Revised Draft
February 6 2002
Lake Victoria Fisheries: Policy Conflicts Created by Predator-Prey Relations
Gardner Brown*,#
Brett Berger**
Moses Ikiara***
*University of Göteborg
University of Washington
#Resources for the Future, University Fellow
**Federal Reserve Board, Washington
***Kenya Institute for Public Policy, Research and Analysis
Please address all correspondence to:
gbrown@u.washington.edu
Lake Victoria Fisheries: Policy Conflicts Created by Predator-Prey Relations
Abstract
Economic efficient resource use is just one policy goal. Foreign exchange earnings,
employment for women and healthy people are other goals promulgated by Tanzania,
Kenya and Uganda in the management of Lake Victoria Fisheries. That these social goals
are in conflict is vividly portrayed in the bioeconomic predator-prey model where
favoring a particular species (goal) reduces the sustainable harvest of another species
(goal). Species interdependence is increasingly important to understand. However,
traditional concepts such as maximum sustained yield take on a new meaning. To be
operational, fishery biologists and other researchers must choose an explicit value for
each fish, a rate of exchange. Data from Kenya are used to estimate the population
dynamics equations and illustrate the opportunity cost of trading one goal for another.
JEL Codes:
Q22 Fishery and Wildlife
Q28 Government Policy
2
Introduction
In the world of practical affairs, managers of renewable resources such as fish
pursue goals other than economic efficiency. Indeed, economic efficiency often seems
not to rank very high in the scale of possible objectives. We begin first with a predatorprey model driven by profit maximization. The model is then extended to demonstrate
clearly how pursuing an additional goal, such as increasing the harvest of one species to
improve foreign exchange balances, for example, can decrease the harvest of another
species whose important public role contributes to the nutritional (protein) requirements
of an indigenous population. The conflict in goal achievement arising from the biological
interdependence between species is illustrated by a case in Lake Victoria, Africa. The
two interdependent species accounting for more than 88 percent of the harvest in a recent
year are Nile perch (lates niloticus) and dagaa (Rastrineabola argentia).
Natural scientists have given some attention to the study of multiple interacting
species [Larkin, 1966, May, 1974, Pielou 1969, Maynard-Smith 1974] but there has been
less study of multiple species in an optimization framework, particularly from a
bioeconomic perspective. Multiple species models are not generous in the results
produced. Clark [1990] concludes that while optimal steady state values often can be
determined, exact solutions for the optimal path to the steady state are not known and
may not exist for some problems. The simpler the model, the more likely clean results
will emerge. Solow’s analysis of a Volterra model is insightful but the model has no
intraspecific competition; i.e. natural resource capital has no diminishing returns. Still, he
3
concludes that even in the two species models, there are no easily obtained qualitative
results resembling those gleaned from single species model.
Wilen and Brown (1986) make some progress in characterizing the solution to a
predator-prey optimization problem. Modest success comes at the cost of assuming a
unidirectional coupled system. The lower organism enhances the growth of the upperlevel organism but the predator has no effect on the prey. This assumption seems to be
rather strong and is not empirically credible for the perch-dagaa population dynamics
studied below. Ragozin and Brown (1985) established the existence of a steady state and
described the unique approach to it for a predator-prey system in which only the predator
species is harvested. It is this model which most closely resembles the model of perch
and dagaa, with harvest of both species set forth below.
In addition to the analysis of multiple objectives, we contribute to the predatorprey literature by deriving some unambiguous comparative static results discussed in Sec
3. Under restrictive but reasonable conditions – met in the Lake Victoria case, a price
increase for the harvested predator (prey) increases (decreases) the stock of both predator
and prey. Just as in the traditional single species model with a stock externality, harvest
can respond either way to price increase. In the case studied, predator (prey) harvest
responds positively to an own price increase if the relative optimal stock is less than or
equal (greater than or equal) to one-half the respective carrying capacity.
[For final draft, see about the case of P2  h1 , h2  ]
2. The Model
Perch (predator) population dynamics is given by
(1.1)
4
dR 
 R  f ( R)  h1  RD ,
dt
f R   0 for R  Rmsy , f R  R  Rmsy  0
where R= the stock of perch at time t; time subscripts are suppressed,
h1 = harvest of perch,
D = the stock of dagaa, and
msy = maximum sustained yield (single stock definition).
The dagaa enter the perch population dynamics linearly.  is the effect of a unit change
in dagaa on the percent growth rate of perch.
In particular, assume the underlying population dynamics for the perch is logistic,
(1.2)
 R
R  r1 R 1    h1  RD ,
 R
where
r1  the intrinsic rate of growth for the perch,
R = the carrying capacity for the perch.
The logistic is introduced for illustrative purposes, but both biological and economic
reasoning require any candidate growth function to exhibit diminishing returns.
Dagaa (prey) population dynamics is specified as
(2.1)
dD 
 D  g ( D)  h2  DR ,
dt
dg
dg
 0 for D  Dmsy ,
0
dD
dD D Dmsy
where
h2 = harvest of dagaa.
Adopting the logistic form again,
5
 D
D  r2 D 1    h2  DR.
 D
(2.2)
In this primitive model, it is assumed that the marginal cost of harvest is constant
(perhaps 0), independent of the population size. The goal in the introductory model is to
maximize the present value of profit ( )

