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Dynamical analysis and optimal harvesting of a stochastic three-species cooperative system with delays and Lévy jumps
- Yuanfu Shao1, 2Email authorView ORCID ID profile,
- Yuming Chen2 and
- Binxiang Dai3
https://doi.org/10.1186/s13662-018-1874-6
© The Author(s) 2018
- Received: 5 September 2018
- Accepted: 8 November 2018
- Published: 19 November 2018
Abstract
A three-species cooperative system with time delays and Lévy jumps is proposed in this paper. Firstly, by comparison method and inequality techniques, we discuss the stability in mean and extinction of species, and the stochastic permanence of this system. Secondly, by applying asymptotic method, we investigate the stability in distribution of solutions. Thirdly, utilizing ergodic method, we obtain the optimal harvesting policy of this system. Finally, some numerical examples are given to illustrate our main results.
Keywords
- Delay
- Lévy
- Stability
- Optimal harvesting
1 Introduction
The Lotka–Volterra model, proposed by Lotka [1] and Volterra [2], is used to describe the evolutionary process in population dynamics, physics, and economics. In the last decades, many modifications of it have been investigated (see, for example, [3–7]).
As we know, time delays are very important in ecosystem models. They may cause populations to fluctuate. Time delays can reflect natural phenomena more authentically. Kuang [22] has pointed out that ignoring time delays means ignoring the reality. Therefore, it is essential to take into account the influence of time delays in the biological modeling [22–26].
Moreover, in ecological managing, harvesting often appears. Since over-harvesting or unreasonable harvesting may cause a number of detrimental effects including ecological destruction and species extinction, it is important for us to study the optimal harvesting policies for sustainable development.
Finally, it has been recognized that single-species or two-species ecological models cannot describe the natural phenomena accurately, and many vital behaviors can only be exhibited by a system with three or more species [27–30].
Our main aims are as follows. Firstly, since time delays, white noises, and Lévy jumps are included in (1.2), it is of great significance to study their effects on dynamics. By use of comparison methods and some inequality techniques, we obtain the stability in mean, extinction of populations, and stochastic permanence.
Secondly, a stochastic model cannot tend to a positive fixed point, i.e., there exists no traditional positive equilibrium state. It is necessary to study the convergence in distribution of solutions. Because of time delays, we cannot apply the traditional method like using the explicit solution by solving the corresponding Fokker–Planck equation. Here we will apply an asymptotic approach to get the convergence.
Lastly, in view of the importance of optimal harvesting policy, by applying the ergodic methods, we get the optimal harvesting effort (OHE) \(C^{*}=(c_{1}^{*},c_{2}^{*},c_{3}^{*})^{T}\) such that the expectation of sustainable yield (ESY) \(Y(C)=\lim_{t\to\infty}\sum_{i=1}^{3}\mathbb{E(}c_{i}x_{i}(t))\) is the maximum and all species are still persistent.
The rest of this paper is organized as follows. Section 2 begins with some notations, definitions, and some important lemmas which are essential in our discussion. Section 3 focuses on the dynamical behavior of (1.2) including persistence, extinction of species, and stochastic permanence. Section 4 is devoted to the stability in distribution. Section 5 considers the existence of optimal harvesting policy and obtains the maximum of ESY (MESY). Some numerical examples are given in Sect. 6 to demonstrate the obtained theoretical results. The paper concludes with a brief conclusion and discussion.
2 Preliminaries
Throughout this paper, we always assume that \(A>0\) and \(A_{i}>0\) (\(i=1\), 2, 3). This means that when there are no stochastic perturbations, a positive equilibrium state exists for model 1.2 (1.2). Furthermore, we always assume that K stands for a generic positive constant whose value may be different at different places.
On the parameters, we make the following assumptions.
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
Now we give some definitions and lemmas which will be used in stating and proving our main results.
