Ecological Modelling 154 (2002) 1 – 7
www.elsevier.com/locate/ecolmodel
The maximum sustainable yield of Allee dynamic system
Zhen-Shan Lin a,*, Bai-Lian Li b
a
Department of Geography, Nanjing Normal Uni6ersity, Nanjing 210097, People’s Republic of China
b
Department of Biology, The Uni6ersity of New Mexico, Albuquerque, NM 87131 -1091, USA
Received 11 April 2001; received in revised form 19 October 2001; accepted 31 October 2001
Abstract
In this study an attempt is made to investigate comprehensively the maximum sustainable yield (MSY) of Allee
population dynamic system. The results show that: (1) because the net production rate of Allee system is a function
of the shape parameter d, all Allee systems which possess different B (B=A/K, here K is the carrying capacity, and
A the parameter of Allee effect) are adapted themselves to the environments by controlling their production rate.
Allee effect does not change the interaction between the system and environment, but it will reduce the MSY; (2) the
MSY of Allee system is approximately one-nineteenth of the height of the allometry curve of body size for all
A/K50.01. However when the net production rate R takes the values from 3 to 15, the MSY of Allee system is also
approximately 1/19 of the height of the allometry curve of body size for all A/K 50.001. © 2002 Elsevier Science
B.V. All rights reserved.
Keywords: Population dynamics; Allee-effect; MSY
1. Introduction
Surplus production models for biomass yield
have a long, useful history in fishery management
(Ricker, 1983). By wedding this management technique to certain results known from the study of
body size scaling or allometry (Calder, 1984;
Damuth, 1987; Charnov, 1991, 1992, 1993a,b),
Charnov (1993a) considered that MSY (in weight)
is approximately 1/6 of C, where C is defined by
K = CW − 0.75 (here K is the carrying capacity and
W the adult body weight), i.e. the maximum
sustainable yield (MSY) is independent of body size
for mammals.
Suppose that population growth is governed by
the equation:
K− N
dN
= r mN
K
dt
d
where N is the population size (adults per unit
area), d is the shape parameter and rm is the
maximum intrinsic rate of increase.
Let u be the relative population size (N/K), the
equation can then be rewritten as:
d(NW)
= Wrm(1−u)duK=f(u, rm, d)
dt
* Corresponding author.
E-mail address: [email protected] (Z.-S. Lin).
(1)
(2)
where d(NW)/dt is the surplus production available
for harvesting from a population held at size N.
0304-3800/02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 8 0 0 ( 0 1 ) 0 0 4 7 9 - 3
Z.-S. Lin, B.-L. Li / Ecological Modelling 154 (2002) 1–7
2
Letting
#f
=0
#u
one obtains the steady state:
or
um =1/(1 +d),
d =(1 − um)/um
(3)
and
MSY = Wrm(1− um)dumK =f(um, rm, d)
(4)
Suppose Rm is the net reproduction rate of the
steady state, from Fowler’s approximation
(Fowler, 1989),
um =0.633−0.187y,
y =loge(loge Rm)
(5)
Considering the maximum intrinsic rate of increase rm and the carrying capacity K, the scaling
laws (Charnov, 1993a) can be generally shown as
following:
rm = A3W
− 0.25
,
K =CW
− 0.75
(6)
where both C and A3 are the parameters, and
A3 =
ln Rm
,
A1 + A2
A2 = h/W
A1 = M − 1/W
0.25
,
0.25
,
A1 +A2 :2.4
(7)
here M − 1 is the mean adult lifespan and h is the
age of first reproduction.
From Eq. (6) we have
rmKW =CA3
(8)
height of allometry curve of the production density body size (MSY:C/6), which means that the
MSY of non-Allee population system is also independent of the shape parameter d, or the MSY is
independent of the internal interaction. Two
things account for the results: Charnov used Eq.
(3) (which is not a function of body size by
Fowler’s rule) to determine the shape parameter,
and Eq. (5) to control um.
However, the interactions of different species
(mammals and others) with the environment are
different in a certain local region, the shape
parameter d will change with different models, i.e.
the MSY is a function of the shape parameter d.
So, Charnov’s results (MSY: C/6) are not suitable for all mammals. On the other hand, it is
known that an Allee effect can occur in many
population systems (Lewis and Kareiva, 1993;
Amarasekare, 1998). Wang et al. (1999) set up a
competitive dynamics of population subject to an
Allee effect and Stephens and Sutherland (1999)
showed some consequences of Allee effect on
behavior, ecology and conservation. We usually
want the maximum sustainable yield of Allee system with the minimum effort. Is MSY of Allee
system also independent of the body size for
mammals and others?
