Dynamical analysis and optimal harvesting of a stochastic three-species cooperative system with delays and Lévy jumps

Assumption 2.1

\(\mathbb{R}_{1}>0\), \(\mathbb{R}_{2}>0\), and \(\mathbb {R}_{3}>0\).

Remark 2.1

Under Assumption 2.1, an easy computation yields \(\frac{\Delta_{ij}}{\tilde{\Delta}_{ij}}>\frac{A_{i}}{\tilde{A}_{i}}\), i, \(j=1\), 2, 3, \(i\neq j\).

Assumption 2.2

\(a_{ii}>\sum_{j=1,j\neq i}^{3}a_{ij}\) and \(a_{ii}>\sum_{j=1,j\neq i}^{3}a_{ji}\), i, \(j=1\), 2, 3.

Assumption 2.3

Definition 2.1

([20, 24])

Definition 2.2

([20])

Lemma 2.1

([20])

Lemma 2.2

If Assumption 2.2 holds, then for any given initial data \(\phi=(\phi_{1},\phi_{2},\phi_{3})\in C([-\tau,0],R_{+}^{3})\), system (1.2) has a unique solution \(x(t)=(x_{1}(t),x_{2}(t),x_{3}(t))^{T}\in R_{+}^{3}\) a.s.

Proof

Remark 2.2

Actually, by Corollary 3.5 of Hu et al. [8], the condition \(2a_{ii}>\sum_{j\neq i,j=1}^{3}a_{ij}\) can ensure that system (1.2) has a unique positive solution for any initial data \(\phi\in C([-\tau ,0],R_{+}^{3})\). But for the proof of the latter global attractivity, we conduct our study always under Assumption 2.2 in this paper.

Lemma 2.3

Proof

The proof is motivated by that of Lemma 4 in [20]. By using Itô’s formula to the function \(e^{t}V(x)\), where V is the function defined in the proof of Lemma 2.2, one can get \(\limsup_{t\to\infty}\mathbb{E}V(x(t))\leq K\) and \(\limsup_{t\to\infty}\mathbb{E}|(x(t))|\leq K\). The rest of the proof is similar to that in [20] and hence is omitted. □

Lemma 2.4

Proof

Theorem 3.1

Proof

(iv) Obviously, when \(r_{i}-b_{i}<0\), by an easy computation, we have \(A_{i}-\tilde{A}_{i}<0\). By Lemma 2.1, it follows from (3.1)–(3.3) that \(\lim_{t\to\infty} x_{1}(t)=\lim_{t\to\infty}x_{2}(t)=\lim_{t\to\infty} x_{3}(t)=0\), which means that all species go extinct. When \(r_{i}-b_{i}>0\), by a similar computation, we have \(A_{i}-\tilde {A}_{i}>0\), \(i=1\), 2, 3. This completes the proof. □

Theorem 3.2

Then model (1.2) is stochastically permanent provided that the following assumption holds.

Assumption 3.1

\(\min_{i=1,2,3}r_{i}-\max_{i=1,2,3}\sigma _{i}^{2}-\int_{Z} \max_{i=1,2,3} (\gamma_{i}-\ln(1+\gamma_{i}(u)) )\lambda(du)>0\).

Proof

Remark 3.1

It is clear from Assumption 3.1 that the Lévy noise is harmful to the permanence of system (1.2). Furthermore, for the case of one species, our obtained result is in accordance with that of Liu and Wang [20].

Theorem 4.1

Model (1.2) is stable in distribution if Assumption 2.2 holds.

Proof

Firstly, let \(x(t)=x(t,\phi)\) and \(\check{x}(t)=x(t,\check{\phi})\) be any two solutions of model (1.2) with initial data \(\phi\in C([-\tau ,0],R_{+}^{3})\) and \(\check{\phi}\in C([-\tau,0],R_{+}^{3})\), respectively. For \(i=1\), 2, 3, we denote \(\bar{x}_{i}(t)=x_{i}(t)-\check{x}_{i}(t)\) and prove \(\lim_{t\to\infty}\mathbb{ E}|\bar{x}_{i}(t)|=0\), that is, system (1.2) is globally attractive.

