EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING

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1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 6, Number 3, Pages c 21 Institute for Scientific Computing and Information EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING TAPAN KUMAR KAR AND KUNAL CHAKRABORTY Communicated by Guofeng Zhang) This paper is dedicated to our authors Abstract. In a fully dynamic model of an open-access fishery, the level of fishing effort expands or contracts according as the net economic revenue to the fishermen is positive or negative. A model reflecting this dynamic interaction between the net economic revenue and the effort in a fishery is called a dynamic reaction model. In the present paper we consider a prey-predator type fishery model with a partial closure of prey species. The steady states of the systems are determined, and the dynamical behavior of both the species is examined. The optimal harvest policy is formulated and solved with the help of Pontryagin s maximal principle. Results are illustrated with the help of numerical examples. Key Words. Prey-predator, dynamic reaction, global stability, bifurcation, optimal harvesting. 1. Introduction Major parts of human population in developing countries earn their livelihood by fishing. But over exploitation of fisheries is a serious and immediate global problem that current management policies struggle to solve. Scientists and researchers consider that new strategies based on a long term approach are necessary for future fisheries management and sustainable development of ecosystem. One potential solution to these problems is the creation of marine reserves in which fishing is restricted. The implementations of marine reserve areas can protect and enhance the stock biomass by protect the species inside the reserve area and increase fish abundance in adjacent areas. From several economic researches it is noted that economists in general underline different problems of fisheries management and provide the solution as implementation of marine reserve. According to them marine reserve must be seen as one of the many tools of fisheries management and the how, where and for what fisheries, reserves are implemented is of great concern. The migration rates of the stock biomass between the protected and unprotected areas provide the way to recover the exploited stock and thus enhance the stock biomass. The use of MPAs is directed towards ecosystem functioning where ecosystems are easily disrupted by fishing efforts, reserves may be a more appropriate option. For the two fisheries as a whole, the creation of certain size protected areas can Received by the editors May 9, Mathematics Subject Classification. 34K2, 92D

2 EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING 319 yield some benefits in terms of overall harvest and resource rent. Conard 1999) observed two possible benefits from the creation of the protected area. First, the overall variation in biomass and as such harvest; and second, it may reduce the costs of management mistakes. Dixon 1993) shown that the objectives of creation of a marine protected area can lead to conflicts between the various economic agents involved with the area. The study of the population dynamics in harvesting is a subject of mathematical bio-economics, which in turn related to the optimal management of renewable resources Clark, 199). Kar and Misra 26) examined the existence of possible steady states along with their local and global stability in a prey-predator fishery with prey reserve. They also examined the possibilities of existence of bionomic equilibrium and provide the optimal harvesting strategy using Pontryagin s maximum principle. Kar and Chaudhuri 23), investigated a dynamic reaction model in the case of a prey-predator type fishery system, where only the prey species is subjected to harvesting, taking taxation as a control instrument. In this paper they critically compare the uniqueness of their developed model with other researchers like Chaudhuri and Johnson 199); Ganguly and Chaudhuri 1995); Anderson and Lee 1986); Krishna et al. 1998) and Pradhan and Chaudhuri 1999). Kar and Matsuda 26) investigated a prey-predator model with Holling type of predation and harvesting of mature predator species. From the study it is evident that using the harvesting effort as control, it is possible to break the cyclic behavior of the system and drive it to a required steady state. A generalized predator-prey system with exploited terms and the existence of eight positive periodic solutions was studied by Zhanga and Tianb 28). Here the continuation theorem of coincidence degree theory was used. A Lotka-Volterra predator-prey system with a single delay was used by Yan and Zhang 28) in their investigation. Here investigation was carried out on the delay as the bifurcation parameter, analysis of the characteristic equation of the linear system, the linear stability of the system etc. and Holf bifurcations were demonstrated. Kar and Matsuda 27) was also studied a harvested prey-predator system with Holling type III response function. Here the effect of harvesting efforts on the prey-predator system was analyzed. The combined harvesting of two competing species was studied in detail by Chaudhuri 1986 & 1988). Although various models on single species fisheries have so far been developed in the fisheries literature but adequate literature in multispecies fisheries is still limited. Mesterton-Gibbers 1987) reported the multispecies harvesting models. Hannesson 22) suggested that even if all rents disappear by assumption, it is possible to identify this is an economic benefit, particularly when the average catch increases. The variability of the catch falls for a range of values of the population migration parameter and variability of growth. Hannesson 1998) investigated what would happen to fishing outside the marine reserve and to the stock size in the entire area as a result of establishing a marine reserve. Three regimes were compared: 1) open access to the entire area; 2) open access to the area outside the marine reserve and 3) optimum fishing in the entire area. The dynamics of a fishery resource system in an adequate environment which consists of two zones such as a free fishing zone and a reserve zone where fishing is strictly prohibited was studied by Kar 26) using a non-linear mathematical model.

