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Journal of Theoretical Biology 307 (2012) 96–103
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Journal of Theoretical Biology
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Impact of maximum sustainable yield on competitive community
Sunčana Geček, Tarzan Legović n
R. Bošković Institute, POB 180, Bijenička 54, HR-10002 Zagreb, Croatia
H I G H L I G H T S
c
c
c
A system of n competing populations under proportional harvesting strategy is analyzed.
When harvesting is selective, before reaching MSY, probably some species will disappear.
In case of nonselective harvesting, before reaching MSY, species with r r eopt will disappear.
a r t i c l e i n f o
a b s t r a c t
Article history:
Received 16 August 2010
Received in revised form
18 April 2012
Accepted 20 April 2012
Available online 8 May 2012
A system of n competing logistical species of the Volterra type under proportional harvesting strategy is
analyzed. In case of selective harvesting, when the effort is adjusted to each species, the optimum effort
may result in the total maximum sustainable yield (TMSY1). When it exists, reaching TMSY1 does not
affect the system stability character, but it does affect the state, and hence some populations may reach
too small a value to persist in nature. If competition is strong, species with smaller biotic potential may
be driven to extinction. In case the system is harvested with a common harvesting effort, such as in
trawler fishery, the total maximum sustainable yield (TMSY2) is smaller than TMSY1, and all the species
with lower or equal biotic potential to the optimum harvesting effort will be driven to extinction.
In this case a call for implementation of the MSY is equivalent to a call for the extermination of some
species and it runs directly against the Convention on Biological Diversity (CBD, 1992). Therefore, all
legal documents advocating MSY in ecosystems starting with the Johannesburg Plan of Implementation
(JPI, 2002) must be urgently retracted and replaced with adaptive management which will respect CBD.
& 2012 Elsevier Ltd. All rights reserved.
Keywords:
Harvesting
Competition model
System of competitive populations
1. Introduction
Schaefer (1954) defined the maximum sustainable yield (MSY)
for one isolated logistic population in a peaceful environment
under proportional harvesting. Since then, harvesting of one
population has been investigated in random (Beddington and
May, 1977; Ludwig, 1979; Abakuks and Prajneshu, 1981;
Rotenberg, 1987; Jensen, 2005; Bousquet et al., 2008) and
periodic environment (Legović and Perić, 1984). The consensus
is that in nature the MSY will be smaller than in the peaceful
environment. This result is expected to have even greater consequences for systems of populations. Namely, random and
periodic perturbations affect populations directly and propagate
through the system, affecting other populations indirectly.
Larkin (1977) was the first to heavily criticize the application
of the MSY concept to populations of organisms in ecosystems.
May et al. (1979) showed that a great caution is needed before the
n
Corresponding author. Tel.:þ 385 1 46 80 230; fax: þ385 1 46 80 242.
E-mail address: [email protected] (T. Legović).
0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.jtbi.2012.04.027
concept is applied to ecosystems. In the Lotka–Volterra preypredator system (where the prey population obeys Mathusian
growth) there is no MSY when a common harvesting effort is
applied to prey and predator (Legović, 2003). Instead, the yield
increases until the predator population is extinct. Using food web
models Matsuda and Abrams (2005) and Walters et al. (2005)
have shown that some species may go to extinction if MSY is
applied to an ecosystem carelessly. Legović (2008) has shown that
ten out of sixteen groups of demersal fish in the Adriatic Sea have
been harvested beyond MSY (i.e., overfished) and that fishing prey
and predator populations with the same harvesting effort to MSY
may lead to extinction of predators. In a food chain or a cascade of
food cycles Legović et al. (2010) have shown that harvesting any
trophic level or a combination of trophic levels to MSY, except the
top one, may lead to extinction of at least one species. Finally,
indiscriminate harvesting of a system of independent populations
to MSY is likely to cause extinction of species with lower biotic
potentials (Jensen, 1991; Legović and Geček, 2010).
The system of n competing populations has been investigated
since the seminal work by Volterra (1931). The stability conditions which lead to persistence of all competing populations have
S. Geček, T. Legović / Journal of Theoretical Biology 307 (2012) 96–103
been given by Strobeck (1973). In two competitors McClanahan
(1995) found that when a superior competitor is harvested while
the inferior one is left unharvested, the superior competitor is
driven to extinction by the inferior competitor once harvesting
raises the shared resource to a level that the inferior competitor
can survive. He has also shown that the MSY of the superior
competitor in the presence of the inferior competitor increases
with the difference between their biotic potentials. Furthermore,
when two competitors are harvested with the same effort, as
effort intensifies a minor competitor with a smaller carrying
capacity may become dominant if its biotic potential is higher
(Legović, 2008).