How the MSY depends on the shape parameter
d will be shown in Section 2, and some extreme
values of Allee population dynamic system will be
discussed in the Sections 3 and 4.
So,
MSY = C(1−um)dum ln Rm/(A1 +A2)
(9)
The term (1− um)dum ln Rm is almost a constant
for Rm in the range 3– 15, which includes all the
vertebrates in Fowler’s (1989) data set. The term
equals 0.37 at Rm =3 and rises to 0.58 at Rm =15
(Charnov, 1993a,b). This means that for mammals (Rm = 3),
MSY :
C
0.4
A1 +A2
(10)
Estimating A1 +A2 for non-primate (Charnov,
1993a,b; units are kg, year) makes A1 +A2 =2.4
or MSY = 0.17C (: C/6). Thus, Charnov (1993a)
found that MSY is approximately 1/6 of the
2. Some extreme values of population growth
model at different shape parameters
Letting
Ym(u)= (1−um)dum
(11)
Y*=
(1− um)dum ln Rm
a
(12)
where Ym(u) shows the effect of environment, and
Y *a is the interaction between environment and
the mammals population.
Some extreme values of the system at different
shape parameters are shown in Table 1.
From Table 1, we find that the MSY increases
with the increase of shape parameter d. Suppose
Z.-S. Lin, B.-L. Li / Ecological Modelling 154 (2002) 1–7
3
there are different species (mammals and others)
in a certain local region, because of their different
interactions with the environment, the shape
Table 1
Some extreme values of population growth model at different
shape parameters
d
um
Rm
Ym(u)
Y *a
MSY
0.3
0.5
1
1.5
2
3
0.769
0.667
0.5
0.4
0.333
0.25
1.621
2.302
7.664
32.342
144.64
2329
0.495
0.385
0.25
0.186
0.148
0.105
0.239
0.321
0.509
0.647
0.736
0.814
0.1C:C/10
0.134C: C/7
0.212C: C/5
0.27C: C/4
0.307C:C/3
0.339C: C/3
Fig. 2. The curves ya1, ya2 and ya3 show, respectively, the ya,
or MSYm of Allee system insensitive to the maximum net
reproductive rate Rm at B= 0.001, 0.01, and 0.1.
Table 2
The MSY of Allee population system when d takes an extreme
value (Eq. (16))
B
Rm
ya
MSYA
0
0.00001
0.0001
0.001
0.01
0.1
5.18
5.18
5.18
5.18
4.5
2.8
0.1277
0.12765
0.12759
0.1273
0.1237
0.1046
0.053C:C/19
0.053C:C/19
0.053C:C/19
0.053C:C/19
0.052C:C/19
0.044C:C/23
Table 3
The MSY of Allee population system when net reproduction
rate Rm =3
Fig. 1. (a) The relation between the relative population size um
and the shape parameter d at the steady state. (b) The relation
between the shape parameter d and the net production rate Rm
at the steady state.
B
Rm
ya
MSYA
0
0.00001
0.0001
0.001
0.01
0.1
3
3
3
3
3
3
0.1265
0.1265
0.126
0.1260
0.1234
0.1046
0.053C:C/19
0.053C:C/19
0.0053C:C/19
0.053C:C/19
0.051C:C/20
0.044C:C/23
parameter d will change with the different models.
So, the MSY of different species are different,
unless they have the same shape character. However, when a species of mammals is at an equilibrium state, d is a constant for the species, then
its MSY is a constant. And the further calculation
shows that Charnov’s results MSY: C/6 is held
Z.-S. Lin, B.-L. Li / Ecological Modelling 154 (2002) 1–7
4
only when d=0.625 and the population size
Nm =umK=0.615K.
Fig. 1 shows the relationships um – d (a) and
d–Rm (b).
When the nonlinear interaction of the system is
enhanced, the value of d increases and um decreases (Fig. 1(a)), then the population size Nm =
umK is reduced. In order to maintain the
ecological equilibrium, the species must increase
its net reproduction rate Rm (Fig. 1(b)) so that the
weaker increases its net reproduction rate. Because the net reproduction rates of many mammals are less than 145 and greater than 1.6, we
consider that power numbers or the shape
parameters of the system usually are in the range
0.3 B dB 2 which tallies with the fact (Charnov,
1993b, p. 103).