3 32 T.K.KAR AND K. CHAKRABORTY 2. The model and its qualitative properties Our study is mainly concerned with a prey-predator system, the ecological set up of which is as follows. There is a prey which is harvested continuously. There is a predator, living on the prey and it is assumed that the predator is not harvested and hence harvesting does not affect the growth of the predator population directly. However, there is a conflict for common resource i.e. prey between predators and harvesting agency though the predators have competition among themselves for their survival. The growth of prey is assumed to be logistic. Let us assume x and y are respectively the size of the prey and predator population at time t. Keeping these in view, the model becomes 1.1) 1.2) dx = rx dy 1 x K ) αxy a + x ht), = dy + βαxy a + x γy2, where r is the intrinsic growth rate of the prey, K is the environmental carrying capacity of prey, α is the maximal relative increase of predation, a is Michaelis- Menten constant, ht) is the harvesting at time t, d is the death rate of predator, β is conversion factor we assume < β < 1, since the whole biomass of the prey is not transformed to the biomass of the predator). Density dependent mortality rate γy 2 describes either a self limitation of consumers or the influence of predation. Self limitation can occur if there is some other factor other than food) which becomes limiting at high population densities. Traditionally, predation term is taken as αxy but due to the fact that predation tends to infinity as prey population goes to infinity with finite and fixed predator population we consider the predation term as αxy/a + x) Holling,1965). Here we point out that lim x αxy a + x = αy. Harvesting has a strong impact on the dynamic evaluation of a population subjected to it. First of all, depending on the nature of the applied harvesting strategy, the long run stationary density of population may be significantly smaller than the long run stationary density of a population in the absence of harvesting. Therefore, while a population can in the absence of harvesting be free of extinction risk, harvesting can lead to the incorporation of a positive extinction probability and therefore, to potential extinction in finite time. Secondly, if a population is subjected to a positive extinction rate then harvesting can drive the population density to a dangerously low level at which extinction becomes sure no matter how the harvester affects the population afterwards. Thus, the study of population dynamics with harvesting is a very interesting subject of mathematical bioeconomics. Therefore, our objective is to provide some results which are theoretically beneficial to maintaining the sustainable development of the prey-predator system as well as keeping the economic interest of harvesting at an ideal level. We take the harvest rate ht) in the form 2) h = mqxe,

4 EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING 321 where q is the catchability co-efficient, E is the effort used to harvest the population and m < m < 1) is the fraction of the stock available for harvesting. Let us extend the model system 1), assuming that fishery effort itself is a dynamic variable that satisfies de 3) = λpmqx c)e, where c is the constant fishing cost per unit effort, p is the constant price per unit biomass of landed fish and λ is stiffness parameter. Thus,the final model becomes 4.1) dx = rx ) αxy a + x mqxe, 1 x K dy βαxy 4.2) = dy + a + x γy2, 4.3) de = λpmqx c)e, with initial conditions x), y), E). 3. Boundedness, boundary equilibria and stability of the system Boundedness of a model guarantees its validity. The following theorem establishes the uniform boundedness of the model system 4). Theorem 1. All the solutions of the model system 4) which start in R 3 + are uniformly bounded. Proof. Let x), y), E) be any solution of the system with positive initial conditions. Now we define the function W = x + y β + E λp. Therefore, time derivative gives dw = dx + 1 dy β + 1 λp = rx 1 x K de ) αxy a + x mqxe d β y + αxy a + x γ β y2 + mqxe c p E = rx r K x2 d β y γ β y2 c p E. Now for each v >, we have dw + vw = rx r K x2 d β y γ β y2 c p E + vx + v β y + v λp E. Taking v = λc, we get dw + vw u, where u = rk 1 + v ) 2 + γ d 4 r 4β γ + v ) 2. γ

