Discrete Dynamics in Nature and Society

Discrete Dynamics in Nature and Society / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 142534 | https://doi.org/10.1155/2010/142534

Yi Ma, Bing Liu, Wei Feng, "Dynamics of a Birth-Pulse Single-Species Model with Restricted Toxin Input and Pulse Harvesting", Discrete Dynamics in Nature and Society, vol. 2010, Article ID 142534, 20 pages, 2010. https://doi.org/10.1155/2010/142534

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

Academic Editor: Manuel De la Sen
Received18 Mar 2010
Accepted03 Jun 2010
Published11 Aug 2010

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.


FunctionEquilibrium (or )

Ricker

Beverton-Holt

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).


Function Interval of stabilityType of bifurcation

Ricker Flip bifurcation
Beverton-Holt Flip bifurcation

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.

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.

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.

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