Bioeconomic Dynamics of a Fishery Modeled as an S-System KRIPASINDHU CHAUDHURI* AND THOMAS JOHNSON Department of Economics and Business, North Carolina State University, Raleigh, North Carolina 27695-M 10 Received 9 August 1989; revised 3 January 1990
ABSTRACT A bioeconomic catch-rate function fishery by imposing
model of a single species fishery is developed by using a realistic and assuming that a regulatory agency controls exploitation of the a suitable tax on the catch. The existence of a nontrivial steady state
and its stability are examined. The optimal harvesting policy is discussed from the viewpoints of variational calculus and control theory. The fishery is then modeled as an S-system examples
by following the recasting techniques of Savageau of the optimal control curves and the yield-effort
executing
the ESSYNS
1.
and Voit 111. Numerical curves are obtained by
[21 algorithm.
INTRODUCTION
In recent years interest has continued in using bioeconomic modeling to gain insight into scientific management of the exploitation of renewable resources like fisheries and forestries. Clark [3] provides an introduction to this field of study. The techniques and issues associated with the dynamic economic models of natural resource exploitation are further developed in [4] and [5]. Exploitation of a natural resource may also involve problems of law enforcement. This happens particularly when a natural resource is severely depleted. In the nineteenth century many important marine fisheries were on the verge of collapse due to over-exploitation. To avert the crisis some countries entered into multilateral agreements in the North Sea Convention of 1882. By the end of 1977 a majority of the coastal nations declared jurisdiction over the sea up to 200 nautical miles from their shores. The
*Permanent 70032, India.
address:
MI4THEM4TZCAL
Department
BIOSCIENCES
of Mathematics,
99:231-249
Jadarpur
University,
Calcutta-
(1990)
OElsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010
231 002%5564/90/%03.50
232
KRIPASINDHU
CHAUDHURI
AND THOMAS
JOHNSON
responsibility for managing coastal fisheries now lies directly with the coastal countries through the establishment of these 200-mile exclusive economic zones in the sea. Some of the issues associated with the choice and the enforcement of optimal governing instruments in regulating fisheries have been discussed by Anderson and Lee 161. The economic aspects of enforcing laws for regulating marine fisheries have been studied by Sutinen and Anderson [7]. Imposition of taxes and license fees, leasing of property rights, and direct control have been proposed as governing instruments. Economists consider taxation to be superior, at least in theory, to other control policies because of its flexibility [3]. Also, “many of the advantages of a competitive economic system can be better maintained under taxation than under other regulatory methods” [31. Even though a tax is not a politically feasible control instrument at this time the analysis of tax control gives a standard against which to compare other controls. For example, licenses and seasons might be used to approximate the effort computed from the model of a tax policy. In this paper a bioeconomic model of a single species fishery is developed using a catch-rate function that is more flexible and that seems to be more realistic than the constant “catch-per-unit-effort hypothesis” [3]. In this model the fish biomass obeys the logistic law of growth [8]. It is assumed that a regulatory agency exercises control over exploitation of the fishery by imposing a tax on the catch. The dynamics of the fishery is modeled by a system of two nonlinear differential equations that incorporate these aspects. The level of the fishing effort changes continuously, and its rate of change at any time is proportional to the perceived rent (net economic revenue to the fisherman) at that instant. It is found that the existence of a nontrivial steady state of the system depends on the BTP’ of the fish species as well as on the other parameters. Conditions for the existence and stability of a nontrivial steady state are derived. The optimal harvesting problem is studied from the viewpoints of variational calculus and control theory. The dynamical system describing the fishery is then modeled as an S-system following the recasting techniques of Savageau and Voit 111. With the tax rate as the control variable, the optimal control curves and the corresponding yield-effort curves are computed using the ESSYNS 121 program on an IBM PC/AT. The set of parameter values used is shown in Table 1. This appears to be the first time that the S-system analysis developed by Savageau and Voit [9-111 has been effectively applied in modeling both the biological and economic dynamics
‘BTP (biotechnical productivity) of a biological species is defined to be the ratio of its intrinsic growth rate (biotic potential) to its catchability and coefficient [3].
S-SYSTEM
FISHERY
233
MODEL
of a bioeconomic system. Savageau and Voit [l] have discussed in detail the advantages of the S-transformation and its related numerical algorithms ESSYNS in solving problems of this type numerically. “It is generally one to two orders in efficiency due reasonable error error tolerances, than with RKF45 rithm” [l].
2.
superior
to existing
methods
for solving this class of problems,
often
of magnitude faster.. The cost of recasting (identified with decreases to the introduction of additional variables) is generally recouped at tolerances and the net improvements are often remarkable: at stringent several problems were solved 50 or 100 times faster with ESSYNS (Runge-Kutta-Fehlberg) and up to 50 times faster than with Gear algo-
FORMULATION The catch-rate
OF THE PROBLEM
function
h(t) usually encountered
in the fishery models is
(1)
h(t) = qE(t)x(t),
where x(t) = size (or biomass) of the fish population at time t, E(t) = fishing effort at time t, and q = catchability coefficient. This form of the catch-rate function is based on the constant “catch-per-unit-effort hypothesis” [3], and it seems to be unrealistic on the following grounds: (1) It assumes a process of random search for (2) It implies that all fish in the population captured. (3) For a fixed population size, the catch-rate effort does so. (4) For a fixed level of effort, the catch-rate population size gets larger and larger.
fish. are equally increases
linearly
as the
increases
linearly
as the
To relax these restrictive assumptions Clark [4] suggested alternative form of the catch-rate function:
h(t)
=
qE(t)x(t) aE(t)+Zx(t)
’
likely to be
the following
(2)
where a, 1, q are all positive constants. In Equation (2) h -+ (q/a)x as E -+ m for a fixed population size and h + (q/l)E as x +m for a fixed level of effort. Thus the catch-rate function in (21 exhibits saturation effects with respect to the fishing effort E as well as to the stock abundance x. This seems to reflect a reasonable proposition. The parameter a is proportional to the ratio of the population level to the catch-rate at higher levels of effort, while the parameter 1 is proportional to the ratio of the effort level to the catch-rate when the
KRIPASINDHU
234
CHAUDHURI
AND THOMAS JOHNSON
biomass level is very high. The catch-rate function (2) is used in the model discussed below. It is also assumed that a regulatory agency exercises control over the fishery by imposing a tax of T( > 0) units of money per unit biomass of landed fish. A negative value of T implies a subsidy to the fishermen. When a fish species undergoes severe depletion and is on the verge of extinction, imposition of a very high tax may be a reasonable regulatory mechanism to save the fishery from collapse. In particular, an assessment of the western North Atlantic swordfish stock [12] reveals that the stock and the yield have fallen much below their 1978 levels. It has been suggested that the spawning stock biomass should be returned to its 1978 level. Achievement of this goal requires a 60% reduction of the 1988 fishing mortality rate. The imposition of a tax on landed swordfish biomass may be one of the several possible ways to reach this management goal. The present model assumes that the economic system has a feedback whereby the level of fishing effort expands or contracts depending on whether the “perceived rent” (net revenue to the fisherman) is positive or negative. Clark [3] carried out this sort of investigation with a single species for fish with a logistic growth of biomass [8] and with the traditional catch-rate function (1). With this perspective we study a fish species whose biomass growth and harvest are governed by the following dynamical system:
dE -&=A
q(P-T)x aE+lx
-
c E I ’
(3)
Here y = biotic potential of the fish species, K = carrying capacity of the fish species, p = the constant price per unit biomass of the landed fish, c = the constant fishing cost per unit effort, and A = the stiffness parameter. A gives the speed with which the effort reacts to the changes in the perceived rent flow. All of these parameters are assumed to be positive. We note, however, that A is a dimensionless constant. Therefore, the net economic revenue to the society is
i@x
___-
cE
aE+lx -c]+T(
iii&).