5 322 T.K.KAR AND K. CHAKRABORTY Applying the theory of differential inequality Birkoff and Rota, 1982), we obtain W x, y, E) u v + W x), y), E)) e vt and for t, W u v. Thus, all the solutions of the system 4) enter into the region B = {x, y, E) : W uv } + ɛ, for any ɛ >. This completes the theorem. In the following lemma we have mentioned the boundary equilibria and their stability of the system 4) and the condition of existence of them. Lemma 1. System 4) always have two boundary equilibrium points P,, ) and P 1 K,, ). The third boundary equilibrium point P 2 ˆx,, Ê) exists if and only if c < Kmpq. When this condition is satisfied, ˆx and Ê are given by ˆx = c mpq and rkmpq c) Ê = Km 2 pq 2. At P,, ) we find the characteristic equation as µ 3 r d cλ)µ 2 dr cdλ+crλ)µ cdrλ =, the roots of which are d, r and cλ. So,P,, ) is unstable. At P 1 K,, ), the characteristic equation is c + Kmpq)λ µ) dr Krαβ ) Kαβµ + dµ + rµ a + K a + K + µ2 =, the roots of which are r, d + Kαβ and c + Kmpq)λ. a + K Consequently,P 1 K,, ) is locally asymptotically stable if Kαβ < da + K) and c < Kmpq. At P 2 ˆx,, Ê), the characteristic equation is d + ˆxαβ ) a + ˆx µ Êm 2 pq 2ˆxλ + rˆxµ ) K + µ2 =, the roots of which are d + ˆxαβ a + ˆx, µ 1 and µ 2. We observe that µ 1 + µ 2 = rˆx K < and µ 1µ 2 = Êm2 pq 2ˆxλ, therefore, P 2 ˆx,, Ê) is locally asymptotically stable if ˆx a + ˆx < d αβ. 4. The interior equilibrium point: existence and stability Lemma 2. The interior equilibrium point P 3 x, y, E ) of the system 4) exists if cαβ > cd + admpq, m > c/kpq and d > αβ. When these conditions are satisfied x, y and E are given by

6 x = EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING 323 c pmq, cαβ cd admpq y = and E = c + ampq)γ rkmpq c) Km 2 pq 2 + The characteristic equation at P 3 x, y, E ) is given by fµ) = µ 3 + a 1 µ 2 + a 2 µ + a 3 =, where a 1 = γy + rx K αx y a + x ) 2, a 2 = γrx y K a 3 = E γm 2 pq 2 x y λ. + aα2 βx y a + x ) 3 αγx y 2 a + x ) 2 + E m 2 pq 2 x λ, αb 1 K We find that a 1 > if γ > B 1 K + r)a + x ) 2, where B 1 = cαβ cd admpq. c + ampq) Let = a 1 a 2 a 3. Therefore, = B 1 + B 4 B 5B 1 γ B 1 B 7 B 2 + B ) 3, γ rkmpq c) where B 2 = Km 2 pq 2, B 3 = B 6 = aα2 βx a + x ) 3 and B 7 = m 2 pq 2 x λ. ) B 1 B 4 + B 6B 1 γ pαadmpq + cd αβ)) c + ampq) 2. γ B 5B1 2 + B 2 + B ) 3 )B 7 γ γ pαadmpq + cd αβ)) c + ampq) 2, B 4 = rx K, B 5 = After a little simplification we get = B 8γ 2 + B 9 γ + B 1 γ 2, where B 8 = B 2 1B 4 + B 1 B B 2 B 4 B 7, B 9 = B 2 1B 6 + B 1 B 4 B 6 + B 3 B 4 B 7 B 2 1B 4 B 5 B 3 1B 5 B 2 1B 4 B 5 B 1 B 2 B 5 B 7, B 1 = B 3 1B 2 5 B 2 1B 5 B 6 B 1 B 3 B 5 B 7. αx a + x ) 2, Now we have the following theorem which ensures the local stability of P 3 x, y, E ). Theorem 2. If P 3 x, y, E αb 1 K ) exists with γ > B 1 K + r)a + x ) 2 and B 8γ 2 + B 9 γ + B 1 >, then P 3 x, y, E ) is locally asymptotically stable. αb 1 K Proof. The condition γ > B 1 K + r)a + x ) 2 implies that a 1 >. a 3 is always positive. Finally, B 8 γ 2 + B 9 γ + B 1 > implies that = a 1 a 2 a 3 >. Hence by Routh Hurwitz criterion, the theorem follows. Theorem 3. Let us define ρx) = r K αy a + x)a + x and assume that the ) positive equilibrium is locally stable. If ρ) >, then the positive equilibrium is

