AMathematical Model for Optimal Management and Utilization of a Renewable Resource by Population

A dynamical model is proposed and analyzed to study the effect of the population on the resource biomass by taking into account the crowding effect. Biological and bionomical equilibria of the system are discussed. e global stability behavior of the positive equilibrium is studied via the output feedback control. An appropriateHamiltonian function is formed for the discussion of optimal harvesting of resource which is utilized by the population using Pontryagin’s Maximum Principal. A numerical simulation is performed on the model to analyze the theoretical results.


Introduction
Renewable resources are a very important source of food and materials which are essential for the growth and survival of the biological population.e continuous and unplanned use of these resources may lead to the extinction of resources and thereby affecting the survival of resource-dependent species.ere has been a considerable interest in the modeling of renewable resources such as �shery and forestry.Dynamic models for the commercial �shing have been studied extensively taking into account the economic and ecological factors [1,2].Based on the works of Clark [1,2], several investigations have been conducted [3][4][5][6][7][8][9][10][11][12].Leung and Wang [13] proposed a simple economic model and investigated the phenomena of nonexplosive �shing capital investment and nonextinctive �shery resources.Chaudhuri [14] proposed a model for two competing �sh species each of which grows logistically.He examined the stability analysis and discussed the bionomical equilibrium and optimal harvesting policy.It was also shown [14] that there is no limit cycle in the positive quadrant.Ragozin and Brown [15] proposed a model in which the prey has no commercial value and the predator is selectively harvested.When the prey and predator both are harvested, then an optimal policy for maximizing the present value and an estimation to the true loss of resource value due to catastrophic fall in stock level has been discussed in detail by Mesterton-Gibbons [16].In another paper, Mesterton-Gibbons [17] proposed a Lotka-Volterra model of two independent populations and studied an optimal harvesting policy.Fan and Wang [18] generalized the classical model of Clark [1,2] by considering the timedependent the Logistic equation with periodic coefficients and they showed that their model has a unique positive periodic solution, which is globally asymptotically stable for positive solutions.ey also investigated the optimal harvesting policies for the constant harvest and the periodic harvest.e optimal harvesting policy of a stage structure problem was studied by Zhang et al. [19].ey found conditions for the coexistence and extinction of species.Song and Chen [20] proposed two-species competitive system and discussed the local and global stability analysis of the positive equilibrium point of the system.ey have also discussed the optimal harvesting policy for the mature population.Dubey et al. [10] proposed a model where the �sh population partially depends on a resource and is harvested.ey examined stability analysis and the optimal harvesting policy with taxation as a control variable.Dubey et al. [11] discussed a model of a �shery resource system in an aquatic environment which was divided into two zones-the free �shing zone and the reserved zone.ey discussed biological and bionomical equilibria and optimal harvesting policy.Dubey et al. [12] also proposed and analyzed an inshoreoffshore �shery model where the �sh population is being harvested in both areas.en they investigated the stability analysis and optimal harvesting policy by taking taxation as a control instrument.Kar et al. [21] considered a two-prey one-predator model where both the preys grow logistically and harvested.Again Kar and Chottopadhayay [22] described a single species model, which has two stages: (i) a mature stage (ii) an immature stage; they discussed the existence of equilibrium points and their stability analysis.ey proved that the optimal harvesting policy is much superior to the MSY policy and optimal paths always take less time than the suboptimal path to reach the optimal steady state.Kar et al. [23] proposed a prey-predator model with a nonmonotonic functional response and both the species are harvested.To obtain the strategies for the management of the system, they used the harvesting effort as a control variable.ey found the stability condition of an interior equilibrium point in terms of harvesting effort and proved that there exists a super critical Hopf bifurcation.
From the above literature and to the best of our knowledge, it appears that harvesting of a renewable resource which is being utilized by a population for its own growth and development has not been considered by taking into account the effect of crowding.Hence, in this paper, we propose a mathematical model for the biological population which is being partially dependent on a renewable resource.is resource is further harvested for the development of the society.
e organization of the paper is as follows.Section 2 describes the development of the model and Section 3 gives a detailed outline of the stability analysis of the system.e bionomical equilibrium and the maximum sustainable yield are presented in Sections 4 and 5, respectively.e output feedback control is given in Section 6 and the optimal harvesting policy in Section 7. A numerical simulation experiment has been presented in Sections 8 and 9 is the concluding remarks followed by references.

