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Discrete Dynamics in Nature and Society
Volume 2010, Article ID 142534, 20 pages
http://dx.doi.org/10.1155/2010/142534
Research Article

Dynamics of a Birth-Pulse Single-Species Model with Restricted Toxin Input and Pulse Harvesting

1College of Science, Shenyang University, Shenyang 110044, China
2Department of Mathematics, Anshan Normal University, Anshan, Liaoning 114007, China
3School of Mathematics and System Sciences & LMIB, Beihang University, Beijing 100083, China

Received 18 March 2010; Accepted 3 June 2010

Academic Editor: Manuel De la Sen

Copyright © 2010 Yi Ma et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a birth-pulses single-species model with restricted toxin input and pulse harvesting in a polluted environment. Pollution accumulates as a slowly decaying stock and is assumed to affect the growth of the renewable resource population. Firstly, by using the discrete dynamical system determined by the stroboscopic map, we obtain an exact 1-period solution of system whose birth function is Ricker function or Beverton-Holt function and obtain the threshold conditions for their stability. Furthermore, we show that the timing of harvesting has a strong impact on the maximum annual sustainable yield. The best timing of harvesting is immediately after the birth pulses. Finally, we investigate the effect of the amount of toxin input on the stable resource population size. We find that when the birth rate is comparatively lower, the population size is decreasing with the increase of toxin input; that when the birth rate is high, the population size may begin to rise and then drop with the increase of toxin input.

1. Introduction

One of the most important results in the economics of natural resources explains why natural biological stocks are often overexploited. An excessively high harvest rate is a consequence of the common property nature of these resources. Because the reproduction rate depends on the size of the stock, open access harvesting leads to unoptimally low stock levels and in some cases to the extinction of the population. Therefore, it is very realistic for decision makers to plan practicable scheme which sustains renewable resources at a good level of productivity and meets economic goals. Economic and biological aspects of renewable resources management have been considered by Clark [1] and other authors [24].

However, human production and consumption activities may also affect the environment in which natural populations are regenerating. Many different industrial and agricultural effluents enter aquatic environments and pose a potential threat to the different organisms. It is reported that China's Three Gorges Dam reservoir has been fouled by pesticides, fertilizers, and sewage, and more than two kilometers of the Yangtze river are critically polluted. The economic importance of the pollution problem is compounded by the fact that, in contrast with overexploitation, it affects both common and private property resource stocks. Such environmental uncertainty also affects the incentive to harvest a resource. The toxin input may be restricted if necessary, in a number of ways, including area of emitting toxin, timing of year. Time-area closures are used extensively to control human activity.

Most existing theories on harvest strategies largely ignore the effects of seasonality and environmental pollution. In 2005, Xu et al. [4] investigated harvesting in seasonal environments and focused on maximum annual yield (M.A.Y.) and population persistence under five commonly used harvest strategies. It concluded that pulse harvesting was the best amongst all the strategies that they had explored with much larger M.A.Y. and mean population size but smaller population variability at M.A.Y. Also they obtained that harvest timing was of large importance to annual yield and population persistence for pulse harvesting. Harvesting too late may overexploit a population risking extinction with much smaller M.A.Y. as well.

The nature of optimal exploitation and its effects on the dynamics of biological populations when the growth process of a specie is subject to random environmental shocks are not very well understood. So in this paper, considering the results in [4], we will study an optimal pulse harvesting problem of a single birth-pulse population in a polluted environment and see how the timing of pulse harvesting affects the maximum annual-sustainable yield? Can we obtain the similar results to those in [4]? How does a birth pulse proportional to the population affect the dynamics of the population? How does the amount of toxin input affect the volume of renewable resource stock?

For these purposes, we suggest an impulsive equation (see [5, 6]) to model the process of birth pulses and pulse harvesting at different fixed times in a polluted environment in Section 2. Since pollution has harmful effects on the growth and quality of resource population stock, here we assume the toxin input is restricted by a certain time. To our knowledge, there have been no results on this problem in the literature. Impulsive equations are found in almost every domain of applied science and have been studied in many investigations [733]. In Section 3, we investigate the dynamics of such a system by using the stroboscopic map for different density-dependent birth pulses, that is, we choose Ricker function and Beverton-Holt function as birth rate function. Recently, the Ricker equation and Beverton-Holt equation and their generalizations have received much attention in the last years [3136]. In Section 4, we study the effects of pulse harvesting time on the maximum annual-sustainable yield. In Section 5, we will see how the amount of toxin input affects the size of stable resource population. In the last section, we conclude our results.