7 324 T.K.KAR AND K. CHAKRABORTY globally asymptotically stable. Proof. To show the global stability of system 4) let us construct a suitable Lyapunov function. We define a Lyapunov function x u x y v y E w E V x, y, E) = L 1 du + L 2 dv + L 3 u v w dw, x y where L 1, L 2 and L 3 are positive constants to be determined in the subsequent steps. It can be easily verified that the function V is zero at the equilibrium P 3 x, y, E ) and is positive for all other positive values of x, y and E. The time derivative of V along the trajectories of 4) is E [ dv = L 1x x ) r 1 x ) αy ] K a + x mqe + L 2 y y ) d + βαx ) a + x γy + L 3 λe E )pmqx c) [ = L 1 r K + αy a + x)a + x ) ] x x ) 2 + [ 1 a + x) + [ L 1 mq + L 3 pmq]x x )E E ) + L 2 [ γ]y y ) 2. βαa L 1 α + L 2 a + x ) ] x x )y y ) Choosing L 1 = 1 and L 2 = ampq + c aβpmq yeilds [ dv = rk + αy a + x)a + x ) and L 3 = 1, a little algebric manipulation p ] x x ) 2 γl 2 y y ) 2. The coefficient of y y ) 2 is always negative. The coefficient of x x ) 2 is ρx) = r K + αy a + x)a + x ) r K + αy aa + x ) = ρ) Thus if ρ) >, then dv/ < and dv/ = if and only if x = x, y = y and E = E. This completes the proof. The following plot of ρx) indicates that ρ) >, for some parameter values. So the assumption makes some sense.

8 EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING x ρx) x Figure1. Variation of ρx) with the size of the prey population corresponding to the numerical values r =.9, K = 1, q =.9, d =.4, a = 1, α =.2, β =.2, γ =.1, λ = 1, p = 1, c = 2 and m = Remark Again, the condition of global stability reduces to γ > Kαpmqcαβ cd admpq) arc + ampq) 2 = γ say). Once the positive equilibrium is locally stable, the predator intraspecific competition γ; plays an important role on its global stability. When γ crosses γ, the positive equilibrium becomes unstable and a Hopf bifurcation may occur. 5. Bifurcation analysis Prey-predator models with constant parameters are often found to approach a steady state in which the species coexist in equilibrium. But if parameters used in the model are changed, other types of dynamical behavior may occur and the critical parameter values at which such transitions happen are called bifurcation points. When a stable steady state goes through a bifurcation will in general either lose its stability or disappear entirely. Even if the system ends up in another steady state the transition to that state will often involve the extinction of one or more levels of the food chain. On the other hand the entire system may survive in a nonstationary state, but further bifurcation may lead to local extinction of species. In order to preserve the system under consideration in its natural state, crossing bifurcation should be avoided and in doing so it is of great importance to determine the critical parameter values at which bifurcation occur. Liu 1994) derived a criterion of Hopf bifurcation without using the eigenvalues of the variational matrix of the interior equilibrium point. We specify below the result for the current purposes. Liu s criterion. If the characteristic equation of the interior equilibrium point is given by λ 3 +a 1 φ)λ 2 +a 2 φ)λ+a 3 φ) =, where a 1 φ), a 3 φ), φ) = a 1 φ)a 2 φ) a 3 φ) are smooth functions of φ in an open interval about φ R such that i)a 1 φ ) >, φ ) =, a 3 φ ) >