Definition 2.1
- (a)
the population \(x(t)\) is said to go to extinction if \(\lim_{t\rightarrow\infty}x(t)=0\);
- (b)
the population \(x(t)\) is said to be stable in mean if \(\lim_{t\to\infty}\langle x(t)\rangle=K\) a.s., where K is a constant.
Definition 2.2
([20])
Lemma 2.1
([20])
- (i)If there exist some constants \(T>0\), \(\lambda_{0}>0\), λ, \(\sigma_{i}\), and \(\lambda_{i}\) such that, for all \(T>0\),then$$\ln Z(t)\leq\lambda t-\lambda_{0} \int_{0}^{t}z(s)\,ds +\sum _{i=1}^{n}\sigma_{i}B_{i}(t) + \sum_{i=1}^{n}\lambda_{i} \int_{o}^{t} \int_{\mathbb{Z}} \ln(1+\gamma_{i}(v)\tilde{N}(ds,dv)\quad\textit{a.s.}, $$$$\textstyle\begin{cases} \langle Z\rangle^{*}\leq\lambda/\lambda_{0}& \textit{a.s. if }\lambda\geq0, \\ \lim_{t\rightarrow\infty} Z(t)=0 &\textit{a.s. if }\lambda< 0. \end{cases} $$
- (ii)If there exist some constants \(T>0\), \(\lambda_{0}>0\), λ, \(\sigma_{i}\), and \(\lambda_{i}\) such that, for all \(T>0\),then$$\ln Z(t)\geq\lambda t-\lambda_{0} \int_{0}^{t}z(s)\,ds +\sum _{i=1}^{n}\sigma_{i}B_{i}(t)+ \sum_{i=1}^{n}\lambda_{i} \int_{o}^{t} \int_{\mathbb{Z}} \ln(1+\gamma_{i}(v)\tilde{N}(ds,dv),\quad \textit{a.s.}, $$$$\langle Z\rangle_{*}\geq\lambda/\lambda_{0} \quad\textit{a.s.} $$
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
3 Dynamical analysis of system (1.2)
Firstly, we give our main results on the stability in mean and extinction of species for model (1.2).
Theorem 3.1
- (i)If \(A_{i}>\tilde{A}_{i}\) for \(i=1\), 2, 3, then$$\lim_{t\to\infty} \bigl\langle x_{i}(t)\bigr\rangle = \frac{A_{i}-\tilde{A}_{i}}{A} \quad\textit{for }i=1,2, 3. $$
- (ii)For \(i \in\{1,2,3\}\), if \(A_{i}<\tilde{A}_{i}\) and \(\Delta_{ij}>\tilde{\Delta}_{ij}\) for all \(j\neq i\), then$$\lim_{t\to\infty} x_{i}(t)=0 \quad\textit{and}\quad \lim _{t\to\infty} \bigl\langle x_{j}(t)\bigr\rangle = \frac{\Delta_{ij}-\tilde{\Delta}_{ij}}{A_{ii}} \quad\textit{for } j\neq i. $$
- (iii)For \(i\in\{1,2,3\}\), if \(r_{i}>b_{i}\) and \(A_{j}<\tilde {A}_{j}\) for all \(j\neq i\), then$$\lim_{t\to\infty} \bigl\langle x_{i}(t)\bigr\rangle = \frac{\beta_{i}}{a_{ii}} \quad\textit{and}\quad \lim_{t\to\infty} x_{j}(t)=0\quad \textit{for }j\neq i. $$
- (iv)
If \(r_{i}< b_{i}\) (\(i=1\), 2, 3), then \(\lim_{t\to\infty} x_{i}(t)=0\), while if \(r_{i}>b_{i}\) (\(i=1\), 2, 3), then the conclusions of case (i) also hold.
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. □
Next we consider the stochastic permanence of system (1.2).
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
4 Stability in distribution
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 \(\mathbb{P}(C([-\tau,0]; R_{+}^{3}))\).