2. Model Formulation

In the absence of pollution, we assume that the size of single species changes according to the following population growth equation: where is the death rate constant, and is a birth rate function with satisfying the following basic assumptions for :();() is continuously differentiable with ;().

Note that ()–() imply that exists for , and () gives the existence of a carrying capacity such that for , and for . Under these assumptions, nontrivial solutions of (2.1) approach as .

Examples of birth functions found in the biological literature that satisfy ()–() are(), with ;(), with and ;() with .

Functions and with are used in fisheries and are known as the Ricker function and Beverton-Holt function, respectively. Function represents a constant immigration rate together with a linear birth term .

For the population model (2.1), it can be postulated that the size of the resource population is affected by the toxin input, and the presence of toxin in the environment decreases the growth rate of population. These lead to the following single population model with toxin input in the polluted environment: where is the density of the species at time ; is the concentration of toxin in the organism at time ; is the concentration of toxin in the environment at time ; is a birth rate function satisfying assumptions The meanings of other parameters are the same as those of model () in [37].

In this paper, we assume that the capacity of the environment is so large that the change of toxin in the environment that comes from uptake and egestion by the organisms can be ignored (, ), and the toxin input is constant (). For the convenience of computation, we merge into one term, still denoted by .

Now, considering the above assumptions and the continuous harvesting policy of the population, we construct the following system: where denotes the harvesting effort.

Model (2.3) has invariably assumed that the population is born throughout the year, whereas it is often the case that births are seasonal or occur in regular pulses. Many large mammal and fish populations exhibit what Caughley [38] termed a “birth pulse” growth pattern. That is, reproduction takes place in a relatively short period each year. In this paper, we take pulse harvesting policy and assume the timing we take to harvest is fixed every year. Since water pollution has harmful effects on the growth and quality of fish stock, here we assume that the toxin input is restricted by a certain time in a closed polluted environment. Now based on the impulsive differential equations, we will develop system (2.3) by introducing periodic birth pulses and pulse harvesting at different fixed times in a polluted environment in which the time of toxin input is restricted. That is, we consider the following system: where the meanings of parameters and are the same as model (2.3); is the quantities of population after the birth pulse, , , . For convenience, here we assume that pulse harvesting occurs only once and the population can reproduce only once in each year; represents the time of beginning toxin input in each year; represents the time of pulse harvesting in each year; represents the exogenous rate of toxin input into the environment at time which is assumed to be a constant; represents pulse harvesting effort at , . In the following section, we will investigate the dynamics of model (2.4).

3. Dynamical Behaviors of System (2.4)

3.1. Stroboscopic Maps of Model (2.4) with Ricker Function and Beverton-Holt Function

We can easily obtain the analytical solution of system (2.4) at the interval : (i)if , (ii)if , with , and denoting the densities of the population, the concentration of the toxin in the organism, and the concentration of the toxin in the environment at time , respectively. For the Ricker function, that is, , (3.1) (or (3.2)) holds on the interval . After each successive birth pulse, more populations are added, yielding where where denotes .

Similarly, for the Beverton-Holt function, that is, we have the following stroboscopic map of system (3.2):

Equations (3.3) and (3.5) are difference equations. They describe the density of the population, the concentration of the toxin in the organism, and the concentration of the toxin in the environment at in terms of values at . We are, in other words, stroboscopically sampling at its pulsing period , . The dynamics of system (3.3) and system (3.5), coupled with system (3.1) (or (3.2)), determine the dynamical behavior of model (2.4) for the Ricker function and for the Beverton-Holt function, respectively. Thus, in the following we will focus our attention on system (3.3) and system (3.5) and investigate the various dynamical behaviors.

The dynamics of these nonlinear models can be studied as a function of any of the parameters. Here we will focus on for the Ricker function and the Beverton-Holt function and expound the changes in the qualitative dynamics of models (3.3) and (3.5) as varies.

3.2. Stability of Nonnegative Equilibria of System (3.3) and System (3.5)

The system (3.3) (or (3.5)) leads to a trivial equilibrium and a unique positive equilibrium if , which is listed in Table 1.

tab1
Table 1: Nontrivial equilibria of the two models with birth pulses.