= net economic revenue to the fisherman + economic revenue to the regulatory agency.
S-SYSTEM
3.
FISHERY
MODEL
THE STEADY
STATE
235
The nontrivial steady state (2, E) obtained X = 0 = _I!?simultaneously is given by
by solving
the equations
i_J[~~+(P-m-dl dP-T)
’
(4)
and
We have the following possibilities: (A) Let a 2 q/-y. This implies that the parameter a is no less than the “biotechnical productivity” [3] of the fish species. (Ai) In the case of taxation we have 0 < T < p so that p - T > 0 and hence, X > 0. (Aii) In the case of subsidy T < 0 so that p - T > 0 and hence x > 0. (Aiii) The case 0 < p < T seems to be irrelevant to a commercial fishery and is not considered in this analysis. However, this case could be relevant for a sport fishery where the process rather than the sale of the catch is the primary objective. However, to make E > 0 requires the additional parametric constraint: q( p - T) > fc. ThusX>O,E>Oprovidedthatu>q/yandq(p-T)>Ic.
(6)
(B) Let a < q/y. This implies that the biotechnical productivity of the fish species exceeds the value of the parameter a. As before, we may argue that p - T > 0. It is found that x and E are both positive if, and only if, the constraint (P-T)(q-ay) q and qr > lc; and (B) ay 0. In the case (B), Det A > 0 if Z2c2 - qr*(q - ay) > 0. Using (B), we may easily prove after a little calculation that the inequality PC2 - qd(
q - ar) > 0
leads to the inequality - ay < 0, which is obviously true. Hence no mathematical inconsistency arises in assuming that 12c2 - qd(q
- ay) > 0.
However, the values of the parameters may be such that l*c* - qr*(q ay) < 0 also holds. We now examine what happens to Det A in this case. After a little simplifcation, we find that Det A = Ac(qa -
lc){lc - 4q - WI) qas-*
Obviously Also
Det A > 0 by virtue of (B). Thus Det A > 0 always.
Trace A = - ---&[acA7r(q7T-zc)+qT*(ay-q)+l*C*]
In case (A), W > 0 and hence Trace A < 0. Thus Det A > 0 and Trace A < 0 in this case.
S-SYSTEM FISHERY MODEL
231
This means that the eigenvalues of the linearized system (8) are either both real and negative or are complex conjugates with negative real parts. We may, therefore, conclude that the steady state (2, E) is either a stable node or a stable focus. The same conclusion holds in case (B) when 12c2 - q2(
q - ay) > 0.
When 12c2 - qn-*(q - ay) < 0 we write
and find that (Bi) Trace A < 0 when acha(qr - lc) > qr2(q - ay)- 12c2 > 0, and (Bii) Trace A > 0 when 0 < uchrr(qr - Zc) < qr’(q - ay>- 12c2. The conclusion is the same as in (A) when (Bi) holds. In case (Bii), the eigenvalues of (8) are either both real and positive or are conjugates with positive real parts. This implies that the steady state (i,E) is either an unstable node or an unstable focus when (B) and (Bii) hold. 4.
OPTIMAL
HARVESTING:
A CONTROL
PROBLEM
The objective of the regulatory agency is to maximize the total discounted net revenues that 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) where 6 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 Equation (3) and the constraint Tmin < T(t)
< Tma”
(10)
on the control variable T(t). One may allow for a subsidy by taking T mh < 0. Since the objective function is still linear in the control variable E, the optimal control will be a “bang-bang” control between T”‘” and T max until it is optional to adopt the singular control. We now construct the Hamiltonian equation;
AE
>
where PI(t) and p2(t) are the adjoint (or co-state) variables.
(11)
238
KRIPASINDHU CHAUDHURI AND THOMAS JOHNSON
Equating
to zero the coefficient
of T in (ll),
CL*(~)qhx(t)E(t)= aE(t) + Ix(t)
we get o
.
Since E(t) > 0 and x(t) > 0, we have
This gives the condition is
and this simplifies
for singular control. The adjoint equation
for p2(t)
to
dt> = e-8t[p_$(!y2],
(13)
since p2(t) = 0. Thus, the shadow price p(t) along the singular path is given by p(t) The adjoint equation
=hl(t)es’=
for p&j,
p-_$yq2.
(14)
namely
d/+ -=_dt
aH ax
reduces to 4%
dt=since j_h2(t) = 0. Using Equations
apqE 2e -*’
aqE2
(aE+IX)2
(uE+I,~)~
1 (15)
(3) and (13) in (15) we get
y[qr,y-c(nE+1x)]
-- ~;(l-~)(aE+~~))=y(l-~)[p-~(~)2].
(16)
S-SYSTEM FISHERY MODEL
239
This is the singular path. After a little simplifcation the steady state singular solution is given by I
PWX) + =6
it can be shown that
{4~(x)12- 4X2WX)j
[4x - ~~(X>12 qlcx2
(
p-
1
(17)
[4x - ~~(X>12 .
This is the same path obtained from the Euler-Lagrange equation in the variational calculus approach. The authors derive Equation (17) in two different ways purely as an academic exercise to illustrate the confluence of the two methods in linear control problems. The resulting equilibrium values are
x=x*, T=T*=p-
lx*F(x*)
E=E*=
4x* - ex*)
’
lcf
(18)
4x* - ax*)
where x = x* is the unique positive root (if one exists) of Equation (17). In terms of the parameters of the system Equation (17) simplifies to the form 2pu543
+ Ky2[ pay 6 - p) + 4up( q - ur) - uk] x2
+zpK*[(q
+ K'[p(6 5.
S-SYSTEM
- ay)(aPs
- lc) + P(4 - w)(q
- y)(q - fly)*-
-W)lx
Ic( q6 - qy + ay*)] = 0.
(19)
ANALYSIS
We now recast the dynamical system (3) into an S-system [l] in order to gain further insight into its dynamic nature and to compute the control path for the taxation policy with the necessary conditions. An S-system is a system of nonlinear ordinary differential equations of the form
240 An
KRIPASINDHU
CHAUDHURI
S-system is a special case of the Generalized
AND THOMAS
JOHNSON
Mass Action System
xi= 5 cqkfixR’,‘It PikiiX:il*iE(l,...,n). j=l j=l k=l
Conversely,
k=l
an S-system is a generalization
of the form
~i=&__iE(l,...,“)’ j=l
which has been called a “Half-system.” Voit, Savageau and Irvine [13] summarize [14-171 with the following statement: “While S-systems, modeling alternatives,
the results of several studies
Generalized Mass Action systems, and Half-systems are all viable S-systems are of particular interest. They provide the only type of
power-law model with nonzero steady states that can be calculated by symbolic, analytical means. Furthermore, analysis of biochemical pathways has shown that S-systems yield more accurate representations than competing Mass Action systems. Although Half-systems have an appealingly simple mathematical structure, their use as power-law approximations of biological systems is limited since they only have steady states in which at least one variable vanishes.”
