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Unformatted text preview: 232 Chapter 10 not always easy to ﬁnd the optimum level of sheep on the pasture. If the optimum num
ber of sheep, T2, in Figure 10.15(b) is exceeded by even a single sheep, then wool pro
duction falls drastically, to a much lower stable level. If that single sheep is then removed,
the system does not return to its previous level, but instead rides the thin red curve in Fig
ure 10.15(c). Only modest increases in wool yield are produced as more sheep are
removed from the pasture until the number of sheep drops to T1. When the number of
sheep is below T1, the wool production takes a giant leap upward. However, wool pro
duction still does not reach the level of production that was achieved at exactly T2 sheep. HARVESTING FISH An Example of the Peruvian Anchovy Peruvian anchovy live in the cool, upwelling, nutrientrich waters along the coast of
Peru and northern Chile. Reproductive maturity is reached at about 1 year of age and
the typical anchovy lives about 3 years. Anchovy occur in large schools and are caught
near the surface with nets. This was the largest ﬁshery in the world until it collapsed in
1972 in association with an El Niﬁo that disrupted these cold currents and diminished
productivity. Fishing was suspended to allow the stocks to recover, but there was no
immediate return of the anchovy. Populations of seabirds that feed on the anchovy also
remained low. Since anchovy have a rapid generation time, we can assume that there
has been adequate time for the population to return to its former level since the ﬁshing
suspension. But, as shown in Figure 10.16, it hasn’t. Why hasn’t it? In the sheep—grazing example, we had complete control over the number of sheep
that were grazed on the pasture. When we harvest trees from a forest or ﬁsh or whales
from the sea, natural forces beyond the control of the forester or the ﬁsher or the Whaler
inﬂuence the size of the resource available for harvesting. However, the theory that we
have already developed will help us see possible outcomes of different resource harvest
ing strategies. Often the goal is to achieve the maximum sustainable yield (MSY). The
word sustainable is an important qualiﬁer since the maximum yield in any single year is
to simply harvest all the resource that exists at that time. However, this is not sustainable
since next year there may be no resource. A bit later, we show how to estimate the MSY,
but for now we accept that the estimated MSY for Peruvian anchovy was about 9 to 10
million tons/year, based on average conditions for several years before the El Niﬁo
episode in 1972. Note that the harvest exceeded MSY in 1970 and in 1966—67. Two common ways of regulating natural resources like ﬁsheries are explained in
Box 10.1. H
U.) H
._: 3 Anchovy catch (million of tons)
\l 1 1955 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89
Year Figure 10.16 . Total catch for the Peruvian anchovy fishery. After the collapse in 1972, fishing was greatly reduced but the population had only
‘1 slightly recovered by 1989. After Krebs 1994. Exploited Resources Box 10.1 Methods of Regulating a Fishery or Other Dynamic Natural Resources As with the sheep, we begin by setting up a growth and loss
expression for ﬁsh: Net growth of a
ﬁsh population in :
a ﬁshing season WU“)  L
K logistic total losses
recruitment minus from (a)
curve of the ﬁsh ﬁshing (per season) (per season). Now we focus on the loss part of this equation, which is the
part that governments can regulate. We may distinguish two
methods for regulating the losses due to ﬁshing. 1. Fixed quota. Total losses to ﬁshing, L, are regulated by
setting a limit, Q, on the number of ﬁsh harvested per
season, regardless of the number of ﬁsh in the popula
tion, as illustrated in the following diagram. Logistic growth of ﬁsh
/ Regulators set the total ﬁshing
loss per season
to Q, regardless
of population
size, N. Here are
three possible
levels of Q. 2. Fixed effort quotas. We can expand the loss part of
Eq. (a) as follows: L 2 (ﬁshing effort)(C)(N). (b) Here the losses, L, are a product of three factors: the amount
of ﬁshing effort (which increases with the amount of time
spent ﬁshing, the number of ﬁshing boats, the area the boats
cover per day, etc.), a parameter of “ﬁsh catchability,” C,
and the size of the ﬁsh population, N. Fish catchability
measures the efﬁciency of each unit of ﬁshing effort. Fish
catchability therefore has units of ﬁsh caught per time
period per effort expended per ﬁsh in the population. We
assume that C is a constant. Then, Eq. (b) says that all else
being equal, the more ﬁsh in the population, the more effort
expended trying to catch those ﬁsh, and the easier the ﬁsh
are to catch, then the more ﬁsh will be caught in a given time
period. Equation (b) also says that the harvested losses repre
sent a type 1 functional response by the consumers—the
people doing the ﬁshing. The numbers of ﬁshers and the
“foraging” effort of each ﬁsher are combined into a single
term: the “ﬁshing effort.” Thus the analog of Figure 10.14 in
this case, where the number of ﬁsh caught increases linearly
with the ﬁsh population size (as by Eq. b), looks something
like the following. The slopes of these
lines = (C )(effort), and
effort is regulated.
