment, in addition to being subject to environmental noises, the trends of biological systems depend not only on the present state but also on the past state, such as the growth period from juvenile to adult in the growth model of biological populations. Such phenomena are called time-delay phenomena. "All species should exhibit time delay" in the real world ^[33] and incorporating time delay into biological systems makes them much more realistic than those without delay, since a species growth rate relies on not only the current state, but also the past state ^[34,35,36]. As we all know, systems with discrete time delays and those with continuously distributed time delays do not contain each other. However, systems with S-type distributed time delays contain both ^[37,38].

Furthermore, with a growing number of toxicant entering into the ecosystem, many species have been extinctive and some of them are on the verge of extinction, environmental pollution has received much attention in international society. Naturally, it is meaningful to estimate environmental toxicity so as to develop optimal harvesting policies.

In the past few decades, stochastic population systems driven by different types of environment noises have received great attention and have been studied extensively. For example, Abbas et al. ^[39] studied the effect of stochastic perturbation on a two-species competitive system by constructing a suitable Lyapunov functional. Han et al. ^[40] investigated two-species Lotka-Volterra delayed stochastic predator-prey systems, with and without pollution. Liu and Chen ^[41] investigated a stochastic delay predator-prey system with Lévy noise in a polluted environment. Zhao and Yuan ^[42] considered the optimal harvesting policy of a stochastic two-species competitive model with Lévy noise in a polluted environment. Liu et al. ^[43] studied the dynamics of a stochastic regime-switching predator-prey system with harvesting and distributed delays.

However, to the best of our knowledge to date, results about stochastic time-delay population system driven by three different types of environment noises have rarely been report. Hence, in this paper we consider the optimization problems of harvesting for the following two stochastic hybrid delay Lotka-Volterra systems with Lévy noise in a polluted environment:

and

where

$\int_{-\tau_{ji}}^{0}x_{i}(t+\theta)\mathrm{d}\mu_{ji}(\theta)$ are Lebesgue-Stieltjes integrals, $\tau_{ji} > 0$ are time delays, $\mu_{ji}(\theta)$, $\theta \in [-\tau, 0]$ are nondecreasing bounded variation functions, $\tau = \max\left\{\tau_{ji}\right\}$. For other parameters in systems (1.7) and (1.8), see Table 1.

As fundamental assumptions, we assume that $W_{1}(t)$, $W_{2}(t)$, $\rho(t)$ and $N$ are independent and $\rho(t)$ is irreducible. Hence, $\rho(t)$ has a unique stationary distribution $\pi = \left(\pi_{1}, \pi_{2}, ..., \pi_{S}\right)$. Our aim is, for each system of (1.7) and (1.8), to get the optimal harvesting effort $H^{*} = \left(h_{1}^{*}, h_{2}^{*}\right)^{\mathrm{T}}$ such that

① Both $x_{1}(t)$ and $x_{2}(t)$ are not extinct;

② The expectation of sustained yield $Y(H) = \lim_{t \rightarrow +\infty}\mathbb{E}\left[\sum_{i = 1}^{2}h_{i}x_{i}(t)\right]$ is maximal.

The rest of this paper is arranged as follows. In Section 2, we study the existence and uniqueness of global positive solution to systems (1.7) and (1.8). For every system, sufficient and necessary conditions for persistence in mean and extinction of each species are obtained in Section 3. We discuss the conditions for global attractivity of the systems in Section 4. In Section 5, sufficient and necessary conditions for the existence of optimal harvesting strategy are established. Furthermore, we give the accurate expressions for the OHE and MESY. Finally, some brief conclusions and discussions are shown in Section 6.

2.   Existence and uniqueness of global positive solution

In this paper, we have three fundamental assumptions for systems (1.7) and (1.8).

