Dynamical analysis of a logistic model with impulsive Holling type-II harvesting
- Qiaoxia Lin^{1},
- Xiangdong Xie^{2}Email authorView ORCID ID profile,
- Fengde Chen^{1} and
- Qifang Lin^{2}
https://doi.org/10.1186/s13662-018-1563-5
© The Author(s) 2018
Received: 13 September 2017
Accepted: 16 March 2018
Published: 27 March 2018
Abstract
A logistic model with impulsive Holling type-II harvesting is proposed and investigated in this paper. Here, the species is harvested at fixed moments. By using the techniques derived from the theory of impulsive differential equations, sufficient conditions for both permanence and extinction of the system are established, respectively. Sufficient conditions which ensure the existence, uniqueness, and global attractivity of a positive periodic solution of the system are obtained. Our study shows that impulsive controls play an important role in maintaining the sustainable development of the ecological system. Compared with the linear impulsive capture or continuous nonlinear-type capture, our study shows that the nonlinear impulsive capture could lead to more complicated dynamic behaviors. Numeric simulations are carried out to show the feasibility of the main results. The results obtained here maybe useful to the practical biological economics management.
Keywords
MSC
1 Introduction
During the last decade, many scholars [1–39] proposed various single or multiple species modeling. Such topics as the existence and stability of the equilibrium, the existence, uniqueness, and stability of the periodic solution or almost periodic solution, the persistence and extinction of the system have been extensively investigated, and many interesting results have been obtained. It brings to our attention that all the models are based on a single species model, while a logistic model is one of the basic single species models, it is the cornerstone of the mathematics biology. On the other hand, the harvest of populations is one of the human purposes to achieve the economic interests. Already, there are many scholars investigating the dynamic behaviors of the population models incorporating the harvesting, see [3, 4, 7, 28–31] and the references cited therein.
For system (1.4), an interesting question is the following: Are the dynamical behaviors of system (1.4) similar to or quite different from those of system (1.5) and system (1.6)? It is generally known that the ecological system will be destroyed with the high capturing intensity or high frequency capture. In order to ensure the economic benefits and sustainable population development, should we control the level of capture strength and cycle?
The paper is organized as follows. In Sect. 2, some sufficient conditions for the existence and global attractivity of system (1.4) are derived; also, under some conditions, the system may admit a unique positive periodic solution. In Sect. 3, we investigate the extinction property of system (1.4). In Sect. 4, several numeric simulations are carried out to illustrate the feasibility of the main results. We end this paper with a brief discussion.
2 Permanence and global attractivity
Theorem 2.1
Proof
Remark 2.1
Remark 2.2
Theorem 2.2
If condition (2.1) holds, then system (1.4) has at least one θ-periodic solution \(x(t)\), for which \(x(0)>0\).
Proof
Owing to r, a and α, β, γ are positive constants, the impulsive system (1.4) is periodic.
The proof of Theorem 2.2 is completed. □
Theorem 2.3
Proof
As a direct corollary of Theorems 2.2 and 2.3, we have the following.
3 Extinction
In this section, we give the following result which indicates that species \(x(t)\) will be driven to extinction.
Theorem 3.1
Proof
As a direct corollary of Theorem 3.1, we have the following.
Corollary 3.1
Corollary 3.2
4 Numeric simulations
Example 4.1
Consider system (1.4) with the following coefficients: \(r=3\), \(a=2\), \(\theta=1\).
Example 4.2
Consider system (1.4) with the following coefficients: \(r=3\), \(a=2\), \(\theta=0.05\).
Remark 4.1
All the analytical calculations are performed in detail in Appendix A.1.
Example 4.3
Take \(r=3\), \(a=2\), \(\alpha=1\), \(\beta=0.2\), \(\gamma=0.5\).
Figure 3, Fig. 4, and Fig. 5 exhibit the effect of harvesting cycle θ. One could easily see that if θ is large enough (\(\theta>\frac{1}{r}\ln\frac{\alpha a+\beta r}{a(\alpha-\gamma )} \)), then (2.1) holds; and consequently, species x is permanent. With the increase in θ, the density of species x is increasing accordingly. If θ is small enough such that \(\theta<\frac{1}{r}\ln\frac{\alpha}{\alpha-\gamma}\), then (3.1) holds, and the species will be driven to extinction. That is, the harvesting cycle can change the persistence and extinction property of the system.
Remark 4.2
All the analytical calculations are performed in detail in Appendix A.2.
Example 4.4
Take \(r=3\), \(a=2\), \(\alpha=1\), \(\beta=0.2\).
Numerical simulations (Fig. 6, Fig. 7) show that if γ is small enough (\(\gamma<\alpha-\frac{\alpha a+\beta r}{ae^{r\theta }} \)), such that (2.1) holds, then the species is permanent. For the fixed θ, as γ gradually decreases, the density of species x is increasing accordingly. If γ is large enough such that \(\gamma>\alpha-\frac{\alpha}{e^{r\theta}}\), then (3.1) holds, the species will be driven to extinction. From this point, the capture intensity γ plays a negative effect on the persistence property of the system. Also, for the fixed θ, as γ is increasing gradually, the time for the species to be extinct becomes shorter.
5 Discussion
In this paper, a logistic model incorporating nonlinear impulsive Holling type-II harvesting is proposed and studied.
Based on the theoretical analysis and numerical simulations, we show that different parameter relationships may result in different dynamical behaviors of the system. From Remarks 4.1 and 4.2, sufficiently small value of θ and sufficiently large value of γ will cause the extinction of the species. Furthermore, high capture frequency θ (Fig. 3) and high capture intensity γ (Fig. 7) accelerate the speed of extinction. If the value of θ is large enough and the value of γ is small enough, the species is permanent (Figs. 4 and 6). Furthermore, with the increase in the harvesting cycle and the decrease in the capture intensity, the density of species x is increasing.
To sum up, to ensure the permanence of the specie, we could increase the period between the capture or decrease the capture strength.
At the end of the paper, we would like to mention that we assume that θ is a positive constant in system (1.4), that is, for all \(k\in Z^{+}\), \(t_{k}-t_{k-1}=\theta\). What would happen if we assume that \(t_{k}\) is a periodic sequence or an almost periodic sequence? We leave this for future investigation.
Declarations
Acknowledgements
The authors are grateful to the reviewers for useful suggestions which improved the contents of this paper. The research was supported by the Natural Science Foundation of Fujian Province (2015J01012, 2015J01019).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that there is no conflict of interests.
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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