Optimal harvesting of prey–predator system with interval biological parameters: A bioeconomic model
Highlights
► We study the prey–predator model under imprecise biological parameters. ► We consider impreciseness as a form of an interval in nature. ► We introduce parametric functional of an interval to study the model. ► We identify the equilibrium points of the model and discuss their stabilities. ► We discuss the bionomic equilibrium and optimal harvesting policy of the model.
Introduction
Mathematical modeling of ecosystems is a study of understanding the mechanisms that influence the growth of species and their existence and stability. Research in the field of theoretical ecology initiated by Lotka [1] and by Volterra [2]. Many researchers widely discussed various aspects of Lotka–Volterra prey–predator model in ecological literature where they considered all the biological parameters precise in nature. But in reality they are not precise in nature. In the present paper we consider imprecise prey–predator harvesting model by considering some of the biological parameters as interval numbers.
Researchers are attracted by the problems of harvesting policy and bio-economic modeling of multispecies fisheries. Biological resources are renewable resources. Among the various renewable resources, the important one is fishery. On the other hand bio-economic models assists natural resource managers in controlling appropriate level of stocks and catches. Different species of fishes are decreased due to enhancement of fishing power, high growth rate of world population and lack of knowledge of the characteristics of exploited species. Clark [3], [4] introduced economic and biological aspects of renewable resources management in the literature of multispecies fisheries. Bhattacharya and Begum [5] discussed the bionomic equilibrium of two species system. Bene et al. [6] analyzed a simple bio-economic model dealing with the management of a marine renewable resource. Dubey et al. [7] studied biological and bionomic equilibria for a model of fishery resource with reserve area. Yunfei et al. [8] investigated harvesting of a phytoplankton zooplankton model and discussed the existence of bionomic equilibria and the optimal harvesting policy. Prey–predator model with combined harvesting model has been extremely discussed by Hannesson [9], Ragogin [10], Chaudhuri [11], Chaudhuri and Roy [12], Samanta et al. [13], along with others. Costa et al. [14] presented Lotka–Volterra and Leslie–Gower predator–prey model and showed global stability of a desired equilibrium population by the application of simple switching controls. Chen and Hsui [15] developed a two-period model and a multi-period model and analyzed the optimal inter-temporal utilization of a finite resource of stock. Also proposed to impose a tax on the harvest rate. Das et al. [16] presented prey–predator harvesting model in presence of toxicity and studied optimal harvesting policy using Pontryagin’s maximal principle. Palma and Olivares [17] determined optimal harvest policy in an open access fishery in which both prey and predator species are subjected to non selective harvesting and the growth rate of prey species is affected by Allee effect. Rebaza [18] investigated boundedness of solutions, existence of bionomic equilibria, existence and stability properties of equilibrium points and periodic solutions of a predator–prey model with continuous threshold prey harvesting and prey refuge. Li and Wang [19] and Li et al. [20] presented optimal harvesting policy for stochastic logistic population model. A bio-economic analysis of the optimal long-term management of a genetic resource in the presence of selective harvesting was developed by Guttormsen et al. [21].
In the above discussion all the researchers have developed the model based on the assumption that the biological parameters are precisely known but the scenario is different in real life. The impreciseness can happen in many ways such as experimental part, the data collection, the measurement process and determining the initial conditions. To overcome these difficulties imprecise model is more realistic in the field of mathematical biology. Bassanezi et al. [22] used fuzzy differential equations to study the stability of fuzzy dynamical systems in population dynamics. Barros et al. [23] analyzed the behavior of population dynamic models which are demographic and environmental fuzziness. Peixoto et al. [24] presented the fuzzy predator–prey model. Guo et al. [25] established fuzzy impulsive functional differential equation using Hullermeier’s approach of a population model. The biological parameter may not be fixed but rather it may vary due to several reasons. So the biological parameters are very sensible and treated as positive imprecise number instead of fixed real number. To handle such type of model with imprecise biological parameters there are different approaches such as stochastic approach, fuzzy approach and fuzzy-stochastic approach. In stochastic approach the imprecise parameters are replaced by random variables with known probability distributions. In fuzzy approach the imprecise parameters are replaced by fuzzy sets with known membership function or by fuzzy numbers. On the contrary, in fuzzy stochastic approach some parameters are taken as fuzzy in nature and rest of the parameters are taken as random variables. However it is very difficult to construct a suitable membership function or a suitable probability distribution for the imprecise biological parameters.