In the neighborhood of (), the dynamics of (3.3) and (3.5) are controlled by the linearization with equal to the linearization counterpart of (3.3) or (3.5) and . (or ) is stable when the absolute values of eigenvalues of are all less than one.

For the trivial equilibrium of (3.3), where and there is no need to calculate the exact form of () as they are not required in the analysis that follows.

The eigenvalues of are , , and ; if then is locally asymptotically stable. In terms of the model parameters, and after a bit of rearranging, for (3.3), inequality (3.8) reads

Similarly, for the trivial equilibrium of (3.5), So if then is locally asymptotically stable.

Thus if inequality (3.9) (or (3.11)) holds true, is stable. For this range of , the population will be extinct. Otherwise, is unstable, and a small population will be increased from .

For the difference equations (3.3) and (3.5), we can also define the intrinsic net reproductive number (the average number of offspring which an individual produces over the course of its lifetime). For (3.3), is given by

For (3.5), is given by

Inequality (3.9) (3.11) can be rewritten as (). That is, if on average, individuals do not replace themselves before they die, then the population is doomed.

Note that when (i.e., ), then . Thus as increases through , passes through the equilibrium at and exchanges stability with it in a transcritical bifurcation.

If , for the linearization of (3.3) about this positive equilibrium , There is no need to calculate the exact form of () as it is not required in the analysis that follows.

The eigenvalues of are , , ; if then is locally asymptotically stable.

Similarly, the linearization of (3.5) about this positive equilibrium is If then is locally asymptotically stable.

The stability of is lost in only one way as increases. Condition (3.15) or (3.17) is violated for . The critical values are listed in Table 2 for each model. A flip bifurcation occurs and the equilibrium loses stability to stable two cycles (see Figure 1).

tab2
Table 2: Critical value of the parameter for each model. must be less than for stability.
fig1
Figure 1: Bifurcation diagrams of (3.3) and (3.5) for the population . Shown is the effect of parameter on the dynamical behavior. Parameter values are (a) Ricker function, . (b) Beverton-Holt function, .

For , equilibrium of system (3.3) (or system (3.5)) is stable. For this range of , trajectories of model (2.4) approach the origin.

For , the equilibrium of system (3.3) (or system (3.5)) is stable. For this range of , trajectories of model (3.3) and (3.5) approach the 1-period solution of model (2.4),(i)If , (ii)if ,

That is, 1-period solution (3.18) (or (3.19))of model (2.4) is locally asymptotically stable. Right at , there is a transcritical bifurcation of periodic solutions. and () pass through each other and exchange stability.

As increases beyond , it passes through a cascade of period-doubling bifurcations that eventually lead to chaotic dynamics and many other complexities, and so 2-period solutions, 4-period solutions,, chaotic strange attractors occur in model (2.4).

Figure 2 gives extinct solution, 1-period solution, 2-period solutions, 4-period solutions, and chaotic strange attractors of model (2.4) for Ricker function and Beverton-Holt function, respectively, and they correspond to trivial equilibrium , unique positive equilibrium , 2-period points, 4-period points, and chaos of model (3.3) and model (3.5), respectively.

fig2
Figure 2: Time series of population in model (2.4) with parameter values (a)–(e) correspond to extinct solution, 1-period solution, 2-period solution, 4-period solution, and chaos for Ricker function, and and respectively; (a')–(e') correspond to extinct solution, 1-period solution, 2-period solution, 4-period solution, and chaos for Beverton-Holt function, and and , respectively.

4. The Effects of Pulse Harvesting Time on the Maximum Annual-Sustainable Yield

Many authors are interested in studying the optimal management of renewable resources, which has a direct relationship to sustainable development. From the point of view of ecological managers, it may be desirable to have a unique positive equilibrium which is asymptotically stable, in order to plan harvesting strategies and keep sustainable development of system. In this section, we will study how the pulse harvesting affects the maximum annual-sustainable yield.

For , the equilibrium of system (3.3) (or system (3.5)) is stable. For this range of , trajectories of model (2.4) approach the periodic solution with period 1, that is, periodic solution (3.18) (or (3.19)) of system (2.4) is locally asymptotically stable.

