Flaaten’s (1991) study on competing species conjectures that a higher price (harvesting costs) of one species yields a lower (greater) own stock-size and a greater (lower) stock-size of the competing species. I show both conjectures are wrong.fishery management, multiple species model, renewable resources

Tibor Neugebauer

English

Research Papers in Economics

  1
 
 
 
 
BIOECONOMICS OF SUSTAINABLE 
HARVEST OF COMPETING SPECIES: A 
COMMENT
∗ 
 
by 
Tibor Neugebauer
☼ 
University Hannover, Germany 
 
Abstract 
Flaaten’s (1991) study on competing species conjectures that a higher price (harvesting costs) of 
one species yields a lower (greater) own stock-size and a greater (lower) stock-size of the 
competing species. I show both conjectures are wrong. 
 
Keywords: fishery management, multiple species model, renewable resources  
JEL-code: Q22, Q57 
 
 
 
 
 
 
 
 
 
 
 
 
                                                           
∗ The author thanks Rögnvaldur Hannesson, Ola Flaaten and Santiago Rubio for valuable comments. 
☼ Corresponding address: Tibor Neugebauer, Universität Hannover, Wirtschaftswissenschaftliche Fakultät, Lehrstuhl 
Finanzmarkttheorie, Königsworther Platz 1, D – 30167 Hannover, Germany, +49 - 511 7625161 (Phone), +49 - 511 
7625240 (Fax), T.Neugebauer@mbox.vwl.uni-hannover.de   2
There exists a growing sense among resource scientists that scientific effort should be directed at 
multiples species, community and ecosystem approaches. (van Kooten and Bulte 2000, p. 217; 
quoted also in Finnoff and Tschirhart 2003)  
1. Introduction 
Growth of a fishery population is determined to a large extent by its environment. Since one 
species only forms part of a complex ecological system including prey, competitors or 
predators, modeling its dynamics with a single differential or difference equation neglects such 
ecological interdependence, and can lead to wrong assessments. In the literature, it was pointed 
out that spillovers that occur from one population onto another might change crucially the 
optimal extraction plan in a fishery (Clark, 1976; Ragozin and Brown, 1985; Hannesson, 1986; 
Conrad and Adu-Asamoah, 1986; Wilen and Brown, 1986; Flaaten, 1991; Tu and Wilman, 
1992; Conrad and Salas, 1993; Stroebele and Wacker, 1995; Flaaten and Stollery, 1996; van 
Ierland and De Man, 1996; Wacker, 1999; Arnason, 2000; Barbier, 2001; Imeson, van den 
Bergh and Hoekstra, 2002; Tschirhart, 2002; Finnoff and Tschirhart, 2003). In contrast to the 
single species model, the optimal solution can involve harvesting of some species at a loss as a 
lower stock increases the profits from the fishery of the interdependent species.  
One frequently quoted paper of this literature which studies the bioeconomics of Gause’s 
(1934) deterministic competing species model is by Flaaten (1991). Flaaten provides six 
theorems of which the last two involve general conjectures on the impact of market parameter 
changes on the long run stock levels. In his Theorem 5 (Theorem 6), Flaaten conjectures that a 
higher price (harvesting costs) of one species will yield a lower (greater) own stock level and a 
greater (lower) stock level of the competing species in the steady state. In this paper, I show that 
the theorems are no general results. In fact, Theorem 5 is reversed if costless harvesting is 
assumed.       3
In the remainder of the paper, I review the competing species model and derive the 
optimal harvesting decision of the sole owner. I follow closely the steps of Flaaten (1991). I 
show where the proof of the theorems is flawed and I provide counter examples for both 
theorems. Finally, I sum up. 
2. The model 
In the notation I follow Flaaten (1991). The growth function of species i is given by the logistic 
production function:  
 
Gi(X1, X2) = riXi (1 - Xi - αiXj), (i,j = 1, 2, i≠j),   (1) 
 
where  Xi  ∈ [0,1] denotes the stock of species i relative to its carrying capacity, αi is the 
dimensionless competition parameter,
1 and ri is the intrinsic growth rate. In contrast to the 
single species model the competition parameter is incorporated to describe by how much the 
living space of species i is affected through the presence of the competing species j. The greater 
the competition parameter is the “flatter” is the curve of the growth function of species i, and 
the lower is the species’ stock level in the biological equilibrium without harvesting. 
Harvesting from both resource stocks is assumed to be independent from each other. 
Each unit of effort E can be dedicated either to catching species 1 or species 2 but not to both at 
the same time. The catch rates are yi(Xi) = Ei Xi (i = 1, 2). Given Schaefer’s harvest function and 
assuming constant costs per unit effort, the unit harvesting costs are ci(Xi) = ci/Xi (i = 1, 2).
2 
Under the assumption of constant prices p1 and p2 of species 1 and 2, respectively, the total 
profit from harvesting the two species is  
 
