A dynamic reaction model of a two-species fishery with taxation as a control instrument: a capital theoretic analysis

Ecological Modelling 121 (1999) 1 – 16 www.elsevier.com/locate/ecolmodel

A dynamic reaction model of a two-species fishery with taxation as a control instrument: a capital theoretic analysis T. Pradhan *, K.S. Chaudhuri Department of Mathematics, Jada6pur Uni6ersity, Calcutta 700 032, India Received 3 July 1998; received in revised form 2 February 1999; accepted 18 March 1999

Abstract In a fully dynamic model of an open-access fishery, the level of fishing effort expands or contracts according as the perceived rent (i.e., the net economic revenue to the fishermen) is positive or negative. A model reflecting this dynamic interaction between the perceived rent and the effort in a fishery, is called a dynamic reaction model. The present paper deals with a dynamic reaction model of a fishery consisting of two competing species, each of which obeys the logistic law of growth. A regulatory agency controls exploitation of the fishery by imposing a tax per unit biomass of the landed fish. It is also assumed that the gross rate of investment of capital in the fishery is proportional to the perceived rent. With this capital theoretic approach, the dynamical system consisting of the growth equations of the two-species and also of the fishing effort is formulated. The existence of its steady states and their stability are then studied using eigen value analysis. The optimal harvest policy is discussed next with the help of Pontryagin’s maximum principle. The results are then illustrated using numerical examples. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Open-access fishery; Dynamic reaction model; Fishing effort; Perceived rent; Taxation; Optimal harvest policy

1. Introduction Bioeconomic modelling of the exploitation of biological resources like fisheries and forestry’s has gained importance in recent years in view of overexploitation of these resources to meet their global needs. The techniques and issues associated with bioeconomic exploitation of these resources have been discussed in details by Colin Clark in his books (Clark, 1976, 1985). Since any marine * Corresponding author. E-mail address: [email protected] (T. Pradhan)

fishery is essentially a multispecies one, exploitation of mixed-species fisheries has started drawing attention of researchers. Although numerous models on single species fisheries have so far appeared in the fishery literature, studies on multi-species fisheries are not yet adequate. The difficulties in the determination of an optimal harvesting policy for a mixed-species fishery are theoretical as well as practical. It is difficult to construct a realistic model of a multi-species community. Even if we succeed in formulating such a model, it is quite likely that the model may not be analytically tractable. It is also difficult to deter-

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T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

mine the optimal harvest policy of a multi-species model having more than one state and (or) control variables. Maximum difficulty lies in the quantitative estimation of the various interaction coefficients in a multi-species ecosystem. In spite of these difficulties, modest attempts have been made from time to time to study multi-species models. Clark (1976) (Art. 9.1, p. 303) discussed the problem of combined or non-selective harvesting of two ecologically independent populations obeying the logistic law of growth. He (Art. 9.3, p. 317) also discussed, with limited success, the problem of independent harvesting of a two-species community. Silvert and Smith (1977) studied some aspects of optimal exploitation of a two-species fishery. The problem of combined harvesting of two competing species were studied in details by Chaudhuri (1986, 1987). Chaudhuri and Saha Ray (1996) also studied combined harvesting of a prey-predator community with some preys hiding in refuges. Multispecies harvesting models were also studied by Ragozin and Brown (1985), Wilen and Brown (1986), Mesterton-Gibbons (1987, 1988) etc. Regulation of exploitation of biological resources has become a problem of major concern nowadays in view of the dwindling resource stocks and the deteriorating environment. Exploitation of marine fisheries naturally involves the problems of law enforcement. The economics of law enforcement in marine fisheries was discussed by Sutinen and Andersen (1985). Several governing instruments are suggested for the choice of a regulatory control variable. These are imposition of taxes and license fees, leasing of property rights, seasonal harvesting, direct control, etc. Various issues associated with the choice of an optimal governing instrument and its enforcement in fishery were discussed by Anderson and Lee (1986). Optimal timing of harvest was adopted as a regulatory device by Kellogg et al., (1988) in their study of the North Carolina Bay Scallop Fishery. Imposition of a tax per unit biomass of the landed fish is a possible governing instrument in the regulation of fisheries. As described by Clark (1976) (Art. 4.6, p. 116), ‘‘economists are particularly attracted to taxation, partly because of its flexibility and partly because

many of the advantages of a competitive economic system can be better maintained under taxation than under other regulatory methods’’. Politicians, however, dislike use of taxation as a control instrument. Nevertheless, the analysis of tax control gives a standard against which we may compare other controls. For example, licenses and seasons might be used to approximate the effort computed from the model of a tax policy. A single species fishery model using taxation as a control measure was first discussed by Clark (1976) (Art. 4.6, p. 116). Chaudhuri and Johnson (1990) extended this model using a catch-rate function, which was more realistic than that in Clark (1976). Ganguly and Chaudhuri (1995) made a capital-theoretic study of a single species fishery with taxation as a control policy. However, no multi-species fishery model has yet been developed with taxation as a control instrument. In almost cent percent of the fishery models discussed so far in the fishery literature, the fishing effort E is assumed to be fixed. However, a fully dynamic, open access model requires that the level of effort should expand or contract according as the perceived rent (i.e., the net economic revenue to the fishermen) is positive or negative. A model reflecting the dynamic interaction between the fishery and the effort is called a dynamic reaction model. Dynamic reaction models have so far been studied by Clark (1976) (Art. 4.6, p. 118), Chaudhuri and Johnson (1990), and Ganguly and Chaudhuri (1995) in the case of single species fisheries. No dynamic reaction model has yet been discussed in the case of multispecies fisheries. The present paper deals with a dynamic reaction model in the case of a two-species fishery, taking taxation as a control instrument. We study the development of a fishery consisting of two competing species obeying the logistic growth law. Capital theoretic approach is adopted to formulate the dynamic fishery model. The existence of the possible steady states along with their local stability is discussed. The various conditions arising in the analysis give different ranges in which the tax must lie to get the desired results. The optimal harvest policy is discussed using the maximum principle due to Pontryagin in Burghes and

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

Graham (1980). Numerical examples are taken up to illustrate how the system works.