Since we only need to consider the annual-sustainable yield in one period, without loss of generality we can choose and the annual-sustainable yield is

Our main purpose is to get an such that reaches its maximum at and study how the maximum annual- sustainable yield changes as varies. Numerical analysis implies that there exists a unique such that reaches its maximum for each fixed and (see Figure 3. for Figure 3(a), reaches its maximum at ; For Figure 3(b), reaches its maximum at ). Also from Figure 3 we can observe that the maximum annual-sustainable yield dramatically depends on the pulse harvesting time, and the maximum annual-sustainable harvest yield is the largest at and the smallest at . It shows that if we harvest immediately after the birth pulse, the largest maximum annual-sustainable harvest yield is obtained and that if we harvest near the time of birth pulse, the maximum annual-sustainable yield is the smallest.

fig3
Figure 3: The annual-sustainable yield of (3.3) and (3.5), showing the relationship between the maximum annual-sustainable and the harvesting timing. Parameter values are . (a) Ricker function, . (b) Beverton-Holt function, .

5. The Effect of the Amount of Toxin Input on the Size of Population

From Figure 4, we can see that when the positive equilibrium is stable for each model, the population size begins to rise and then drops with the increase of toxin input. In the following, we will study how the amount of toxin input influences the population size when the equilibrium is stable.

fig4
Figure 4: Bifurcation diagrams of (3.3) and (3.5) for the population . Shown is the effect of parameter on the dynamical behavior. Parameter values are (a) Ricker function, . (b) Beverton-Holt function, .

For Ricker function, define . is equivalent to . So if , there exists a unique positive equilibrium of system (3.3). From Figure 4(a), we know if the amount of toxicant input is larger than a threshold , then this positive equilibrium is stable.

Lemma 5.1. For any , one has

Proof. Let , then ; so we have , which implies that is the maximum of . This completes the proof.

Definition 5.2. The Lambert W-function is defined to be a multivalued inverse of the function satisfying .

For convenience, we denote it by W. First of all, the function has the positive derivative if . Define the inverse function of restricted to the interval as which is monotonic increasing. Similarly, we define the inverse function of restricted to the interval as which is monotonic decreasing. In view of the nature of this study, both and will be employed only for because both functions are real value for in this interval. For more details of the concepts and properties of the Lambert W-function, see [3941].

Theorem 5.3. Assume . If , the population size will be decreasing with the increase of toxin input. If and, the population size will be increasing with the increase of toxin input. the population size will be decreasing with the increase of toxin input.

Proof. Since we have (i)If , from Lemma 5.1, we have So which implies that Thus, we have which means that the population size will be decreasing with the increase of toxin input. (ii)If , we have . Both and are meaningful.AIf , then and So, which implies that and .BIf , then and So, which implies that and . So if , the population size will be increasing with the increase of toxin input.CIf , then So, which implies that and . So if the population size will be decreasing with the increase of toxin input.
his completes the proof.

By similar analysis, we can obtain similar conclusions for Beverton-Holt model.

For Theorem 5.3, we give the following biological implication.

Remark 5.4. When the birth rate of resource population is comparatively lower, resources will be relatively abundant, and the size of the population will decrease with the increase of the amount of toxin input. That will make the pulse birth rate ( for Ricker function and for the Beverton-Holt function) of the population grow and the internal competition harsh. At this moment, the increase of the number of toxin will weaken the internal competition and prompt the size of the population to expand. However, when the number of toxin increases to some degree, the number of the population born in an impulsive period will be smaller than the total of the number of natural death and death resulting from toxin, and so the size of the population will decline.

6. Conclusion

Environmental pollution in the last decades has received a great deal of attention from several researchers. The main objective of the present paper is to study the dynamics of a birth-pulse single-species model with the restricted toxin input and pulse harvesting in a closed polluted environment. We have obtained the complete expression for the 1-period solution and the threshold conditions for their stability. We show the relationship both between pulse harvesting time and the maximum annual-sustainable yield and between the amount of toxin input and the stable equilibrium population size. Our results show that the best time of harvesting is immediately after the birth pulse and that when the birth rate of the population is comparatively lower, the population size will be decreasing with the increase of the amount of toxin input; they also show that then with the increase of birth rate, the stable population size may begin to rise and then drop with the increase of toxin input. From the viewpoint of biology, the mathematical results are full of biological meanings and can be used to provide reliable foundations for making decisions. To protect the population from extinction and maintain the quality of the recourse population, human activity must be controlled to restrict the toxin input to a certain extent. Numerical simulations which we have performed also show that birth pulse and pulse harvesting make the single-species model in a restricted polluted environment we consider more complex and dominated by periodic and chaotic solutions.