An ecosystem approach requires that sustainability of all
species is ensured. On the other hand, MSY maintains only the
sustainability of the yield at the maximum level. Since the two
concepts may be contradictory to each other, the question is:
What consequences do we expect from applying MSY to a system
of n competing populations of the Volterra type?
The equation of a logistic population dynamics subject to
proportional harvesting (Schaefer’s model, 1954) is
ð1Þ
where r is the biotic potential (per capita birth rate minus death
rate when N 5K), K is the carrying capacity, e is the harvesting
effort. Harvesting effort is computed from the efficiency of
harvesting tools, numbers of tools and the time they spend
harvesting.
The yield is : Y ¼ eN:
The solution to
N(t¼ 0)¼ N0 40 is
ð2Þ
the
Eq.
(1)
with
the
initial
NðtÞ ¼ AK=½1þ ðAK=N 0 1Þexp ðArtÞ
condition
ð3Þ
where A¼(1 e/r).
The solution approaches positive equilibrium value
Nn ¼ Kð1e=rÞ
ð4Þ
when t-N (for every N0) if and only if the harvesting effort is
smaller than the biotic potential (eor).
The yield in the equilibrium is a quadratic function of e
Y n ¼ eKð1e=rÞ:
2
2
3. A system of n competing populations and its stability
First consider the system of two competing species of the
Volterra type with population numbers N1 and N2:
dN 1 =dt ¼ r 1 N 1 ð1ðN1 þ aN2 Þ=K 1 Þ
ð6Þ
dN 2 =dt ¼ r 2 N 2 ð1ðN2 þ bN 1 Þ=K 2 Þ
ð7Þ
where r1 and r2 are biotic potentials, K1 and K2 are the carrying
capacities of the first and the second species respectively. Coefficients a and b are the per capita competition parameters of the
second species on the first and of the first species on the second,
respectively. The competition is of the interference type (Kot, 2001).
The system has four equilibrium values
a)
b)
c)
d)
n
n
¼0, N2(1)
¼0;
total extinction of both N1 and N2: N1(1)
n
n
extinction of N1: N1(2) ¼0, N2(2) ¼K2;
n
n
extinction of N2: N1(3)
¼K1, N2(3)
¼0;
n
n
persistence of N1 and N2: N1(4) ¼(K1 aK2 )/F, N2(4)
¼(K2 bK1 )/F
where F ¼1 ab.
It is well known (Kot, 2001) that the necessary and sufficient
n
n
conditions for (N1(4)
, N2(4)
) to be a stable node are
2. A review of MSY of one isolated population in peaceful
environment
dN=dt ¼ rNð1N=KÞeN,
97
ð5Þ
Since d Y/de o0, it has the maximum at e¼eopt ¼r/2, when
Y ¼rK/4¼MSY.
The MSY is obtained when the population arrives at Nn ¼K/2.
If the population drops below K/2 we call it an overfished
population or an overharvested stock. Of course, one may harvest
with a greater harvesting effort than r/2, but then the yield will
tend to a value which is smaller than MSY and the population will
tend to an equilibrium value Nn oK/2, that is, to an overfished
stock. Finally, as e approaches r, Nn approaches zero, i.e., the
population tends to extinction.
In an area of interest, fishing may be limited according to two
different strategies: limiting fishing effort, as discussed above (i.e.,
limiting the number of fishing tools and the time interval when
fishing is performed) or limiting the yield (often called quota). In
case of limiting the quota, the MSY does not exist because the
resulting Nn ¼K/2 is unstable. Hence, although often used, this
strategy is eliminated from further consideration.
K 1 aK 2 4 0
and
K 2 bK 1 4 0:
ð8Þ
The conditions (8) imply: F ¼1 ab 40.
To interpret the conditions (8), let us examine why would
N2 in a system characterized by a ¼ b ¼0.5, (K1 ¼300)/(K2 ¼100) 4
(a ¼0.5) but (K2 ¼100)/(K1 ¼300) o(b ¼0.5) tend towards extinction? With a coefficient b, N1 suppresses N2 and hence with
sufficiently large K1 with respect to K2, the impact of competition
of N1 on N2 is too intense and N2 tends to extinction because dN2/
dt o0, while N1-Nn1 ¼K1.
Note
1) In the above model of two competitive logistic species the
niches of the two species do not overlap entirely, i.e., the
competition is not complete. If that were the case, then
a ¼ b ¼1 and the competitive exclusion of one species
would occur.
2) Competition intensity is not the only factor which determines
whether one species will be forced to extinction. The first and
the second condition in (8) state that the environment plays a
crucial role too (i.e., the ratio of K1 to K2).
Consider a general system of n competing populations of the
Volterra type
2 0
1
3
n
X
ð9Þ
dN i =dt ¼ r i N i 41@
aij N j A=K i 5,i ¼ 1,. . .,n
j¼1
where aii ¼1; r1,y,rn and K1,y,Kn are biotic potentials and
carrying capacities of the n species, respectively; aij (iaj) is the
competition coefficient of the j-th species on the i-th species.