3. The maximum sustainable yield of Allee system
when d takes an extreme value
Suppose that population growth with an Allee
effect is governed by the equation
N
dN
=Nrm 1−
K
dt
d
N
−B ,
K
B = A/K
(13)
i.e.
d(NW)
=WrmK(1− u)du(u − B)
dt
(14)
When u satisfies the following condition:
u 2Am(2+d)−uAm(2+ B +Bd) +B = 0
(15)
or
Table 4
MSY of Allee population system when Rm = 15
B
Rm
ya
MSYA
0
0.00001
0.0001
0.001
0.01
0.1
15
15
15
15
15
15
0.125
0.125
0.125
0.1242
0.1204
0.097
0.052C:C/19
0.052C:C/19
0.052C: C/19
0.052C: C/19
0.050C:C/20
0.04C: C/25
d=
(1− uAm)(2uAm − B)
uAm(uAm − B)
(16)
Eq. (14) will possess the largest d(NW)/dt, then
the maximum sustainable yield:
MSYA =
C
(1−uAm)duAm(uAm − B)ln Rm
A1 + A2
(17)
Here Eqs. (5)–(8) are used to deduce Eq. (17).
In order to compare with Charnov’s (1993a)
results about non-Allee system, we first use Eq.
(5) to determine uAm, then Eq. (16) to determine d
just as Charnov’s (Charnov, 1993a).
Letting
y(u)= (1− uAm)duAm(uAm − B)
(18)
ya=(1−uAm)duAm(uAm − B)ln Rm = y(u)ln Rm
(19)
Fig. 2 shows the relation between ya and Rm at
different B.
Comparing with Charnov’s Figure 1 (1993a),
some differences between Population Model and
Allee Population Model are found:
1. the MSY of Allee population system is also
independent of the body size for mammals
(and others), and it is less than that of population model;
2. the MSY of non-Allee system monotonically
increases with the net production rate, and
there is a maximum value in the MSY of Allee
system. When the net production rate is greater
than the extreme point, the MSY of Allee
system will monotonically decrease with the
net production rate. Larger B will induce faster
decrease in MSY.
Tables 2–4 show some MSY values of Allee
population dynamic system in different conditions.From Tables 2–4, (1) the MSY of Allee
population system is only 1/3 of that of non-Allee
system (C/6, Charnov, 1993a); (2) when d takes
an extreme value, i.e.
d=
(1− uAm)(2uAm − B)
,
uAm(uAm − B)
the MSY of Allee system is approximately 1/19 of
Z.-S. Lin, B.-L. Li / Ecological Modelling 154 (2002) 1–7
5
Fig. 3. The relation between effect of environment y(u) and the relative population size u of Allee system; where y(u) is noted as
yi (u), and i =1, 2, 3, 4, 5 indicates, respectively, d= 0.5, 1, 1.5, 2, and 3. (a) B =0.1; (b) B =0.01.
the height of the allometry curve of population
density body size for all B 50.01, and when net
reproduction rate Rm takes the values from 3 to
15, the MSY of Allee system is also approximately 1/19 of the height of the allometry curve of
population density body size for all B5 0.001.
Z.-S. Lin, B.-L. Li / Ecological Modelling 154 (2002) 1–7
6
4. Some extreme values of Allee system at
different shape parameters
In Section 3 the maximum sustainable yield of
Allee system has been discussed when d takes an
extreme value. Where d is determined by uAm (Eq.
(16)) which is determined by Rm (Eq. (5)). That
means the independent variable of the system is
R. However, the MSY is determined by u which is
associated with the largest d(NW)/dt. So u is
exactly the independent variable of the system.
Fig. 3 shows the variations of y(u) with u in
different d when B = 0.1 (a) and B = 0.01 (b).
According to Fig. 3, Eqs. (5), (15) and (16),
some extreme values of Allee system when B =
Table 5
Some extreme values of Allee system at different shape
parameter when B = 0.1
d
um
Rm
y(u)
ya
MSYA
0.5
1
1.5
2
3
0.82
0.68
0.59
0.51
0.41
1.445
2.177
3.52
6.89
27
0.25
0.126
0.076
0.05
0.026
0.092
0.098
0.096
0.097
0.086
0.038C:C/26
0.041C: C/24
0.04C: C/24
0.04C: C/24
0.038C:C/28
Table 6
Some extreme values of Allee system at different shape
parameter when B =0.01
d
um
Rm
y(u)
ya
MSYA
0.5
1
1.5
2
3
0.82
0.68
0.59
0.51
0.41
1.445
2.177
3.52
6.89
27
0.282
0.146
0.082
0.061
0.034
0.104
0.114
0.103
0.118
0.112
0.043C:C/23
0.048C: C/21
0.043C: C/23
0.049C: C/20
0.047C:C/21
Table 7
Some extreme values of Allee system at different shape
parameter when B =0.001
d
um
Rm
y(u)
ya
MSYA
0.5
1
1.5
2
3
0.82
0.68
0.59
0.51
0.41
1.445
2.177
3.52
6.89
27
0.285
0.148
0.0829
0.062
0.034
0.105
0.115
0.112
0.12
0.112
0.044C:C/23
0.048C:C/21
0.047C: C/21
0.05C: C/20
0.047C: C/21
Fig. 4. The x axis is Rm, and y axis is d. The curves show the
relation, respectively, between the shape parameter d and net
production rate Rm (at the steady state) of Allee system when
B=0.01, 0.001, and 0.0001.