Our results about harvesting and those in [4] all conclude that harvest timing is of large importance to annual yield, whether it is pulse harvesting or open/closed piecewise continuous-time harvesting. Harvesting too late may overexploit a population with much smaller maximum annual yield.

There is still a tremendous amount of work to do. From the point of reality, it would be interesting to study the emerging dynamical behavior of a stage structure birth-pulse single-species model with restricted toxin input and pulse harvesting for mature resource population. But it is very difficult to discuss this system, because the stage-structure makes the dynamics of the system very complicate. We hope this issue will be well addressed in the near future, and we leave it for the subject matter of our future research.

Acknowledgments

This work is supported by National Natural Science Foundation of China (10971001, 10971009, and 10871017) and is also supported by Excellent Staffs Support Project of Universities and Colleges in Liaoning.

References

  1. C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 2nd edition, 1990. View at Zentralblatt MATH · View at MathSciNet
  2. L. H. R. Alvarez, “Optimal harvesting under stochastic fluctuations and critical depensation,” Mathematical Biosciences, vol. 152, no. 1, pp. 63–85, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. T. Mitra and S. Roy, “Optimal exploitation of renewable resources under uncertainty and the extinction of species,” Economic Theory, vol. 28, no. 1, pp. 1–23, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. Xu, M. S. Boyce, and D. J. Daley, “Harvesting in seasonal environments,” Journal of Mathematical Biology, vol. 50, no. 6, pp. 663–682, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. Baĭnov and P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, vol. 66 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, UK, 1993. View at MathSciNet
  6. V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific, Singapore, 1989. View at MathSciNet
  7. B. Liu, L. Chen, and Y. Zhang, “The effects of impulsive toxicant input on a population in a polluted environment,” Journal of Biological Systems, vol. 11, no. 3, pp. 265–274, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  8. B. Liu, Y. Zhang, and L. Chen, “Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control,” Chaos, Solitons and Fractals, vol. 22, no. 1, pp. 123–134, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Z. Liu, R. Tan, and L. Chen, “Global stability in a periodic delayed predator-prey system,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 389–403, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. Z. Liu and L. Chen, “Periodic solution of a two-species competitive system with toxicant and birth pulse,” Chaos, Solitons and Fractals, vol. 32, no. 5, pp. 1703–1712, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Gao, Z. Teng, and D. Xie, “Analysis of a delayed SIR epidemic model with pulse vaccination,” Chaos, Solitons and Fractals, vol. 40, no. 2, pp. 1004–1011, 2009. View at Publisher · View at Google Scholar · View at Scopus
  12. T. Zhang and Z. Teng, “Pulse vaccination delayed SEIRS epidemic model with saturation incidence,” Applied Mathematical Modelling, vol. 32, no. 7, pp. 1403–1416, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. H. Gu, H. Jiang, and Z. Teng, “BAM-type impulsive neural networks with time-varying delays,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 3059–3072, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. X. Song and Z. Xiang, “The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects,” Journal of Theoretical Biology, vol. 242, no. 3, pp. 683–698, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  15. X. Zhou, X. Song, and X. Shi, “Analysis of competitive chemostat models with the Beddington-DeAngelis functional response and impulsive effect,” Applied Mathematical Modelling, vol. 31, no. 10, pp. 2299–2312, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  16. X. Song, M. Hao, and X. Meng, “A stage-structured predator-prey model with disturbing pulse and time delays,” Applied Mathematical Modelling, vol. 33, no. 1, pp. 211–223, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. X. Meng and L. Chen, “Permanence and global stability in an impulsive Lotka-Volterra N-species competitive system with both discrete delays and continuous delays,” International Journal of Biomathematics, vol. 1, no. 2, pp. 179–196, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  18. X. Meng and L. Chen, “The dynamics of a new SIR epidemic model concerning pulse vaccination strategy,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 582–597, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Jiao and L. Chen, “Global attractivity of a stage-structure variable coefficients predator-prey system with time delay and impulsive perturbations on predators,” International Journal of