Mathematical Model
Let us consider a renewable resource growing logistically in a habitat.en the dynamics of this resource biomass is governed by where  is the density of resource biomass,  0 is its intrinsic growth rate,  1 is an interspeci�c interference coe�cient, and  =  0 / 1 is the carrying capacity.
Let  be the density of the population at any time   0 which utilizes the resource biomass  for its own growth and development.us, the intrinsic growth rate and carrying capacity of the resource biomass will depend on the density of population.Hence, we assume that  0 and  in (1a) are functions of .us, (1a) reduces to where  0 is a positive constant.
We consider the following assumptions.
(i) e intrinsic growth rate  0  is a decreasing function of and it satis�es We take a particular form of  0  as (ii) e carrying capacity  is also a decreasing function of  and it satis�es We take a particular form of  as Let us denote  =  0 / 0 and  2 =  1  0 / 0 .en (1b) can be rewritten as Now, we consider a population of density  which is also growing logistically.We assume that the growth rate and carrying capacity of the population depend on the resource biomass density.If the resource biomass density increases, the growth rate and carrying capacity of population also increase.us, the dynamics of population is governed by the following differential equation: where  is the intrinsic growth rate of population,  is its carrying capacity in the absence of resource biomass,  1 and  2 are the growth rates of population in the presence of resource biomass.We assume that the resource biomass is harvested with the harvesting rate ℎ = , where  is a positive constant and in �shery resource it is known as catchability coe�cient, and  is the harvesting effort, which is a control variable.
Keeping the above aspect in view, the dynamics of the system can be governed by the system of the following differential equations: In Section 3, we present the stability analysis of model (8a)-(8b).

Stability Analysis
First of all, we state the following lemma which is a region of attraction for the model system (8a)-(8b).

Lemma 1. e set
is a region of attraction for all solutions initiating in the interior of the positive quadrant, where e above lemma shows that all the solutions of model (8a)-(8b) are nonnegative and bounded, and hence the model is biologically well behaved.
e proof of this lemma is similar to Freedman and So [24], Shukla and Dubey [25], hence omitted.Now, we discuss the equilibrium analysis of the present model.It can easily be checked that model (8a)-(8b) has four non-negative equilibria, namely, e equilibrium points  0 and  2 always exist.For the existence of equilibrium point  1 , we note that   is given by is shows that  1 exists if For the fourth equilibrium point  * , we note that  * and  * are the positive solutions of the following algebraic equations: Substituting the value of , from (13b) into (13a), we get a cubic equation in  as follows: where e above equation ( 14) has a unique positive solution  =  * , if the following inequality holds: Aer knowing the value of  * , the value of  * can then be calculated from the relation From ( 16), we note that for the coexistence of resource biomass and population together in a habitat, the intrinsic growth rate of the resource biomass must be larger than a threshold value.is threshold value depends upon the carrying capacity of the population and the harvesting effort.Now we discuss the local and global stability behavior of these equilibrium points.For local stability analysis, �rst we �nd variational matrices with respect to each equilibrium point.en by using eigenvalue method and the Routh-Hurwitz criteria, we can state eorems 2 to 5. eorem 2. (i) If the equilibrium point  1    0) exists, then  0 is always unstable in the - plane.
(ii) If the equilibrium point  1    0) does not exist, then  0 is a saddle point with stable manifold locally in the -direction and with unstable manifold locally in the -direction.eorem 3. e equilibrium point  1    0), whenever it exists, is a saddle point with stable manifold locally in the -direction and with unstable manifold locally in the -direction.eorem 4. (i) If    1  + ), then  2 0 ) is a saddle point with unstable manifold locally in the -direction and stable manifold locally in the -direction.
In the next theorem, we are able to �nd a su�cient condition for  * to be globally asymptotically stable.eorem 6.Let the following inequality holds in Ω: en the interior equilibrium  * is globally asymptotically stable with respect to all solutions initiating in the interior of the region Ω de�ned in Lemma 1.
Proof of eorem 6. Proof of this theorem is given in the Appendix.
Our next result shows that the model under consideration cannot have any closed tra�ectories in the interior of the �rst quadrant.
is shows that Δ  does not change sign and is not identically zero in the positive quadrant of the - plane.By Bendixon-Dulac criteria, it follows that the system (8a)-(8b) has no closed trajectory, and hence no periodic solution in the interior of the positive quadrant of the - plane.

Bionomical Equilibrium
In this section, we study the bionomical equilibrium of the model system (8a)-(8b).Bionomical equilibrium is the level at which the total revenue (TR) obtained by selling the harvested biomass in an economic equilibrium case is equal to the total cost (TC) to the harvested biomass, that is, the economic rent is completely dissipated.e net economic revenue at time  is given by where  is price per unit biomass and  is utilized cost per unit resource biomass.e bionomical equilibrium is  ∞  ∞   ∞   ∞ , where  ∞   ∞ , and  ∞ are the positive solutions of Solving (22a), we get It is clear that us the bionomical equilibrium  ∞  ∞   ∞   ∞  exists under condition (23).