5 Optimal harvesting effort
Theorem 5.1
- (i)
Suppose that \(\beta_{1}|_{C=G}>0\), \(\beta_{2}|_{C=G}>0\), \(\beta_{3}|_{C=G}>0\), \(A_{1}|_{C=G}>\tilde{A}_{1}|_{C=G}\), \(A_{2}|_{C=G}>\tilde {A}_{2}|_{C=G}\), \(A_{3}|_{C=G}>\tilde{A}_{3}|_{C=G}\). Then the OHE is \(C^{*}=G\) and the MESY is \(Y^{*}=G^{T}A^{-1}(L-G)\).
- (ii)
If conditions of (i) do not hold, then the OHE does not exist.
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. □
6 Numerical simulations
System (1.2) is permanent with \(\sigma_{1}=\sigma _{2}=\sigma_{3}=0.1\), \(\gamma_{1}(u)=0.2\), \(\gamma_{2}(u)=1\), \(\gamma_{3}(u)=3\). (a) shows the stability of the equilibrium state of the corresponding deterministic system while (b), (c), (d) sketch the time series of \(x_{1}(t)\), \(x_{2}(t)\), and \(x_{3}(t)\), respectively
System (1.2) is stochastically permanent when \(\gamma_{1}(u)=0.2\), \(\gamma_{2}(u)=1\), \(\gamma_{3}(u)=3\)
The attractivity of system (1.2) when \(\sigma_{1}=\sigma _{2}=\sigma_{3}=0.1\), \(\gamma_{1}(u)=0.2\), \(\gamma_{2}(u)=1\), \(\gamma_{3}(u)=3\)
Distributions of \(x_{1}(t)\), \(x_{2}(t)\), \(x_{3}(t)\) when \(\sigma _{1}=\sigma_{2}=\sigma_{3}=0.1\), \(\gamma_{1}(u)=0.2\), \(\gamma_{2}(u)=1\), \(\gamma_{3}(u)=3\)
The MESY \(Y^{*}=0.2165 \) of (1.2) when \(\sigma_{1}=\sigma _{2}=\sigma_{3}=0.1\), \(\gamma_{1}(u)=0.2\), \(\gamma_{2}(u)=1\), \(\gamma_{3}(u)=3\)
Next we consider the effects of Lévy jumps by letting all parameters be the same as above except the parameters \(\gamma_{i}(u)\), \(i=1\), 2, 3.
\(x_{1}\) dies out while \(x_{2}\) and \(x_{3}\) persist when \(\gamma_{1}(u)=1.4\), \(\gamma_{2}(u)=1\), \(\gamma_{3}(u)=3\), and the other parameters are the same as those for Fig. 1
Both \(x_{1}\) and \(x_{3}\) die out while \(x_{2}\) persists when \(\gamma_{1}(u)=2.5\), \(\gamma_{2}(u)=1\), \(\gamma_{3}(u)=3.5\), and the other parameters are the same as those for Fig. 1
All species go to extinction with \(\gamma_{1}(u)=2.5\), \(\gamma_{2}(u)=2\), \(\gamma_{3}=3.5\), and the other parameters are the same as those for Fig. 1
7 Discussion and conclusion
In this paper, we consider a three-species stochastic cooperative system with time delays and Lévy noise. Theorem 3.1 gives some sufficient conditions on the stability in mean and extinction of each population. Theorem 3.2 gives sufficient conditions on permanence. The stability in distribution of each species is covered in Theorem 4.1. Finally, by use of ergodic method, the optimal harvesting policy is established in Theorem 5.1.
Actually, time delays are considered in our research. They bring much difficulty for us, and we successfully defined some complex functionals to overcome it. Though any concrete effects of time delays are not reflected in our obtained results, we believe that it is very interesting to further explore them. This will be our future work.
Declarations
Acknowledgements
This work was done during Dr. Shao’s visit of Wilfrid Laurier University. He would like to thank the faculty and staff of the Department of Mathematics for the hospitality. Simultaneously, the authors would like to thank the editor and the anonymous reviewer for their valuable comments, which helped to improve the presentation of the paper.
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.
Competing interests
The authors declare that they have no competing interests in this paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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