Journal of Environmental Accounting and Management
Fishery Type Based PreyPredator Optimal Harvesting Model
under Intuitionistic Fuzzy Environment
Journal of Environmental Accounting and Management 11(3) (2023) 307327  DOI:10.5890/JEAM.2023.09.005
M. Mukherjee$^{1}$, D. Pal$^{2}$, S. K. Mahato$^{1}$
$^{1}$ Department of Mathematics, SidhoKanhoBirsha 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 LotkaVolterra 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
preypredator 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 preypredator model is analyzed by using
BendixonDulactest. Furthermore, numerical simulations are presented in
tabular form and graphical form to support the theoretical results.}
[\hfill Preypredator 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.]
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\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.
Preypredator interaction is the best way to represent fishery management
mathematically. Preypredator 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,
preypredator problems are very interesting field of research for many
researchers. Liu and Huang \cite{16} presented harvesting strategies of
preypredator system with the help of Holling type IV functional responses. Sarkar et al \cite{17} studied preypredator mathematical system with different kinds of functional response.
Choi and Kim \cite{18} proposed LotkaVolterra based constant
harvesting problem. Dynamical performance of a preypredator system with
constant predator harvesting is presented by Mondal and Samanta \cite{19}. Li et al \cite{20} presented differential algebraic
preypredator system with nonlinear prey harvesting. Chakraborty et al \cite{21} studied a preypredator stage structured optimal harvesting
model and investigated bifurcation phenomenon. Liao et al \cite{22}
considered a preypredator 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 preypredator 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 preypredator 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 preypredator 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 preypredator 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 preypredator 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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