A Stage-Structured Prey-Predator Fishery Model In The Presence Of Toxicity With Taxation As A Control Parameter of Harvesting Effort

In this paper we have considered stage-structured fishery model in the presence of toxicity, which is diminishing due to the current excessive use of fishing efforts resulting in devastating consequences. The purpose of this study is to propose a bio-economic mathematical model by introducing taxes to the profit per unit biomass of the harvested fish of each species with the intention of controlling fishing efforts in the presence of toxicity. We obtained both boundary and interior equilibrium points along with the conditions ensuring their validity. Local stability for the interior equilibrium point has been found by the trace-determinant criterion and global stability has been analyzed through a suitable Lyapunov function. We have also obtained the optimal harvesting policy with the help of Pontryagins maximum principle. Lastly, numerical simulation with the help of MATLAB have been done and thus, the results of the formulated model have been established.


Introduction
Overfishing of commercial fish species is a grave problem due to the rapid growth of industrialization and population.Various surveys have indicated that there has been a rapid decline of fish species.Given the economic importance of the fishery, management measures aiming at controlling fishing efforts are needed for sustainability of the species.Possible control instruments for regulating harvesting efforts as were pointed out by [3] could be taxation, license fees, lease of property rights, seasonal harvesting, fishing period control, creating reserve zones and many more depending on the nature of the fishery.Open access is the condition where access to the fishery (for the purpose of harvesting fish) is unrestricted; i.e., the right to catch fish is free and open to all.So the taxation method could apply as the efficiency method.Authors [1]- [8] have suggested that governments or fishing regulatory authorities can use taxation as an effective control instrument to regulate the extent of fishing efforts.[10] proposed a mathematical model to study the growth and exploitation of a schooling fish species by imposing a tax on the catch to control the overexploitation of fish species.[2] discussed a dynamical model for a single species fishery, which depends partially on logistically growing resource with functional response and taxation as a control instrument to protect fish population from overexploitation.[9] studied a fishery model containing predator fish and prey fish in which the predator was the commercial fish by including spawning periods and taxation.[5] studied a dynamic model for fishery resource with reserve area and taxation.[4] further analyzed a non-linear mathematical model to study the dynamics of an inshore-offshore fishery under variable harvesting by considering taxation as the control instrument [9]- [19], [22].Moreover, the effect of toxic substance on ecological communities have posed huge problems on environment.Such type of problems in mathematical modeling were studied by scholars.In the real world, almost all animals have the stage-structure of immature and mature.However, no attempt has been made to study the optimal taxation policy of a stage-structured three species fishery in the presence of toxicity in which they interact in a prey-predator manner and all species being subjected to harvesting [20]- [21].

Mathematical Model
In this section, we discuss the mathematical model describing the stage-structured predator-prey model where each fish species-prey, immature predator and mature predator is infected by toxic materials released by some external sources like factories, industries, etc. and also subjected to harvesting.The model is developed as under: 1.In the Lotka Volterra prey-predator model, the prey population is growing logistically at the rate r 1 with carrying capacity L in the absence of predator species.So we consider the first term of prey species as r 1 x 1 (1 − x 1 L ).The mature predator consumes the prey species at the rate β .We suppose that the mature predator attack the prey at the rate of β x 1 .In this model, we have also considered different harvesting rates of each species rather than the same harvesting rate.Also, the effect of toxicity on the prey population is measured by γ 1 x 2 1 .Since, shows that there is a growth in the production of toxic materials of the prey population and more of the prey species consume the toxic substance.Thus, the dynamics of prey population is governed by: where, is the stock biomass of prey species at time t, is the stock biomass of mature predator species at time t, r 1 is the intrinsic growth rate of prey species, β is the rate of interaction of prey with mature predator, γ 1 is the coefficient of toxicity to the prey species, q 1 is the catchability coefficient of prey species, is the fishing effort of prey population.
2. The dynamics of immature predator is governed by: where, is the stock biomass of immature predator population at time t, is the stock biomass of mature predator population at time t, α is the growth rate of immature predator because of mature predator, µ is the death rate of predator species, γ 2 is the coefficients of toxicity to the immature predator, δ is the conversion rate of immature predator population to mature predator population, q 2 is the catchability coefficient of immature predator species, E 2 =E 2 (t) is the fishing effort of immature predator population.
3. The dynamics of mature predator is governed by: where, is the stock biomass of prey population at time t, is the stock biomass of immature predator population at time t, is the stock biomass of mature predator population at time t, θ is the conversion rate from prey to predator, γ 3 is the coefficients of toxicity to the mature predator, q 3 is the catchability coefficient of mature predator species, is the fishing effort of mature predator species.
4. We impose taxations on the fishing efforts so as to sustain fishing of the species.Thus E 1 E 2 and E 3 are dynamic variables i.e. time dependent.Let p 1 , p 2 and p 3 be the fixed selling price per unit population of prey, immature predator and mature predator species respectively and let c 1 , c 2 and c 3 be the fixed cost of harvesting per unit effort for the prey, immature predator and mature predator species respectively.Thus, the economic revenue for the species will be: Let τ 1 > 0, τ 2 > 0 and τ 3 > 0 be the taxes imposed per unit prey population, immature predator population and mature predator population harvested respectively.The net economic revenue is obtained by the introduction of taxes to the fixed selling price per unit population of fish species.Hence, the above equations become: 5. Thus, using (2.1), (2.2), (2.3) and above equations, the system of equations become: ) (2.14) ϕ j for j=1,2,3 are adjustment coefficients.