3.1. Equilibrium states
The system (9) can have at most 2n equilibrium states. One
total extinction state and n states where only one equilibrium
population is greater than zero always exist, there may be at most
n!/(2!(n 2)!) combinations of equilibrium states where only two
populations are greater than zero, n!/(3!(n 3)!) combinations of
equilibrium states where only three populations are greater than
zero, etc., n equilibrium states where all except one population
are greater than zero and finally one equilibrium state where all
98
S. Geček, T. Legović / Journal of Theoretical Biology 307 (2012) 96–103
3.2. Extinction in a competition model
populations are greater than zero. This last state is called the total
persistence state.
The total persistence state exists when the solution to the
following system of linear equations:
If the coefficients of the competition model between n ¼2
species satisfy inequalities
K ¼ AN n
K 1 aK 2 40
ð10Þ
and
K 2 bK 1 o 0,
i.e., N n ¼ A1 K exists and has all components positive.
In the above
then for any system solution (N1(t), N2(t)) with N1(t0)40 and
N2(t0)40 for some t0 40,
N n ¼ ½N n1 ,. . .,Nnn T ,K ¼ ½K 1 ,. . .,K n T
ðN 1 ðtÞ,N 2 ðtÞÞ-ðK 1 ,0Þ
and
A ¼ ½aij :
If the matrix A is invertible, the equilibrium value for the i-th
species is given by Cramer’s rule
Nni ¼ det Ai =det A, i ¼ 1,. . .,n
where Ai is the matrix formed by replacing the i-th column of A
by the column vector K.
The necessary and sufficient conditions that the system (9) has
a stable equilibrium with all the Ni 40 are: det Ai 40 and n 2
additional conditions arising from Routh–Hurwitz Criterion for
the characteristic polynomial of the Jacobian matrix
J¼ diag (ri Nni /Ki) A
to have all roots with negative real part (Strobeck, 1973).
The conditions det Ai 40 imply det A40 (Strobeck 1973,
Appendix A).
In general, the criteria for n species to coexist together depend
on the values ri, Ki, and aij, but there are some cases where only
conditions on Ki and aij are needed. If the matrix A is symmetric
and positive definite, then the Jacobian matrix will be stable.
Therefore, in that case, requirements det Ai 40 establish the set of
sufficient conditions.
An example of a symmetric positive definite matrix is the
competition matrix arising from the ecological niche theory
(May and MacArthur, 1972). According to May and MacArthur,
the one-dimensional resource spectrum sustains a series of species,
each of which has a preferred position in the spectrum and a
characteristic variance about the mean, as given by the utilization
function. If all species utilization functions are the usual bell-shaped
Gaussian curves with constant width w and separation d along the
resource continuum, and if the resource spectrum is such that at the
equilibrium all populations are equal, then the competition matrix
^2
A¼[a(i j) ] is symmetric and positive definite for all 0o a ¼
2
exp( d /4w2)o1. This means that that the stability sets no limit
to the species packing. But, the smallest eigenvalue of A (lmin E4Op
(w/d) exp( p2w2/d2)), which determines the stability character, can
be very narrow. In case of substantial niche overlap, lmin tends to
zero faster than any finite power. This results in long damping times
for the perturbed trajectory.
In a more general case, if the utilization function of the i-th
species is described by the probability density distribution fi(x)
with the mean x0,i (center of the niche) and variance s2i (niche
breadth), then the coefficient of competition between i-th and j-th
species
Z
aij ¼ f i ðxÞf j ðxÞ dx
characterizes the degree of the niche overlap. The quadratic form
!2
Z X
Z
n
n
n
X
X
aij yi yj ¼
f i ðxÞyi f j ðxÞyj dx ¼
f i ðxÞyi dx
i,j ¼ 1
i,j ¼ 1
as t-N. Ahmad and Lazer (1996, 1998) studied the extension of
this phenomenon to the case n42. They have shown that if the
coefficients satisfy certain inequalities, then given a solution with
positive initial values, the n-th component of the solution
approaches zero, while the rest approach the solution to a smaller
system.
More specifically, Ahmad and Lazer (1996) proved that if the
system coefficients satisfy
Ki 4
n
X
aij K j ,1 r ir n1
j¼1
j ai
then there exists a solution N1 ¼Nn1,y, Nn 1 ¼Nnn 1 to the system
of linear equations
Ki ¼
n1
X
aij N j
j¼1
with Nn1,y, Nnn 1 4 0. In addition, if det A 40 and the relation
Kn o
n1
X
anj Nnj
ð11Þ
j¼1
holds, then for any system solution (N1(t), N2(t),y, Nn 1(t), Nn(t))
with Ni(t0)40, 1ri rn for some t0 40
ðN 1 ðtÞ,N 2 ðtÞ,. . .,N n1 ðtÞ,N n ðtÞÞ-ðN n1 ,Nn2 ,. . .,Nnn1 ,0Þ
as t-N.
