Nonlinear Analysis: Theory, Methods & Applications
Optimal impulsive harvesting on non-autonomous Beverton–Holt difference equations
Introduction
The optimal utilization and sustainable development of renewable natural resources have been important issues for a long time. The exploitation of fish populations has provided classic examples of the disastrous effects of overexploitation of a renewable resource [18], a recent example being the collapse in 1992 of the Atlantic Cod Gadus morhua fishery in eastern Canada [24]. Therefore, it is important to develop management policies that maximize biomass yield and avoid overexploitation to maintain sustainable fisheries. Thus, the adoption of an ecosystem-based approach for fishery management, the goal of which is to rebuild and sustain populations, species, biological communities, and marine ecosystems at high levels of productivity and biological diversity, has often been recommended [10], [17], [29], [31].
A classic discrete model describing population growth is the Beverton and Holt equation [7] where is population size at time and and are constants. In some bottom-feeding fish populations, like North sea plaice Pleuronectes platessa and haddock Melanogrammus aeglefinus studied by Beverton and Holt [7], recruitment appears to be essentially unaffected by fishing, and this is true over a wide range of fishing effort. These species have very high fertility rates and very low survivorship to adulthood.
Most exploited fish resources experience some form of environmental seasonality, which often entails periodic fluctuations in the availability of resources. Therefore, a logical modification of a simple difference equation is the addition of periodic temporal variations, i.e. we have the following periodic Beverton–Holt equation: and there exists a positive integer such that . The existence and stability of a periodic solution of system (1.2) has been investigated by Clark and Gross [11].
We can extend the Beverton–Holt model (1.2) to the case when the recruitment is affected by some harvesting (e.g. fishing or hunting). Let us assume that there is an impulsive sequence at which the system is subject to a perturbation which causes decreases (only is left at the -th intervention, where is harvesting effort and is the value before the perturbation). So we have the following impulsive Beverton–Holt model: where is the size after -th impulsive perturbation. In order to keep the -periodicity of model (1.3), we assume that there exists a positive integer such that with . From a biological point of view, the positive integer means that there exist impulsive moments each period or frequency of intervention.
Since impulsive differential equations incorporate elements of continuous and discrete systems, the methods and results for system (1.3) can be used to investigate the following periodic logistic equation with impulsive harvesting. where , , and is continuous for and exist and , where . Suppose the system (1.5) is -periodic, that is, there exists such that
The model equations (1.3) ((1.5)) describe the variation of the population number of an isolated species in a periodically changing environment. The intrinsic rate of change is related to the periodically changing possibility of regeneration of the species, and the carrying capacity of the system is related to the periodic change of the resources maintaining the evolution of the population. The jump condition reflects the possibility of an impulse effect on the population, and the positive integer in the Eq. (1.4) ((1.6)) denotes the times of impulsive harvest within each period (e.g. one month or one year). Therefore, the models presented here are more general than models in which continuous harvest strategies are examined [10], [17], [29]. This is one of the advantages of systems with impulsive harvest strategies in the sense that in the real world the effects of human actions on fish resources are seasonal or occur in regular pulses. The other advantage of this type of model is that we can investigate the relationship between the intensity and frequency of harvesting, and further examine the effects of the optimal timing of harvesting and seasonal environments on the maximum sustainable yield (MSY). Equations of this kind are found in almost every domain of applied sciences (see [4], [3]). Some impulsive equations have been recently introduced to population ecology [22], [5], [2], [12], [27], [28], [29], [30] and chemotherapeutic treatment of disease [21], [25].
Recently, much attention has been paid to the Beverton–Holt equations, and their corresponding continuous systems [6], [15], [14]. In particular, Berezansky and Braverman [6] investigated the asymptotic properties of the impulsive Beverton–Holt difference equation, and applied their results to the impulsive non-autonomous logistic equations due to the close connection between discrete and continuous models. The present paper investigates how impulsive harvesting affects the growth of populations and what impulsive harvesting policies will maximize the biomass yield sustainably. Moreover, these results are applied to the impulsive logistic equations. This decision problem with multiple objectives is a complex and difficult task even if the dynamics of single species are known accurately. As far as we know, there have been no previous studies on the optimal impulsive harvesting problem for non-autonomous periodic difference systems.
We show here (in Section 2) that there exists a unique positive periodic solution of (1.3) which is globally asymptotically stable if suitable sufficient conditions hold true, and the effects of the intensity and frequency of harvesting on population persistence are investigated numerically. We examined the optimal management policy with MSY as a management objective, i.e. we obtained the optimal harvesting effort and the corresponding optimal population level (see Section 3) by using technique analysis methods and discrete optimal control theory. In Section 4, some interesting results concerning the effects of seasonal environments and harvest timing on the MSY are proposed and examined. It is a logical extension of this to apply our results, obtained for the Beverton–Holt equation, to a periodic logistic equation with impulsive harvesting (Section 5), and, further, we compare the optimal impulsive harvesting policy with an optimal continuous harvesting policy by using theoretical proof and numerical methods, and then discuss the advantages of an optimal impulsive harvesting policy. Finally, some biological conclusions are given and possible developments are discussed (Section 6).
Section snippets
Periodic solutions of a Beverton–Holt model with impulsive effects
The main purpose of this section is to investigate the existence of periodic solutions of a Beverton–Holt model with impulsive effects. First, let us consider the -periodic Beverton–Holt model without impulsive effects, i.e. with . Model (2.1) can be used to mimic a population’s response to seasonal fluctuations in its environment or to mimic a population with several discrete life-cycle stages [8], [26].
It follows from the work of Clark and Gross [11]
Optimal impulsive harvesting policy on Beverton–Holt equations
On the basis of the results obtained in Section 2, it is now possible to derive some key features of the optimal harvesting policy. That is, we aim to work out the optimal impulsive harvesting effort which maximizes the sustainable yield of the system (2.5) under condition (2.6). To this end, we define the set and . Without loss of generality, we only need to investigate our problem in any one period due to the periodicity. For
The effects of harvest timing and seasonal fluctuations on the MSY
The main purpose of this section is to investigate how the harvest timing and seasonal fluctuations affect the optimal harvest efforts and the MSY. It is well known that in fisheries management programmes it is common practice to permit exploitation only during a specified harvest season, the open season, whilst during the rest of the period, the closed season, no harvesting is allowed. So the threshold points where we start and stop harvesting are crucial for fisheries management programmes,
Applications to the logistic differential equation with impulsive harvest
Now consider the logistic differential equation with impulsive harvest, i.e. with where . The system (5.1) has an explicit solution in every interval without an impulse , i.e. If we define then it follows from (5.3) that
Discussion
Whenever a natural population is exploited there is a risk of overexploitation: too many individuals are harvested and the population is driven into biological jeopardy or economic insignificance, even to extinction. To determine the best way to exploit a population, it is necessary to know what the consequences will be of different exploitation strategies. But in order to know these consequences, we first need some understanding of the dynamics of a population in the presence or absence of
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