The recasting of the original system (31 as an S-system (2) is performed by following the procedure developed by Savageau and Voit [ll. By introducing the new variables x1=x, x2 = E, x3= K-x, x4 =aE +1x, the dynamical system (3) transforms to:
/t1=
Xx1x3 - 4x1x2x4-1~
i2
=
A(P
i3
=
4x,x2x4-1
i4
=
ah( p - T)qxIx2x4-’ - aAcxz + %x1x3 - lqx1xzx41.
-
Thx1x2x4-‘-
Acxn, (20)
-
$x1x31
The system (20) represents the power-law form [18] of the original system (3). However, it is not an S-system because the fourth equation consists of four terms. This sum is reduced by defining x4 as the product of two new variables, x4
then differentiating
the product,
=
x5x6
3
identifying
(21)
the appropriate
terms in the
S-SYSTEM FISHERY MODEL
241
fourth equation of (20) for each new derivative, and permutting the indices (5 -+ 4, 6 + 5). The final S-system form of the original system (3) becomes:
i1=
$X1X3 - 4XlX2X2X~‘~
~2=A(p-~)qX1X2X41X;1-~CX2~
(22)
i3=qX1X2X41x5-1~XIX3r ~4=ahq(p-T)X~X2X~1X;2-~~~X2X;1~
lY i5 = EX1X3X4 The constraints
l-
lqx1x2x42x;1.
among the variables
x4x5=
x3=K-x17 x4X5 =x6
are:
(=
ax2+lxl,
(23)
old X4).
Although the S-system (22) contains five variables, two of them (xi and x2) are the original variables x and E. Thus the constraints provided by the definition of new variables and the assignment of initial values to x4 and ,Q restrict the system behavior to a manifold of the same dimension as in the original system. 6.
NUMERICAL
EXAMPLES
As pointed out by Savageau and Voit [l], the general S-system possesses no finite closed-form solution in terms of elementary functions. Hence one has no alternative but to look for efficient numerical solutions in dealing with S-systems. The set of numerical algorithms called ESSYNS developed by Irvine, Savageau, and Voit [2] yields efficient numerical solutions of S-systems. We have executed ESSYNS with an IBM PC/AT for our S-system (22) with the set of parameter values given in Table 1. Our objective is to construct a map for the approach to the steady state given by the singular control. In order to do this the S-system (22) must first be solved separately for the extreme values T,, and T_ of T. Logical choices are T,, = 0 and T_ = p. But T_ = p makes the steady state (x, J!?) indeterminate. Hence T,, may be assigned to a value sufficiently close to p but not equal to p. We take T,, = 15.9 when p = 16. The
TABLE 1 y = 0.05, p = 16,
K=looo, n = 20,
4 = 1, 1 = 2,
h = 0.002, c = 0.1.
242
KRIPASINDHU
CHAUDHURI
MINIMUM
200 -
L o
AND THOMAS
JOHNSON
TAX CURVES
A:
[999,(O)
6:
[500.10~
c:
(100.10)
ISO-
k u IOO-
50-
o-
0 I I I, I I I * I ., 400 tm 800
c 0
200
A_ , I 1000
POPULATION FIG. 1. Minimum-tax
trajectories
of the system represented
by (3).
minimum-tax trajectories obtained by executing ESSYNS with the parameter values in Table 1 are shown in Figure 1 below. By adopting a similar procedure, the maximum-tax trajectories shown in Figure 2 are also generated. We have used STATGRAPHICS’ software to improve the quality of the graphs produced by ESSYNS. This is done by simply transferring the data generated through ESSYNS to STATGRAPHICS. These two packages are compatible and can be executed with IBM PC/AT compatible microcomputers. However, the quality of the graphs also depends on the type of output from STATGRAPHICS. We used PrintAPlot3 to make an EPSON:LQ/850 printer simulate a plotter. Next the nonlinear algebraic Equation (19) is solved for the optimal (singular control) equilibrium values by executing the NLSYS.DOC procedure in GAUSS4 to obtain the results: x* = 140, for the parameter
E* = 86,
T* - 14.58
(241
values in Table 1.
‘STATGRAPHICS is a registered trademark of Statistical Graphics STSC Inc., 2115 East Jefferson Street, Rockville, Maryland 20853, USA. 3PrintAPlot is a registered trademark of Insight Country Club Drive, Mora Za, California 94556, USA. 4GAUSS
is a registered
East, Kent, Washington
trademark
98042, USA.
of Aptech
Development
Systems,
Corporation
Corporation,
1024
Inc., 26250 196th Place South
243
S-SYSTEM FISHERY MODEL
250 ’
2co-
k 0
MAXIMUM TAX ,“‘,“-1”‘r”‘I
C
CiJRVES
A:
(374.100)
8:
(2AO.135)
_
C:
(81.200)
:
ISO-
k Lu IOO-
so-
POPULATION
FIG. 2. Maximum-tax trajectories of the system represented by (3).
A novel feature of an S-system is that it is time reversible. Exploiting this feature we used ESSYNS to construct the minimum-tax and maximum-tax trajectories that pass through the equilibrium point P(x*, E*), as shown in Figure 3. The approach path of the control variable T(t), generated through ESSYNS, is exhibited in Figure 3. If the initial point (x0, E,) is a point such as B or D on one of the maximum-tax and minimum-tax curves through point P(x*, E*), then the indicated policy is simply to use the appropriate tax T,,_ or Tti in order to drive the system to P(x*,E*), where the tax switches to the optimal singular control tax T*. If the system begins at a point below the DPB curve such as at the completely unexploited state A (999, lo), the minimum-tax curve evolving from that point is followed until it meets the maximum-tax path PB. At this intersection, T switches from Tmin to Tmaxand drives the system to P. We cannot take x0 = K = 1000 because x,(O) = K - x0 vanishes, and in an S-system no dependent variable can take a zero value. If the fishery is initially overexploited at a point above the DPB curve such as at C (30, 115), then the indicated taxation policy is to impose the maximum-tax to reduce the level of excessive effort and to build-up the fish population. The system then follows the maximum-tax path evolving from C until it meets the minimum-tax path. At this intersection, T switches from Tmaxto Tmin and drives the system to P.
244
KRIPASINDHU OPTIMAL
CHAUDHURI
CONTROL
AND THOMAS JOHNSON
CURVES
200-
& 0
150-
::
-
w
100so-
POPULATION
FIG. 3.
Optimal
tax
paths corresponding to the parameter values given in Table 1.
YIELD-EFFORT
CURVES
2505
200 -
o02068
1~.,1,1~1,.~I.I.I~~~I.~~I~.~I 10
12
14
16
YIELD
FIG. 4. Optimal yield-effort curves corresponding Table 1.
to the parameter values given in
245
S-SYSTEM FISHERY MODEL
We now incorporate mation
the yield into the calculation
by using the transfor-
9X1X2 x6=
(3)
x4x5
in the ESSYNS algorithm. With the recasting catch rate as in Equation (2). The yield-effort ing to the control paths (Figure 3) are ESSYNS with the parameter values in Table 7.