Assumption 1. $r_{j}(i) > 0$, $a_{jk}(i) > 0$ and there exist $\gamma_{j}^{*}(i) \geq \gamma_{j*}(i) > -1$ such that $\gamma_{j*}(i) \leq \gamma_{j}(\mu, i) \leq \gamma_{j}^{*}(i)$ $(\mu \in \mathbb{Z})$, $\forall i \in \mathbb{S}$, $j, k = 1, 2$. Hence, for any constant $p > 0$, there exists $\mathcal{C}_{j}\left(p\right) > 0$ such that

Remark 1. Assumption 1 implies that the intensity of Lévy noise is not too big to ensure that the solution will not explode in finite time (see, e.g., ^{[42,44,45,46,47]}).

Assumption 2. $0 < k_{i} \leq g_{i}+m_{i}$ $(i = 1, 2)$, $\sup_{t \in \mathbb{R}_{+}}u(t) \leq h$.

Remark 2. Assumption 2 means $0 \leq C_{i}(t) < 1$ $(i = 1, 2)$ and $0 \leq C_{E}(t) < 1$, which must be satisfied to be realistic because $C_{1}(t)$, $C_{2}(t)$ and $C_{E}(t)$ are concentrations of the toxicant (see Lemma 2.1 in ^[48]).

Assumption 3. The limit of $u(t)$ when $t \rightarrow +\infty$ exists, i.e., $\lim_{t \rightarrow +\infty}u(t) = u^{E}$.

Lemma 1. (Lemma 4.2 in ^[49]) If Assumption 3 holds, then

To study the long-term dynamics of a stochastic population system, we first study the existence and uniqueness of global positive solution to the system.

Theorem 1. For any initial condition $(\xi_{1}, \xi_{2})^{\mathrm{T}} \in \mathcal{C}([-\tau, 0], \mathbb{R}_{+}^{2})$, system (1.7) (or system (1.8)) has a unique global solution $(x_{1}(t), x_{2}(t))^{\mathrm{T}} \in \mathbb{R}_{+}^{2}$ on $t \in [-\tau, +\infty)$ a.s. Moreover, for any constant $p > 0$, there exists $K_{i}(p) > 0$ such that

Proof. The proof is standard and hence is omitted (see e.g., ^[50]).

3.   Persistence in mean and extinction

Before studying the persistence in mean and extinction of systems (1.7) and (1.8), we first present the following lemma.

Lemma 2. Denote $o(t) = \left\{f(t) \, |\, \lim_{t\rightarrow +\infty}\frac{f(t)}{t} = 0\right\}$. Suppose $Z(t)\in \mathcal{C}(\Omega\times [0, +\infty), \mathbb{R}_{+})$(^[51]).

$(\mathfrak{i})$ 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

3.1.   Predator-prey system

Denote

To begin with, let us consider the following stochastic auxiliary system:

Lemma 3. For system (3.6):

$(\mathfrak{a})$ If $\Sigma_{1}-r_{11}C_{1}^{E}-h_{1} < 0$, then $\lim_{t \rightarrow +\infty}X_{i}(t) = 0$ a.s. $(i = 1, 2)$.

$(\mathfrak{b})$ If $\Sigma_{1}-r_{11}C_{1}^{E}-h_{1} \geq 0$, $\Sigma_{2}-\frac{A_{21}}{A_{11}}r_{11}C_{1}^{E}-r_{22}C_{2}^{E}-\frac{A_{21}}{A_{11}}h_{1}-h_{2} < 0$, then

$(\mathfrak{c})$ If $\Sigma_{1}-r_{11}C_{1}^{E}-h_{1} \geq 0$, $\Sigma_{2}-\frac{A_{21}}{A_{11}}r_{11}C_{1}^{E}-r_{22}C_{2}^{E}-\frac{A_{21}}{A_{11}}h_{1}-h_{2} \geq 0$, then

Proof. By Itô's formula and the strong law of large numbers, we compute

where

$\mathfrak{Case} \, \, (\mathfrak{i}):$ $\Sigma_{1}-r_{11}C_{1}^{E}-h_{1} < 0$. Then $\lim_{t \rightarrow +\infty}X_{1}(t) = 0$ a.s. Hence, for $\forall \epsilon \in (0, 1)$ and $t\gg 1$,

which implies $\lim_{t \rightarrow +\infty}X_{2}(t) = 0$ a.s.