Later, Ahmad and Lazer (1998) showed that det A40 is not
the necessary condition, but that the inequality (11) is almost
necessary and sufficient for the n-th species to approach extinction in the sense that if the condition (11) is changed to
Kn 4
n1
X
anj Nnj
j¼1
then for any system solution (N1(t), N2(t),y, Nn 1(t), Nn(t)) with
Ni(t0)40, 1ri rn for some t0 4 0
inf t Z t0 N i ðtÞ 4 0,1 ri r n
and the persistence of all species is maintained.
4. MSY for a system of two competing populations
If the two populations are subjected to their own proportional
harvesting effort, e1 and e2 respectively (i.e., the selective harvesting),
the equations become
dN 1 =dt ¼ r 1 N 1 ½1ðN 1 þ aN2 Þ=K 1 e1 N1
ð12Þ
dN 2 =dt ¼ r 2 N 2 ½1ðN 2 þ bN 1 Þ=K 2 e2 N2 :
ð13Þ
i¼1
is always positive definite, and the system (9) is dissipative in the
sense of Volterra (Svirezhev and Logofet, 1983). For dissipative
systems, the existence of the total persistence state is necessary
and sufficient condition for its stability throughout the positive
orthant (Volterra, 1931; Svirezhev and Logofet, 1983; Hofbauer
and Sigmund, 1998; Logofet, 1993).
The only equilibrium point in which both populations may be
positive is
Nn1 ¼ ½CaD=F
ð14Þ
Nn2 ¼ ½DbC=F
ð15Þ
where C ¼K1 (1 e1/r1) and D ¼K2(1 e2/r2), F ¼1 ab.
S. Geček, T. Legović / Journal of Theoretical Biology 307 (2012) 96–103
The necessary and sufficient conditions that the system
(12)–(13) has a stable equilibrium with Nn1 40 and Nn2 40 are
C 40,D 4 0,CaD 4 0,DbC 4 0:
ð16Þ
The conditions (16) imply F ¼det A ¼1 ab 40.
Now, in addition to the ratio of K1 to K2, the persistence of
populations will depend on e1, r1, e2, r2, i.e., in general, the region
where the persistence of both populations is guaranteed is
reduced because more conditions on independent parameters
need to be satisfied. However, there are cases when the competitive system cannot persist because conditions (8) are not
satisfied, but with moderate harvesting of either or both species
(depending on values K1, K2, r1 and r2), the community persistence is possible.
The total yield in equilibrium is a quadratic function in both
e1 and e2
Y t ¼ fe1 ½CaD þe2 ½DbCg=F:
ð17Þ
The total maximum sustainable yield is obtained when Yt is
maximized with respect to e1 and e2. From qYt/qe1 ¼0 and qYt/
qe2 ¼0, and under the condition that the matrix of second
derivatives (the Hessian) is negative definite, Yt is maximized
when
e1opt ¼ ðR H þ P LÞ=ðG LH2 Þ
and
e2opt ¼ ðP H þ R GÞ=ðG LH2 Þ,
ð18Þ
where G ¼ 2K 1 =r 1 ,H ¼ aK 2 =r 2 þ bK 1 =r 1 ,L ¼ 2K 2 =r 2 ,P ¼ K 1 aK 2 ,
R ¼ K 2 bK 1 :
In case GL H2 40, the Hessian matrix of Yt (i.e., the matrix
of second derivatives) is negative definite, the harvesting
efforts e1opt and e2opt are positive, and Yt attains the maximum
at (e1opt, e2opt). If, additionally, Nn1(e1opt, e2opt) and Nn2( e1opt, e2opt)
are positive, then TMSY exists
TMSY ¼ e1opt ½Cðe1opt ÞaDðe2opt Þ þ e2opt ½Dðe2opt ÞbCðe1opt Þ =F:
ð19Þ
In case both a and b tend to zero and hence the species are
becoming independent, the condition holds and we have e1optr1/2, e2opt-r2/2 and TMSY-(r1K1 þ r2K2)/4. In case both a and b
tend to one, and hence it is becoming increasingly unlikely for
species to persist even without harvesting, the condition
obviously does not hold and TMSY does not exist.
In between the two above extremes lies a parameter region
where unique e1opt and e2opt produce the TMSY. However, since
99
conditions on the existence of the maximal value of Yt are
external, they do not necessarily imply Nni 40, i¼1,2 nor the
stability of the equilibrium.