2. Formulation of the problem Let us consider two competing fish populations whose growths obey the following dynamical system:

   

dx x =rx 1− −a1xy dt K dy y = sy 1− − b1xy dt L

(1)

where r, s, a1, b1, K, and L are all positive constants. Here r and s represent the biotic potentials and K and L the carrying capacities of the two species. This model assumes the existence of an external resource that supports each species, according to the logistic law of growth, in the absence of the other species. The interaction terms − a1xy and − b1xy indicate that the two species compete with each other for the use of the common resource. Here, a1x may be interpreted, in a wider sense, to be the trophic function or the functional response of the second species to the density of the first. Similar interpretation holds for b1y also. We assume that both the species are subjected to a nonselective harvesting effort governed by the differential equations:

   

dx x = rx 1− −a1xy − q1Ex dt K dy y = sy 1− − b1xy − q2Ey L dt

(2)

where q1 and q2 are the catchability coefficients of the two species and E denotes the effort devoted to their combined harvesting. Here we take E as a dynamic (i.e., time-dependent) variable governed by the equations: E(t) =aQ(t),

05 a 5 1

(3)

and dQ =I(t)− gQ(t), dt

Q(0) = Q0

(4)

3

where I(t) is the gross investment rate (expressed in physical terms) at time t. Q(t) is the amount of capital invested in the fishery at time t and g is the (constant) rate of depreciation of the capital. The fishing effort E at any time is assumed to be proportional to the instantaneous amount of the capital invested. If Q(t) represents, for example, the number of standardized fishing vessels available to the fishery at time t, it is then quite reasonable to assume that E(t) should be proportional to Q(t) and that the maximum effort capacity equals the number of vessels available (for a= 1). The case a=0 indicates a situation where no fishing effort is expended although fishing vessels may be available. This reflects the condition of an over exploited fishery; the fishing vessels must be kept inoperative because the fish population is already severely depleted. These are the justifications in favor of Eq. (3). Eq. (4) models the possible adjustments of the level of capital. A regulatory agency controls exploitation of the fishery by imposing a tax t(\ 0) per unit biomass of the landed fish. Any subsidy to the fishermen may be interpreted as a negative value of t.The net economic revenue to the fishermen (perceived rent) is E{q1(p1 − t)x+ q2(p2 − t)y− c}. where p1 is the constant price per unit biomass of the first species, p2 is the constant price per unit biomass of the second species and c is the constant cost per unit of harvesting effort. We assume that the gross rate of investment of capital is proportional to this ‘perceived rent’. Thus we have: I= b{q1(p1 − t)x+ q2(p2 − t)y− c}E,

0 5b5 1 (5)

Eq. (5) asserts that the maximum investment rate at any time equals the perceived rent (for b= 1) at that time. The case b= 0 may be used only in the situation of negative perceived rent, when the capital stocks are perfectly ‘non-malleable’ (i.e., when disinvestment of capital assets is not feasible). If the fishery operates on a loss and at the same time, the capital stocks are perfectly ‘malleable’ (i.e., disinvestment of capital is allowed),

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

4

the sole owner of the fishery is benefited by allowing a continuous disinvestment of the capital assets. In this case, I B0 and b \0 since negative investment means disinvestment. By virtue of Eq. (5), Eq. (3) and Eq. (4) yield the result: dE =[ab{q1(p1 −t)x +q2(p2 −t)y −c} − g]E. dt

x¯ =

Ks(r− a1L) rs− a1b1KL

and y¯ =

Lr(s− b1K) rs− a1b1KL

when E= 0.Now, x¯ \ 0 and y¯ \ 0 when either (6)

The fishermen and the regulatory agency are actually two different components of the society at large. Hence, the revenues earned by them are the revenues accrued to the society through the fishery. The net economic revenue to the society is: (p1q1x+p2q2y−c)E = E{q1(p1 −t)x +q2(p2 −t)y −c} +t(q1x +q2y)E which equals the net economic revenue to the fishermen (perceived rent) plus the economic revenue to the regulatory agency. Therefore, we have the following system of equations:

   

(a)

r \ a1 L

and

s \b1, K

r B a1 L

and

s B b1, K

or (b)

i.e., rsB a1b1KL holds. Therefore, the equilibrium point P3 exists if either (a) or (b) holds. These are the conditions for the existence of the non-trivial steady state of an unexploited (E=0) system.When y= 0, we have from Eq. (7): x¯ =

cab+g \0 abq1(p1 − t)

since p1 \ t in the case of taxation. t being negative in the case of subsidy, x¯¯ will be always positive.

!

dx x = rx 1− −a1xy − q1Ex, dt K

cab+ g Now E( \ 0 if tB p1 − . Kabq1

dE =[ab{q1(p1 −t)x +q2(p2 −t)y −c} − g]E. dt

Therefore, P4 exists if: (7) 0B tBp1 −

3. The steady states

and

The steady states of the system of Eq. (7) are given by:

p1 \

P1(K, 0, 0),

P3(x¯, y¯, 0),

P4(x¯¯ , 0, y¯¯ ),

cab+ g Kabq1

(8)

cab+ g for taxation. Kabq1

!