9 326 T.K.KAR AND K. CHAKRABORTY ) d ii), dφ φ=φ then a simple Hopf bifurcation occurs at φ = φ. Now we analyse the bifurcation of the model system assuming γ as the bifurcation parameter. Theorem 4. If P 3 x, y, E αb 1 K ) exists with γ > B 1 K + r)a + x ) 2 andb 1 <, then a simple Hopf bifurcation occurs at the unique positive value γ = γ = B 9 + B 2 9 4B 8B 1 2B 8. Proof. The characteristic equation of P 3 x, y, E ) is given by µ 3 + a 1 γ)µ 2 + a 2 γ)µ + a 3 γ) =. We observe that a 3 > for all positive values of γ. Now γ) = a 1 γ)a 2 γ) a 3 γ) = B 8γ 2 + B 9 γ + B 1 γ ) =. Furthermore, ) d = B 9 dγ γ=γ γ 2 2B 1 B 2 γ 3 = 9 4B 8 B 1 >. 2B 8 Hence, by Liu s criterion, the theorem follows. γ 2 and it is easy to see that Remark Since γ ) = and d /dγ) γ=γ >, γ) > for γ > γ and according to the Routh Hurwitz criterion, P 3 x, y, E ) is locally asymptotically stable for γ > γ. Furthermore, according to the criterion a simple Hopf bifurcation occurs at γ = γ and for decreasing < γ < γ, it approaches a periodic solution. 6. Optimal Harvesting Policy In commercial exploitation of renewable resources the fundamental problem from the economic point of view, is to determine the optimal trade-off between present and future harvests. If we look at the problem it is observed that the marine fishery sectors become more important not only for domestic demand but also from the imperatives of exports. In this section our objective is to maximize the total discounted net revenues from the fishery. Symbolically our strategy is to maximize the present value J given by 5) J = e δt pmqx c)e where δ is the instantaneous annual discount rate. Now we wish to find out the path traced out by x t), y t), E t) ) with the optimal fraction m so that if prey predator populations and the harvesting effort are kept along this path, then the regulatory authority will be assumed to achieve its objective.

10 EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING 327 The problem 5), subject to population equations 4) and control constraints m min m m max, can be solved by applying Pontryagin s maximum principle. The Hamiltonian of this control problem is H = pmqx c)e + λ 1 [rx 1 x K + λ 3 [λpmqx c)e] ) αxy a + x mqxe ] [ + λ 2 dy + βαxy ] a + x γy2 where λ 1 t), λ 2 t) and λ 3 t) are the adjoint variables. The Hamiltonian must be maximized for m [m min, m max ]. Assuming that the control constraints are not binding i.e. the optimal solution does not occur at m min or m max ), we have singular control Clark, 199) given by H 6) m = pqxe λ 1qEx + λ 3 pqexλ =. We intend to derive here an optimal equilibrium solution of the problem. Since we are considering an equilibrium solution, x, y and E are to be treated as constants in the subsequent steps. Now the adjoint equations are dλ 1 7) 8) dλ 1 = δλ 1 H x = δλ 1 [pmqe + λ 1 r 2xr K aαy a + x) 2 mqe ] +λ 3 pmqeλ, dλ 2 = δλ 2 [λ 1 dλ 2 = δλ 2 H y αx a + x + λ 2 dλ 3 = δλ 3 H E d + αβx a + x 2γy dλ 3 9) = δλ 3 [pmqx c) λ 1 mqx + λ 3 pmqx c)λ], Let us solve the adjoint equations7),8)and9), we get ) ) aαβy + λ 2 a + x) 2 )], pmqe + 1 δ 1) λ 1 = pmqx c)λpmqe ) = A 1 say) δ αx y a+x ) + rx 2 K + aα2 x βy a+x ) 3 δ+γy ) + λpm2 q 2 x E δ αx 11) λ 2 = a + x )δ + γy ) A 1, 12) λ 3 = 1 δ [pmqx c) A 1 mqx ]. From the equations6),1)and12),we get the singular path as follows: 13) [pqx c) A 1 mqx]λpqxe A 1 δqex =. Let m be a solution if it exists) of 13). Using this value of m in Lemma 2 we obtain the optimal equilibrium solution.