PRACTICAL
into the S-system x6 gives the curves (Figure 4) correspondthen generated by executing 1.
APPLICATION
In discussions of this problem an advisor to fishery regulators questioned the practicality of the “bang\bang” optimal control. One approach is to see how much the control trajectory would depart from that where the final tax-rate was imposed at the beginning. The suboptimal paths given by adopting the final (singular) tax are shown in Figure 5 for the two examples shown in Figure 3. The corresponding suboptimal yield-effort paths are shown in Figure 6. From these examples it seems that this suboptimal policy might reduce adjustment costs for a fishery starting from a low effort point such as A. A complete evaluation of the lost social return is a matter for further analysis and can be examined by adding to the S-system a OPTIMAL 250 ’
CONTROL
I * 8r I CQP:
’
”
I
CURVES ’
SINGULAR
”
TAX
I,
*7I
CURVE
POPULATION
FIG. 5. Optimal and singular tax curves corresponding in Table 1.
to the parameter values given
243
KRIPASINDHU CHAUDHURI AND THOMAS JOHNSON YIELD-EFFORT 250-’
’
’
’
CURVES ’
’
’
’
‘-
FIG. 6. Optimal and singular yield-effort curves corresponding to the parameter values given in Table 1.
variable time:
that will record
i?
the accumulated
=
PqxIx2k‘;
‘x5-
social returns
k6
-
up to the given
cx2x6,
Here x7 will give the accumulation of value as in the objective function (9). The suboptimal singular tax curves are shown along with the optimal control paths in Figure 5. The yield-effort curves corresponding to Figure 5 are shown in Figure 6. Since it is difficult to estimate the population size correctly, the control curves (Figure 3 and Figure 5) may not be very useful to the regulatory agency. On the other hand the yield-effort curves (Figure 4 and Figure 6) may be easily used by managers to regulate the taxation policy in a near optimal manner. The above procedure for computing approximate optimal population-effort and yield-effort paths can be used in the management of any fishery once the dynamical system describing the fishery is reducible to an S-system and the parameters associated with the fishery are identified and estimated. We have shown here how the S-system analysis and the ESSYNS algorithm can be used as powerful tools in fishery management. 8.
DISCUSSION
We have developed a bioeconomic single species fishery using a realistic
model for the management of a catch-rate function (2). The fish
S-SYSTEM FISHERY MODEL
241
species is governed by the logistic law of growth. A regulatory agency exercises control over the exploitation of the fishery by imposing a suitable tax of T per unit biomass of landed fish. A positive value of T amounts to taxation, while a negative value of T implies subsidy. Taxation may serve as a useful tool for preventing over-exploitation of a natural resource. As we have pointed out, the imposition of a tax on landed swordfish biomass may be one of the ways to regenerate the stock in U.S. swordfish fisheries to the 1978 level. As a tool of control economists consider taxation to be superior in theory to other tools, such as the imposition of license fees and catch limits. Our model takes the fishing effort E as a dynamic variable and assumes that the level of E expands or contracts accordingly as the instantaneous perceived rent is positive or negative. The rate of change of E is proportional to the perceived rent at any time. Taking all these factors into account the dynamics of the fishery is modeled by the nonlinear differential equation system (3). The conditions for the existence of a nontrivial steady state (X, El of the system (3) are derived in both case (A), when the BTP (biotechnical productivity) of the fish species does not exceed the value of the parameter a, and case (B), when the BTP exceeds a. It is found that the steady state is either a stable node or a stable focus in case (A). The same conclusion also holds in cases (Bl) and (Bi>. But the steady state becomes an unstable node or an unstable focus when (B) and (Bii) hold. The optimal harvesting policy problem is then recast as an optimum control problem and analyzed by using the well-known conditions of the maximum principle [19]. The shadow price along the singular path and the equation for the singular path are obtained. It is seen that the steady state singular solution actually follows the same equilibrium path as that deduced by using variational calculus. The most important and novel feature of the present paper is the application of the S-system analysis [9-111 to the bioeconomic problem (3). The system (3) is first recast into the economical nonlinear form of an S-system by following the techniques of Savageau and Voit [l]. We then execute the ESSYNS algorithm [2] with IBM PC/AT to compute: (i) the minimum tax trajectories (Figure 1); (ii) the maximum tax trajectories (Figure 2); (iii) the computed approach path for the control variable T (Figure 3); (iv) the yield-effort curves corresponding to (iii) above (Figure 4); (v) the computed control paths, as well as the singular paths (Figure 5); (vi> the yield-effort curves corresponding to (v) above (Figure 6). Computations are for the particular set of parametric values given in Table 1. This process of computing control paths that satisfy the necessary conditions and then graphing yield-effort curves can be adopted for the
248
KRIPASINDHU
CHAUDHURI
AND THOMAS JOHNSON
management of any natural resource. We have shown how the S-system analysis can be successfully applied in bioeconomic modeling. This work was performed during the tenure of the Fulbright Fellowship awarded to the first author by the Indo-U.S. Sub-Commission on Education and Culture under Grant #:88-11221. The authors thank the authorities of North Carolina State University (Raleigh, USA), the Council for International Exchange of Scholars (Washington, D. C.), the University Grants Commission (New Delhi, India), and Jadarpur University (Calcutta, India) for their help and cooperation in doing this work. The authors are also thankful to Ms. Scott for typing the manuscrtpt.
REFERENCES 1 M. A. Savageau and E. 0. Voit, Recasting nonlinear differential equations as S-systems: a canonical nonlinear form, Math. Biosci. 87:83-115 (1987). 2 D. H. Irvine, M. A. Savageau, and E. 0. Voit, The User’s Guide to ESSKW, accompanying the interactive program ESSYNS (Evaluation and Simulation of Synergistic Systems) for analysis of mathematical models expressed in S-system form, copyrighted by the authors (1989). 3 C. W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources, John Wiley & Sons, New York (1976). 4 C. W. Clark, Mathematical models in the economics of renewable resources, SIAM 5 6
Review 21:81-99 (1979). C. W. Clark and J. D. DePree, A simple linear model for the optimal exploitation of
renewable resources, Appl. Math. Optim. .5:181-196 (1979). L. G. Anderson and D. R. Lee, Optimal governing instrument, operation level and enforcement in natural resource regulation: the case of the fishery, Amen’. .I. Agr. Econ. 68:678-690
(1986).
7 J. G. Sutinen and P. Andersen, The economics of fisheries law enforcement, Economics
8
Land
61~387-397 (1985).
P. F. Verhulst, Notice sur la loi que la population suit dans son accroissemant, Mathematique et Physique l&112-121 (1838). M. A. Savageau, Mathematics of organizationally complex systems, Biomed. B&him.
Correspondance
9
Acta 44~839-844 (1985).
10 E. 0. Voit and M. A. Savageau, S-system analysis of biological systems, in V. Capasso, E. Gorsso and S. L. Paveri-Fontana (Eds.), Lecture Notes in Biomathematics 57:517-524
(1985).