The Maximum Sustainable Yield
e maximum sustainable yield (MSY) of any biological resource biomass is the maximum rate at which it can be harvested and any larger harvest rate will lead to the depletion of resource eventually to zero.In the absence of any population, the value of MSY is given by [1] If the resource biomass is subjected to the harvesting by a population, the sustainable yield is given by We note that us, ℎ MSY   *  *  +  2  *  * , when  *   −    * 2 +  2  * .
From the above equations, it is interesting to note that when  *  , then  *  2 and ℎ MSY  4  ℎ  MSY .is result matches to Clark [1].If ℎ  ℎ MSY , then it denotes the overexploitation of the resource and if ℎ < ℎ MSY , then the resource biomass is under exploitation.

Output Feedback Control
e habitat under our consideration consists of a resource biomass which is utilized by a population.e resource biomass is being harvested and the harvesting effort    is considered as an input.e harvesting effort is applied to the stock and produces a yield    per unit effort.We assume that the total yield  per unit effort is subject to the constraint  min <  <  max .en our objective is to construct an output feedback control     +  in such a way that the steady state  *  *   *  is globally asymptotically stable for the closed-loop system.en model equations (8a)-(8b) can be written as in the vector matrix differential equation form Now, under an analysis similar to Louartassi et al. [26] and Mazoudi et al. [27], one can prove the following theorem.eorem 9. �or any constant �shing e�ort  0 , there exists a   0 such that  * is globally asymptotically stable through the output feedback control law    −  * , where  *   * .

Optimal Harvesting Policy
In this section, we discuss the optimal management of a renewable resource in the presence of population which is to be adopted by the regulatory agencies to protect the resource and to ensure the survival of the population with a sustainable development.e present value  of a continuous time stream of revenues is given by where  is the instantaneous rate of annual discount.us our objective is to max , subject to the state equations 8a -8b and to the control constraints 0 ≤  ≤  max .
For this purpose, we use Pontryagin's Maximum Principle.e associated Hamiltonian function is given by where  1 and  2 are adjoint variables and    −  −  −  1  is called a switching function.e optimal control  which maximizes  must satisfy the following conditions: 0, when   < 0, that is, e usual shadow price is  1   and the net economic revenue on a unit harvest is  − .is shows that if the shadow price is less than the net economic revenue on a unit harvest, then    max and if the shadow price is greater than the net economic revenue on a unit harvest, then   0. When the shadow price equals the net economic revenue on a unit harvest, that is,   0, then the Hamiltonian  becomes independent of the control variable , that is,   0.
is is a necessary condition for the singular control  *  to be optimal over control set 0 <  * <  max .
us, the optimal harvesting policy is when   0, we have Now, in order to �nd the path of singular control, we utilize the Pontrygin's Maximum Principle.According to this principle, the adjoint variables  1 and  2 must satisfy e above equations can be rewritten as Using (13a)-(13b), the above equations can be rewritten as Equation (38) can again be written as where Solving (38), we get We note that when   , then the shadow price  2   is bounded if  0 = 0. us, we have Now from (37), we get where Equation ( 43) yields Again for the shadow price  1   to be bounded, we must have  1 = 0. us, Equations ( 34) and (46) yield Substituting the values of  1 ,  1 ,  2 , and  2 into (47), we get Hence, solving (13a)-(13b) with the help of (48), we get an optimal solution (  ,   ) and the optimal harvesting effort  =   .