"

In the case of subsidy, we must have:

P2(0, L, 0), P5(0, y¯¯ *, E( *)

"

r cab+g Now E( = 1− . q1 Kabq1(p1 − t)

dy y = sy 1− − b1xy − q2Ey, dt L

P0(0, 0, 0),

i.e., rs\a1b1KL

and

P6(x*, y*, E*). We now study the different conditions under which these steady states exist. From Eq. (7), we have:

tBmin 0, p1 −

cab+ g . Kabq1

Similarly, when x= 0, we have: y¯¯ *=

cab+ g \0 Kabq2(p2 − t)

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

since p2 \t in the case of taxation and

!

x*={K(q2a1L− q1s)(cab + g)

"

− KLq2ab(p2 − t) (rq2 − sq1)}

cab + g s 1− . E( * = Labq2(p2 −t) q2

/{ab[Lq2(p2 − t) (Kq1b1 − rq2)

cab +g . Now E( *\0 if tBp2 − Labq2

+Kq1(p1 − t)(q2a1L− q1s)]} +L(cab + g)(Kq1b1 − rq2)}

cab +g 0 BtBp2 − Labq2 p2 \

/{ab[Lq2(p2 − t) (Kq1b1 − rq2)

(9)

+ Kq1(p1 − t)(q2a1L− q1s)]}

cab + g for taxation. Labq2



+ Kq1s(p1 − t) (a1L− r)}



/{ab[Lq2(p2 − t) (Kq1b1 − rq2)

Therefore, both the equilibria P4 and P5 exist in the case of taxation if



+ (cab + g) (rs− KLa1b1)}

cab +g for E( * \0. Labq2

0 B tBmin p1 −



cab + g cab + g , p2 − . Kabq2 Labq2

+ Kq1(p1 − t)(q2a1L− q1s)]}

K\ max

cab +g ab

Bmin {Kp1 q1, Lp2 q2}in the case of taxation. In the case of subsidy, the first inequality remains the same while the second one becomes: cab+g \ max {Kp1 q1, Lp2 q2}. ab The non-trivial equilibrium P6(x*, y*, E*) is obtained by solving the following system of equations:

   

x − a1y1 −q1E = 0 K

(10)

y − b1x− q2E = 0 s 1− L

(11)

r 1−

and

!

"

rq1 s , q1b1 b1

and

L\ max

!

"

q1s r , . q 2a 1 a 1 (H1)

In the fishery literature, the ratio of biotic potential to the catchability coefficient is called (Clark, 1976) the biotechnical productivity (BTP). Accordingly, we write (BTP)x =

r q1

and

s (BTP)y = . q2

Depending on whether (BTP)x \ orB (BTP)y, we have the following two cases: Case I: If (BTP)x \ (BTP)y, then y*\ 0. However, x*\ 0 [ t\ p2 −

cab+ g q2a1L− q1s · Lq2ab rq2 − sq1 (16)

and

ab{q1(p1 −t)x + q2(p2 −t)y −c} − g=0. (12)

Solving (10), (11), and (12), we have:

(15)

We now study the existence of the nontrivial steady state P6(x*, y*, E*) in the following cases:

Also t\0 [

(14)

E*={ab{rq2L(p2 − t) (Kb1 − s)

As before, in the case of subsidy, y¯¯ * is always positive and we must have t Bmin 0, p2 −

(13)

y*= {KLabq1(p1 − t) (rq2 − sq1)

Therefore, P5 exists if

and

5

E*\0 [ 0B tB {ab{rq2Lp2(Kb1 − s) + sq1Kp1(a1L− r)} − (cab + g) (KLa1b1 − rs)} /{ab{rq2L(Kb1 − s)+ sq1K(a1L− r)}}

(17)

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

6

Case II: However,

If

(BTP)x B(BTP)y,

x* \ 0.

cab +g Kq1b1 −rq2 · . Kq1ab sq1 −rq2

y*\ 0 [ t \p1 −

E*\0 if the inequality (17) holds.

!

then

"

rq2 s K B min , q1b1 b1

and

!

(18) and

"

q1s r L Bmin , . q2a1 a1 (H2)

Again, we have the following two cases: Case I: If (BTP)x \(BTP)y, then x* \0. However, y*\ 0 if the inequality (18) holds and E*\0 if the inequality (17) holds. Case II: If (BTP)x B(BTP)y, then y* \ 0. Also x*\ 0 if the inequality (16) holds and E* \ 0 if (17) holds.

!

K \ max

"

rq2 s , q1b1 b1

and

!

L Bmin

q1s r , q2a1 a1

"

(H3) In this case, P3(x¯,y¯, 0) does not exist. But it is natural to expect that the unexploited system should possess a nontrivial steady state. Therefore, (H3) does not reflect a realistic situation. If there is no competition between the two fish species, we have a1 = b1 =0 and in this case, the interior steady state P6(x*, y*, E*) is given by: x* =

Kq1s(cab + g)+ KLq2ab(p2 −t) (rq2 −sq1) ab{Lq 22r(p2 −t) +Kq 21s(p1 −t)} (13a)

y*=

Lq2r(cab +g)− KLq2ab(p1 −t) (rq2 −sq1) ab{Lq 22r(p2 −t) +Kq 21s(p1 −t)} (14a)

and E* =

ab{rq2sL(p2 − t) + Kq1sr(p1 − t)} − rs(cab+ g) ab{Lq 22r(p2 − t) + Kq 21s(p1 − t)} (15a)

Now we consider two cases: Case I: If (BTP)x \(BTP)y, then x* \0. However,

E*\0 [ 0 BtB

ab(p2q2L+ p1q1K)− (cab +g) ab(q2L+ q1K) (17a)

y*\0 [ t\ p1 −

cab+ g q2r · Kq1ab rq2 − sq1

(18a)

Case II: If (BTP)x B (BTP)y, then y*\ 0. However, x*\0 [ t\ p2 −

q1s cab+ g · Lq2ab sq1 − rq2

(16a)

and E*\ 0 if (17a) holds. Thus, the procedure to examine the existence of the nontrivial steady state for the two competing fish species would be as follows: 1. Check first which one of the conditions (H1) and (H2) holds for the known carrying capacities of the given species. 2. Check next the BTPs of the two species. 3. Select a proper tax-range consistent with (i) and (ii). In the case of two ecologically independent species, one has to check only their BTPs and then to select an appropriate tax-range. Moreover, comparing (16) with (16a) and (18) with (18a), it is found that the regulatory agency has to impose more tax to regulate harvesting of competing fish species. This outcome is consistent with biological implications of harvesting competing species. Since the species are competing for a common source of food, their growth is restricted due to limitation of the food supply. As a result, they are vulnerable to heavy exploitation. It is, therefore, necessary to impose higher taxes on their catch.