$\mathfrak{Case \, \, (ii):}$ $\Sigma_{1}-r_{11}C_{1}^{E}-h_{1} \geq 0$. Consider the following auxiliary function:

Then $X_{2}(t) \leq \widetilde{X_{2}(t)}$ a.s. By Itô's formula, for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Thanks to Lemma 2 and the arbitrariness of $\epsilon$, for arbitrary $\gamma > 0$,

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

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

Lemma 4. For system (1.7), $\limsup_{t \rightarrow +\infty} t^{-1} \ln x_{i}(t) \leq 0$ a.s. $(i = 1, 2)$.

Proof. Thanks to Lemma 3 and (3.9), system (3.6) satisfies $\lim_{t \rightarrow +\infty}t^{-1}\ln X_{i}(t) = 0$ a.s. $(i = 1, 2)$. From the stochastic comparison theorem, we obtain the desired assertion.

Lemma 5. For system (1.7), if $\lim_{t \rightarrow +\infty}x_{1}(t) = 0$ a.s., then $\lim_{t \rightarrow +\infty}x_{2}(t) = 0$ a.s.

Proof. The proof of Lemma 5 is similar to that of Lemma 3 $(\mathfrak{a})$ and here is omitted.

Theorem 2. For system (1.7), define $\Theta_{1} = \Sigma_{1}-r_{11}C_{1}^{E}-h_{1}$, $\Theta_{2} = \frac{|A_{2}|}{A_{21}}$.

$(\mathfrak{i})$ If $\Theta_{2} > 0$, then

$(\mathfrak{ii})$ If $\Theta_{1} > 0 > \Theta_{2}$, then

$(\mathfrak{iii})$ If $0 > \Theta_{1}$, then $\lim_{t \rightarrow +\infty}x_{i}(t) = 0$ a.s. $(i = 1, 2)$.

Proof. Clearly, $\Theta_{1} > \Theta_{2}$. Thanks to (3.14), for $\forall \gamma > 0$,

By Itô's formula and (3.18), we deduce

$\mathfrak{Case} \, \, (\mathfrak{i}):$ $\Theta_{2} > 0$. According to system (3.19), we compute

Based on Lemma 4, for $\forall \epsilon \in (0, 1)$ and $t\gg 1$,

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

By (3.22), $x_{2}(t)$ is not extinct. Based on Lemma 5, $x_{1}(t)$ is not extinct either. In view of Lemma 2 and the arbitrariness of $\epsilon$, we obtain

According to (3.23) and system (3.19), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Thanks to Lemma 2 and the arbitrariness of $\epsilon$, we obtain

Combining (3.22) with (3.25) yields

Combining (3.26) with system (3.19) yields that for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

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

$\mathfrak{Case} \, \, (\mathfrak{ii}):$ $\Theta_{1} > 0 > \Theta_{2}$. In view of (3.20), we deduce

By Lemma 5, $\lim_{t \rightarrow +\infty} x_{2}(t) = 0$ a.s. Thus, for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Thanks to Lemma 2 and the arbitrariness of $\epsilon$, we obtain

$\mathfrak{Case} \, \, (\mathfrak{iii}):$ $0 > \Theta_{1}$. By system (3.19), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

So, $\lim_{t \rightarrow +\infty}x_{1}(t) = 0$ a.s. From Lemma 5, $\lim_{t \rightarrow +\infty}x_{2}(t) = 0$ a.s.

Remark 3. If $\mathbb{S} = \left\{1 \right\}$, $h_{i} = 0$, $r_{ii} = 0$, $\mu_{ii}(\theta)$ are constant functions defined on $[-\tau, 0]$, $a_{ij} = 0$ ($i \neq j$) and $\mu_{ij}(\theta)$ are defined as follows:

then system (1.7) becomes

Hence, Theorem 2 contains Theorem 1 in ^[52] and Theorem 2 in ^[53] as a special case.