Numerical Simulations
In order to investigate the dynamics of the model system (8a)-(8b) with the help of computer simulations, we choose the with initial condition: (0) =  and (0) = .
For this set of parameters, condition (16) for the existence of the interior equilibrium is satis�ed.is shows that  * ( * ,  * ) exists and it is given by  * = 0.4790,  * = 102.310.
us the interior equilibrium point  * ( * ,  * ) is locally asymptotically stable.It is noted that condition (18) in eorem 6 is not satis�ed for the set of parameters chosen in (49).Since condition ( 18) is just a sufficient condition for  * to be globally asymptotically stable, no conclusion can be drawn at this stage.e behavior of  and  with respect to time  is plotted in Figure 1 for the set of values of parameter chosen in (49).From this �gure, we see that the density of population increases, whereas the density of resource biomass decreases with respect to time, and both settle down at its equilibrium level.For the set of values of parameters given in (51), it may be noted that the positive equilibrium  * ( * ,  * ) exists and it is given by  * = 17.4311,  * = 117.0579.
It may be cheeked that in this case (for the values of parameters given in (51)), condition (18) in eorem 6 is satis�ed.is shows that  * ( * ,  * ) is locally as well as globally asymptotically stable in the interior of the �rst quadrant.In Figure 2, we have plotted the behavior of  and  with different initial values.Figure 2 shows that all the trajectories starting from different initial points converge to the point  * (17.4311, 117.0579).is shows that  * (17.4311, 117.0579) is globally asymptotically stable.
It may be noted here that  2 and  2 are important parameters governing the dynamics of the system.erefore, we have plotted the behavior of  and  with respect to time for different values of  2 in Figure 3 and for different values of  2 in Figure 4.
From Figure 3(a), we note that  decreases as  2 increases.If  2 is very small, then  initially increases slightly and then decreases and converges to its equilibrium level.If  2 increases beyond a threshold value, then  always decreases and settles down at its equilibrium level.Figure 3(b) shows that  also decreases as  2 decreases.is is due to the fact that with the increase in  2 , the equilibrium level of  decreases and since the population  is dependent on the resource biomass , thus  also decreases.However, it is interesting to note here that  always increases with respect to time  and �nally attains its equilibrium level.
From Figure 4(a), it is noted that  decreases as  2 increases.Figure 4(b) shows that  increases as  2 increases.It may also be noted that for all positive values of  2 ,  increases continuously with time and �nally settles down at its steady state.However, the resource biomass  initially increases for some time, then decreases continuously and �nally gets stabilized at its lower equilibrium level.is shows that if the population utilizes the resource without any control, then the resource biomass decreases continuously and it may be doomed to extinction.In order to see the qualitative behavior of the optimal harvesting resource, we solve (13a), (13b), and (48) for the same set of values of parameters as given in (51) with the additional values as  = 0.001 and  = 5. en we get the optimal equilibrium levels as given below: We observe that  is a very important parameter which governs the dynamics of the system.e behavior of  with respect to time is shown in Figure 5 and the behavior of  with respect to time is shown in Figure 6 for different values of . Figure 5 shows that if  is less than its optimal level (  = 146.201),then  increases and settles down at its equilibrium level.If     , then  decreases and settles down at a lower equilibrium level.If  is large enough, then  tends to zero. Figure 6 shows that if     , and then  increases with time and �nally obtains its equilibrium level.If     , then  again increases with time, but it settles down at a lower equilibrium level.is is due to the fact that when     , then  decreases and hence  also decreases.is suggests that the harvesting effort  should be always kept less than   (optimal harvesting level) so that both the resource and the population can be maintained at an optimal level.

Conclusions
In this paper, a mathematical model has been proposed and analyzed to study the effect of harvesting of a renewable resource.It has been assumed that the resource is being utilized by a population for its own growth and survival.e  resource and the population both are growing logistically.e existences of equilibria and stability analysis have been discussed with the help of the stability theory of ordinary differential equations.It has been shown that the positive equilibrium  * (whenever it exists) is always locally asymptotically stable.But  * is not always globally asymptotically stable.However, we have found a sufficient condition under which  * is globally asymptotically stable.is condition gives a threshold value of the speci�c growth rate () of resource biomass.If  is larger than this threshold value, then  * is globally asymptotically stable.Using Bendixon-Dulac criteria, it has been observed that the model system has no limit cycle in the interior of the positive quadrant.We have also discussed the bionomical equilibrium of the model and it has been observed that the bionomical equilibrium of the resource biomass does not depend upon the growth rate and carrying capacity of the population.An analysis for sustainable yield (ℎ) and maximum sustainable yield (ℎ MSY ) has been carried out.It has been shown that if ℎ > ℎ MSY , then the resource biomass will tend to zero and if ℎ < ℎ MSY , then the resource biomass and population may be maintained at desired level.e global asymptotic stability behavior of the positive equilibrium has also been studied through the output feedback control method.
We have constructed a Hamiltonian function and then using Pontryagin's Maximum Principle, the optimal harvesting policy has been discussed.An optimal equilibrium solution has been obtained.e results are validated based on the numerical simulation.Here we observed that  2 ,  2 and  are important parameters governing the dynamics of the system.It has been found that if  2 increases, then the resource biomass and the population both decrease.However, if  2 increases, then the population density increases and the resource biomass density decreases.A threshold value for the optimal harvesting effort   has been found theoretically as well as numerically.It has been shown that the harvesting effort  should be always kept less than   to maintain the resource and the population at an optimal equilibrium level. eorem

F 1 :
Plot of  and  versus  for the values of parameters given in (49).followingset of values of parameters (other set of parameters may also exist):

F 3 :
Plot of  and  with respect to time  for different values of  2 , and other values of parameters are the same as given in (51).

F 4 :
Plot of  and  with respect to time for different values of  2 with  2 = 0.0001 and other values of parameters are the same as in Figure2.

18 F 5 :
Plot of  versus  for different values of .

F 6 :
Plot of  versus  for different values of .

7 .
System (8a)-(8b) cannot have any limit cycle in the interior of the positive quadrant.Proof.Let       is a continuously differential function in the interior of the positive quadrant of - plane.