4. Local stability The eigenvalues of the variational matrix V(0, 0, 0) are r(\0), s(\ 0) and − (cab + g)(B 0) (Appendix I). Hence, the trivial equilibrium point P0(0, 0, 0) is an unstable equilibrium point. Near P0, the two species x and y grow, while the effort declines. The eigenvalues of the variational matrix V(K, 0, 0) are − r, s− b1K and abq1(p1 − t)K− (cab +g) (see appendix II). The third eigen value

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

is negative if t\p1 − (cab +g)/Kabq1. Then P1 is asymptotically stable only if s/K B b1 and t \ p1 − (cab +g)/Kabq1. The eigenvalues of the variational matrix V(0, L, 0) are r − a1L, −s and abq2(p2 − t)L −(cab +g). We see that P2 is asymptotically stable only if: r B a1 L

t\p2 −

and

cab +g . Labq2

One of the eigenvalues of the variational matrix V(x¯,y¯,0) is ab{q1(p1 −t)x¯ + q2(p2 −t)y¯ } − (cab +g) (see appendix III).This eigen value is positive or negative according to whether tBor\

!

1 cab +g p1q1x¯ +p2q2y¯ − q1x¯ + q2y¯ ab

"

The other two eigenvalues of V(x¯,y¯,0) are given by the roots of the following quadratic equation in l: KLl 2 +(rLx¯ +sKy¯ )l + x¯ y¯ (rs −a1b1LK) = 0. (20) rLx¯ +sKy¯ KL

B0 always and product of the roots =

x¯ y¯ (rs −a1b1LK) . KL

If the condition (a), which is one of the necessary and sufficient conditions for the existence of P3, holds, then the roots of Eq. (20) are real and negative or complex conjugates having negative real parts. Thus P3 is asymptotically stable only when the condition (a) together with the condition t\

!

1 cab + g p1q1x¯ +p2q2y¯ − q1x¯ + q2y¯ ab

"

holds.

When the condition (b) holds, the product of the roots of (20) is negative and hence one of the roots must be positive. Therefore, P3 is unstable if the condition (b) holds or the condition tB

!

1 cab + g p1q1x¯ +p2q2y¯ − q1x¯ + q2y¯ ab

"

From the above inequalities, it is clear that when the condition (a) holds, then P3 is asymptotically stable inside the region q1(p1 − t)x+ q2(p2 − t)yB

cab+ g ab

in the first quadrant of the x−y plane. One of the eigenvalues of the variational matrix is V(x¯, 0, E(( ) is s− b1x¯ − q2E( (see appendix IV). The other two eigenvalues are the roots of the following quadratic equation in l: l2+

rx¯¯ l+ abx¯E( q 21(p1 − t)= 0 K

In this equation, sum of the roots = − (19)

In Eq. (20), sum of the roots = −

7

holds.

(21) rx¯ B0 K

and product of the roots = abx¯E(( q 21(p1 − t)\ 0. Therefore, the roots of (21) are either real and negative or complex conjugates with negative real parts. Therefore P4, is asymptotically stable or unstable according to whether sBb1x¯¯ + q2E(( or s\ b1x¯¯ + q2E(( holds. Similarly, P5 is asymptotically stable or unstable according to whether rBa1y¯¯*+q2E(( *

or

r\ a1y¯¯*+q2E(( * holds.

That is, P4 is asymptotically stable outside the region b1x+ q2E\ s in the first quadrant of the x− E plane and P5 is asymptotically stable outside the region a1y+ q2E\ r in the first quadrant of the y−E plane.Now we study the stability of the nontrivial equilibrium point P6(x*, y*, E*).The characteristic equation of the variational matrix V(x*, y*, E*) is given by (see appendix V) l 3 + m1l 2 + m2l+ m3 = 0 where m1 =

sy* rx* + \0 L K

m2 =

rsx*y* + abq 22(p2 − t)E*y*− a1b1x*y* KL +abq 21(p1 − t)E*x*

(22)

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

8

and m3 =

abE*x*y* {q1K(p1 −t) (sq1 −Lp2a1) LK + q2L(p2 − t) (rq2 −Kq1b1)}

(23)

If (H2) holds, then m3 \0, i.e., (H2) is the sufficient condition for m3 \0. Again



m1m2 − m3 =





sx*y* 2 rs −a1b1KL rx *2y* rs − a1b1KL + KL K KL L +



sabq 22(p2 −t)E*y* 2 rabq 21(p1 −t)E*x* 2 + L K

+abq1q2E*x*y*{a1(p1 −t) +b1(p2 −t)}. Here also, if (H2) holds, then m1m2 −m3 \0, i.e., (H2) is also the sufficient condition for m1m2 − m3 \0. Therefore, by Routh – Hurwitz stability criterion in Murray (1993), we can say that (H2) is the sufficient condition for local asymptotic stability of the non-trivial steady state P6(x*, y*, E*). When the condition (H1) holds, m3 is always negative because the tax t is less than the market price of the fish species in the case of taxation. Therefore, the interior equilibrium P6(x*, y*, E*) is always unstable when the condition (H1) holds because the Routh– Hurwitz (Murray 1993) criterion is not fulfilled.One of the conditions under which the boundary equilibrium P1(K, 0, 0) is unstable is s/K \b1, which is a part of (a), a condition necessary for the local stability of the planar equilibrium P3(x¯, y¯, 0).Similarly, one of the unstability conditions for the boundary equilibrium P2(0, L, 0) is r/L \a1, which is also a part of (a), a necessary condition for the local stability of the plannar equilibrium P3(x¯, y¯, 0). Thus, the unstability conditions of both the boundary equilibria P1(K, 0, 0) and P2(0, L, 0) directly affect the stability of the planar equilibrium P3(x¯, y¯, 0). We may draw a similar conclusion when:

! 