Remark 4. If $\mathbb{S} = \left\{1 \right\}$, $h_{i} = 0$, $\gamma_{i}(\mu, 1) = 0$, $\mu_{ii}(\theta)$ are constant functions defined on $[-\tau, 0]$, $a_{ij} = 0$ ($i \neq j$) and $\mu_{ij}(\theta)$ are defined as follows:

then system (1.7) becomes

Hence, Theorem 2 contains Theorems 4.1, 4.2, 5.1 and 5.2 in ^[40] as a special case.

3.2.   Competitive system

Denote

To begin with, let us consider the following stochastic auxiliary system:

Lemma 6. For system (3.36):

Proof. Thanks to Itô's formula and the strong law of large numbers, we obtain

$\mathfrak{Case} \, \, (\mathfrak{i}):$ $\Xi_{1} < 0$. Then $\lim_{t \rightarrow +\infty} X_{1}(t) = 0$ a.s. Consider the following auxiliary system:

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

Therefore, for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

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

So, for $\forall \gamma > 0$, we have

Combining (3.37) with (3.41) yields that for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Thanks to Lemma 2 and the arbitrariness of $\epsilon$, we obtain:

$\mathfrak{Case} \, \, (\mathfrak{ii}):$ $\Xi_{1} \geq 0$. Then,

Combining (3.37) with (3.43) yields that for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

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

Hence, for $\forall \gamma > 0$, (3.41) is true. According to system (3.37) and (3.41), for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

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

Lemma 7. For system (1.8), $\limsup_{t \rightarrow +\infty}t^{-1}\ln x_{i}(t) \leq 0$ a.s. $(i = 1, 2)$.

Proof. Thanks to Lemma 6 and (3.37), system (3.36) satisfies $\lim_{t \rightarrow +\infty}t^{-1}\ln X_{i}(t) = 0$ a.s. $(i = 1, 2)$. From the stochastic comparison theorem, we obtain the desired assertion.

Theorem 3. For system (1.8):

Proof. By Itô's formula, we compute

According to system (3.46) and Lemma 2, we obtain:

By system (3.46), we compute

By Lemma 7 and (3.47), we obtain that for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

Making use of Lemma 2 yields

On the one hand, combining system (3.46) with (3.49) yields

On the other hand, in view of systems (3.46) and (3.49), we deduce that if $\Delta > 0$, $\Delta_{1} \geq 0$ and $\Delta_{2} \geq 0$, then for $\forall \epsilon \in (0, 1)$ and $t \gg 1$,

According to Lemma 2 and the arbitrariness of $\epsilon$, we obtain

Thus, Theorem 3 (4)–(5) follows from combining (3.49) with (3.51).

Remark 5. If $\mathbb{S} = \left\{1 \right\}$, $h_{i} = 0$, $r_{ii} = 0$ and $\mu_{ij}(\theta)$ are constant functions defined on $[-\tau, 0]$, then system (1.8) becomes

Hence, Theorem 3 contains Theorem 4 in ^[17] as a special case.

Remark 6. If $\mathbb{S} = \left\{1 \right\}$, $h_{i} = 0$, $r_{ii} = 0$, $\mu_{ii}(\theta)$ are constant functions defined on $[-\tau, 0]$, $a_{ij} = 0$ ($i \neq j$) and $\mu_{ij}(\theta)$ are defined as follows:

then system (1.8) becomes

Hence, Theorem 3 contains Theorem 1 in ^[53] as a special case.

4.   Global attractivity

Assumption 4. $2 a_{jj} > \sum_{i = 1}^{2}A_{ij}$ $(j = 1, 2)$.

Theorem 4. Under Assumption 4, system (1.7) (or system (1.8)) is globally attractive.