Equilibrium points
The steady state solutions are obtained from the following system of equations: Thus, From (3.17) we get, Thus, i.e., International Scientific Publications and Consulting Services Thus, the system has the following 18 possible equilibrium points: , 0, 0, 0, 0, 0) where, p 1 > τ 1 .
where p 2 > τ 2 . where where where International Scientific Publications and Consulting Services , 0, 0, 0) where where where where where where where International Scientific Publications and Consulting Services

Local stability
We investigate the local stability of the equilibrium points.Here, the trace-determinant criteria is used.An equilibrium point is locally stable if the Jacobian matrix evaluated at that point has a positive determinant and negative trace.The Jacobian matrix of the system is a 6X6 matrix given by: where, Therefore, International Scientific Publications and Consulting Services Thus, the jacobian at the equilibrium point P 18 is given by: where, International Scientific Publications and Consulting Services ] Thus, trace is given by: trace and determinant is given by: det[J(P 18 )] = −n 14 * n 41 * −n 52 * n 63 * n 25 * −n 36 > 0 Thus, the equilibrium point P 18 is locally stable.

Global stability
Global stability is investigated through a suitable Lyapunov function.
Thus, the time derivative is given by: International Scientific Publications and Consulting Services where, Now, substituting the values of G, H, I, K, L, R in dV dt , we get: which is: where, and Thus, we get, Therefore, the equilibrium point P 18 is globally stable.

Optimal Harvesting Policy
In this section, we analyze the optimal harvesting policy for the system in (2.10-2.15)so as to maximize the total discounted net revenue using taxation as a control parameter.The present value 'J' of a continuous time-stream of revenues is : where δ =instantaneous rate of annual discount.Thus, our objective is to maximize 'J' subject to the equations in (2.10-2.15)and to the control parameters: τ min j < τ < τ max j for j=1,2,3.
To find the optimal equilibrium, Pontryagin's maximum principle is used.The associated Hamiltonian function is given by: where, λ 1 , λ 2 , λ 3 , λ 4 , λ 5 and λ 6 are adjoint variables in terms of time.
Hamiltonian 'H' should be maximized for τ(t) ∈ [τ min j , τ max j ] where j=1,2,3.We assume that the control constraints are not binding(that is, the optimal solution does not occur at τ(t) = τ min j or τ max j for j=1,2,3).Thus, we have singular control given by: Applying (4.26), we obtain, By Pontryagin's maximum principle, again we have: Considering (4.29a) we get, Substituting the value of λ 3 from above, we get: where, Now, employing an I.F.= e −A 1 t to solve (4.30) resulted into: where T 1 is a constant of integration.
Hence, (4.37) can be rewritten as, Now, considering 4.29c) we get, Substituting the values of λ 1 and λ 2 , we get, where, Now, employing an I.F.= e −D 1 t to solve (4.39) resulted into the following, where T 3 is a constant of integration.
2. We have also plotted the fish population     We have developed the mathematical model involving stage structure, taxation policy and effect of toxicity on fish species.We have found out eighteen equilibrium points out of which one is interior and the others are boundary equilibrium points.For the interior equilibrium point, we have proved that it is locally stable using trace-determinant criteria.Also by using Lyapunov function, we have proved that the interior equilibrium point is globally stable when S 1 > S 2 .We have also found out the optimal harvesting policy by using Pontryagin's Maximum Principle.Finally, we have verified our results with the help of numerical simulations by using MATLAB software to solve our system of equations.

( 4
.48) International Scientific Publications and Consulting Services

x 1 ,
x 2 , x 3 with respect to time for different values of harvesting efforts E 1 , E 2 , E 3 which are shown in Fig 5,6,7 and 8 and observed that as the harvesting efforts increases, the fish population decreases and gradually move towards extinction.International Scientific Publications and Consulting Services

Figure 1 :Figure 2 :
Figure 1: Effect of taxation on the system

Figure 3 :Figure 4 :
Figure 3: Effect of taxation on the system

Figure 5 :Figure 6 :
Figure 5: Effect of harvesting efforts on the system

Figure 7 :Figure 8 :
Figure 7: Effect of harvesting efforts on the system