1 cab + g p q x¯ + p2q2y¯ − q1x¯ +q2y¯ 1 1 ab B tB min p1 −

"

n

cab +g cab +g , p2 − . Kabq1 Labq2

Again, one of the stability conditions for the boundary equilibrium P1(K, 0, 0) being t\ p1 − cab+ g , the planar equilibrium P4(x¯, 0, E( ) does Kabq1 not exist when the boundary equilibrium P1(K, 0, 0) is asymptotically stable. Similarly, the planar equilibrium (0, y¯ *, E( *) does not exist when the boundary equilibrium P2(0, L, 0) is locally asymptotically stable. If the condition (H1) holds, the interior steady state P6(x*, y*, E*) is unstable and then the boundary equilibria P1(K, 0, 0) and P2(0, L, 0) are locally asymptotically stable if t\p1 −

cab+ g Kabq1

t\ p2 −

and

cab+ g , respectively. Labq2

In the case (H2), the conditions s/b1 \ K and r/a1 \ L imply that the boundary equilibria P1(K, 0, 0) and P2(0, L, 0) are both locally unstable while the interior equilibrium P6(x*, y*, E*) is stable. For the non-competing fish species, we have m1 =

sy* rx* + \ 0, L K

m2 =

rsx*y* + abq 22(p2 − t)E*y* KL + abq 21(p1 − t)E*x*\ 0

and m3 =

abE*x*y* 2 {q 1Ks(p1 − t) + q 22Lr(p2 − t)}\ 0 LK

After a little calculation, we have m1m2 − m3 =

rs 2x*y* 2 r 2sx* 2y* + KL 2 K 2L +

sabq 22(p2 − t)E*y* 2 L

+

rabq 21(p1 − t)E*x* 2 \ 0. K

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

Therefore, by Routh – Hurwitz criterion of stability in Goh (1980), it is clear that the interior steady state P6 for the non-competing fish species is always locally asymptotically stable.

Now the adjoint equations are (H dl1 =− dt (x

!

= −e − dtp1q1E− l1 r− 5. Optimal harvest policy

J=



e − dt{p1q1x +p2q2y −c}E dt

(26)

0

where d denotes the instantaneous annual rate of discount. Our objective is to determine a tax policy t= t(t) to maximize J subject to the state Eq. (7) and the constraint tmin 5t(t) 5 tmax

(27)

on the control variable t(t). One may allow for a subsidy by taking tmin B0. We apply Pontryagin’s maximum principle in Burghes and Graham (1980) to obtain the optimal equilibrium solution to this control problem. The Hamiltonian of this control problem is H=e

− dt

{p1q1x + p2q2y − c}E

!   ! 

+l1(t) rx 1 −

x − a1xy − q1Ex K

y + l2(t) sy 1 − − b1xy − q2Ey L

2rx − a1y− q1E K

"

+ l2b1y

The objective of the regulatory agency is to maximize the total discounted net revenues that the society derives from the fishery. Symbolically, this objective amounts to maximizing the present value J of a continuous time-stream of revenues given by:

&

9

=−

(H (y

= − e − dtp2q2E+ l1a1x

!

2sy − b1x− q2E L

− l2 s−

"

(31)

and dl3 (H =− dt (E [ e − dt(p1q1x+ p2q2y− c)− l1q1x− l2q2y= 0 by (29). (32) Now from (30) and (32), we have dl1 = P0e − dt + Q0l1 (say) dt where and

" "

(30)

P0 = Q0 =



b1 (p q x+ p2q2y− c)− p1q1E q2 1 1



2r b1q1 − x+ a1y+ q1E− r. K q2

The solution of this linear equation is l1(t)= K0eQ0t −

+l3(t)[ab{q1(p1 − t)x + q2(p2 − t)y − c} − g]E

(28) where l1(t), l2(t), and l3(t) are adjoint variables. Since H is linear in t, the condition that the Hamiltonian H must be a maximum for t such that tmin BtB tmax is Ht =0. Now Ht = 0 [ l3{−ab(q1x +q2y)}E =0 [ l3 =0. (29)

P0 e − dt Q0 + d

The shadow price l1(t)edt is bounded as t“ if K0 = 0. Then l1(t) b1 (p q x+ p2q2y− c)− p1q1E q 1 1 =− 2 e − dt 2r b1q1 − x+ a1y+ q1E− r+ d K q2



Similarly,



T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

10

a1 (p1q1x+ p2q2y −c) − p2q2E q1 l2(t) = − e − dt 2s a1q2 − y +b1x1 +q2E − s +d L q2





Py+Q+

where,

Therefore, from (32), we have (p1q1x+p2q2y−c) b1 (p1q1x+ p2q2y −c)− p1q1E q + 2 q 1x 2r b1q1 − x +a1y +q1E − r+ d K q2 a1 (p1q1x +p2q2y −c) − p2q2E q1 q 2y = 0 + 2s a1q2 − x +b1y +q2E − s+ d L q1









This gives the equation to the singular path.Now for the optimal equilibrium solution we have from (7), E=

!  " !  "

1 x r 1− −a1y q1 K

and E=

(34)

1 y s 1− −b1x q2 L

(35)

From (34) and (35) we have K(q2a1L− q1s) L(q1b1K−q2r)

=

!