When r1 ¼r2, K1 ¼K2 and a ¼ b, the TMSY exists and it is the
global maximum (Fig. 1a and b). As the competition coefficients
increase, the area of species coexistence narrows and the TMSY
value decreases toward the MSY of one population.
In the next example (Fig. 2) we investigate how TMSY affects
populations characterized by a considerable difference in biotic
potential (r1 ¼1, r2 ¼5). For a relatively small value of ab
(ab ¼0.06) TMSY is obtained (Fig. 2a), both populations are
positive and stable. When the competition intensity on the
second species increases from b ¼0.1 to b ¼0.4, the expression
(18) gives positive eopt values but there is no TMSY such that both
populations are positive (Fig. 2b). Instead MSY of the second
species (with a higher biotic potential) is obtained. For a ¼0.6,
b ¼ 0.8 the expression (18) gives negative values for e1opt and e2opt.
Obviously, TMSY has turned into MSY of the second population
for which eopt ¼2.5 (Fig. 2c).
The above examples show that TMSY exists if species have
similar biotic potentials, carrying capacities and competition
coefficients and if they do not compete intensively. All of these
conditions are hardly ever met in nature. With the difference
among parameters increasing, TMSY may become smaller than
MSY of one of the populations or it may not even exist.
4.1. Equal harvesting efforts (e1 ¼e2 ¼e)
In this case, which resembles trawler fishing and hunting
where hunters shoot at both competitors whenever they have a
chance, the harvesting effort on both populations is the same.
The maximization of Yt given by (17) is obtained when
eopt ¼ 1=2ðA þ BÞ=ðA=r 1 þ B=r 2 Þ
where
ð20Þ
A ¼ K 1 ð1bÞ,B ¼ K 2 ð1aÞ:
If the value of eopt ends up contradicting conditions (16), one of
the two species will tend to extinction. This is because increasing
harvesting effort moves the equilibrium point toward one of the
axes, i.e., toward extinction of one population. On the other hand,
the system does not admit a limiting cycle if the only equilibrium
inside the positive quadrant has moved to one of the axes.
Suppose r1 ¼r2 ¼r. Then eopt ¼ r/2 and conditions (16) reduce
to conditions (8). In this case, the stability character of the
system of two competitive species is unaffected by harvesting.
However, the rate of return to the equilibrium is smaller. As the
Fig. 1. The yield (Y) as a function of the harvesting efforts (e1, e2) for r1 ¼ r2 ¼1, K1 ¼ K2 ¼ 100 and equal competition coefficients: (a) a ¼ b ¼ 0.2 and (b) a ¼ b ¼0.6. Solid lines
border the area where both Nn1 and Nn2 are positive (K1(1 e1/r1) 4 aK2(1 e2/r2) and K2(1 e2/r2) 4 bK1(1 e1/r1)). The pair (Nn1(e1opt, e2opt), Nn2(e1opt, e2opt)) is a stable
equilibrium point. At the TMSY, neither N1 nor N2 are overfished. For smaller competition coefficients, TMSY value is higher and the equilibrium population is positive for
larger domain of harvesting efforts (e1, e2). For higher competition coefficients the value of TMSY ¼rK/(2(1 þ a)) is closer to a single population MSY ¼rK/4.
100
S. Geček, T. Legović / Journal of Theoretical Biology 307 (2012) 96–103
Fig. 2. The yield (Y) as a function of the harvesting efforts for r1 ¼ 1, r2 ¼5, K1 ¼ K2 ¼ 100 and different competition coefficients. Solid lines border the area where both Nn1 and
Nn2 are positive. (a) For a ¼ 0.6, b ¼0.1 the yield attains the global maximum and the pair (Nn1(e1opt, e2opt), Nn2(e1opt, e2opt)) is a stable equilibrium point (e1opt ¼0.4763,
e2opt ¼2.5120). (b) For a ¼ 0.6, b ¼0.4 there is no optimal harvesting effort such that both species persist. The optimal harvesting efforts estimated from (18) are
e1opt ¼0.8912 and e2opt ¼2.6586. The Hessian matrix is negative definite and Y attains a maximum, but at that maximum, the population Nn1 with smaller r has gone to
extinction. (c) For a ¼ 0.6, b ¼ 0.8, the expression (18) gives negative values for e1opt and e2opt.
difference between r1 and r2 increases, eopt moves toward the
larger of the two biotic potentials and the equilibrium population
of the species with lower biotic potential decreases more than in
the previous case. It is clear that if eopt Zmin {r1,r2} the species
with smaller biotic potential tends to extinction. However, examination of conditions (16) shows that for increasing competition
between species, extinction occurs even if eopt is smaller than min
{r1,r2}. For example, if K1 ¼K2, r1 ¼1, r2 ¼2 and a ¼ b ¼0.5, the
optimum harvesting effort which produces TMSY is eopt ¼0.666. It
turns out that with this effort, condition C/D4 a is not satisfied and
N1 tends to extinction. Hence, TMSY becomes MSY of the second
population and then eopt is not given by the expression (20) but it is
simply eopt ¼r2/2¼1. In case K1 ¼200 and K2 ¼100 or a ¼ b r0.4
both species persist while TMSY is reached.