Proof. Let $(x_{1}(t; \phi), x_{2}(t; \phi))^{\mathrm{T}}$ and $(x_{1}(t; \phi^{*}), x_{2}(t; \phi^{*}))^{\mathrm{T}}$ be, respectively, the solution to system (1.7) (or system (1.8)) with $\phi$ and $\phi^{*} \in \mathcal{C}([-\tau, 0], \mathbb{R}_{+}^{2})$, we only need to show

Define

By Itô's formula, we derive

Based on (4.3), we obtain

By (4.4), we deduce

Define $H_{i}(t) = \mathbb{E}\left[\left|x_{i}(t; \phi^{*})-x_{i}(t; \phi)\right|\right]$ $(i = 1, 2)$. Then for any $t_{1}, t_{2} \in [0, +\infty)$,

Denote $\max_{i \in \mathbb{S}}r_{j}(i) = r_{j}^{*}$, $\max_{i\in \mathbb{S}}|\sigma_{j}(i)| = \sigma_{j}^{*}$, $\sup_{s \geq-\tau}C_{j}(s) = C_{j}^{*}$, $\max_{i\in \mathbb{S}}\sup_{\mu \in \mathbb{Z}}|\gamma_{j}(\mu, i)| = \gamma_{j}^{*}$. Based on Hölder's inequality, for $t_{2} > t_{1}$ and $p > 1$, we deduce

In view of Theorem 7.1 in ^[54], for $p \geq 2$, we obtain

From Hölder's inequality, we derive

According to Hölder's inequality, we get

According to the Kunita's first inequality in ^[55], for $p > 2$, we get

By Theorem 1 and (4.7)–(4.11), we deduce that for $p > 2$ and $|t_{2}-t_{1}| \leq \frac{1}{2}$,

where

Combining (4.7) with (4.12) yields

Then, for any $\epsilon > 0$, there exists $\delta(\epsilon) = \min\left\{\frac{\epsilon^{p}}{2^{p}M}, \frac{1}{2}\right\}$ such that for any $t_{2} > t_{1}$ satisfying $|t_{2}-t_{1}| < \delta(\epsilon)$, we have $\left|H_{j}(t_{2})-H_{j}(t_{1})\right| < \epsilon$. Therefore, (4.1) follows from (4.5), (4.14) and Barbalat's conclusion in ^[56].

5.   Optimal harvesting strategy

Now, let consider the optimal harvesting problem of systems (1.7) and (1.8).

5.1.   Predator-prey system

Theorem 5. For system (1.7), define

$(\mathfrak{i})$ $\mathit{\text{If}}\;$

then the optimal harvesting strategy exist. Moreover, $H^{*} = \left(h_{1}^{*}, h_{2}^{*}\right)^{\mathrm{T}}$ and

$(\mathfrak{ii})$ $\mathit{\text{If}}\;$ one of the following conditions holds, then the optimal harvesting strategy does not exist:

$(\mathfrak{a})$ $\Theta_{1}|_{h_{1} = h_{1}^{*}} < 0$;

$(\mathfrak{b})$ $\Theta_{1}|_{h_{1} = h_{1}^{*}} > 0 > \Theta_{2}|_{h_{1} = h_{1}^{*}, \, \, h_{2} = h_{2}^{*}}$;

$(\mathfrak{c})$ $h_{1}^{*} < 0$ or $h_{2}^{*} < 0$;

$(\mathfrak{d})$ $4A_{11}A_{22}-(A_{12}-A_{21})^{2} < 0$.

Proof. Thanks to (2.3), there exists $\mathrm{C} = \sum_{i = 1}^{2}K_{i}(p) > 0$ such that

By Theorem 3.1.1 in ^[57], $(x_{1}(t), x_{2}(t), \rho(t))^{\mathrm{T}}$ has an invariant measure $\nu(\cdot \times \cdot) \in \mathbb{R}_{+}^{2} \times \mathbb{S}$. From Theorem 3.1 in ^[58], $\nu(\cdot \times \cdot)$ is unique. Thanks to Theorem 3.2.6 in ^[59], $\nu(\cdot \times \cdot)$ is ergodic. Hence, we have

Let

On the one hand, from Theorem 2 $(\mathfrak{i})$, for every $H \in \mathcal{U}$, we obtain

On the other hand, if the OHE $H^{*}$ exists, then $H^{*} \in \mathcal{U}$.