Here are three
different effort levels. Regulation is effected by limiting the “effort” term in Eq.
(b). This may be done by restricting the number of ﬁshing
permits, the length of the ﬁshing season, or restricting
regions where ﬁshing is allowed. Variations in ﬁshing effort
from year to year will lead to different catches, which will
trace out the recruitment curve (assuming that it remains
constant from year to year). For example, the following dia
gram shows ﬁve different efforts in ﬁve different years
applied to a ﬁsh population and the resulting catches for
each. Catch 2
dN Catch 3 7t— Catch 1 Catch 4 The following diagram shows those same ﬁsh catches plot—
ted against the effort values instead of against ﬁsh popula
tion size, N. 233 234 Chapter 10 Predicted maximum
}/ catch = MSY
Catch 2 Fish caught Catch 3 season Catch 1 , Preéictedbptimum, ‘ f .
effort: abovethis tiaeﬂah :
catchdeclines ‘_ _ « This plot also yields a hump—shaped curve, like the logistic
curve. This is a key result since this ﬁgure shows that, even
without knowing the absolute ﬁsh population size, N, the
optimum effort and the MSY can still be estimated simply
by the relative position of the yield—effort curve. The opti
mum effort and the MSY are shown with arrows in the pre
ceding ﬁgure. If, additionally, the catchability constant, C, is
known, then ﬁsh population size, N, and the corresponding
r and K may also be estimated from the yield—effort curve
(we work through some examples later in this chapter). The catch versus Peruvian anchovy data and a ﬁtted
curve are shown in Figure 10.17. Note that during this
period the anchovy did not appear to be severely overex
ploited. Obviously several complications may affect all this.
Fishing effort is normally modiﬁed from year to year—but
with climatic changes, the recruitment curve, and the r and
K for the ﬁsh population are also changing yearly. Addition
ally, the ﬁsh population may not grow according to a logis—
tic equation. A full model might require knowledge of the
age structure of the ﬁsh population. Densitydependence Total catch (million tons/year) t—‘NWkaOVOO 51015 20 25 30 35 4O 45 50 55 60
Total ﬁshing eﬂ'ort (million trips) Figure 10.17 The relationship between total fishing effort and total catch
for the Peruvian anchovy fishery. The effects of humans
and sea birds are combined in these data. The parabola is
based on a continuous logistic equation fitted to these data.
Arrows indicate the estimate for MSY and appropriate
fishing effort. After Krebs (1994) from Boerema and
Gulland (1973). might be different for different ages, and the functional form
of the densitydependence could contain several higher
order terms. Recall the more complicated singlespecies
models that we explored in Chapter 5. Finally, the catcha
bility, C, may not be a constant but instead a function of
effort and/or ﬁsh population size, N. In this case, we would
not have type 1 functional response curves but perhaps type
2 or type 3 curves. As we have already shown, the latter can
yield break points and alternative stable states. Finally, the
ﬁsh population is also being exploited by several natural
predators, and we usually have little knowledge of their
abundance and functional responses. The task for government resource regulators is to prevent a tragedy of the com
mons by keeping the ﬁshing industry from “killing the goose that laid the golden egg.”