                                                           
1 Throughout the paper we will assume that αi∈(0,1), i=1,2. This is a necessary condition for having an interior, 
positive stable steady state without harvesting (see Flaaten [7, 166]). 
2 The unit cost of harvesting species i accounts for i’s catchability quotient.   4
π(X1, X2) = b1(p1, X1) y1 + b2(p2, X2) y2,      (2) 
 
where bi(pi, Xi) = pi - ci(Xi) (i = 1, 2) denotes the net profit per unit harvest of species i.  
The management problem of the competing species problem involves the maximization 
of the present value (PV) of the total profit from both fisheries subject to the two equations of 
motion. Thus, the externality of the production of species 1 is internalized in the production of 
species 2, and vice versa. The planner faces the following two-control problem:  
 
max PV=∫
∞
−
0
2 1 ) , ( dt X X e
tπ
δ  
s.t.  2 , 1 ), ( )) ( ), ( ( ) ( 2 1 = − = i t y t X t X G t x i i i &      (3) 
 
where δ denotes the discount rate and the dot in the constraints indicates time derivatives. The 
stocks change in time due to the difference between growth and catch. Assuming an interior 
solution the following equation pair determines the optimal solution to the planner’s 
maximization problem, 
 
b1(X1
*) = 1/δ [b1(X1
*) G11 + b2(X2
*) G21 – c1’(X1
*) G1(X1
*, X2
*)] 
b2(X2
*) = 1/δ [b1(X1
*) G12 + b2(X2
*) G22 – c2’(X2
*) G2(X1
*, X2
*)]   (4) 
 
where Gji(X1
*, X2
*) = ∂Gi/∂Xj
* (i, j = 1, 2) is the derivative of species j’s growth function with 
respect to i’s steady state stock level. In the steady state, the total profit of the marginal unit 
extracted from the stock (on the left hand side) must equal the discounted marginal rent of this 
unit (on the right hand side), which reflects the present value of all future losses that result 
through harvesting from a lower stock. In the steady state, growth must equal catch, yi = Gi. 
Hence, we can rewrite (4) using (2) to obtain   5
 
) 2 , 1 (
) , ( 1
) (
*
*
2
*
1 * =
∂
∂
= i
X
X X
X b
i
i i
π
δ
.       (5) 
Flaaten (1991) starts from equation (5) to analyze how equilibrium stock levels change when 
prices or costs change. Following his steps we differentiate with respect to p1 to yield the 
following  
 








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
1
*
2 12
1
*
1 11
1
1
1
*
1
1 1
1
p
X
p
X
p p
X
b b x p π π
π
δ
      (6) 








∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
1
*
2 22
1
*
1 21
1
2
1
*
2
2
1
p
X
p
X
p p
X
b x π π
π
δ
      (7) 
where  
i
i
ip p
b
b
∂
⋅ ∂
=
) (
, 
*
) (
i
i
ix
X
b
b
∂
⋅ ∂
= , 
i
i
X ∂
⋅ ∂
=
) ( π
π , and 
Xj X i
ij
∂ ∂
⋅ ∂
=
) (
2π
π .    (8) 
 
Rearranging equations (6) and (7) with respect to stock-price derivatives we obtain 
 
1
1
1
1
*
2 12
1
*
1
1
11 ) (
p
b
p
X
p
X
b p x ∂
∂
=
∂
∂
+
∂
∂
−
π
δ π δ π      (9) 
1
2
1
*
2
2
22
1
*
1 21 ) (
p p
X
b
p
X
x ∂
∂
− =
∂
∂
− +
∂
∂ π
δ π π .      ( 1 0 )  
 
Applying Cramer’s rule, equations (9) and (10) can be solved with respect to the stock-price 
derivatives, yielding the following equations, 
 
| |
) )( (
1
2
12
2
22
1
1
1
1
*
1
D
p
b
p
b
p
X
x p ∂
∂
+ −
∂
∂
−
=
∂
∂
π
π δ π
π
δ
     ( 1 1 )    6
| |
) ( ) ( 1
1
1
21
1
2
11
1
1
*
2
D
b
p p
b
p
X
p x δ
π
π
π
π δ −
∂
∂
+
∂
∂
−
=
∂
∂
     ( 1 2 )
 
where     
x
x
b
b
D
2
22 21
12
1
11
δ π π
π δ π
−
−
= .     ( 1 3 )  
 