   e  t P1 h1  P2 h2 dt ,
0
where   the discount rate,
Pi  the unit profit of harvested fish, i  1(perch), 2(dagaa).
It is understood that harvest cannot exceed a certain maximum
0  h1  h1 max
0  h2  h2 max,
at any moment, an assumption made for mathematical convenience.
Other objective functions are possible. For example, those concerned about
managing perch and dagaa might value foreign exchange at a unit value of v1 . Since most
perch are exported the annual amount of foreign exchange (F) is simply
F  P1h1 .
Then the annual payoff of perch harvesting is
P1  v1 P1 h1.
Similarly, since dagaa is an artisanal fishery and dagaa form a substantial fraction of the
total protein consumed by dwellers around lake Victoria, planners might reasonably place
a social value on nutrition that exceeds the price of dagaa since a diminution of protein
gives rise to opportunity costs such as public health costs and other social concerns such
6
as absenteeism. If the social health benefit of a unit of dagaa is v 2 , then the annual payoff
from harvesting dagaa is
P2  v2 h2 .
In both of these important cases, the social goals of increased foreign exchange
and improved health formally are the same as an increase in the price of perch and dagaa
respectively. Thus we can have a richer measure of welfare, yet have some economy in
model formulation by remembering that the prices for perch and dagaa are placeholders
for more than the mere ex-vessel price of fish. We will give concrete examples in the
empirical section of how introducing multiple objectives changes the harvesting policy.
Forming the current value Hamiltonian for the simple maximization problem,
(3)
H  P1h1  P2 h2  1  f ( R)  h1  RD   2 g ( D)  h2  DR ,
with 1 and  2 the adjoint variables for (1.1) and (2.1).
For a maximum of H, either we use the following bang bang controls,
(3.1)
h1  0 if 1  P1 ,
h1  h1 max if 1  P1 ,
(3.2)
h2  0 if 2  P2 ,
h2  h2 max if 2  P1 ,
or singular controls when
(3.3)
1  P1 ,
(3.4)
2  P2 .
The adjoint equations are
(4.1)
7
1  1  1  f ( R)  D  2 D
(4.2)
2  2  2 g ( D)  R  1R
In a steady state interior equilibrium, from (4.1) (4.2), (3.3) and (3.4)
(5.1)
  f ( R)  D 
P2
D,
P1
(5.2)
  g ( D)  R 
P1
R.
P2
In equilibrium, according to the right-hand side of (5.1) and (5.2), the real marginal rate
of return on each species has to earn the market rate of return (  ).
It will be useful in subsequent analysis to rewrite (5.1) and (5.2) as
(5.1')
  f ( R)  P1  P2 
D
,
P1
(5.2')
  g ( D)  (P1  P2 )
R
,
P2
The terms to the right of f ( R) and g ( D) arise from the predator-prey relation and can
be thought of as a biological technical externality. These additional terms vanish when
the species’ interaction coefficients,     0. Just as the traditional stock externality
(when unit harvest cost falls as stock increases) has economic parameters so too does the
technical externality with the common terms, P1  P2 , in each equation. When
P1  P2  0 , the technical externality is positive meaning that both optimal steady state
perch and dagaa stocks are larger than for a model with independent stocks     0.
The technical externality is increasing in P1 and decreasing as P2 increases. Note for
future reference that when
8
P1  P2
P2
 0, an increase in
P2 , the price of dagaa, makes P1  P2 smaller, and contributes to reducing the optimal
dagaa stock as a result of an increase in the price of dagaa.
Rewrite (4.1) for 1  0 as
(6)
1 
1  f ( R)  D   2 D
.