Proof of $(\mathfrak{i})$. Based on the first condition of (5.2), we obtain that $\mathcal{U}$ is not empty. According to (5.7), for $H = (h_{1}, h_{2})^{^{\mathrm{T}}} \in \mathcal{U}$, we have

Let $\varrho(\cdot \times \cdot)$ be the stationary probability density of system (1.7), then we get

Noting that system (1.7) has a unique ergodic invariant measure $\nu(\cdot \times \cdot)$ and that there exists a one-to-one correspondence between $\varrho(\cdot \times \cdot)$ and $\nu(\cdot \times \cdot)$, we deduce

In view of Eqs (5.5), (5.8), (5.9) and (5.10), we deduce

Solving $\frac{\partial Y^{*}(H)}{\partial h_{1}} = \frac{\partial Y^{*}(H)}{\partial h_{2}} = 0$ yields

Define the Hessian matrix $\Lambda$ as follows:

Thanks to $-2A_{22} < 0$ and $4A_{11}A_{22}-(A_{12}-A_{21})^{2} > 0$, $\Lambda$ is negative definite. Thus, $Y^{*}(H)$ has a unique maximum, and the unique maximum value point of $Y^{*}(H)$ is $H^{*} = (h_{1}^{*}, h_{2}^{*})^{^{\mathrm{T}}}$. Hence, (5.3) follows from (5.11).

Proof of $(\mathfrak{ii})$. First, from Theorem 2 $(\mathfrak{iii})$, under condition $(\mathfrak{a})$, the optimal harvesting strategy does not exist. Next, let us show that the optimal harvesting strategy does not exist, provided that either $(\mathfrak{b})$ or $(\mathfrak{c})$ holds. The proof is by contradiction. Suppose that the OHE is $\widetilde{H^{*}} = \left(\widetilde{h_{1}^{*}}, \widetilde{h_{2}^{*}}\right)^{^{\mathrm{T}}}$. Then $\widetilde{H^{*}} \in \mathcal{U}$. In other words, we have

On the other hand, since $\widetilde{H^{*}} = \left(\widetilde{h_{1}^{*}}, \widetilde{h_{2}^{*}}\right)^{^{\mathrm{T}}} \in \mathcal{U}$ is the OHE, then $\left(\widetilde{h_{1}^{*}}, \widetilde{h_{2}^{*}}\right)^{^{\mathrm{T}}}$ must be the unique solution to system $\frac{\partial Y^{*}(H)}{\partial h_{1}} = \frac{\partial Y^{*}(H)}{\partial h_{2}} = 0$. Hence, $\left(h_{1}^{*}, h_{2}^{*}\right)^{^{\mathrm{T}}} = \left(\widetilde{h_{1}^{*}}, \widetilde{h_{2}^{*}}\right)^{^{\mathrm{T}}}$. Therefore, the Eq (5.14) becomes

which contradicts with both $(\mathfrak{b})$ and $(\mathfrak{c})$.

Now we are in the position to prove that if the following condition holds, then the optimal harvesting strategy does not exist (i.e., prove $(\mathfrak{d})$):

From the first condition of (5.16), we obtain that $\mathcal{U}$ is not empty. Hence (5.11) is true. $-2A_{22} < 0$ implies that $\Lambda$ is not positive semidefinite. The second condition of (5.16) indicates that $\Lambda$ is not negative semidefinite. Namely, $\Lambda$ is indefinite. Thus, $Y^{*}(H)$ does not exist extreme point. So the OHE does not exist.

The proof is complete.

Remark 7. If $\mathbb{S} = \left\{1 \right\}$, $r_{ii} = 0$, $\gamma_{i}(\mu, 1) = 0$, $\mu_{ii}(\theta)$ are constant functions defined on $[-\tau, 0]$, $a_{ij} = 0$ ($i \neq j$) and $\mu_{ij}(\theta)$ are defined as follows:

then system (1.7) becomes

Hence, Theorem 5 contains Theorem 1 in ^[60] as a special case.

Remark 8. If $r_{ii} = 0$, $\gamma_{i}\left(\mu, \rho(t)\right) = 0$, $\mu_{ii}(\theta)$ are constant functions defined on $[-\tau, 0]$ and $a_{ij} = 0$ ($i \neq j$), then system (1.7) becomes

Therefore, Theorem 2 and Theorem 5 contains, respectively, Theorem 1 and Theorem 2 in ^[12] as a special case.