11 E. 0. Voit and M. A. Savageau, Equivalence between S-systems and Volterra-systerns, Math. Biosci. 78:47-55 (1986). 12 Report of the NMFS Swordfish Stock Assessment Workshop (March 20-24, 1989), National Marine Fisheries Service, Southeast Fisheries Center, Miami Laboratory, 1989 (unpublished). 13 E. 0. Voit, M. A. Savageau, and D. H. Irvine, An introduction to S-systems, in Canonical Nonlinear Modeling: S-system Approach to Understanding Complexity, E. 0. Voit, ed., Dept. of Biometry, Medical University of South Carolina, Charleston (in press).
S-SYSTEM
FISHERY
249
MODEL
14
M. A. Savageau,
15
metabolic control theory. 1. Fundamental similarities and differences. Math. Biosci. 86:127-145 (1987). A. Sorribas and M. A. Savageau, A companion of variant theories of intact biochemical systems. 1. Enzyme-enzyme interaction and biochemical systems theory. Math. Biosci. 94:161-193 (1989).
E. 0.
Voit,
and
D. H. Irvine,
Biochemical
systems
theory
and
16
A. Sorribas and M. A. Savageau, A companion of variant theories of intact biochemical systems. 2. Flux oriented and metabolic control theories. Math. Biosn’. 94:195-238 (1989).
17
A. Sorribas and M. A. Savageau, Strategies for representing metabolic pathways within biochemical systems theory: Reversible pathways. Math. Biosci. 94:239-269 (1989). M. A. Savageau and E. 0. Voit, Power-law approach to modeling biological systems I. Theory, J. Ferment. Technol. 60:221-228 (1982). G. Leitmann, An Introduction to Optimal Control, McGraw-Hill, New York (1966).
18 19
ABSTRACT A bioeconomic catch-rate function fishery by imposing
model of a single species fishery is developed by using a realistic and assuming that a regulatory agency controls exploitation of the a suitable tax on the catch. The existence of a nontrivial steady state
and its stability are examined. The optimal harvesting policy is discussed from the viewpoints of variational calculus and control theory. The fishery is then modeled as an S-system examples
by following the recasting techniques of Savageau of the optimal control curves and the yield-effort
executing
the ESSYNS
1.
and Voit 111. Numerical curves are obtained by
[21 algorithm.
INTRODUCTION
In recent years interest has continued in using bioeconomic modeling to gain insight into scientific management of the exploitation of renewable resources like fisheries and forestries. Clark [3] provides an introduction to this field of study. The techniques and issues associated with the dynamic economic models of natural resource exploitation are further developed in [4] and [5]. Exploitation of a natural resource may also involve problems of law enforcement. This happens particularly when a natural resource is severely depleted. In the nineteenth century many important marine fisheries were on the verge of collapse due to over-exploitation. To avert the crisis some countries entered into multilateral agreements in the North Sea Convention of 1882. By the end of 1977 a majority of the coastal nations declared jurisdiction over the sea up to 200 nautical miles from their shores. The
*Permanent 70032, India.
address:
MI4THEM4TZCAL
Department
BIOSCIENCES
of Mathematics,
99:231-249
Jadarpur
University,
Calcutta-
(1990)
OElsevier Science Publishing Co., Inc., 1990 655 Avenue of the Americas, New York, NY 10010
231 002%5564/90/%03.50
232
KRIPASINDHU
CHAUDHURI
AND THOMAS
JOHNSON
responsibility for managing coastal fisheries now lies directly with the coastal countries through the establishment of these 200-mile exclusive economic zones in the sea. Some of the issues associated with the choice and the enforcement of optimal governing instruments in regulating fisheries have been discussed by Anderson and Lee 161. The economic aspects of enforcing laws for regulating marine fisheries have been studied by Sutinen and Anderson [7]. Imposition of taxes and license fees, leasing of property rights, and direct control have been proposed as governing instruments. Economists consider taxation to be superior, at least in theory, to other control policies because of its flexibility [3]. Also, “many of the advantages of a competitive economic system can be better maintained under taxation than under other regulatory methods” [31. Even though a tax is not a politically feasible control instrument at this time the analysis of tax control gives a standard against which to compare other controls. For example, licenses and seasons might be used to approximate the effort computed from the model of a tax policy. In this paper a bioeconomic model of a single species fishery is developed using a catch-rate function that is more flexible and that seems to be more realistic than the constant “catch-per-unit-effort hypothesis” [3]. In this model the fish biomass obeys the logistic law of growth [8]. It is assumed that a regulatory agency exercises control over exploitation of the fishery by imposing a tax on the catch. The dynamics of the fishery is modeled by a system of two nonlinear differential equations that incorporate these aspects. The level of the fishing effort changes continuously, and its rate of change at any time is proportional to the perceived rent (net economic revenue to the fisherman) at that instant. It is found that the existence of a nontrivial steady state of the system depends on the BTP’ of the fish species as well as on the other parameters. Conditions for the existence and stability of a nontrivial steady state are derived. The optimal harvesting problem is studied from the viewpoints of variational calculus and control theory. The dynamical system describing the fishery is then modeled as an S-system following the recasting techniques of Savageau and Voit 111. With the tax rate as the control variable, the optimal control curves and the corresponding yield-effort curves are computed using the ESSYNS 121 program on an IBM PC/AT. The set of parameter values used is shown in Table 1. This appears to be the first time that the S-system analysis developed by Savageau and Voit [9-111 has been effectively applied in modeling both the biological and economic dynamics
‘BTP (biotechnical productivity) of a biological species is defined to be the ratio of its intrinsic growth rate (biotic potential) to its catchability and coefficient [3].
S-SYSTEM
FISHERY
233
MODEL
of a bioeconomic system. Savageau and Voit [l] have discussed in detail the advantages of the S-transformation and its related numerical algorithms ESSYNS in solving problems of this type numerically. “It is generally one to two orders in efficiency due reasonable error error tolerances, than with RKF45 rithm” [l].
2.
superior
to existing
methods
for solving this class of problems,
often
of magnitude faster.. The cost of recasting (identified with decreases to the introduction of additional variables) is generally recouped at tolerances and the net improvements are often remarkable: at stringent several problems were solved 50 or 100 times faster with ESSYNS (Runge-Kutta-Fehlberg) and up to 50 times faster than with Gear algo-
FORMULATION The catch-rate
OF THE PROBLEM
function
h(t) usually encountered
in the fishery models is
(1)
h(t) = qE(t)x(t),
where x(t) = size (or biomass) of the fish population at time t, E(t) = fishing effort at time t, and q = catchability coefficient. This form of the catch-rate function is based on the constant “catch-per-unit-effort hypothesis” [3], and it seems to be unrealistic on the following grounds: (1) It assumes a process of random search for (2) It implies that all fish in the population captured. (3) For a fixed population size, the catch-rate effort does so. (4) For a fixed level of effort, the catch-rate population size gets larger and larger.