Biomathematics, vol. 1, no. 2, pp. 197–208, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. Jiao, X. Meng, and L. Chen, “Global attractivity and permanence of a stage-structured pest management SI model with time delay and diseased pest impulsive transmission,” Chaos, Solitons and Fractals, vol. 38, no. 3, pp. 658–668, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Tang, R. A. Cheke, and Y. Xiao, “Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 12, pp. 2311–2341, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  22. S. Tang, Y. Xiao, and R. A. Cheke, “Multiple attractors of host-parasitoid models with integrated pest management strategies: eradication, persistence and outbreak,” Theoretical Population Biology, vol. 73, no. 2, pp. 181–197, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Tang and R. A. Cheke, “Models for integrated pest control and their biological implications,” Mathematical Biosciences, vol. 215, no. 1, pp. 115–125, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. H. Zhang, P. Georgescu, and L. Chen, “An impulsive predator-prey system with Beddington-DeAngelis functional response and time delay,” International Journal of Biomathematics, vol. 1, no. 1, pp. 1–17, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Sun and L. Chen, “Permanence and complexity of the eco-epidemiological model with impulsive perturbation,” International Journal of Biomathematics, vol. 1, no. 2, pp. 121–132, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. X. Liu and L. Chen, “Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impulsive perturbations on the predator,” Chaos, Solitons and Fractals, vol. 16, no. 2, pp. 311–320, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. R. Q. Shi, X. J. Ji, and L. S. Chen, “Existence and global attractivity of positive periodic solution of an impulsive delay differential equation,” Applicable Analysis, vol. 83, no. 12, pp. 1279–1290, 2004. View at Publisher · View at Google Scholar
  28. H. Liu and L. Li, “A class age-structured HIV/AIDS model with impulsive drug-treatment strategy,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 758745, 12 pages, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. C. J. Wei and L. S. Chen, “Global dynamics behaviors of viral infection model for pest management,” Discrete Dynamics in Nature and Society, vol. 2009, Article ID 693472, 16 pages, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  30. K. Liu and L. Chen, “On a periodic time-dependent model of population dynamics with stage structure and impulsive effects,” Discrete Dynamics in Nature and Society, vol. 2008, Article ID 389727, 15 pages, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. S. Tang, R. A. Cheke, and Y. Xiao, “Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 12, pp. 2311–2341, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  32. M. G. Roberts and R. R. Kao, “The dynamics of an infectious disease in a population with birth pulses,” Mathematical Biosciences, vol. 149, no. 1, pp. 23–36, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  33. S. Tang and L. Chen, “Density-dependent birth rate, birth pulses and their population dynamic consequences,” Journal of Mathematical Biology, vol. 44, no. 2, pp. 185–199, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. M. De la Sen and S. Alonso-Quesada, “A control theory point of view on Beverton-Holt equation in population dynamics and some of its generalizations,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 464–481, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. S. Elaydi and R. J. Sacker, “Periodic difference equations, population biology and the Cushing-Henson conjectures,” Mathematical Biosciences, vol. 201, no. 1-2, pp. 195–207, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. V. L. Kocic, “A note on the nonautonomous Beverton-Holt model,” Journal of Difference Equations and Applications, vol. 11, no. 4-5, pp. 415–422, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. T. G. Hallam, C. E. Clark, and G. S. Jordan, “Effects of toxicants on populations: a qualitative approach. II. First order kinetics,” Journal of Mathematical Biology, vol. 18, no. 1, pp. 25–37, 1983. View at Google Scholar
  38. G. Caughley, Analysis of Vertebrate Populations, John Wiley & Sons, New York, NY, USA, 1977.
  39. R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey, and D. E. Knuth, “On the Lambert W function,” Advances in Computational Mathematics, vol. 5, no. 4, pp. 329–359, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. R. M. Corless, G. H. Gonnet, D. E. G. Hare, and D. J. Jeffrey, “Lambert's function in Maple,” Maple Technical Newsletter, vol. 9, no. 1, pp. 12–22, 1993. View at Google Scholar
  41. D. A. Barry, P. J. Culligan-Hensley, and S. J. Barry, “Real values of the W-function,” ACM Transactions on Mathematical Software, vol. 21, no. 2, pp. 161–171, 1995. View at Publisher · View at Google Scholar · View at MathSciNet