Equations (11) and (12) coincide with equations (54) and (55) in Flaaten only if ∂π
i/∂p1 = 0 (i 
=1, 2). Since ∂π
1/∂p1 = G11 = r1 (1 - 2X1
*
 - α1X2
*) and ∂π
2/∂p1 = G12 = - α1 r1 X1
*
 < 0 are different 
from zero, the proofs to Theorem 5 and Theorem 6 in Flaaten are wrong. In fact, we can find an 
illustrative example which reverses Theorem 5 as the following observation reveals. 
 
Observation If harvesting in the competing species model (with αi>0, i=1,2) is costless the 
following holds:  0
1
*
1 ≥
∂
∂
p
X
, and  0
1
*
2 ≤
∂
∂
p
X
. 
 
As shown in Flaaten, it follows from the second order condition for the existence of an 
interior solution that the denominator to both equation (11) and (12) is positive, i.e. |D| > 0, and 
the second derivative of the profit function with respect to the own stock, π
ii = - 2 ri  pi < 0  (i = 
1, 2), is negative. Note bip = 1, and ci = 0 (i = 1, 2) implies bix = 0. Taking into account π
ij = - α1 
r1 p1 - α2 r2 p2 < 0, and G12 < 0, the proof reduces thus to showing that the marginal growth rate 
in the steady state exceeds the interest rate, i.e., G11 ≥ δ. This is trivially satisfied, because if the 
interest rate is greater than the marginal growth rate the optimal management decision is to 
extinguish the species. At the corner solution the stock-price derivatives are zero. If both stocks 
are positive the inequalities are strict. A higher price of species 1 implies its higher relative 
efficiency. In other words, if there are no extraction costs and 1’s price increases the planner   7
cares less about the spillovers from species 1 to species 2 and more about the externality that 
affects species 1. 
Figure 1 illustrates for given parameter values (α1 = α2 = c1 = c2 = δ = ½ and r1 = r2 = 1) 
that Flaaten’s Theorem 5 is not verified either if positive costs are allowed. Figure 1 displays 
the signs of the derivatives for the price ranges p1, p2 ∈ [1, 10]. Note that prices in this range are 
at least double as high as costs. In the northwest area of Figure 1 where the competing species’ 
price is sufficiently greater than the own species’ price, i.e., approximately when p2 > - 2.75 + 
3.75 p1 ≥ 1, derivatives are in line with Flaaten’s theorem. However, for other price ratios the 
theorem must be rejected. 
 
---- Insert Figure 1 about here ---- 
 
As pointed out above, the proof to Flaaten’s Theorem 6 has the identical flaw as the 
proof to Theorem 5. Theorem 6 states that the steady state stock level of one species is (i) 
increasing in own harvesting costs and (ii) decreasing in the harvesting costs of the competing 
species. Using the same parameter values and price ranges as in Figure 1, Figure 2 depicts the 
signs of the derivatives with respect to harvesting costs. In the south east area of Figure 2, i.e., 
approximately where 1 ≤ p2 < - 1/6 + 2.4 p1, the second part of the theorem must be rejected. 
 