fish. are equally increases
linearly
as the
increases
linearly
as the
To relax these restrictive assumptions Clark [4] suggested alternative form of the catch-rate function:
h(t)
=
qE(t)x(t) aE(t)+Zx(t)
’
likely to be
the following
(2)
where a, 1, q are all positive constants. In Equation (2) h -+ (q/a)x as E -+ m for a fixed population size and h + (q/l)E as x +m for a fixed level of effort. Thus the catch-rate function in (21 exhibits saturation effects with respect to the fishing effort E as well as to the stock abundance x. This seems to reflect a reasonable proposition. The parameter a is proportional to the ratio of the population level to the catch-rate at higher levels of effort, while the parameter 1 is proportional to the ratio of the effort level to the catch-rate when the
KRIPASINDHU
234
CHAUDHURI
AND THOMAS JOHNSON
biomass level is very high. The catch-rate function (2) is used in the model discussed below. It is also assumed that a regulatory agency exercises control over the fishery by imposing a tax of T( > 0) units of money per unit biomass of landed fish. A negative value of T implies a subsidy to the fishermen. When a fish species undergoes severe depletion and is on the verge of extinction, imposition of a very high tax may be a reasonable regulatory mechanism to save the fishery from collapse. In particular, an assessment of the western North Atlantic swordfish stock [12] reveals that the stock and the yield have fallen much below their 1978 levels. It has been suggested that the spawning stock biomass should be returned to its 1978 level. Achievement of this goal requires a 60% reduction of the 1988 fishing mortality rate. The imposition of a tax on landed swordfish biomass may be one of the several possible ways to reach this management goal. The present model assumes that the economic system has a feedback whereby the level of fishing effort expands or contracts depending on whether the “perceived rent” (net revenue to the fisherman) is positive or negative. Clark [3] carried out this sort of investigation with a single species for fish with a logistic growth of biomass [8] and with the traditional catch-rate function (1). With this perspective we study a fish species whose biomass growth and harvest are governed by the following dynamical system:
dE -&=A
q(P-T)x aE+lx
-
c E I ’
(3)
Here y = biotic potential of the fish species, K = carrying capacity of the fish species, p = the constant price per unit biomass of the landed fish, c = the constant fishing cost per unit effort, and A = the stiffness parameter. A gives the speed with which the effort reacts to the changes in the perceived rent flow. All of these parameters are assumed to be positive. We note, however, that A is a dimensionless constant. Therefore, the net economic revenue to the society is
i@x
___-
cE
aE+lx -c]+T(
iii&).
= net economic revenue to the fisherman + economic revenue to the regulatory agency.
S-SYSTEM
3.
FISHERY
MODEL
THE STEADY
STATE
235
The nontrivial steady state (2, E) obtained X = 0 = _I!?simultaneously is given by
by solving
the equations
i_J[~~+(P-m-dl dP-T)
’
(4)
and
We have the following possibilities: (A) Let a 2 q/-y. This implies that the parameter a is no less than the “biotechnical productivity” [3] of the fish species. (Ai) In the case of taxation we have 0 < T < p so that p - T > 0 and hence, X > 0. (Aii) In the case of subsidy T < 0 so that p - T > 0 and hence x > 0. (Aiii) The case 0 < p < T seems to be irrelevant to a commercial fishery and is not considered in this analysis. However, this case could be relevant for a sport fishery where the process rather than the sale of the catch is the primary objective. However, to make E > 0 requires the additional parametric constraint: q( p - T) > fc. ThusX>O,E>Oprovidedthatu>q/yandq(p-T)>Ic.
(6)
(B) Let a < q/y. This implies that the biotechnical productivity of the fish species exceeds the value of the parameter a. As before, we may argue that p - T > 0. It is found that x and E are both positive if, and only if, the constraint (P-T)(q-ay) q and qr > lc; and (B) ay 0. In the case (B), Det A > 0 if Z2c2 - qr*(q - ay) > 0. Using (B), we may easily prove after a little calculation that the inequality PC2 - qd(
q - ar) > 0
leads to the inequality - ay < 0, which is obviously true. Hence no mathematical inconsistency arises in assuming that 12c2 - qd(q
- ay) > 0.
However, the values of the parameters may be such that l*c* - qr*(q ay) < 0 also holds. We now examine what happens to Det A in this case. After a little simplifcation, we find that Det A = Ac(qa -
lc){lc - 4q - WI) qas-*
Obviously Also
Det A > 0 by virtue of (B). Thus Det A > 0 always.
Trace A = - ---&[acA7r(q7T-zc)+qT*(ay-q)+l*C*]
In case (A), W > 0 and hence Trace A < 0. Thus Det A > 0 and Trace A < 0 in this case.
S-SYSTEM FISHERY MODEL
231
This means that the eigenvalues of the linearized system (8) are either both real and negative or are complex conjugates with negative real parts. We may, therefore, conclude that the steady state (2, E) is either a stable node or a stable focus. The same conclusion holds in case (B) when 12c2 - q2(
q - ay) > 0.
When 12c2 - qn-*(q - ay) < 0 we write
and find that (Bi) Trace A < 0 when acha(qr - lc) > qr2(q - ay)- 12c2 > 0, and (Bii) Trace A > 0 when 0 < uchrr(qr - Zc) < qr’(q - ay>- 12c2. The conclusion is the same as in (A) when (Bi) holds. In case (Bii), the eigenvalues of (8) are either both real and positive or are conjugates with positive real parts. This implies that the steady state (i,E) is either an unstable node or an unstable focus when (B) and (Bii) hold. 4.
OPTIMAL
HARVESTING:
A CONTROL
PROBLEM
The objective of the regulatory agency is to maximize the total discounted net revenues that 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) where 6 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 Equation (3) and the constraint Tmin < T(t)
< Tma”
(10)
on the control variable T(t). One may allow for a subsidy by taking T mh < 0. Since the objective function is still linear in the control variable E, the optimal control will be a “bang-bang” control between T”‘” and T max until it is optional to adopt the singular control. We now construct the Hamiltonian equation;
AE
>
where PI(t) and p2(t) are the adjoint (or co-state) variables.
(11)
238
KRIPASINDHU CHAUDHURI AND THOMAS JOHNSON
Equating
to zero the coefficient
of T in (ll),
CL*(~)qhx(t)E(t)= aE(t) + Ix(t)
we get o
.
Since E(t) > 0 and x(t) > 0, we have
This gives the condition is
and this simplifies
for singular control. The adjoint equation
for p2(t)
to
dt> = e-8t[p_$(!y2],
(13)
since p2(t) = 0. Thus, the shadow price p(t) along the singular path is given by p(t) The adjoint equation
=hl(t)es’=
for p&j,
p-_$yq2.
(14)
namely
d/+ -=_dt
aH ax
reduces to 4%
dt=since j_h2(t) = 0. Using Equations
apqE 2e -*’
aqE2
(aE+IX)2
(uE+I,~)~
1 (15)
(3) and (13) in (15) we get
y[qr,y-c(nE+1x)]
-- ~;(l-~)(aE+~~))=y(l-~)[p-~(~)2].
(16)
S-SYSTEM FISHERY MODEL
239
This is the singular path. After a little simplifcation the steady state singular solution is given by I
PWX) + =6
it can be shown that
{4~(x)12- 4X2WX)j
[4x - ~~(X>12 qlcx2
(
p-
1
(17)
[4x - ~~(X>12 .