In this paper, we consider one prey one predator harvesting model and develop a solution procedure that is able to calculate equilibrium points, bionomic equilibrium points, and the optimal harvesting policy of the model where at least one biological parameters of the model is an interval number. We present the parameters as an interval in parametric function form and then study the parametric model [26], [27], which is called parametric prey–predator model (PPPM). A parametric mathematical program is formulated to find the different behavior of the model (such as different biological equilibrium points, bionomic equilibrium points and optimal harvesting policy) for different value of parameter. The proposed procedure is more effective and interesting since we get different behavior of the model using functional form of an interval parameter for different value of parameters. We develop a new procedure to study the prey–predator harvesting model using interval valued technique. The main advantage of the proposed procedure is that we can presented different characteristic of the model in a single framework.
The rest of this paper is formulated as follows: in Section 2, we discuss some prerequisite mathematics on interval numbers. In Section 3, we present prey–predator harvesting model with precise biological parameters. Section 4 briefly states the prey–predator harvesting model with imprecise parameters. Then parametric mathematical model is formulated for study the different behavior of the model. We discuss the dynamical behavior of the imprecise model in Section 5. Bionomic equilibrium of the model is presented in Section 6. In Section 8 we have presented optimal harvesting policy of the model. Numerical examples are considered in Section 8. Finally, in Section 9, some conclusions are drown from the discussion.
Section snippets
Prerequisite mathematics
An interval number is represented by closed interval and defined by . Where R is the set of real numbers and are the left and right limit of the interval number respectively. Also every real number can be represented by the interval number , for all . Definition 1 Interval-valued function Let , and consider the interval . From a mathematical point of view, any real number can be represented on a line. Similarly, we can represent an internal by a function. If the interval
Prey–predator model
We consider a simple model of prey–predator interaction with Malthusian growth proposed by Lotka [1] and Volterra [2]where and are all positive. Here denotes the size of the prey population and denotes the size of the predator population. It is assumed that the prey reproduction is influence by the predator only while the predator reproduction is limited by the amount of prey caught. It is also assumed that the prey population grows exponentially with a
Imprecise prey–predator model
By the construction of the prey–predator model the parameters such as prey population growth rate (r), predator population decay rate (s) and predation coefficients are positive in nature and are considered precise. Intuitively if any of the parameters are imprecise, furthermore when any parameter of the right hand side of Eqs. (3), (4) is interval number rather than a single value, then it is not so straight forward to convert equations to the standard from like (3), (4). For an
Dynamical behavior of the harvesting model
The equilibrium points of the parametric model (7), (8) is given by steady state equationsAfter algebraic calculation, we get the trivial and non trivial equilibrium points and where and are given byClearly and if .
Therefore we find that the existence of non-trivial equilibrium of Eqs. (7), (8) is totally dependent on the biotechnical productivity
Bionomic equilibrium of the imprecise prey–predator model
To discuss the bionomic equilibrium of the imprecise prey–predator model, we consider the parameters such asThe net economic rent or net revenue () at any time is given bywhere , i.e., and represent the net revenues for the prey and predator
Optimal harvesting policy for the harvesting model
In this section, our objective is to maximize, the objective functional form of the harvesting model, with the instantaneous annual rate of discount is as follows:subject to the state constraints (7), (8) and the control constraintsHamiltonian for the problem is given bywhere and are adjoint variables.
Numerical examples
In this section, two numerical examples are given to illustrate the proposed methodology presented in this paper. Example 1 Let us consider a set of artificial values of parameters as follows in appropriate units: , , , , , , , and .
The trivial equilibrium point always exist for all values of . The non trivial equilibrium points and the eigenvalues of variational matrices at the corresponding points of equilibrium are given
Conclusions
Prey–predator harvesting model has undergone different development in theoretical and practical applications in the field of biomathematics. Most of the researchers have developed the prey–predator harvesting model based on the assumption that the biological parameters are precisely known but the scenario is different in real life situation. In this paper, we developed a method to find the biological equilibrium points, bio-economic equilibrium points and optimal harvesting policy when some
Acknowledgements
The authors would like to express their gratitude to the Editor and Referees for their encouragement and constructive comments in revising the paper.
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