FOOD CHAIN MODEL WITH LOGISTIC GROWTH AND SELECTIVE OPTIMAL HARVESTING UNDER FUZZY ENVIRONMENT

: A multispecies food chain harvesting model is formulated based on Lotka-Voltera model with three species which are affected not only by harvesting but also by the presence of prey, predator and the super predator. In order to understand the dynamics of the system, it is assumed that the all three species follows the logistic law of growth. Further, there is demand for prey predator species in the market and hence selective harvesting of two species is performed. We derive the condition for global stability of the system using a suitable Lyapunov function. The possibility of existence of bioeconomic equilibrium is discussed. The optimal harvest policy is studied and the solution is derived under imprecise inflation in fuzzy environment using Pontryagin’s maximal principle. Finally some numerical examples are discussed to illustrate the model.

: the common catching effort.
vi. 12 : predator response rates towards the prey. vii. 21 : the rate of conversion of prey to predator. viii.
23 : super-predator response rates towards the predator.
ix. 32 : the rate of conversion of predator to super-predator. x.
: constant fishing cost per unit effort.

INTRODUCTION
In recent days the important part of research on biological modelling is the bioeconomic modelling of exploitation of biological resources such as fisheries and forestry's etc. In the literature, there are some single species [2,3] models in fisheries. But, in reality, marine fisheries consist of multispecies of which one may be prey and others predators and super predators which make a complex ecological food chain. Moreover, both prey and predators are eaten by different sections of people in the society and also used as different medicinal ingredients, so all three species have the demand in the market. Thus, the biological as well as economical study of exploitation of multi-species is now an emerging field of research in society. Also as this field of research includes from fisherman to scientist of all subjects, so now-a-days ecological modelling is very vast area for researchers.
Initially, Clark [5,6] introduced this type idea with the technique to approaching for the result.
Normally the main objective of the study of multi-species marine fisheries problems is to investigate the conditions/constraints for bionomic equilibrium of the species and to determine the optimum harvesting policy of the species in order to maximize the present value of the revenues earned from them without disturbing the ecological balance amongst the species.
Initially in this field of study Clark [5] first presented an optimal equilibrium policy for the harvesting of two independent species. Later using this concept, Chaudhuri [1,2,3] formulated and solved the optimal control problem for combined harvesting of two competing species in deterministic environment. Later Chaudhuri and Saha Ray [4], Mesterton-Gibbons [7], Kar & 3 FOOD CHAIN MODEL Chaudhuri [12] and some others studied the two species prey-predator fishery models for optimal harvesting of both the species. There are only few fisheries models with three species-prey, predator and super predator with harvesting. Recently Kar and Chaudhury [9] considered a model with two competing prey and one predator. Some more investigations on biological food-chain models [8,13,14,15,16,17,18,19,20] have also been reported recently. However, till now none considered food chain model with logistic growth and harvesting for three species-one prey, one predator and one super predator with harvesting fuzzy environment. As mentioned above, in the world, there are some communities who eat even super-predators. Moreover, these may now-adays be used for some other purposes also i.e. for medicines, etc. They also did not consider the optimal harvesting policy in fuzzy environment taking imprecise inflation and discount rates for food-chain system.
In this paper, an optimal harvesting of three species food chain-the first one is a prey, the second is a predator and third is a super predator which feeds on predator is formulated. The logistic growth of all three species is assumed and selective harvesting of prey and predator species is considered. The local stability, global stability and the bioeconomic equilibrium of the system are studied and the necessary conditions/constraints are derived. Taking the inflation and discount into account and considering these to be imprecise in nature cf. Maiti and Maiti [10], the problem is formulated as an optimal control problem for maximum return of revenue and solved for optimum harvesting of the species using Pontryagin's maximal principle. Lastly, some numerical experiments and simulations are depicted to illustrate the model.

MODEL FORMULATION
Let us consider three marine fish species for example Scoliodon sorakwa (shark), letes calcaifera (bhetki), sardinella longicepts (sardine) which make a food chain system and prey-predator are subjected to harvesting continuously. In this system we assume that the super predator (shark) feeding on predator species only and there is no competition between the species. Here the predator which lives on prey and super predator which lives on predator both these species grow according to the logistic growth along with the prey species (i.e. the population density of each species is resource limited).
The catch rate functions 1 and 2 are based on CPUE (CATCH-PER-UNIT EF-FORT).

LOCAL STABILITY
To discuss the local stability of the system first we need to construct the variational matrix ( , , ) corresponding to the system (1).
The variational matrix ( , , ) is given by:

BIONOMIC EQUILIBRIUM
The term bionomic equilibrium of a biological system is the combination of biological equilibrium as well as economic equilibrium. As we already know that the biological equilibrium is obtained by solving = = = 0. Also the economic equilibrium is said to be achieved when TR (the total revenue obtained by selling the harvested biomass) equals TC (the total cost for the effort devoted to harvesting).
Which indicates that for three species of a food chain also an infinite inflation leads to complete dissipation of economic revenue. This result was also initially investigated by Clark [5] in a combined harvesting of two species and recently by Chaudhuri [1] and also by Kar & Chaudhuri [9].
Similarly, also we get, Therefore from (55) and (56) we have As, we have = ( −1 ), = 1, 2, 3. So, from (57) is of ( −1 ) and hence is a decreasing function of (≥ 0). Therefore, we can conclude that = 0 (that is the economic environment when inflation rate it and discount rate are equal) and which gives the maximization of .

CONCLUSION
In this paper we formulated a food chain model of three species, prey, predator and super predator with logistic law of growth and selective harvesting of prey and predator species is considered.
The existence and stability of this system under possible steady states are investigated. The possibility of existence of bioeconomic equilibrium and global stability has been discussed and optimal harvesting policy is investigated with imprecise inflation and discount using Pontryagin's 13 FOOD CHAIN MODEL Maximal principle. Finally, the model is illustrated with the help of a numerical example and MATLAB simulations.