If the figurative owner of predator capital sells a unit for harvest, the rental rate of 1 is
earned. The owner bears two future changes in perpetuity so they are capitalized by  .
First is the marginal contribution of that fish to the fishery whose unit economic value is
1 . Second, drawing down perch by one unit increases the marginal productivity of dagaa
by  D , which is valued at its shadow value of  2 each period to the figurative owner of
the prey. Thus the owner of perch or society in general would gain
 2 D
by the

dimunition of perch by one unit. A similar economic interpretation is possible by
rewriting (5.2) usefully.
Using the specific functional forms for the population dynamics in (1.2) and (2.2)
in combination with (4.1) and (4.2) in steady state, yields
(7.1)
R
R    R D  P1  P2 
1   

,
2  r1  2r1 
P1

(7.2)
D
D
  D R  P1  P2 
1   

 .
2  r2  2r2 
P2

(7.1) and (7.2) illustrate that the perch steady state population (R) is a linear and positive
function of dagaa (D) and the steady state population of dagaa is a linear and positive
function of perch, for P1  P2  0.
9
Solving, yields
(8.1)
D* 
(8.2)
R* 
D 2(r2   )r1 P2  R (P1  P2 )( r1   )
4r1 r2 P2  D R P1  P2  P11
2
R 2(r1   )r2 P1  D (P1  P2 )( r2   )
4r1r2 P1  D R (P1  P2 ) 2 P21
It will be useful to determine maximum sustained yield (msy) for perch and dagaa. This
is not a trivial determination since Rmsy and Dmsy are not independent. In fact, one must
choose weights for perch and dagaa harvest to obtain a solution as we show in appendix
1. Assuming that the harvest of each stock is weighted by its price, then
(9.1)
Rmsy 
(9.2)
Dmsy 
R 2r1 r2 P1  D r2 P1  P2 
4r1 r2 P1  D R P1  P2  P21
2
D 2r2 r1 P2  R r1 P1  P2 
4r1 r2 P2  D R P1  P2  P11
2
Note that the solutions to the MSY problems are just the equilibrium levels for  = 0.
That is, Rmsy  R * and Dmsy  D * when the discount rate is zero. Moreover,
D *  Dmsy
R *  Rmsy
whenever
(i)
P1  P2  0
(ii)
r1 , r2   1
because
It is reasonable to assume that r1   , r2   ; the intrinsic rate of growth for each species is greater than
the discount rate for the general case. If this is not true, then in the traditional “lumped parameter” model,
with no stock externalities, when the predator-prey interaction term explicitly has been omitted, it would
1
10
(10.1) Rmsy  R*  
R  2r2 P1  D( P1   P2 ) 
4r1r2 P1  DR ( P1   P2 ) 2 P21
(10.2) Dmsy  D*  D*  
>0
D  2r1 P2  R ( P1   P2 ) 
4r1r2 P2  DR  P1   P2  P11
2
> 0.
When conditions (i) and (ii) hold, positive D* and R* require positive denominators in
both (8.1) and (8.2), an implication used subsequently.
3. Policy Analysis through Comparative Statics
Introductory remarks warned us that not many analytical results can be easily
extracted from a predator-prey model. However, new insights characterizing changes in
steady state predator-prey equilibrium due to parameter changes are presented below. We
would like to know how the optimal harvest of dagaa or perch vary with changes in
parameters such as a price change in perch or dagaa. Such an exercise in comparative
statics is useful for policy analysis. Since foreign exchange is positively related to the
price of perch, then improving foreign exchange earnings is connected to changes in the
optimal harvest of dagaa and its impact on nutrition levels, for example. We reach such
conclusions in steps.
How do the optimal stocks, and harvest of perch and dagaa respond to changes in
the price of perch or dagaa? The comparative statics results are worked out in Appendix 2
and are summarized below. There are ambiguous results when P1  P2  0. Therefore,
assume P1  P2  0.
For a change in P1,
For a change in P2,
pay to extinguish the species because they are relatively unproductive capital. At no population level is the
11
dR
dD
 0,
 0,
dP1
dP1
dD
dR
 0,
 0.
dP2
dP2
Earlier we identified the predator prey technical biological externality in the
steady state equations (5.1') and (5.2') where the common factor is P1  P2 . We noted
that when P1  P2  0 , this pushed the optimal solution for R and D to the right of
  f R  g D. See Figure 1. Thus when P1 increases, it increases the positive
technical biological externality which increases optimal R. The reasoning is the same for
the positive effect of P1 on optimal D in (5.2'). The population of both perch and dagaa
decrease when P2 increases because this exogenous change decreases the positive
technical biological externality.
Additionally, if R* 
1
R,
2
dh2
dh
 0, 2  0 ;
dP2
dP1
and if D* 
1
D,
2
dh2
dh
 0, 2  0.
dP2
dP1
For all other cases, the signs are indeterminate. We can provide no economic intuition for
these results. We know, for example, that R*  Rmsy from (10.1) and if R* increased due
to an increase in P1 in a single species model, h1 would increase. Here the condition for
dh1
1
1
 0 is that R*  R and in this model Rmsy  R as is the case in the single
2
2
dP1
species model. See Appendix 1.
real rate of return greater than the return on other forms of capital.
12
4. Illustrative Solutions
We have developed estimates of the parameters for the predator-prey model using
data that vary from highly credible to speculative. Nevertheless, quantitative solutions do
have merit for several reasons. First, we demonstrate numerically the minimum
parameters necessary to reach any conclusions about harvest levels for perch and dagaa.