=

! 

G= B

 

 

r b1q1 − , K q2 r b1q1 − + d, K q2

a p I= 1(p1q1A+ p2q2)+ 2 (s+ b1LA), q1 L p2 a1 J= (p1q1B− C)− (sL− b1LB), q1 L s aq M= − 1 2, L q1 P= p1q1A+ p2q2

and

Q= p1q1B− C

+ q1(Cy+ D)(Ay + B)(My +d)

(37)

+ q2y(Iy+ J)(Fy + G)= 0

(38)

+ (PFd +MFQ+ PMG+q1CAd + q1AMD + q1CMB+ q2IG+q2 JF)y 2

"

+ (PGd + FQd +MQG+q1CBd + q1ADd + q1MBD+ q2GJ)y+ QGd + q1DBd = 0.

1 {(rK − Br)−(Ar +a1K)y} q1K

Also, E=



1 Ay+ B r 1− −a1y q1 K

b1 p1 (p1q1B− C)+ (Kr− Br), q2 K

(36)

From (34) and (36), we have E=

D=

or (PFM+ q1CAM+ q2IF)y 3

K(q1s − q2r) B= q1b1K−q2r

and

p1 b1 (p1q1A+ p2q2)+ (Ar+ a1K), q2 K

or (Py+ Q)(Fy +G)(My +d)

x =Ay+ B where A=

C=

F=A (33)

Cy+ D Iy+ J · q1(Ay+ B)+ · q2y= 0 Fy+G My+ d (41)

(39)

"

y 1 s 1− − b1(Ay +B) L q2

1 {(sL −B1LB) − (s +b1LA)y} q2L

From (33), (36), (39), and (40) we have

(40)

(42)

Let ld be a positive root (if it exists) of equation (42). Using this value of y in (36) and (39), we have xd and Ed, respectively. Thus, the optimal equilibrium point (xd, yd, Ed ) is obtained for a particular value of d. In terms of these optimal equilibrium values, the optimal value of t is given by td =

ab(p1q1xd + p2q2yd )− (cab + g) ab(q1xd + q2yd )

(43)

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

11

6. Numerical examples

6.1. Example 1 Let r =5.5, s = 8, K =80, L =100, q1 =0.005, q2 = 0.007, a =0.07, b =0.03, p1 =16, p2 =20, a1 = 0.05, b1 = 0.02, c =5, g= 0.01, and d = 0.1 in appropriate units. As the regulatory agencies are always interested in the interior steady states, we first study the conditions of existence and stability of the steady state P6(x*, y*, E*).These parameter values satisfies the condition (H2)

! !

" "

i.e., KBmin

rq2 s , ={385,400} q1b1 b1

and L B min

q1s r , ={114.29,110} q2a1 a1

Here and

(BTP)x = (BTP)y =

r =1100 q1

s = 1142.86 q2

Hence (BTP)y \(BTP)x. Therefore, P6 exists if (16) and (17) hold, i.e., in order to ensure existence of the nontrivial steady state P6(x*, y*, E*), we have to select tax t such that −26.48Bt B6.36. If there is no case of subsidy, we take tmin =0 and tmax =6 (say). Now taking the minimum tax tmin =0 in (13), (14), and (15), we have x* = 4.85, y*= 66.96, and E*= 363.77, respectively. But for the maximum tax tmax =6, we have x* =8.56, y*=95.25, and E* =29.89, i.e., P6(4.85, 66.96, 363.77) and P6(8.56, 95.25, 29.89) are the steady states for minimum and maximum taxation respectively. From these results, it is clear that if the fishermen have to pay no tax, then they use a large amount of effort compared to the case when the fishermen have to pay the maximum tax. As a result, the steady state values of the two species for the case of tmin = 0 are much less than those for the case of tmax =6. Since these parameter values satisfy the condition (H2), we see that P6(x*, y*, E*) is locally asymptotically stable.

Fig. 1. Time course display of x and y for tmin =0.

The computer analyzed results for the time course display of the two species x and y and the phase space trajectory for tmin = 0 and tmax = 6 using these parameter values are shown in Figs. 1–4. We have also checked that for these parameter values, the steady states P0(0, 0, 0), P1(80, 0, 0), and P2(0, 100, 0) are always unstable. If the regulatory agency selects the tax t such that 05 t5 6 in order to ensure existence of the interior steady state P6, then the steady state P3(8.89, 97.78, 0) is unstable and P4 does not exist. For the

Fig. 2. Time course display of x and y for tmax =6.

12

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

tion are found to be 66.17695, 17.32012, − 1.337375. The third root is discarded because yd cannot be negative. If we take yd = 17.32012, then from (36), we have xd = − 1.662896, which is not possible. So we take yd = 66.17695. Using this value of yd in (36) and (39), we have xd = 4.74468 and Ed = 372.99115, respectively. Therefore, (4.74468, 66.17695, 372.99115) is the optimal equilibrium solution for the parameter values taken above. Also, from (43), we have the optimal value of the tax t, in this case as td = − 0.24141B0. We thus have a case of subsidy.

6.2. Example 2 Fig. 3. Trajectories in the three-dimensional phase space for tmin = 0.

minimum tax tmin = 0, P5(0, 69.73, 345.97) is asymptotically stable, whereas for the maximum tax tmax = 6, P5(0, 99.61, 4.44) is unstable. Again, using these parameter values, Eq. (42) becomes 0.0001776y 3 −0.0145916y 2 +0.1837318y +0.2722412=0

(44)

By using the Newton – Raphson method in Atkinson (1994), the roots yd of this cubic equa-

Fig. 4. Trajectories in the three-dimensional phase space for tmax =6.