This is the same path obtained from the Euler-Lagrange equation in the variational calculus approach. The authors derive Equation (17) in two different ways purely as an academic exercise to illustrate the confluence of the two methods in linear control problems. The resulting equilibrium values are
x=x*, T=T*=p-
lx*F(x*)
E=E*=
4x* - ex*)
’
lcf
(18)
4x* - ax*)
where x = x* is the unique positive root (if one exists) of Equation (17). In terms of the parameters of the system Equation (17) simplifies to the form 2pu543
+ Ky2[ pay 6 - p) + 4up( q - ur) - uk] x2
+zpK*[(q
+ K'[p(6 5.
S-SYSTEM
- ay)(aPs
- lc) + P(4 - w)(q
- y)(q - fly)*-
-W)lx
Ic( q6 - qy + ay*)] = 0.
(19)
ANALYSIS
We now recast the dynamical system (3) into an S-system [l] in order to gain further insight into its dynamic nature and to compute the control path for the taxation policy with the necessary conditions. An S-system is a system of nonlinear ordinary differential equations of the form
240 An
KRIPASINDHU
CHAUDHURI
S-system is a special case of the Generalized
AND THOMAS
JOHNSON
Mass Action System
xi= 5 cqkfixR’,‘It PikiiX:il*iE(l,...,n). j=l j=l k=l
Conversely,
k=l
an S-system is a generalization
of the form
~i=&__iE(l,...,“)’ j=l
which has been called a “Half-system.” Voit, Savageau and Irvine [13] summarize [14-171 with the following statement: “While S-systems, modeling alternatives,
the results of several studies
Generalized Mass Action systems, and Half-systems are all viable S-systems are of particular interest. They provide the only type of
power-law model with nonzero steady states that can be calculated by symbolic, analytical means. Furthermore, analysis of biochemical pathways has shown that S-systems yield more accurate representations than competing Mass Action systems. Although Half-systems have an appealingly simple mathematical structure, their use as power-law approximations of biological systems is limited since they only have steady states in which at least one variable vanishes.”
The recasting of the original system (31 as an S-system (2) is performed by following the procedure developed by Savageau and Voit [ll. By introducing the new variables x1=x, x2 = E, x3= K-x, x4 =aE +1x, the dynamical system (3) transforms to:
/t1=
Xx1x3 - 4x1x2x4-1~
i2
=
A(P
i3
=
4x,x2x4-1
i4
=
ah( p - T)qxIx2x4-’ - aAcxz + %x1x3 - lqx1xzx41.
-
Thx1x2x4-‘-
Acxn, (20)
-
$x1x31
The system (20) represents the power-law form [18] of the original system (3). However, it is not an S-system because the fourth equation consists of four terms. This sum is reduced by defining x4 as the product of two new variables, x4
then differentiating
the product,
=
x5x6
3
identifying
(21)
the appropriate
terms in the
S-SYSTEM FISHERY MODEL
241
fourth equation of (20) for each new derivative, and permutting the indices (5 -+ 4, 6 + 5). The final S-system form of the original system (3) becomes:
i1=
$X1X3 - 4XlX2X2X~‘~
~2=A(p-~)qX1X2X41X;1-~CX2~
(22)
i3=qX1X2X41x5-1~XIX3r ~4=ahq(p-T)X~X2X~1X;2-~~~X2X;1~
lY i5 = EX1X3X4 The constraints
l-
lqx1x2x42x;1.
among the variables
x4x5=
x3=K-x17 x4X5 =x6
are:
(=
ax2+lxl,
(23)
old X4).
Although the S-system (22) contains five variables, two of them (xi and x2) are the original variables x and E. Thus the constraints provided by the definition of new variables and the assignment of initial values to x4 and ,Q restrict the system behavior to a manifold of the same dimension as in the original system. 6.
NUMERICAL
EXAMPLES
As pointed out by Savageau and Voit [l], the general S-system possesses no finite closed-form solution in terms of elementary functions. Hence one has no alternative but to look for efficient numerical solutions in dealing with S-systems. The set of numerical algorithms called ESSYNS developed by Irvine, Savageau, and Voit [2] yields efficient numerical solutions of S-systems. We have executed ESSYNS with an IBM PC/AT for our S-system (22) with the set of parameter values given in Table 1. Our objective is to construct a map for the approach to the steady state given by the singular control. In order to do this the S-system (22) must first be solved separately for the extreme values T,, and T_ of T. Logical choices are T,, = 0 and T_ = p. But T_ = p makes the steady state (x, J!?) indeterminate. Hence T,, may be assigned to a value sufficiently close to p but not equal to p. We take T,, = 15.9 when p = 16. The
TABLE 1 y = 0.05, p = 16,
K=looo, n = 20,
4 = 1, 1 = 2,
h = 0.002, c = 0.1.
242
KRIPASINDHU
CHAUDHURI
MINIMUM
200 -
L o
AND THOMAS
JOHNSON
TAX CURVES
A:
[999,(O)
6:
[500.10~
c:
(100.10)
ISO-
k u IOO-
50-
o-
0 I I I, I I I * I ., 400 tm 800
c 0
200
A_ , I 1000
POPULATION FIG. 1. Minimum-tax
trajectories
of the system represented
by (3).
minimum-tax trajectories obtained by executing ESSYNS with the parameter values in Table 1 are shown in Figure 1 below. By adopting a similar procedure, the maximum-tax trajectories shown in Figure 2 are also generated. We have used STATGRAPHICS’ software to improve the quality of the graphs produced by ESSYNS. This is done by simply transferring the data generated through ESSYNS to STATGRAPHICS. These two packages are compatible and can be executed with IBM PC/AT compatible microcomputers. However, the quality of the graphs also depends on the type of output from STATGRAPHICS. We used PrintAPlot3 to make an EPSON:LQ/850 printer simulate a plotter. Next the nonlinear algebraic Equation (19) is solved for the optimal (singular control) equilibrium values by executing the NLSYS.DOC procedure in GAUSS4 to obtain the results: x* = 140, for the parameter
E* = 86,
T* - 14.58
(241
values in Table 1.
‘STATGRAPHICS is a registered trademark of Statistical Graphics STSC Inc., 2115 East Jefferson Street, Rockville, Maryland 20853, USA. 3PrintAPlot is a registered trademark of Insight Country Club Drive, Mora Za, California 94556, USA. 4GAUSS
is a registered
East, Kent, Washington
trademark
98042, USA.
of Aptech
Development
Systems,
Corporation
Corporation,
1024
Inc., 26250 196th Place South
243
S-SYSTEM FISHERY MODEL
250 ’
2co-
k 0
MAXIMUM TAX ,“‘,“-1”‘r”‘I
C
CiJRVES
A:
(374.100)
8:
(2AO.135)
_
C:
(81.200)
:
ISO-
k Lu IOO-
so-
POPULATION
FIG. 2. Maximum-tax trajectories of the system represented by (3).