Second, we provide estimates of the desired harvest and stock levels for perch and dagaa
for Kenya using the best available information. Rational management of these two
fisheries has to be guided by some empirical understanding of the biological interactions
between these two fisheries and the underlying economics governing them. Policy
makers can differ about the management goals for the two fisheries. Fishery biologists
can disagree with the biological structure and parameter estimates of the model as can
economists disagree with the economic anatomy. However, this model demonstrates the
need for a basic data set and it is intended to provide the basis for building a more
credible model. Third, changing the parameters gives a sense of which parameters play
the most sensitive role in changing the solution values for the harvest of perch and dagaa.
Parameter estimates
P1  Kshs 21.45  the ex-vessel price per kg for perch ,
P2  Kshs 5.9  the ex-vessel price kg for dagaa .
The prices are based on recent unpublished data developed by Ikiara. We assume
  .03  the discount rate.
r1  0.15  the intrinsic rate of growth for perch provided by Schindler.
r2  0.8  the intrinsic rate of growth for dagaa provided by Manyala.
The species interactions coefficients are
13
 = .0000002,
 = -.0000007.
While fishery biologists agree that there is a predator-prey relationship between these two
most important fisheries in Lake Victoria, no research on the interaction coefficients is
available. The order of magnitude of the interaction coefficients is governed by the fact
that if the interaction term is larger than
1
1
for the perch or
for the dagaa, the
D
R
predator-prey component in the dynamic population equations induce changes in the
stock larger than the size of the stock. The particular values selected are based on relative
values provided by fishery biologists in lake Victoria but scaled down to meet required
order of magnitude values just mentioned.
There are no estimates for the carrying capacity parameters but they can be
estimated indirectly. Fishery biologists have beliefs about maximum sustained yield and
informal discussion yielded the following estimates:
h1msy  55,000 tonnes for perch,
h2 msy  80,000 tonnes for dagaa
Given these values we derived
R  620,336 tonnes,
D  882,805 tonnes,
using the technique described in Appendix 1 which also yields
Rmsy  343,825 tonnes
Dmsy  465,518 tonnes
14
Steady State Economic Optimum
Using equations (8.1), (8.2), (1.2) and (2.2) when D  R  0,
h1*  47,963 tonnes,
h2*  101,800 tonnes,
R *  280,272 tonnes,
D *  444,508 tonnes.
Recent harvest levels were 98,280 tonnes of perch and 65,520 tonnes of dagaa in 1995,
amounts substantially in excess of optimal values.
If we could magically reach the economic optimum instantaneously, the total
revenue equals Kshs 1.63 billion per year or a present value in perpetuity of about Kshs
54.3 billion. See Table 1. If we were at Rmsy and Dmsy , the total revenue is Kshs 20
million more per year. See Table 2. However, at Rmsy and Dmsy it makes economic good
sense to harvest down the stocks to R * and D * , earning a lump sum of Kshs 1.49 billion
and put it in the bank at .03 rate of interest to capture an annual stream in perpetuity of
about Kshs 44.6 million. This strategy is a superior choice by about Kshs 24 million in
additional revenue per year, a gain of about 1.3 percent (See the first three rows of Table
1). It may strike some one as a little surprising that the economic optimum and the msy
solution are so close. That is because the discount rate of 3 percent is close to zero, which
is the implied discount rate of the msy solution, together with the fact that harvest is
unresponsive to changes in the stocks in the neighborhood of Rmsy and Dmsy .
15
Optimum Path to the Steady State
Unfortunately the two fisheries are neither at the economic optimum nor at msy
stock levels because this is a free entry fishery. The approximate location of the two
fisheries is not known. Since recent harvest levels in Kenya exceed our estimated
optimum and there is widespread concern about over fishing, there is no reason to believe
that the fishery is in a free entry equilibrium at these levels. There are no estimates of
stock, so, there is no way to know confidently where to begin the path to the steady state.
Parenthetically, there is no current research agenda which would enable researchers to
reliably estimate the population levels of perch and dagaa.
Suppose we start at one-half or one-fourth of msy levels for perch and dagaa.
How long does it take to reach the optimum steady state values and what is the present
value of this policy? Starting at one-fourth msy, it takes about 6 years to reach the
optimum which yields a present value of about Kshs 47 billion. See Table 1. Starting at
one-half msy stock levels, it takes about 3 years to reach the optimum which has a
present value of about Kshs 51 billion.
A free entry steady state at one-fourth or one-half msy, yields annual revenues of
about Kshs 720 million and Kshs 1.23 billion or about Kshs 24 billion and Kshs 41
billion respectively, in present value terms. Moving to the optimum from these two initial
free entry equilibria would increase the present value of total revenue in these fisheries by
Kshs 23 billion for the one-fourth msy starting point and Kshs 9 billion for the one-half