11 328 T.K.KAR AND K. CHAKRABORTY Thus we have established the existence of an optimal equilibrium solution that satisfies the necessary conditions of the maximum principle. 7. Numerical examples As the problem is not a case study, the real world data are not available for this model. We, therefore, take here some hypothetical data with the sole purpose of illustrating the results that we have established in the previous sections. In order to ensure the existence of bifurcation let us consider the parameters of the system as m =.5, r =.8, q =.1, d =.1, a = 1, α =.75, β =.75, K = 1, λ = 1, p = 1, c = x,y & E 12 1 E t 3 2 y x Figure2. Solution curves of prey population, predator population and fishing effort with the increasing time when γ = Figure3. Phase plane trajectories of biomass and fishing effort beginning with different initial levels when γ = x,y & E 12 1 E t 3 2 y x Figure4. Solution curves of prey population, predator population and fishing effort with the increasing time when γ = Figure5. Phase plane trajectories of biomass and fishing effort beginning with different initial levels when γ =

12 EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING x,y & E 12 1 E t 3 2 y 1 1 x 2 3 Figure6. Solution curves of prey population, predator population and fishing effort with the increasing time when γ = Figure7. Phase plane trajectories of biomass and fishing effort beginning with different initial levels when γ = If we consider the value of γ = then it is observed from the Figure 2&3 that P 3 x, y, E ) is locally asymptotically stable and the populations x, y and fishing effort E converge to their steady states in finite time. Now if we gradually decrease the value of γ, keeping other parameters fixed, then by Theorem 4 we have got a critical value γ = such that P 3 x, y, E ) loses its stability as γ passes through γ. Figures 4&5 clearly show the result. It is also noted that if we consider the value of γ = , then it is evident from Figures 6&7 that the positive equilibrium P 3 x, y, E ) is unstable and there is a periodic orbit near P 3 x, y, E ). Example1 In order to ensure the existence of the optimal equilibrium we use the following values: r =.9, K = 1, q =.9, d =.4, a = 1, α =.2, β =.2, γ =.1, λ = 1, δ =.1, p = 1, c = 2. Using these parameter values from optimal harvesting policy we have found the optimal value of m = and the corresponding optimal equilibrium is , , 11.45).

13 33 T.K.KAR AND K. CHAKRABORTY Prey biomass Predator biomass Fishing effort x,y & E 1 8 E t y x Figure8. Solution curves of prey,predator populations and fishing effort with the increasing time when m = Figure9. Phase plane trajectories of biomass and fishing effort beginning with different initial levels when m = Example2 In this example we use the parameter values as: r =.3, K = 1, q =.15, d =.32, a = 15, α =.45, β =.45, γ =.5, λ = 3, δ =.1, p = 8, c =.5. Using these parameter values from optimal harvesting policy we have found the optimal value of m = and the corresponding optimal equilibrium is , , ) Prey biomass Predator biomass Fishing effort 2 15 x,y & E 1 8 E t 3 2 y x Figure1. Solution curves of prey, predator populations and fishing effort with the increasing time when m = Figure11. Phase plane trajectories of biomass and fishing effort beginning with different initial levels when m = Concluding remarks This paper deals with a prey-predator type fishery model with subject to prey harvesting only. One real example is the Antarctic krill whale fishery. While there is a moratorium on the killing of whales, the Antarctic krill population is being increasingly harvested over the recent years. As the krill population is the main source of food for whale, large krill catches are bound to affect the growth of both the krill and whale populations. So it is necessary to control krill harvesting by