Secondly, denote by \(p(t,\phi, dy)\) the transition probability of the process \(x(t,\phi)\) and denote by \(P(t,\phi,R_{+}^{3})\) the probability of event \(x(t,\phi)\in R_{+}^{3}\) with the initial data \(\phi\in C([-\tau,0]; R_{+}^{3})\). Let \(\mathbb{P}(C([-\tau,0]; R_{+}^{3}))\) be the space of all probability measure on \(C([-\tau,0];R_{+}^{3})\). Then it follows from Lemma 2.3 and Chebyshev’s inequality that the family \(\{p(t,\phi, dy)\}\) is tight, that is, for any arbitrarily given \(\varepsilon>0\), there exists a compact subset \(\mathcal{K}\subseteq R_{+}^{3}\) such that \(P(t, \phi,\mathcal {K})\geq1-\varepsilon\). We prove that the series \(\{p(x(t,\phi,R_{+}^{3}))\}\) is Cauchy in \(\mathbb{P}(C([-\tau,0]; R_{+}^{3}))\).

Theorem 5.1

Proof

By Theorem 4.1, (1.2) only has an invariant measure \(\nu(\cdot)\). By Corollary 3.4.3 of Prato and Zabczyk [32], \(\nu(\cdot)\) is strong mixing. By Theorem 3.2.6 in [32], \(\nu(\cdot)\) is ergodic. Hence \(\lim_{t\to\infty}t^{-1}\int_{0}^{t}C^{T}x(s)\,ds=\int_{R_{+}^{3}}C^{T}\nu(dx)\).

(ii) By contradiction, suppose that there exists an OHE \(\tilde{C}^{*}=(\tilde{C}_{1}^{*},\tilde{C}_{2}^{*},\tilde{C}_{3}^{*})^{T}\). Then \(\tilde{C}^{*}\in\varXi\), which implies that \(\beta_{1}|_{C=\tilde{C}^{*}}>0\), \(\beta_{2}|_{C=\tilde{C}^{*}}>0\), \(\beta_{3}|_{C=\tilde{C}^{*}}>0\), \(A_{1}|_{C=\tilde{C}^{*}}>\tilde{A}_{1}|_{C=\tilde{C}^{*}}\), \(A_{2}|_{C=\tilde{C}^{*}}>\tilde{A}_{2}|_{C=\tilde{C}^{*}}\), \(A_{3}|_{C=\tilde{C}^{*}}>\tilde{A}_{3}|_{C=\tilde{C}^{*}}\), \(g_{1}\), \(g_{2}\), \(g_{3}\geq0\). Further, \(\tilde{C}^{*}\) is a solution of (5.1). In view of the uniqueness of its solutions, \(\tilde{C}^{*}=G\), which means that \(\beta_{1}|_{C=G}>0\), \(\beta_{2}|_{C=G}>0\), \(\beta_{3}|_{C=G}>0\), \(A_{1}|_{C=G}>\tilde{A}_{1}|_{C=G}\), \(A_{2}|_{C=G}>\tilde{A}_{2}|_{C=G}\), \(A_{3}|_{C=G}>\tilde{A}_{3}|_{C=G}\). Clearly, they contradict with the assumption. The proof is completed. □

Funding

This work was supported partially by the Natural Science Foundation of Guangxi (2016GXNSFAA380194), the Visiting Program of the Education Department of Guangxi, NSERC, NSF of China (51479215, 11871475 and 11861027).

Authors’ contributions

YS carried out all studies and drafted the manuscript. YC conceived of the study and participated in its design and coordination and helped to draft the manuscript. BD participated in the design of the study and performed the simulation analysis. All authors read and approved the final manuscript.