A novel feature of an S-system is that it is time reversible. Exploiting this feature we used ESSYNS to construct the minimum-tax and maximum-tax trajectories that pass through the equilibrium point P(x*, E*), as shown in Figure 3. The approach path of the control variable T(t), generated through ESSYNS, is exhibited in Figure 3. If the initial point (x0, E,) is a point such as B or D on one of the maximum-tax and minimum-tax curves through point P(x*, E*), then the indicated policy is simply to use the appropriate tax T,,_ or Tti in order to drive the system to P(x*,E*), where the tax switches to the optimal singular control tax T*. If the system begins at a point below the DPB curve such as at the completely unexploited state A (999, lo), the minimum-tax curve evolving from that point is followed until it meets the maximum-tax path PB. At this intersection, T switches from Tmin to Tmaxand drives the system to P. We cannot take x0 = K = 1000 because x,(O) = K - x0 vanishes, and in an S-system no dependent variable can take a zero value. If the fishery is initially overexploited at a point above the DPB curve such as at C (30, 115), then the indicated taxation policy is to impose the maximum-tax to reduce the level of excessive effort and to build-up the fish population. The system then follows the maximum-tax path evolving from C until it meets the minimum-tax path. At this intersection, T switches from Tmaxto Tmin and drives the system to P.
244
KRIPASINDHU OPTIMAL
CHAUDHURI
CONTROL
AND THOMAS JOHNSON
CURVES
200-
& 0
150-
::
-
w
100so-
POPULATION
FIG. 3.
Optimal
tax
paths corresponding to the parameter values given in Table 1.
YIELD-EFFORT
CURVES
2505
200 -
o02068
1~.,1,1~1,.~I.I.I~~~I.~~I~.~I 10
12
14
16
YIELD
FIG. 4. Optimal yield-effort curves corresponding Table 1.
to the parameter values given in
245
S-SYSTEM FISHERY MODEL
We now incorporate mation
the yield into the calculation
by using the transfor-
9X1X2 x6=
(3)
x4x5
in the ESSYNS algorithm. With the recasting catch rate as in Equation (2). The yield-effort ing to the control paths (Figure 3) are ESSYNS with the parameter values in Table 7.
PRACTICAL
into the S-system x6 gives the curves (Figure 4) correspondthen generated by executing 1.
APPLICATION
In discussions of this problem an advisor to fishery regulators questioned the practicality of the “bang\bang” optimal control. One approach is to see how much the control trajectory would depart from that where the final tax-rate was imposed at the beginning. The suboptimal paths given by adopting the final (singular) tax are shown in Figure 5 for the two examples shown in Figure 3. The corresponding suboptimal yield-effort paths are shown in Figure 6. From these examples it seems that this suboptimal policy might reduce adjustment costs for a fishery starting from a low effort point such as A. A complete evaluation of the lost social return is a matter for further analysis and can be examined by adding to the S-system a OPTIMAL 250 ’
CONTROL
I * 8r I CQP:
’
”
I
CURVES ’
SINGULAR
”
TAX
I,
*7I
CURVE
POPULATION
FIG. 5. Optimal and singular tax curves corresponding in Table 1.
to the parameter values given
243
KRIPASINDHU CHAUDHURI AND THOMAS JOHNSON YIELD-EFFORT 250-’
’
’
’
CURVES ’
’
’
’
‘-
FIG. 6. Optimal and singular yield-effort curves corresponding to the parameter values given in Table 1.
variable time:
that will record
i?
the accumulated
=
PqxIx2k‘;
‘x5-
social returns
k6
-
up to the given
cx2x6,
Here x7 will give the accumulation of value as in the objective function (9). The suboptimal singular tax curves are shown along with the optimal control paths in Figure 5. The yield-effort curves corresponding to Figure 5 are shown in Figure 6. Since it is difficult to estimate the population size correctly, the control curves (Figure 3 and Figure 5) may not be very useful to the regulatory agency. On the other hand the yield-effort curves (Figure 4 and Figure 6) may be easily used by managers to regulate the taxation policy in a near optimal manner. The above procedure for computing approximate optimal population-effort and yield-effort paths can be used in the management of any fishery once the dynamical system describing the fishery is reducible to an S-system and the parameters associated with the fishery are identified and estimated. We have shown here how the S-system analysis and the ESSYNS algorithm can be used as powerful tools in fishery management. 8.
DISCUSSION
We have developed a bioeconomic single species fishery using a realistic
model for the management of a catch-rate function (2). The fish
S-SYSTEM FISHERY MODEL
241
species is governed by the logistic law of growth. A regulatory agency exercises control over the exploitation of the fishery by imposing a suitable tax of T per unit biomass of landed fish. A positive value of T amounts to taxation, while a negative value of T implies subsidy. Taxation may serve as a useful tool for preventing over-exploitation of a natural resource. As we have pointed out, the imposition of a tax on landed swordfish biomass may be one of the ways to regenerate the stock in U.S. swordfish fisheries to the 1978 level. As a tool of control economists consider taxation to be superior in theory to other tools, such as the imposition of license fees and catch limits. Our model takes the fishing effort E as a dynamic variable and assumes that the level of E expands or contracts accordingly as the instantaneous perceived rent is positive or negative. The rate of change of E is proportional to the perceived rent at any time. Taking all these factors into account the dynamics of the fishery is modeled by the nonlinear differential equation system (3). The conditions for the existence of a nontrivial steady state (X, El of the system (3) are derived in both case (A), when the BTP (biotechnical productivity) of the fish species does not exceed the value of the parameter a, and case (B), when the BTP exceeds a. It is found that the steady state is either a stable node or a stable focus in case (A). The same conclusion also holds in cases (Bl) and (Bi>. But the steady state becomes an unstable node or an unstable focus when (B) and (Bii) hold. The optimal harvesting policy problem is then recast as an optimum control problem and analyzed by using the well-known conditions of the maximum principle [19]. The shadow price along the singular path and the equation for the singular path are obtained. It is seen that the steady state singular solution actually follows the same equilibrium path as that deduced by using variational calculus. The most important and novel feature of the present paper is the application of the S-system analysis [9-111 to the bioeconomic problem (3). The system (3) is first recast into the economical nonlinear form of an S-system by following the techniques of Savageau and Voit [l]. We then execute the ESSYNS algorithm [2] with IBM PC/AT to compute: (i) the minimum tax trajectories (Figure 1); (ii) the maximum tax trajectories (Figure 2); (iii) the computed approach path for the control variable T (Figure 3); (iv) the yield-effort curves corresponding to (iii) above (Figure 4); (v) the computed control paths, as well as the singular paths (Figure 5); (vi> the yield-effort curves corresponding to (v) above (Figure 6). Computations are for the particular set of parametric values given in Table 1. This process of computing control paths that satisfy the necessary conditions and then graphing yield-effort curves can be adopted for the
248
KRIPASINDHU
CHAUDHURI
AND THOMAS JOHNSON
management of any natural resource. We have shown how the S-system analysis can be successfully applied in bioeconomic modeling. This work was performed during the tenure of the Fulbright Fellowship awarded to the first author by the Indo-U.S. Sub-Commission on Education and Culture under Grant #:88-11221. The authors thank the authorities of North Carolina State University (Raleigh, USA), the Council for International Exchange of Scholars (Washington, D. C.), the University Grants Commission (New Delhi, India), and Jadarpur University (Calcutta, India) for their help and cooperation in doing this work. The authors are also thankful to Ms. Scott for typing the manuscrtpt.
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