14 EFFORT DYNAMICS IN A PREY-PREDATOR MODEL WITH HARVESTING 331 introducing some regulatory mechanism. Imposition of a reserve is applied on krill harvesting. As stated above, we have adopted reserve as a controlling instrument to regulate harvesting of prey. We have derived suitable policies for reserve. The model also incorporates a fully dynamic interaction between the fishing effort and the perceived rent. In the optimally managed fishery, marine reserve may or may not increase fisheries rent. In many cases, marine reserves of any size would not be optimal at all. When positively sized marine reserves would be optimal, the introduction of marine reserves may or may not increase fisheries rents, depending on whether the actual size of marine reserves is sufficiently close to the optimal or not. It is important to realize that even when marine reserves of a certain size are optimal, their actual imposition may not hit the target. The problem of setting the correct marine reserves is qualitatively similar to the setting of the correct TAC or fisheries tax rate. The main difference is that the knowledge base for setting marine reserves is probably substantially weaker than that for these more traditional controls. Acknowledgments Research of T. K. Kar is supported by the Council of Scientific and Industrial ResearchCSIR), India Grant no. 2516)/ 8 / EMR-II dated References [1] Anderson L. G. and Lee D. R., Optimal governing instrument, operation level and enforcement in natural resource regulation: The case of the fishery, American Journal of Agricultural Economics, ) [2] Birkoff G. and Rota G.C., Ordinary Differential Equations. Ginn;1982. [3] Chaudhuri K. S. and Johnson T., Bioeconomic dynamics of a fishery modeled as an S-system, Mathematical Biosciences, 99199) [4] Chaudhuri K. S., A bioeconomic model of harvesting a multispecies fishery, Ecol. Model., ) [5] Chaudhuri K. S., Dynamic optimization of combined harvesting of a two-species fishery, Ecol. Model., ) [6] Clark C. W., Mathematical Bioeconomics: The optimal Management of Renewable Resources, 2nd ed., John Wiley and Sons, New York, 199. [7] Conrad J.M., The bioeconomics of marine sanctuaries, Journal of Bioeconomics, 11999), [8] Dixon J.A., Economic benefits of marine protected areas, Oceanus, 363)1993) [9] Ganguly S. and Chaudhuri K. S., Regulations of a single species fishery by taxation, Ecological Modelling, )51-6. [1] Hannesson R., Marine Reserves: What would they accomplish? Marine Resource Economics, ) [11] Hannesson R., The economics of marine reserves, Natural Resource Modeling, 1522) [12] Holling C.S., The functional response of predators to prey density and its role in mimicry and population regulation, Memories of Entromological Society of Canada, )1-6. [13] Kar T. K. and Chaudhuri K. S., Regulation of a prey-predator fishery by taxation: A dynamics reaction model, Journal of Biological Systems, 112) 23) [14] Kar T.K. and Misra S., Influence of prey reserve in a prey-predator fishery, Nonlinear Analysis, 6526) [15] Kar T. K., A model for fishery resource with reserve area and facing prey-predator interactions, Canadian Applied Mathematics Quarterly, 144) 26), [16] Kar T.K.and Matsuda H., Controllability of a harvested prey-predator system with time delay, Journal of Biological Systems, 142) 26)

15 332 T.K.KAR AND K. CHAKRABORTY [17] Kar T.K. and Matsuda H., Global dynamics and controllability of a harvested prey-predator system with Holling type III functional response, Nonlinear Analysis: Hybrid Systems, 127) [18] Krishna S. V., Srinivasu P. D. N. and Kaymakcalan B., Conservation of an ecosystem through optimal taxation, Bulletin of Mathematical Biology, 61998) [19] Liu, W. M., Criterion of Hopf bifurcations without using eigenvalues, J. Math. Anal. Appl ) [2] Mesterton-Gibbons M., On the optimal policy for combined harvesting of independent species, Natural Resource Modelling, 21987) [21] Pradhan T. and Chaudhuri K. S., A dynamic reaction model of two species fishery with taxation as a control instrument: A capital theoretic analysis, Ecological Modelling, )1-16. [22] Yan X.P. and Zhang C.H., Hopf bifurcation in a delayed Lokta-Volterra predator-prey system, Nonlinear Analysis: Real World Applications, 928) [23] Zhanga Z. and Tianb T., Multiple positive periodic solutions for a generalized predator-prey system with exploited terms, Nonlinear Analysis: Real World Applications, 928) Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah ,West Bengal, India tkar1117@gmail.com Department of Mathematics, MCKV Institute of Engineering, 243 G.T.Road N), Liluah, Howrah-71124, West Bengal, India kc mckv@yahoo.co.in.