---- Insert Figure 2 about here ---- 
 
3. Summary 
In the paper, it has been shown that two essential theorems of Flaaten’s (1991) well-known 
paper on optimal management of competing species are wrong. These theorems make a claim 
about how the equilibrium stock in the sole owner fishery moves when prices or costs are   8
changed. A higher price of one species does not imply necessarily a lower bioeconomic 
equilibrium stock as Flaaten conjectured. This result is important since it shows that the 
competing species model does not behave the same as the single species model.
3 Particularly in 
the case of costless harvesting the optimal resource management decision might induce more 
extraction of the competing species in order to enhance the stock of the more valuable species. 
Furthermore, the steady state stock level is not necessarily decreasing in the harvesting costs of 
the competing species as conjectured in Flaaten’s Theorem 6. It turns out that for some values 
the opposite is true. Finally, I cannot reject both conjectures of Flaaten’s Theorem 6 since the 
steady state stock level is increasing in own harvesting costs for all values I considered. 
However, I am not able to provide a proof in favor of the theorem either. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                                           
3 Flaaten 1991 shows that even harvesting one species at a loss can be optimal in the long-run equilibrium.   9
4. References 
Arnason, R., 2000, Economic instruments for achieving ecosystem objectives in fisheries 
management, ICES J. Marine Science 57(3), 742-751. 
Barbier, E.B., 2001, A note on the economics of biological invasions, Ecological Economics 
39(2), 197-202. 
Clark, C., 1976, Mathematical Bioeconomics, John Wiley & Sons, New York. 
Conrad, J. M., and R. Adu-Asamoah, 1986, Single and multispecies systems: The program of 
tuna in the eastern tropical Atlantic, Journal of Environmental Economics & Management 
13, 50-68. 
Conrad, J. M., and R. Salas, 1993, Economic strategies for coevolution: timber and butterflies in 
Mexico, Land Economics 69, 404-415.  
Finnoff, D. and J. Tschirhart, 2003, Harvesting in an eight-species ecosystem, Journal of 
Environmental Economics & Management 45(3), 589-611. 
Flaaten O., 1991, Bioeconomics of sustainable harvest of competing species, Journal of 
Environmental Economics & Management 20, 163-180. 
Flaaten, O., and K. Stollery, 1996, The economic costs of biological predation, Environmental 
& Resource Economics 8, 75-95.  
Gause, G. F., 1934, The struggle for existence (Hafner, New York/London - reprinted 1969). 
Hannesson, R., 1983, Optimal harvesting of ecologically interdependent fish species, Journal of 
Environmental Economics & Management 10, 329-345. 
van Kooten G.C. and E.H. Bulte, 2000, The Economics of Nature. (Blackwell Publishers, 
Malden, Mass). 
Van Ierland, E.C. and N.Y.H. De Man, 1996, Ecological engineering: First steps towards 
economic analysis, Ecological Engineering 7(4), 351-371.   10
Imeson, R.J., Van den Bergh, J.C.J.M. and J. Hoekstra, 2002, Integrated models of fisheries 
management and policy, Environmental Modelling & Assessment 7(4) 259-271.  
D.L. Ragozin, and G Brown, 1985, Harvest policies and nonmarket valuation in a predator - 
prey system, Journal of Environmental Economics & Management 12, 155-168. 
W.J. Stroebele, and H. Wacker, 1995, The economics of harvesting predator-prey systems, 
Journal of Economics 61, 65-81. 
J. Tschirhart, 2002, Resource competition among plants: From optimizing individuals to 
community structure. Ecological Modelling 148, 191-212. 
P.N.V. Tu, and E.A. Wilman, 1992, A generalized predator-prey model: Uncertainty and 
management, Journal of Environmental Economics & Management 23, 123-138. 
H. Wacker, 1999, Optimal harvesting of mutualistic ecological systems, Resource & Energy 
Economics 21, 89-102. 
J. Wilen, and G. Brown, 1986, Optimal recovery paths for perturbations of trophic level 
bioeconomic systems, Journal of Environmental Economics & Management 13, 225-234.   11
Figure 1. Sign of price derivatives 
 
Parameter values: α1= α2 =c1=c2 = δ = 0.5, r=1.5, p1, p2 ∈ [1,10] 
 
Figure 2. Sign of costs derivatives  
 
Parameter values: α1= α2 =c1=c2 = δ = 0.5, r=1.5, p1, p2 ∈ [1,10] 
 

A generalized predator-prey model: Uncertainty and management,

A note on the economics of biological invasions,

Bioeconomics of sustainable harvest of competing species,

Ecological engineering: First steps towards economic analysis,

Economic instruments for achieving ecosystem objectives in fisheries management,

Economic strategies for coevolution: timber and butterflies in Mexico,

Harvest policies and nonmarket valuation in a predator -prey system,

Harvesting in an eight-species ecosystem,

Mathematical Bioeconomics,

Optimal harvesting of ecologically interdependent fish species,

Optimal harvesting of mutualistic ecological systems,

Optimal recovery paths for perturbations of trophic level bioeconomic systems,

Resource competition among plants: From optimizing individuals to community structure.

Single and multispecies systems: The program of tuna in the eastern tropical Atlantic,

The economic costs of biological predation,

The economics of harvesting predator-prey systems,

The Economics of Nature.

The struggle for existence (Hafner,

BIOECONOMICS OF SUSTAINABLE HARVEST OF COMPETING SPECIES: A COMMENT

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BIOECONOMICS OF SUSTAINABLE HARVEST OF COMPETING SPECIES: A COMMENT

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