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Journal of Environmental Accounting and Management 11(3) (2023) 307--327 | DOI:10.5890/JEAM.2023.09.005

M. Mukherjee$^{1}$, D. Pal$^{2}$, S. K. Mahato$^{1}$

$^{1}$ Department of Mathematics, Sidho-Kanho-Birsha University, Purulia, West Bengal, 723104, India

$^{2}$ Chandrahati Dilip Kumar High School (H.S.), Chandrahati, West Bengal, 712504, India

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Abstract

Depending upon Lotka-Volterra model along with biological parameters, intuitionistic fuzzy in nature, this paper deals with harvesting system of prey species and predator species. Our aim is to analyze the prey-predator model whose numerical values are imprecise in nature. To get rid of the vagueness, we use the concept of triangular intuitionstic fuzzy numbers. We first develop the crisp model under some important assumptions. Then the crisp model is transformed into intuitionistic fuzzy model by using the concept of Hukuhara derivative and then it is crispified with the help of Yager's Ranking method. We investigate existence of the equilibrium points of above mentioned crispified system and the condition of the local stability of those points is obtained by analyzing the eigen values of the variational matrix. Then the condition of global stability is described by using suitable Lyapunov function. The economic aspects along with the harvesting policies at optimal stage are described. The existence of the limit cycle of the fuzzy prey-predator model is analyzed by using Bendixon-Dulac-test. Furthermore, numerical simulations are presented in tabular form and graphical form to support the theoretical results.} [\hfill Prey-predator model \par \hfill Optimal harvesting\par \hfill Stability \par\hfill Intuitionistic fuzzy\par \hfill Yager's Ranking][Dimitri Volchenkov][12 December 2021][14 June 2022][1 July 2023][2023 L\&H Scientific Publishing, LLC. All rights reserved.] \maketitle %\thispagestyle{fancy} \thispagestyle{firstpage} \renewcommand{\baselinestretch}{1} \normalsize \section{Introduction} The study of mathematical modeling in ecology becomes a great success in ecological science. As a result, the study of mathematical modeling enriched day by day. The journey of the theoretical ecology was started by Lotka \cite{1} and Volterra \cite{2}. Now a day, due to the increasing human population, exploitation of natural resources such as fisheries and forestry is a global problem to protect our ecosystem. Therefore, main objectives of the scientists and researchers should be to find the proper way to utilize the natural resources so that there is a balance between the developments of human society and smooth running of ecosystem. Prey-predator interaction is the best way to represent fishery management mathematically. Prey-predator systems (Clark \cite{3}, Fan and Wang \cite{4}, Zhang et al \cite{5}, Kar \cite{6}, Shih and Chow \cite{7}, Kim \cite{8}, Pei et al \cite{9}, Dong et al \cite{10}, Khajanchi \cite{11}, Khajanchi \cite{12}, Khajanchi and Banerjee \cite{13}, Sarkar and Khajanchi \cite{14}, Saha and Samanta \cite{15}) are basically represented by ordinary differential equations. Recently, prey-predator problems are very interesting field of research for many researchers. Liu and Huang \cite{16} presented harvesting strategies of prey-predator system with the help of Holling type IV functional responses. Sarkar et al \cite{17} studied prey-predator mathematical system with different kinds of functional response. Choi and Kim \cite{18} proposed Lotka-Volterra based constant harvesting problem. Dynamical performance of a prey-predator system with constant predator harvesting is presented by Mondal and Samanta \cite{19}. Li et al \cite{20} presented differential algebraic prey-predator system with nonlinear prey harvesting. Chakraborty et al \cite{21} studied a prey-predator stage structured optimal harvesting model and investigated bifurcation phenomenon. Liao et al \cite{22} considered a prey-predator diffusion model with harvesting and studied dynamical behaviour of the model. However, it is admissible that the biological coefficients used in mathematical formulation of the ecological phenomenon/food chain model etc. are not always precisely known due to various reasons. In reality, almost every species in the universe is implanted in sporadically fluctuating atmospheres. For example, the reproduction rate of various bacteria throughout the day is deeply affected due to temperature differences in aquatic as well as terrestrial environments. Again, in the same environment, light concentration regulates photosynthesis process throughout the periods. Therefore, it is necessary to recognize the imprecise biological parameters that control the dynamical behaviour of the specified community. So, the interaction among the species and its dynamical behaviour are immensely affected by these imprecise/vague parameters. Also, deterministic ecological modeling approaches need definite biological coefficients. But in reality, many model parameters can only be estimated crudely. To handle such crudely estimated parameters, many researchers take