Hence, for species which differ in biotic potentials for a factor of
two and with approximately the same carrying capacities, it is already
possible that the optimum harvesting effort, which gives TMSY,
induces extinction of the population with a lower biotic potential.
In continuation to the example in Fig. 2, we investigate the
coexistence of two species which differ considerably in biotic
potentials and competition coefficients. In case of selective
harvesting when the effort is adjusted to each species, the total
maximum sustainable yield (TMSY1) is attained at (e1opt,e2opt) and
both species are preserved (Fig. 2a). In case the system is
harvested with a common harvesting effort, the TMSY does not
exist (Fig. 3). Instead, the MSY of the second species is obtained
when the first species has already gone to extinction.
In the next case, species with the same biotic potential, different
carrying capacity and a small value of ab ¼0.04 (a ¼ b ¼ 0.2) are
investigated. TMSY exists and (Nn1(eopt), Nn2(eopt)) is a stable equilibrium point (Fig. 4). Populations do not switch dominance and at
the TMSY point, neither N1 nor N2 are overfished.
Consider an example with even smaller value of ab ¼0.02, but
where the effect of the second population on the first has
decreased to a ¼0.1. Biotic potential of the first population is
smaller but its carrying capacity is higher. TMSY exists and
(Nn1(eopt), Nn2(eopt)) is a stable equilibrium point (Fig. 5). Populations switch their dominance as e increases toward eopt. Now the
MSY of the second population is higher than TMSY when both
populations are present.
In case carrying capacities are the same (K1 ¼K2 ¼100),
biotic potentials differ r1 ¼1, r2 ¼2, and a ¼ b ¼0.2, TMSY exists
and (Nn1(eopt), Nn2(eopt)) is a stable equilibrium point (Fig. 6a).
An increase in competition to a ¼ b ¼0.4, still produce TMSY but
Fig. 3. The equilibrium population values (Nn1 and Nn2) and the yield (Y) as
functions of the common harvesting effort (e) for r1 ¼ 1, K1 ¼ 100, r2 ¼5, K2 ¼ 100,
a ¼0.6, b ¼0.1. The yield starts as an increasing function of the harvesting effort,
and when Nn1 has gone to zero, it continues as a parabola. There is no optimal
harvesting effort, because for eopt ¼0.6633, estimated from (20), the population Nn1
with smaller r tends to extinction. The MSY ¼r2 K2/4¼ 125 is composed of Nn2 only.
it will not be recognized in practice because it is smaller than MSY of
the second population. In case the competition is asymmetric
(a ¼0.2 and b ¼1) the optimum harvesting effort results in MSY of
the second population which has the higher biotic potential (Fig. 6c).
Obviously, given the same effort, the yield decreases with
increasing competition among species (compare Fig. 6a and b).
If the competition is strong enough, TMSY may turn into the
MSY of one population (Fig. 6b). In case biotic potentials,
carrying capacities, or competition coefficients differ considerably, even if it exists, TMSY may not be the global maximum.
Instead, MSY of one population (when the other species is
extinct) may be higher.
5. MSY for a general system of competitive populations
When the system of n competing species of the Volterra type is
subjected to a proportional harvesting, population Eq. (9) become
2 0
1
3
n
X
4
@
A
dN i =dt ¼ r i N i 1
aij N j =K i 5ei Ni ,i ¼ 1,. . .,n:
ð21Þ
j¼1
S. Geček, T. Legović / Journal of Theoretical Biology 307 (2012) 96–103
Fig. 4. The equilibrium population values (Nn1 and Nn2) and the yield (Y) as
functions of the harvesting effort (e) for r1 ¼ r2 ¼ 1, K1 ¼100, K2 ¼50, a ¼ b ¼0.2.
(Nn1(eopt), Nn2(eopt)) is a stable equilibrium point. Populations do not switch
dominance. At the TMSY, neither N1 nor N2 are overfished.
Constants e1,y,en are harvesting efforts on respective
populations.
There may be again at most 2n equilibria of which there may
be only one total persistence state
1
n
N ¼A
B
ð22Þ
T
where B¼[K1(1 e1/r1), y, Kn(1 en/rn)] .
