# Dynamical analysis of a logistic model with impulsive Holling type-II harvesting

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## 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

Permanence Extinction Holling type-II harvesting Impulse## MSC

34C25 92D25 34D20 34D40## 1 Introduction

During the last decade, many scholars [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 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, 29, 30, 31] and the references cited therein.

*t*,

*r*represents the intrinsic growth rate. \(a=\frac{r}{K}\) is usually referred to as the density dependent rate, the positive constant

*K*is the environmental saturation level or carrying capacity. Traditionally, one may assume that the harvest rate \(h(t)\) is a constant

*h*or under the catch-per-unit effort hypothesis \(Ex(t)\); consequently, the logistic equation with harvesting takes the form [3, 4]:

*E*denotes the harvesting effort. Two types of solution behavior may be seen in system (1.2). If \(h< r^{2}/(4a)\), the harvesting is subcritical and solutions either tend to a positive limiting value as \(t\rightarrow\infty\), or, depending on the value of initial value, may tend to zero in finite time. If \(h>r^{2}/(4a)\), the harvesting is supercritical and solutions always tend to zero in finite time.

*x*and would incorporate a saturation effect for enough large

*x*; one option is to replace

*h*in (1.2) with a Holling type-II functional response.

*η*,

*E*,

*μ*,

*ν*are positive parameters that are used for the catchability coefficient of the species, the effort devoted to their nonselective harvesting, each proportional to the radio of the stock-level to the catch-rate at higher levels of effort and each proportional to the radio of the effort level to the catch-rate at higher stock-levels. One could refer to [5, 6, 7] for more detailed information on term (1.3). As \(E\rightarrow\infty\), the limiting value of \(h(t)\) becomes a function of

*x*only and \(h(t)\rightarrow(\eta/\mu)x\). When \(x\rightarrow\infty\), \(h(t)\rightarrow(\eta/\nu)E\). One can easily observe that the catch-rate function in (1.3) embodies saturation efforts with respect to the effort level as well as stocking abundance. Term (1.3) can be translated into another form as follows:

*x*. There are some particular forms on system (1.4). System (1.4) with \(\beta=\gamma=0\), \(\alpha\neq0\) is reduced to the logistic equation 1.1. System (1.4) with \(\alpha=1\), \(\beta=0\) is reduced to the linear impulsive logistic equation [2]:

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

*Assume that*

*holds*,

*then for any positive solution*\(x(t)\)

*of system*(1.4),

*where*

## Proof

*Y*. According to the comparison theorem and [8], for any \(T>0\), we obtain

## 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.

*θ*-periodic solution \(x(t)\) of (1.4) with \(x(0^{+})=x_{0}>0\). For \(t\in(0,\theta]\), we have

*θ*-periodic solution of equation (1.4) for which

*θ*-period solution

The proof of Theorem 2.2 is completed. □

## Theorem 2.3

## Proof

*ε*, from Theorem 2.1, there exists enough large \(T>0\) such that

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

*If the assumption*

*holds*,

*then the species*

*x*

*will be driven to extinction*,

*that is*,

*for any positive solution*\(x(t)\)

*of system*(1.4), \(x(t)\rightarrow0\)

*as*\(t\rightarrow+\infty\).

*Here*,

## Proof

As a direct corollary of Theorem 3.1, we have the following.

## Corollary 3.1

*Assume*\(\beta=\gamma=0\),

*in this case*,

*system*(1.4)

*degenerates to system*(1.1).

*Assume further that*\(r<0\),

*then any positive solution*\(x(t)\)

*of system*(1.1)

*That is*,

*the species*

*x*

*will be driven to extinction when the intrinsic growth rate is negative*.

## 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

*r*,

*a*and

*α*,

*β*,

*γ*(i.e., the population nature coefficients and capture intensity are fixed), if the harvesting cycle

*x*will be driven to extinction in system (1.4), that is, too intensive harvesting could lead to the extinction of the species.

All the analytical calculations are performed in detail in Appendix A.1.

## Example 4.3

Take \(r=3\), \(a=2\), \(\alpha=1\), \(\beta=0.2\), \(\gamma=0.5\).

*x*will be driven to extinction.

*x*will be driven to extinction. Numeric simulation (Fig. 3) supports this finding.

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

*r*,

*a*and

*α*,

*β*,

*θ*(i.e., the population nature coefficients and harvesting cycle are fixed), if the capture intensity

*x*will be driven to extinction in system (1.4), that is, this harvesting cycle cannot be sustained.

All the analytical calculations are performed in detail in Appendix A.2.

## Example 4.4

Take \(r=3\), \(a=2\), \(\alpha=1\), \(\beta=0.2\).

*x*will be driven to extinction.

*x*will be driven to extinction. Numeric simulation (Fig. 7) supports this finding.

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.

## Notes

### 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.

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