the help of stochastic method (Beddington and May \cite{23}, Li et al \cite{24}, Wu et al \cite{25}, Liu and Wang \cite{26}, Liu and Wang \cite{27}). In stochastic methodology, the vague parameters are replaced by random variables with recognized probability distribution functions. But in reality, it is undoubtedly difficult task to identify proper probability distribution functions for all vague parameters. Therefore, this approach may lead the model more complex. To avoid stated disadvantage of the stochastic methodology, we have another technique to represent the imprecise parameters known as fuzzy approach (Zadeh \cite{28}). The best advantage of the fuzzy approach is that the decision maker can describes vague parameters as precisely as a decision maker will be capable to define it. Therefore, fuzzy biological modeling approach is more acceptable than deterministic modeling approach. Researchers are attracted by the advantages of the fuzzy modeling approach. Few researchers have already applied the concept of fuzzy set theory in the field biomathematics. Fuzzy population dynamical system with vague parameters and variables is proposed by Bassanezi et al \cite{29}. Barros et al \cite{30} considered life expectancy system with fuzzy nature of the parameters. Fuzzy parameters based prey-predator interaction is considered by Peixoto et al \cite{31} and they observed the dynamics of the system under fuzziness. Mizukoshi et al \cite{32} proposed a prey-predator model under fuzzy initial conditions. Fuzzy logistic growth model and fuzzy Gompertz growth model are extensively studied by Guo et al \cite{33}. Das et al \cite{34} offered a disease control prey-predator model with uncertain fuzzy parameters. Das et al \cite{35} presented an epidemic model in fuzzy environment. Pal et al \cite{36} studied proportional harvesting model under fuzzy uncertain environment. Also, Pal et al \cite{37}, Pal and Mahapatra \cite{38}, Pal et al \cite{39} and Sharma and Samanta \cite{40} proposed harvesting models with imprecise biological parameters. From the above literature survey we observe that some researchers considered ecological model in terms of interval numbers, fuzzy numbers, stochastic approach and the combinations of these. But as per the authors concerned, no one has considered the ecological model in terms of intuitionistic fuzzy numbers (Mitchell \cite{41}) to handle the uncertain biological parameters. Fuzzy numbers are the generalization of the real numbers whereas the intuitionistic fuzzy numbers are the generalization of the fuzzy numbers and so the uncertainty can be expressed more appropriately by intuitionistic fuzzy numbers. So, we desire to formulate our proposed model in intuitionistic fuzzy environment. In the present research work, we propose one prey and one predator model with logistic growth with vague biological coefficients. We also assume that both the species are subjected to different harvesting effort by the harvesting agencies. At first, we have considered a crisp harvesting model under some important assumptions. Then the crisp model is changed into intuitionistic fuzzy model. We present the imprecise coefficients of our model by the intuitionistic fuzzy numbers rather than triangular fuzzy numbers to make our model more general. If we harvest both species unlimitedly then both the species goes to extinct. So, we perform optimal control harvesting policy to control the species as well as the ecosystem. In the present paper, fuzzy prey-predator system with harvesting is presented by a pair of nonlinear intuitionistic fuzzy differential equations by using the concept of Hukuhara derivative. Then the intuitionistic fuzzy model is crispified by the Yager's ranking index. Our aim of the present paper is to analyze its dynamical behavior, economical features, and finally optimal harvesting strategies of the proposed system under intuitionistic fuzzy version. A proper example of the proposed system is Bhetki fish (Lates Calcaifera) and its prey Sardine fish (Sardinella Longiceps). In the present article, Lates calcarifer (Bhetki) is treated as predator and Sardinella longiceps (Sardine) as prey. Bhetki fish lives on consuming Sardine fish and both prey species and predator species are harvested by the harvesting agencies. The organization of the present paper is as follows. In section 2, we discussed some useful and relevant mathematical definitions. Model formulations of the proposed model along with assumptions and notations are delivered in section 3. In section 4, theoretical studies of intuitionistic fuzzy model is presented. Economical features of the intuitionistic fuzzy harvesting model are constructed in section 5. The existence of limit cycle of intuitionistic fuzzy prey-predator model is studied in section 6. Section 7 illustrates the numerical simulations. Finally, we draw a conclusion in section 8. \begin{figure

Acknowledgments

The authors would like to express their gratitude to the Editor Dimitri Volchenkov, and the anonymous Referees for their encouraging and constructive comments for the improvement of the manuscript.

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