The total yield in equilibrium is
Yt ¼
n
X
ei N ni
ð23Þ
i¼1
From the n normal equations qYt/qei ¼0, i¼1,y,n and if the
Hessian is negative definite, one can find eopt when Yt is maximized. The new equilibrium is Nn(ei,opt) and the total maximum
sustainable yield is
TMSY ¼
n
X
ei,opt N ni ðei,opt Þ:
101
Fig. 5. The equilibrium population values (Nn1 and Nn2) and the yield (Y) as
functions of the harvesting effort (e) for r1 ¼ 1, K1 ¼100, r2 ¼4, K2 ¼ 50, a ¼0.1,
b ¼ 0.2. The yield starts as an increasing function of the harvesting effort, and
when Nn1 has gone to zero, it continues as a quadratic function of only Nn2
population. (Nn1(eopt), Nn2(eopt)) is a stable equilibrium point. Populations switch
dominance as e increases. TMSY is achieved when both populations persist. If the
global MSY is enforced, the first population goes to extinction, while the second
persists at Nn2 ¼ K2/2.
The relevant question is: will harvesting the system of n
competing populations with eopt (when the yield will tend to
TMSY) ensure Nni (eopt)40, for all i¼1, y, n?
If (ri, Ki) are approximately the same as (rj, Kj), for all i a j,
and all the competition coefficients are approximately the same
(which is rarely the case in nature), the stability character of
the system of n populations is unlikely to change. Otherwise,
there exists a possibility that at least the population with the
smallest biotic potential, carrying capacity, or both will be
driven to extinction. Since in nature species differ widely in
their ri and Ki parameters, there is a high probability of having
more than a few species driven to extinction as a consequence
of enforcing TMSY.
ð24Þ
i¼1
Even if all equilibrium populations are positive and if the
equilibrium is stable, due to harvesting some populations may
decrease dramatically. Populations that were already small before
harvesting, may become too small to persist in nature because of
demographic stochasticity. More often, however, at least one
equilibrium population will turn to be zero or negative. Namely,
if there exist a species with low r and K values (hence contributing little to Yt but affecting through competition a species with
higher biotic potential), Yt may be higher if this species is
eliminated and therefore eopt will be adjusted to a value that
drives this species to extinction.
5.1. Equal harvesting efforts (e1 ¼e2 ¼y¼en ¼e)
In this case
Yt ¼ e
n
X
6.1. Prospect of adjusting the quota to TMSY
In this paper only the impact of proportional harvesting
strategy on a competitive community to achieve TMSY is analyzed. Adjusting quota to TMSY is not analyzed because it is well
known that this strategy does not admit a stable MSY even in a
single population governed by the logistic growth (Beddington
and May, 1977). Hence, for adjusting quota to reach the value
equivalent to TMSY using proportional strategy, the conclusion
would have been trivial: no population in a competitive community would persist.
6.2. Harvesting to TMSY and extinction of species
Nni ðeÞ
ð25Þ
i¼1
2
2
Since Yt is a quadratic function in e, if q Yt/qe o0, only one
maximum will exist. The maximum can be obtained from setting
qYt/qe¼0 and solving the corresponding linear equation for eopt.
The resulting total maximum sustainable yield is
TMSY ¼ eopt
6. Discussion
n
X
i¼1
Nni ðeopt Þ:
ð26Þ
A competitive community is characterized with smaller population values than in the case without competition. Harvesting
further diminishes the populations. Harvesting to TMSY means
adjusting harvesting effort to the most productive species. If the
resulting eopt is higher than, or equal to, the biotic potential of, say
m( on) species in the community, certainly these m species will
be driven to extinction. In addition, there may be species whose
biotic potential is higher than eopt and which would also be driven
to extinction due to existing competition of other species.
102
S. Geček, T. Legović / Journal of Theoretical Biology 307 (2012) 96–103
Fig. 6. The equilibrium population values (Nn1 and Nn2) and the yield (Y) as functions of the harvesting effort (e) for r1 ¼ 1, K1 ¼ K2 ¼ 100, r2 ¼2 with different combinations of
competition coefficients: (a) for a ¼ b ¼ 0.2, (Nn1 (eopt), Nn2 (eopt)) is a stable equilibrium point. The yield starts as an increasing function of the harvesting effort, and when Nn1
has gone to zero, it continues as a parabola of only Nn2 population. While e is increasing toward eopt, switching does not occur. The global TMSY is achieved while both
populations persist. For e4 8/9, Nn1 goes to zero. At the TMSY, Nn1 is overfished. (b) For a ¼ b ¼ 0.4, (Nn1 (eopt), Nn2 (eopt)) is a stable equilibrium point. While e is increasing,
switching does not occur. TMSY exists where both populations persist, but this TMSY will not be recognized in practice, since soon after the first population has gone to
extinction, the yield increases toward a global MSY which is composed of the second population only. (c) For a ¼ 0.2, b ¼ 1 the optimal harvesting effort is eopt ¼ r2/2 ¼ 1 but
it does not correspond to both positive equilibrium population values. For e48/9 the population Nn1 with smaller r goes to extinction. While e is increasing the dominance
is switched. The MSY is composed of the second population only (Nn2 ¼ K2/2).