0.1, 0.01, and 0.001 are shown in Tables 5– 7,
respectively.
Form Tables 5–7 it is found that: when d is the
independent variable about the system, both um
and Rm are independent of B, but the MSY is
related to B. For any exact d, the MSY of Allee
system will increase with the decrease of B. The
fact that the net production rate is a function of d
shows that all Allee systems possessing different B
adapt themselves to the environment by controlling their production rate. Allee effect does
not change the interaction between the system and
environment, but it will reduce the MSY.
Fig. 4 shows the relation between d and R of
Allee system when B takes different values.
From Fig. 4, three curves of d under different B
values (0.0001, 0.001, 0.01) are almost coincident,
which means that Allee effect does not change the
interaction between the system and environment.
5. Conclusions
This paper has discussed the MSY of Allee
population dynamic system. The most significant
findings are as follows:
1. By calculating and simulating, it is found that
the MSY of non-Allee population growth increases with the increase of shape parameter d,
and Charnov’s results about MSY: C/6 is
held only when d= 0.625.
Z.-S. Lin, B.-L. Li / Ecological Modelling 154 (2002) 1–7
2. When the nonlinear interaction of the system is
enhanced, the value of the shape parameter d
increases and the relative population size um at
the steady state decreases, then the population
size Nm =umK is reduced. In order to maintain
the ecological equilibrium, the species must
increase its net reproduction rate Rm. So the
weaker always has a big net reproduction rate.
3. Because the net reproduction rates of many
mammals are between 145 and 1.6, the power
numbers or the shape parameters of the system
are in the range 0.3B d B2, which tallies with
the fact.
4. Because the net production rate of Allee system
is a function of d, all Allee systems which possess
different B are adapted themselves to the environment by controlling their production rate.
Allee effect does not change the interaction
between the system and environment, but it will
reduce the MSY.
5. The MSY of Allee system is approximately
one-nineteenth of the height of allometry curve
of the population density body size for all
B50.001. However, when R takes the values
from 3 to 15, the MSY of Allee system is also
approximately 1/19 of the height of allometry
curve of the population density body size for all
B5 0.001. So we consider that as the Allee effect
relates to populations at low densities it is
difficult to see how it can be relevant to the
MSY.
Acknowledgements
This research has been supported in part by
7
Ministry of Science and Technology of China under
grant NKPDBS G1998040900-part 1, and US National Foundation under grant DEB-94-11976.
References
Amarasekare, P., 1998. Allee effects in metapopulation dynamics. Am. Nat. 152, 298 – 302.
Calder, W.A., 1984. Size, Function and Life History. Harvard University Press, Cambridge, MA.
Charnov, E.L., 1991. Evolution of life history variation
among female mammals. Proc. Natl. Acad. Sci. USA 88,
1134 – 1137.
Charnov, E.L., 1992. Allometric aspects of population dynamics: a symmetry approach. Evol. Ecol. 6, 307 – 311.
Charnov, E.L., 1993a. Is maximum sustainable yield independent of body size for mammals (and other)? Evol.
Ecol. 7, 309 – 311.
Charnov, E.L., 1993b. Life History Invariants: Some Exploration of Symmetry in Evolutionary Ecology. Oxford
University Press, New York, p. 1 – 158.
Damuth, J., 1987. Interspecific allometry of population density of mammals and other animals: the independence of
body mass and population energy use. Biol. J. Linn. Soc.
31, 193 – 246.
Fowler, C.W., 1989. Population dynamics as related to rate
of increase per generation. Evol. Ecol. 2, 197 – 204.
Lewis, M.A., Kareiva, P., 1993. Allee dynamics and the
spread of invading organism. Theor. Popul. Biol. 43,
141 – 158.
Ricker, W.E., 1983. Computation and interpretation of biological statistics of fish population. Bull. Fish. Res.
Board Can. 191, 1 – 382.
Stephens, P.A., Sutherland, W.J., 1999. Consequences of the
Allee effect for behaviour, ecology and conservation.
TREE 14, 401 – 405.
Wang, G., Liang, X., Wang, F., 1999. The competitive dynamics of populations subject to an Allee effect. Ecol.
Model. 124, 183 – 192.