It is in the best interest of each individual to prevent the overexploitation of the com
mon ﬁshery that they collectively depend on, yet at any particular time it is in the short
term economic best interest of each ﬁsher to maximize his or her catch. Thus we have
the classic problem of what is best in the short term for the individual, if followed by
all individuals, leads to the collapse of the ﬁshery and thus a bad situation for all who
depend on it for their livelihood. As we have indicated, one possible way to prevent the collapse of the ﬁshery is for
governments to impose a ﬁxedquota harvest (Box 10.1). This imposes a limit on short
term proﬁts so as to maximize longer term proﬁts and the continued viability of the nat
ural resource. For example, regulatory agencies could put a limit on the number of ﬁsh
that can be caught per season. The ﬁshing season would close after the limit is reached. Exploited Resources 235 Figure 10.18 Growth of a logistically growing
resource under a fixed
harvesting quota. In the central red region, thelogistie
, ‘growthfof thelresource exceeds the ,imposed’harvesung mortality, Q, so the
, net growth of the population is positive. d_N .
dt Fixed
harvesting
quota, Q
0
0 x \ ,1 / Population size, N
\ \ ,
\ / However, in these regions of N, the
net growth rate is negative.
(3)
The em sto create; three equilibrium
3 points, twostable and one unstable.
d_N
dt
Fixed
harvesting
quota, Q ’ Unstable // Stable equilibrium
/ equilibrium /
0 x
X P 1 t' ' N
Stable / 0 N*2 Nakl opu a non Size,
equilibrium }_—___>
at N = 0 I I For any population in this range of N, and with the ﬁxed quota, Q, depicted, However, if population size falls the population size, N, will eventually
1nto thls low range of N, then the stabilize at N*1. population size will collapse further to
0 unless harvesting is reduced. (b) ble, and therefore the yield at N 1* is sustainable. However, the one to the left is unsta
ble, and therefore the yield associated with N2* would be unsustainable at the same
quota level. If the quota were increased, the horizontal line would march up the y axis,
giving higher allowed yields. The MSY occurs when the quota is raised to the point that
it just equals the peak of the logistic hump, as depicted in Figure 10.19. For logistic recruitment, the population size will be at K/2 when the MSY is
reached. Therefore the MSY will equal MSY=d—N at £=rK/2(K—£)
dt 2 K 2 (101)
15
4 . It shouldn’t take long to see a major problem with regulation for a ﬁxed—quota level
Qmax. If Qmax is exceeded in a single year, perhaps because it was misidentiﬁed by gov
ernment regulators or because environmental conditions change so that the recruitment
parabola becomes lower, then resource levels will be depressed the following year to
a level below K/2. Now, assuming the same quota level, the ﬁsh population will 236 Chapter 10 Figure 10.19 A much higher fixedquota harvest.
This harvest will produce the MSY. Figure 10.20 Harvesting at the maximum
sustainable level for four different
initial population sizes: 90, 99, 120,
and 200. For this example, r = 1, K = 200, and Qmax is 50. Since K / 2 = 100, any population slipping
below N = 100 eventually crashes
unless Q is reduced. Semistable
equilibrium point Fixed /
harvesting
quota, Q, is
(ii—1;] now at Qmax.
0
0 N*1= K/2 = MSY Population size, N
1‘»
I——I \ . _ .
f For any population in th1s range
. _ . . of N, then with the quota, Qmax,
If the populatlon Size falls Into this depicted, the population size, N,
low range of N, then with this quota. will eventually stabilize at N><lv
Qmax, the population size, N, will
collapse further to 0.
200 N0 = 120 and 200 0 50 100 150 200
Time approach the stable equilibrium of zero, sliding the ﬁsh population into extinction.