6.3. A relationship between TMSY and CBD
It is expected that adjusting harvesting effort to one indiscriminative value, characteristic of fishery using trawlers, in order
to reach TMSY will be even more devastating for the competitive
community than a community of independent populations
(Legović and Geček, 2010). Even if TMSY is achieved, less
productive species are not likely to be preserved. By becoming
extinct from the competitive community, they do not interfere
with the most productive species on which TMSY depends. Hence,
by achieving TMSY one not only ignores the less productive
species, but works directly against their survival and therefore
against the CBD (1992).
6.4. Indiscriminative versus selective harvesting practices
It has been found earlier (Walters et al., 2005) that nonselective gear (trawling) is harmful to the ecosystem and that one
should change fishing towards selective harvesting. This is in
partial agreement with our results that selective harvesting to
TMSY is less likely to drive species to extinction than a common
harvesting effort.
6.5. A comparison of TMSY impact on a competitive community to
other ecosystem models
Competitive communities are ‘‘horizontally’’ structured
(Svirezhev and Logofet, 1983) within one trophic level. While
they are present in most ecosystems and at most trophic levels, in
addition, ecosystems are structured including several trophic
levels, so the question of the comparison between effects of TMSY
on competitive community with other ecosystem models arises.
Matsuda and Abrams (2005) analyzed a model of six species
including five trophic levels and two prey species. They found
that reaching TMSY usually causes extinction of species. In a food
chain and in a cascade of food cycles, harvesting any species to
MSY which is not a top predator, or harvesting a combination of
species to TMSY is likely to cause extinction of all species in
trophic levels above the lowest harvested species (Legović et al.,
2010). Hence we expect that harvesting to TMSY ecosystems with
more trophic levels will produce extinction of species similarly to
a competitive community.
Perhaps the simplest comparison is harvesting to TMSY a
competitive community versus a community of independent
species. If the optimum harvesting effort required to reach TMSY
is greater or equal to the biotic potential of m species (mon,
where n is the number of species in the community of independent species), these m species will become extinct. Similar,
although somewhat less restrictive result may be expected in a
community characterized by facultative mutalistic interactions
(Legović and Geček, 2012). In the competitive community with
the same distribution of biotic potentials of participating species,
generally we expect a greater number of species to end up in
extinction.
6.6. Simulation of a competitive community
Dynamics of a competitive community may be analyzed using
any of the general purpose simulation packages such as R project
(odesolve), Scilab, Simile, Stella, Madonna and many others
(for a comprehensive list see Legović and Benz, 2009). Since
reaching MSY, or more generally TMSY, is a control problem one
may use Scilab which includes control system design and analysis
modules or a dedicated software written for Matlab, such as
GPOPS (Rao et al., 2008).
7. Conclusions
There are two extreme cases of proportional harvesting of
species in a competitive community: (a) adjusting harvesting
effort to each species, i.e., using selective harvesting, and (b)
indiscriminate harvesting where the harvesting effort is approximately the same for all the species, like in trawler fishing. In the
case (a), TMSY may exist with all species present if competition is
weak and Ki/ri ratios are not too different between species.
However, these conditions are unlikely to be met in nature, and
hence most often some species will be lost. Furthermore, reaching
TMSY may threaten the persistence of species whose populations
were small before commencement of harvesting. They may
diminish to a value which does not allow them to persist in
nature, because a critical population level for survival is needed
for each species.
In the case (b), when TMSY exists, all species will frequently
not be present in the yield, i.e., some will be driven to extinction.
The most likely candidates for extinction are species with smallest biotic potential or carrying capacity or both. During harvesting, the intensity of competition among species works against the
S. Geček, T. Legović / Journal of Theoretical Biology 307 (2012) 96–103
persistence of the system. The analyzed cases considered the
difference in biotic potential only for a factor of two to five and
have already shown that TMSY does not necessarily mean
persistence of species. In nature, the differences in biotic potentials and the carrying capacities of competing species are much
larger, hence species extinction as a consequence of application of
TMSY may be widespread.
The main message is: the maximum sustainable yield is an
external condition which does not hold any guarantee that all the
species will persist. To the contrary, calling for application of MSY
(which in our case is equivalent to TMSY) is most likely a call for
the extermination of species in a competitive community in
nature, where the biotic potentials and the carrying capacities
differ widely. Therefore, in order to urgently stop the path from
overfishing to extinction of species, one must first retract all
declarations which call for the application of MSY in ecosystems,
and then start applying adaptive management in agreement with
the CBD.
Acknowledgments
This work was funded by the Croatian Ministry for Science,
Education and Sport.
We are very grateful to the editor and to four excellent
reviewers for a number of useful suggestions.
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