msy starting point.
Table 1. Present Value of Revenue Streams Given Initial and Terminal Points
16
Path
NPV (Kshs)
Start and stay at ( Rmsy , Dmsy )
From ( Rmsy , Dmsy ) to
55,058,346,564
( R* , D* )
55,789,989,139
1.33%
%Change
Start and stay at
( R* , D* )
54,314,764,924
From (0.5* Rmsy , 0.5* Dmsy ) to
*
*
50,555,118,458
( R* , D* )
47,141,260,190
(R , D )
From (0.25* Rmsy , 0.25* Dmsy ) to
5. Multiple Objectives
First, consider foreign exchange. More than one-half the tonnage of Kenya’s fish
harvest is attributable to perch in recent years and in some years, more than 60 percent of
the perch catch has been processed for export, mostly by foreign owned firms financed
by the World Bank and non-governmental organizations. [Ikiara] As pointed out earlier,
clearly foreign exchange earnings are so very directly related to the price of perch that a
linear relation is a good approximation. Thus putting special emphasis on foreign
exchange is like assuming a price increase for perch. We know from the earlier
comparative statics analysis when the price of perch increases, that both perch and dagaa
stocks increase, perch harvest increases and dagaa harvest decreases under the empirical
conditions of the Kenya fisheries in question.
Specifically, assume a 20 percent increase in the price of perch so that the perch
piece of the objective function is 1.2 P1h1. The steady state supply response is inelastic
for both species. The harvest of perch increases by 9 percent (about 4,000 tonnes), and
this increases steady state revenues by about Kshs 94.4 million. Foreign exchange
earnings increase by 9 percent as well. The harvest of dagaa decreases by 13 percent, or
about 13,440 tonnes, which reduces steady state ex-vessel revenues by Ksh 79.3 million.
17
There is additionally the impact of this loss on employment for women and the loss of
protein dagaa provide. Although actual steady state revenues increase annually, this could
only be achieved by stopping fishing for about a year [Brett], so the loss of revenue in
present value terms is about Kshs 1 billion.
Dagaa is a relatively cheap source of animal protein. Declining harvest of dagaa
constitutes a public health concern because nearly 50 percent of the rural population
contiguous to Lake Victoria consume less than WHO’s recommended minimum of 2250
calories per adult equivalent day [Ikiara].
The greatest fraction of urban people suffering food deficiency in Kenya live in
Kisumu, the center of the fishing industry. More generally, there is ample documentation
of pronounced protein deficiency among the people living around the lake, especially
children [Ikiara]. Shortfalls in calorie intake have public consequences such as increased
morbidity and its associated public health costs as well as lowered mental capabilities
when caloric loss is incident on children. Since much of the perch are exported or
supplied to the tourist industry, one way to acknowledge the public benefit of increased
caloric consumption is to put more weight on the harvest of dagaa, v2 h2 . Then dagaa’s
contribution to the objective function is P  v2 h2 and can be analyzed as if the price of
dagaa has increased. We have established from the comparative statics analysis that an
increase in P2 increases the dagaa harvest and decreases the perch harvest. These changes
increase women’s employment, another national goal, because it has been estimated that
75 percent of artisanal (dagaa) fish traders are women [Ogutu, G.E.M, 1988], and perch
processing activities are relatively more capital intensive.
18
To account for the contribution of dagaa to reducing public health costs and
improving women’s employment, suppose v2=0.2 which is equivalent to a 20 percent
increase in the price of dagaa or the welfare weight put on a kg of dagaa. Equilibrium
perch harvest and revenues fall about 8 percent as a consequence of this change and
dagaa harvest and actual revenue increases by about 11 percent. The difference in present
value between this option and the original solution is kshs _______. Against this loss of
income must be compared the marginal improvement in public health benefits and in
women’s employment, not known.
Selecting welfare weights for social goals is a notorious difficult prospect. To see
if the welfare weight of 0.2 makes sense, policy makers could compare the national
income cost of increasing dagaa protein (by about 11,000 tonnes per year)______ and
compare it to the alternative cost of equivalent protein, assuming no extra value is placed
on women’s employment for this calculation.
Self Financing an Optimal Fishery
Suppose we want to provide a subsidy, s, while the harvesters are not fishing and
a tax  while they are fishing. The tax is designed to recover the cost of the subsidy in
present value terms.
Suppose, we start at ½ MSY levels. It takes 3 years to get to steady state and I
will assume, for illustration, that there is no harvest. We want to subsidize for the
opportunity cost of forgone fishing but we only know revenues. Suppose that the
opportunity cost is a fraction, , of revenues,  = s.
Revenue at ½ MSY is Kshs 1.23 X 109. So the opportunity cost, basis 3 years
3
from now, is   1.23 x10 9 e rt dt.  83.75 x10 9.
0
19
In steady state, revenues are Kshs 1.63 billion annually. So the tax revenues in present