5.2.   Competitive system

Theorem 6. For system (1.8), define

($\mathcal{A}_{1}$) If

then the optimal harvesting strategy exists. Moreover, $H^{*} = \left(h_{1}^{*}, h_{2}^{*}\right)^{\mathrm{T}}$ and

($\mathcal{A}_{2}$) If one of the following conditions holds, then the optimal harvesting strategy does not exist:

$({\bf B}_{1})$ $\Xi_{1}|_{h_{1} = h_{1}^{*}} < 0$, $\Xi_{2}|_{h_{2} = h_{2}^{*}} < 0$;

$({\bf B}_{2})$ $\Xi_{1}|_{h_{1} = h_{1}^{*}} < 0$, $\Xi_{2}|_{h_{2} = h_{2}^{*}} \geq 0$;

$({\bf B}_{3})$ $\Xi_{1}|_{h_{1} = h_{1}^{*}} \geq 0$, $\Xi_{2}|_{h_{2} = h_{2}^{*}} < 0$;

$({\bf B}_{4})$ $\Delta \geq 0$, $\Delta_{1}|_{h_{1} = h_{1}^{*}, \, h_{2} = h_{2}^{*}} < 0$, $\Xi_{2}|_{h_{2} = h_{2}^{*}} < 0$;

$({\bf B}_{5})$ $\Delta \geq 0$, $\Delta_{2}|_{h_{1} = h_{1}^{*}, \, h_{2} = h_{2}^{*}} < 0$, $\Xi_{1}|_{h_{1} = h_{1}^{*}} < 0$;

$({\bf B}_{6})$ $\Delta \geq 0$, $\Delta_{1}|_{h_{1} = h_{1}^{*}, \, h_{2} = h_{2}^{*}} < 0$, $\Xi_{2}|_{h_{2} = h_{2}^{*}} \geq 0$;

$({\bf B}_{7})$ $\Delta \geq 0$, $\Delta_{2}|_{h_{1} = h_{1}^{*}, \, h_{2} = h_{2}^{*}} < 0$, $\Xi_{1}|_{h_{1} = h_{1}^{*}} \geq 0$;

$({\bf B}_{8})$ $h_{1}^{*} < 0$ or $h_{2}^{*} < 0$;

$({\bf B}_{9})$ $4A_{11}A_{22}-\left(A_{12}+A_{21}\right)^{2} < 0$.

Proof. The proof of Theorem 6 is similar to that of Theorem 5 and hence is omitted.

Remark 9. If $\mathbb{S} = \left\{1 \right\}$, $r_{ii} = 0$, $\mu_{ii}(\theta)$ are constant functions defined on $[-\tau, 0]$, $a_{ij} = 0$ ($i \neq j$) and $\mu_{ij}(\theta)$ are defined as follows:

then system (1.8) becomes

Hence, Theorem 3 and Theorem 6 contain, respectively, Lemma 2.3 and Theorem 4.1 in ^[47] as a special case.

6.   Conclusions and discussion

In this paper, we study the stochastic dynamics of two hybrid delay Lotka-Volterra systems with harvesting and jumps in a polluted environment. The main results include five theorems. Theorem 2 and Theorem 3 establish sufficient and necessary conditions for persistence in mean and extinction of each species. In Theorem 4, sufficient conditions for global attractivity of the systems are obtained. Theorems 5 and 6 provide sufficient and necessary conditions for the existence of optimal harvesting strategy. Furthermore, we obtain the accurate expressions for the OHE and MESY in Theorems 5 and 6. Our results show that the dynamic behaviors and optimal harvesting strategy are closely correlated with both time delays and environmental noises.

Some interesting questions deserve further investigation. On the one hand, it would be interesting to consider the stochastic hybrid delay food chain model with harvesting and jumps in a polluted environment. On the other hand, it is interesting to investigate the influences of impulsive perturbations on the systems. One may also propose some more realistic systems, such as considering the generalized functional response. We will leave these investigations for future work.

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.