Let r= 5.5, s= 8, K= 450, L= 200, q1 = 0.005, q2 = 0.007, a= 0.5, b= 0.03, p1 = 16, p2 = 20, a1 = 0.05, b1 = 0.02, c =5, g= 0.01, and d= 0.1 and in appropriate units. These parameter values satisfies the condition (H1). i.e.,

K\ max

and

L\ max

! !

" "

rq2 s , = {385,400} q1b1 b1 q1s r , = {114.29,110} q 2a 1 a 1

Here also (BTP)y \ (BTP)x. Therefore, P6 exists if (17) and (18) hold, i.e., in order to ensure existence of the nontrivial steady state P6(x*, y*, E*), we have to select tax t such that 10.91 B tB 13.41. Now we take tmin = 11 and tmax = 13.4. For tmin = 11, we find from (13), (14), and (15) that x*= 202.68, y*= 9.52, E*= 509.38.For tmax = 13.4, we have x*= 351.24, y*= 23.82, E*= 3.20, i.e., P6(202.68, 9.52, 509.38) and P6(351.24, 23.82, 3.20) are the steady states for minimum and maximum taxation, respectively. This case also establishes the tendency of the fishermen to maintain a higher effort level at a lower tax level. These two interior steady states are always unstable as the parameters satisfy the conditions (H1). For these parameter values, the steady states P0(0, 0, 0) and P3(352.17, 23.91, 0) also are unstable. If the regulatory agency selects the tax t such that 115t 513.4 in order to ensure the existence of the interior steady state P6(x*, y*, E*), the steady states P1(450, 0, 0) and P2(0, 200, 0) are

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

also unstable. Again for tmin =11, the steady states P4(226.67, 0, 545.93) and P5(0, 89.95, 628.87) are asymptotically stable. For tmax = 13.4, the steady states P4(435.90, 0, 34.47) and P5(0, 122.66, 441.97) are also asymptotically stable. Using these parameter values, Eq. (42) becomes 0.0103051y 3 −0.0401842y 2 −0.9792724y +3.1446492=0 The roots yd of this cubic equation are 10.26, 3.13, −9.49. But the third root is discarded because yd cannot be negative. For the two positive roots, we have from (36) and (39), two optimal equilibrium solutions as (210.41, 10.26, 483.04) and (136.37, 3.13, 735.83). From (43), we find that the corresponding optimal values of the taxes are 11.21 and 8.07, respectively.

7. Discussion The present paper deals with a problem of nonselective harvesting of a mixed-species (twospecies) fishery, focusing attention on the use of taxation as an optimal governing instrument to control exploitation of the fishery. Ranges for taxation corresponding to the various steady states and also for the optimal equilibrium solution are determined. The regulatory agency is to impose a tax within such a range appropriate to the desired objective. Thus, the results of this paper may find potential applications in regulating commercial exploitation of a mixed species fishery. The most important feature of the present model is that it assumes a fully dynamic interaction between the fishing effort and the perceived rent in the case of two competing fish species. There are abundant instances of competing fish species in the marine ecosystem. In the Antarctic ecosystem, both the whales and the seals feed upon the immense krill population. The Atlantic herring competes with species like mackerel, pollock, shad, etc., for the use of common food

13

sources. This is the first ever dynamic reaction model on a multispecies fishery. This type of attempt in modelling multispecies fishery is important in view of the fact that a fishing vessel in the ocean can hardly exploit a single species of fish. Tuna purse seiners catch both yellowfin and skipjack tuna; pelagic whale fleets catch blue, fin, sei, minke, humpback and sperm whales; salmon trawlers in the northeast Pacific exploit five species of salmon; in the northwest Atlantic, trawlers exploit commercially over thirty species of ground fish; in tropical fisheries, the catch of a day’s trawling may contain as many as two hundred species. Thus multispecies modelling is essential in the context of ecologically interrelated marine fish species. However, management of multispecies fisheries is rendered difficult both by the complexity of the situation and by the irreducible uncertainty regarding ecological interrelations in the ocean. The main difficulty in applying the models of the present type to management of commercial fisheries lies in the estimation of various parameters associated with the model. Sufficient information is seldom available to formulate quantitative predictions of the effects of given levels of catch. Keeping these difficulties in view, Clark (1985) outlined the objective of bioeconomic modelling in the following lines: ‘It should be clear that no amount of bioeconomic modelling is going to solve the problems of multispecies fisheries. The best that can be hoped for is that a useful classification of the issues, and perhaps some insights into appropriate management principles will emerge’.

Acknowledgements The first author wishes to express thanks to the authorities of Jadavpur University, Calcutta and the University Grants Commission, New Delhi for giving their infrastructural and financial support to do this work. The authors also express their deepest sense of gratitude to the three anonymous reviewers for their valuable suggestions to improve the paper.

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

14

Appendix A Variational matrix of the system of equations (7) is

Æ 2rx Ç Ãr − −a y −q E à −a 1x − q 1x 1 1 K à à 2sy à V(x, y, E)= à −b 1y s− − b 1x− q2E − q 2y à à L Ã Ã È abq 1(p1 −t)E abq 2(p2 −t)E ab{q 1(p1 − t)x+ q2(p2 − t)y− c}− g É (45) Therefore,

Ær 0 Ç 0 0 V(0, 0, 0)= Ã0 s à È0 0 −(cab +g)É The eigen values of this variational matrix are r( \ 0), s(\0)

and

−(cab +g) ( B 0).

Appendix B From (45), we have

Æ−r Ç a 1K −q 1K V(K, 0, 0) = Ã 0 s −b 1K 0 Ã È 0 abq 1(p1 −t)K − (cab +g)É 0 The eigen values of V(K, 0, 0) are −r, s − b1K and abq1(p1 −t)K −(cab +g).