value three years from now are   1.63x10 9 e  rt dt   50.33x10 9.
0
Solving for ,
 = .07.
Disregarding political feasibility and transaction costs, the estimated maximum tax
necessary to compensate the losers in order to achieve an optimal fishery is 7 percent. If
canoes, nets and labor in the fisheries have alternative employment opportunities, the tax
would be less.
20
Table 2. Equilibrium Values Given Various Parameter Values
Rmsy
101,802
MSY
Rev/yr. (Kshs)
1,651,750,397
Optimum(*)
Rev/yr. (Kshs)
1,629,442,948
52,391
88,362
1,887,700,465
1,835,204,801
96,480
42,658
114,740
1,386,663,103
1,420,375,760
50,865
93,749
43,654
112,523
1,754,410,304
1,727,368,854
478,606
60,660
56,866
53,447
84,734
1,533,560,044
1,467,493,692
259,088
437,504
55,000
80,000
45,302
108,536
1,651,750,397
1,612,092,709
465,518
237,904
430,501
55,000
80,000
42,483
115,003
1,651,750,397
1,589,785,260
465,518
195,536
416,494
55,000
80,000
36,373
127,135
1,651,750,397
1,530,298,729
h1
msy
80,000
47,963
59,538
61,863
422,399
49,946
249,858
425,668
506,397
315,403
343,825
465,518
343,825
343,825
280,272
55,000
497,712
309,018
471,319
304,464
438,327
242,637
P2 +20%=7.08
311,959
442,424
P2 -20%=4.72
  .04
  .05
  .07
381,343
P1 +20%=25.74
374,461
P1 -20%=17.16
21
*
h2
343,825
h2
msy
h1
R*
D*
444,508
Dmsy
465,518
Parameters\Levels
Standard
*
Figure 1
Optimal Solution for Perch When P1  P2  0
f R   (P1  P2 )
D
P1
f (R )

0
(5.1')
22
R
R*
  f ( R )  P1  P2 
D
P1
References
C.W. Clark (1976), Mathematical Bioeconomics, New York: Wiley.
C.W. Clark (1990), Mathematical Bioeconomics, New York: Wiley.
O. Flaaten (1991), “Bioeconomics of Sustainable Harvest of Competing Species,”
Journal of Environmental Economics and Management, 20: pp. 163-80.
P.A. Larkin (1966), “Exploitation in a Type of Predator-Prey Relationship,” J. Fish. Res.
Canad, 23:pp. 349-56.
R. May (1974), Model Ecosystems, Princeton NJ: Princeton University Press.
J. Maynard-Smith (1974), Mathematical Ideas in Biology, Cambridge: Cambridge
University Press.
G.E.M. Ogutu (ed.) (1988), Artisanal Fisheries of lake Victoria, Kenya: Options for
Management, Production and Marketing, Proceedings of a workshop held in
Kisumau, Kenya.
E.C. Pielou (1969), An Introduction to Mathematical Ecology, New York: Wiley.
D.L. Ragozin and G. Brown. (1985), “Harvest Policies and Nonmarket Valuation in a
Predator – Prey System,” Journal of Environmental Economics and Management,
12: pp. 155-68.
R. Solow (1976), “Optimal Fishing with a Natural Predator,” in (R. Grieson, Ed.), Public
and Urban Economics, Essays in Honor of W.S. Vickrey, Lexington, Mass:
Lexington Books.
J. Wilen and G. Brown (1986), “Optimal Recovery Paths for Perturbations of Trophic
Level Bioeconomic Systems,” Journal of Environmental Economics and
Management 13: pp. 225-34.
23
Appendix 1. Calculation of Maximum Sustainable Yield
The maximum sustainable yield is defined as the maximum of a weighted sum, where the
weights are P1 and P2 (in this context P1 and P2 are two weights and not necessarily the
prices). If the weights are each set to 1, then the problem is one of maximizing
sustainable harvest in tons. Alternatively, adopting an economic perspective and
weighting the harvests by prices, the problem becomes one of maximizing sustainable
instantaneous revenue flow.
Ph
1 1  P2 h2
Maximize
over
h1 , h2 , R, D
subject to:
R0
D0
(A.1)
 R
h1  r1 R 1     RD
 R
 D
h2  r2 D 1     RD
 D
(ie. R  0)
(ie. D  0)
For the above problem, r1 , r2 ,  ,  , R, and D are taken as given. Substituting the
constraints into the objective function, (A.1) becomes a problem only in terms of the
stock levels. The solutions to this problem are
Dmsy 
Rmsy 
D  2r2 r1P2  Rr1 ( P1   P2 ) 
(A.2)
R  2rr2 P1  Dr2 ( P1   P2 ) 
(A.3)
4r1r2 P2  DR ( P1   P2 ) 2 P11
4r1r2 P1  DR ( P1   P2 ) 2 P21
Substituting (A.2) and (A.3) for D and R respectively in the harvest functions gives h1
and h2 in terms of the parameters values. It is therefore possible, given the msy harvest
levels, to implicitly solve for two parameter values. It so happens that the carrying
24
capacities are quadratic functions of the other variables and therefore two solutions are
possible. It was found, however, that given our specific harvest levels, and the other
parameter values, only 1 reasonable solution exists when the weights are equal to the
prices.
25
Appendix 2. Derivation of Comparative Status Results
How do the optimal stocks and harvest of perch and dagaa respond to changes in the
price of each species? Differentiate (7.1) and (7.2) with respect to P1 :
D D   P1   P2  R DR   