Appendix C From (45), we have

Æ rx¯ Ç Ã− Ã − a 1x¯ −q 1x¯ Ã K Ã V(x¯, y¯, 0)= Ã sy¯ Ã − b 1y¯ − − q 2y¯ Ã Ã L Ã Ã È 0 0 ab{q 1(p1 −t)x¯ +q2(p2 − t)y¯ }−(cab + g)É

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

The characteristics equation of this matrix is

à rx¯ à à − − l −a x¯ à −q x ¯ 1 1 à K à sy¯ à Ã=0 − b 1y¯ − −l − q 2y¯ à à L à à à 0 0 ab{q 1(p1 −t)x¯ + q2(p2 −t)y¯ }− (cab + g)−l à or, [ab{q1(p1 −t)x¯ + q2(p2 −t)y¯ } −(cab + g) − l] [KLl 2 + (rLx¯ +sKy¯ )l+ x¯ y¯ (rs−a1b1LK)]= 0.

Appendix D From (45), we have

Æ rx¯ − −a 1x¯¯ à K ( ¯ ( V(x, 0, E )= à à 0 s− b 1x¯¯ −q2E(( Èabq 1(p1 −t)E(( abq 2(p2 −t)E((

Ç

−q 1x¯¯Ã

à à É

0 0

The characteristic equation of V(x¯¯ , 0, E(( ) is

à rx¯¯ − a 1x¯¯ à − −l K à à 0 s −b 1x¯¯ −q2E(( − l Ãabq 1(p1 −t)E(( abq 2(p2 −t)E((

!

Ã

−q 1x¯¯ Ã

Ã=0 0 Ã −l Ã

"

rx¯¯ or, (s − b1x¯¯ − q2E(( − l) l 2 + l +abx¯¯ E(( q 21(p1 −t) = 0. K

Appendix E From (45), we have

Æ Ç rx* Ã − − a 1x* − q 1x*Ã K Ã Ã V(x*, y*, E*)= Ã sy* Ã −b 1y* − − q 2y* Ã Ã L Ã Ã Èabq 1(p1 −t)E* abq 2(p2 −t)E* 0 É

15

T. Pradhan, K.S. Chaudhuri / Ecological Modelling 121 (1999) 1–16

16

The characteristic equation of V(x*, y*, E*) is

à à − rx* − l − a 1x* K à sy* à −b 1y* − −l à L à Ãabq 1(p1 −t)E* abq 2(p2 −t)E* or, l 3 +



 !

à à à Ã=0 − q 2y* à à −l Ã

− q 1x*

sy* rx* 2 rsx*y* + l + + abq 22(p2 −t)E*y*−a1b1x*y*+ abq 21(p1 − t)E*x*}l L K KL

r s + abq 22(p2 −t)E*x*y* + abq 21(p1 −t)E*x*y* −abq1q2E*x*y*{a1(p1 − t)+ b1(p2 − t)}= 0. K L Clark, C.W., 1985. Bioeconomic Modelling and Fisheries Management. John Wiley and Sons, New York. Ganguly, S., Chaudhuri, K.S., 1995. Regulation of a singlespecies fishery by taxation. Ecol. Model. 82, 51 – 60. Goh, B.S., 1980. Management and Analysis of Biological Populations. Elsevier Scientific Publishing Company, Amsterdam. Kellogg, R.L., Easley, J.E., Johnson, T., 1988. Optimal timing of harvest for the North Carolina Bay Scallop Fishery. Am. J. Agri. Econ. 70, 50 – 62. Mesterton-Gibbons, M., 1987. On the optimal policy for combined harvesting of independent species. Nat. Resour. Model. 2, 109 – 134. Mesterton-Gibbons, M., 1988. On the optimal policy for the combined harvesting of predator and prey. Nat. Resour. Model. 3, 63 – 90. Murray, J.D., 1993. Mathematical Biology. Springer, Berlin. Ragozin, D.L., Brown, G., 1985. Harvest policies and nonmarket valuation in a predator-prey system. J. Environ. Econ. Manag. 12, 155 – 168. Silvert, W., Smith, W.R., 1977. Optimal exploitation of a multispecies community. Math. Biosci. 33, 121 – 134. Sutinen, J.G., Andersen, P., 1985. The economics of fisheries law enforcement. Land Econ. 61, 387 – 397. Wilen, J., Brown, G., 1986. Optimal Recovery paths for purturbations of trophic level bioeconomic systems. J. Environ. Econ. Manag. 13, 225 – 234.

References Anderson, L.G., Lee, D.R., 1986. Optimal governing instrument, operation level and enforcement in natural resource regulation: the case of the fishery. Am. J. Agri. Econ. 68, 678 – 690. Atkinson, K., 1994. Elementary Numerical Analysis. John Wiley and Sons, New York. Burghes, D.N., Graham, A., 1980. Introduction to Control Theory Including Optimal Control. John Wiley and Sons, New York. Chaudhuri, K.S., 1986. A bioeconomic model of harvesting a multispecies fishery. Ecol. Model. 32, 267–279. Chaudhuri, K.S., 1987. Dynamic optimization of combined harvesting of a two-species fishery. Ecol. Model. 41, 17– 25. Chaudhuri, K.S., Johnson, T., 1990. Bioeconomic dynamics of a fishery modelled as an S-system. Math. Biosci. 99, 231– 249. Chaudhuri, K.S., Saha Ray, S., 1996. On the combined harvesting of a prey-predator system. J. Biol. Syst. 4, 373– 389. Clark, C.W., 1976. Mathematical Bioeconomics: the Optimal Management of Renewable Resources. John Wiley and Sons, New York.

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