 
P1 2r2 
P2
 P1 2r2  P2 
(11)
D
R  P1  P2  D RDP2




2
P1 2r1 
P1
 P1 2r1 P1
(12)
Substituting (12) into (11),
D D   P1   P2   R   P1   P2  D RD  P2  DR




 


P1 2r2 
P2
P1
2r1 P12  2r2 P2
  2r1 
 P1
and rearranging,
RDD  P2  P1   P2  DR

D
4r1r2 P12 P2
2r2 P2
.

2
P1
 RD  P1   P2  
1 



4r1r2 PP
1
2


(13)
Assume  P1   P2   0 which makes the numerator  0 . Since
 4r r P  DR  P   P  P   0 for R
1
2
1 2 1
1
2
*
2
 0 , the denominator is positive. Therefore
D
0
P1
Then from (12):
R R   P1   P2  D RD P2


 0.


P1 2r1 
P1
2r1P12
 P1
Recall
h1  r1 R 
26
r1 2
R   RD ,
R
(14.1)
h2  r2 D 
r2 2
D   RD .
D
(14.2)
Differentiating (14.1) with respect to P1 and rearranging terms,
h1
R
r R R
R
D
 r1
2 1
D
R
,
P1
P1
R P1
P1
P1
 R
h1  
R
D
.
  r1 1  2    D 
R
P1  
R
P1
 P1
Thus,
(15.1)
R 1
h1
 0 if  , and undetermined otherwise.
R 2
P1
Differentiating (14.2) with respect to P2 and rearranging terms,
 D
h2  
D
R
.
  r2 1  2    R 
 D
P1  
D
P1
 P1
Thus,
D 1
h2
 , and undetermined otherwise.
 0 if
D 2
P1
In order to sign
R D h1
h
,
,
, and 2 , begin by differentiating (7.1) and (7.2)
P2 P2 P2
P2
R
R   P1   P2  D RD    






P2 2r1 
P1
 P2 2r1  P1 
(16)
D
D  P1  P2  R D RP1




2
P2 2r2 
P2
 P2 2r2 P2
(17)
Substituting (17) into (16):
R
R   P1   P2   D   P1   P2  R DR P1  RD 



 


2 

P2 2r1 
P1
P2
  2r2 
 P2 2r2 P2  2r1 P1
27
(15.2)
2
RDR P1  P1   P2  RD
R  RD  P1   P2  
1 


2

P2 
4r1r2 PP
4r1r2 PP
2r1P1
1
2
1
2


R

P2

RDR P1  P1   P2  RD 

2
4r1r2 PP
2r1 P1
1 2
2
 RD  P1   P2  
1 



4r1r2 PP
1
2


(18)
Again assume  P1   P2   0 which makes the numerator negative. Since
 4r r P  DR  P   P  P   0 for D
1
2
1 2 1
1
2
*
2
 0 , the denominator is positive. Therefore
R
 0.
P2
Then from (17),
D D   P1   P2  R DR P1


0


2
P2 2r2 
P2
 P2 2r2 P2
Recall
h1  r1 R 
r1 2
R   RD ,
R
(19.1)
h2  r2 D 
r2 2
D   RD .
D
(19.2)
Differentiating (19.1) with respect to P2 and rearranging terms,
h1
R
r R R
R
D
 r1
2 1
D
R
,
P2
P2
R P2
P2
P2
 R
h1  
R
D
  r1 1  2    D 
R
.
P2  
R
P2
 P2
Thus,
28
R 1
h1
 0 if  , and undetermined otherwise.
R 2
P2
(20.1)
Differentiating (19.2) with respect to P2 and rearranging terms,
 D
h2  
D
R
.
  r2 1  2    R 
 D
P2  
D
P2
 P2
Thus,
29
D 1
h2
 , and is undetermined otherwise.
 0 if
D 2
P2
(20.2)