Therefore harvesting will need to be suspended or greatly reduced until the resource
level can return to a level greater than R* so that the quota Qma1x will again produce a
stable equilibrium. An example of a logistic population being harvested at the MSY is
shown in Figure 10.20 for K = 200. The initial ﬁsh population size is shown at four
alternative levels, and for each the population is followed over time. In this example, the two populations that dipped below N = 100 (= K/ 2) fell to
extinction. The MSY can be approached only from population sizes above K/Z. If reg
ulators take quick action and reduce the harvest rate as soon as they notice that the ﬁsh
population is being overexploited, it will rebound, as Figure 10.21 illustrates. Why didn’t the Peruvian anchovy population rebound like this after ﬁshing was
suspended? The anchovy had not gone extinct but were at very low levels. One of our
assumptions may be wrong, or there may be more going on then is accounted for by
this simple logistic model of ﬁsh growth. We will see how this question might be
answered a little later in the chapter. In real systems we typically have little knowledge of the actual level of the resource
population, let alone the exact shape of the recruitment curve, and yearly variations in
climate inﬂuence the shape of this curve anyway. Thus it is easy to overestimate Qmax,
setting off a population decline and thus producing less than optimal longterm yields
(May et al. 1979). A ﬁxed—effort harvesting scheme avoids some of the stability pitfalls
of the ﬁxed—quota method, as demonstrated in Boxes 10.2 and 10.3). A ﬁxed—effort Figure 10.21 200
This is the same example as in Figure 10.20 for an initial population size of 90. The fish population is on its way to a crash. Now, however, at time 15, the harvest rate is reduced from 50 to 15. With this adjustment the fish N 100
population recovers. Box 10.2 In Figure 10.22, we develop a curve similar to that in Figure
10.15 for a type 1 functional response instead of a type 3
response. This situation corresponds to the ﬁxed—effort regula
tion of a ﬁshery discussed in Box 10.1. We label the resource
population size R and the consumer population as predators R
For example, governments could regulate the number of ﬁsh—
ermen, P, as a means of controlling ﬁshing effort. Solution:
Now we plot the relationship between the levels of R at
equilibria (0n the x axis) and the number of predators, P, in Exploited Resources 237 Fish Population Harvest Rate 50 100 1 50 200
Time Figure 10.23. The dots on the x axis in Figure 10.22 are car
ried over here for each level of predator, and projected (by
dashed lines) to the appropriate number of predators. Then a
line has been drawn to connect these dots. Figure 10.23 is
the analog of Figure 10.12 for a type 1 functional response.
The ﬁnal step is to produce the analog of Figure 10.15.
The open squares on the y axis in Figure 10.22 show the
equilibrium level of dR/dt associated with the indicated
predator numbers on each total consumption curve. These
values are plotted on the y axis in Figure 10.24 against the
corresponding predator numbers on the x axis. Logistic prey
recruitment The straight
\ lines are the
total
consumption
rates for
different
numbers of predators, P. Equilibrium levels of prey for these
different numbers of predators Resources, R Figure 10.22 238 Chapter 10 , . . V . . (Hf
Line of cqurhbna E02“ prey: ~ [A :1 (3
{if Predators, P
DJ
0 H—
CU! LII Prey population, R
Figure 10.23 Box 10.3 Derive an algebraic expression for the graphical relationship between dR/dt and predator numbers that was developed in
Box 10.2. Solution: Given a type 1 functional response of predators, with a pop—
ulation size 'P and a resource population R that grows logis
tically in the absence of harvesting, we may write £2 rR(l—£)—aPR
dt K gar—’EH The first term The second term is the (C)
is the logistic consumption rate of R by P
equation for R. predators, each with encounter rate a: a type 1 functional response. Predator productivity
(proportional to dR/dt when R reaches an equilibrium) 5 1015 30 60
Predators, P Figure 10.24 We let R* be the equilibrium level of R and solve for R* by
setting Eq. (c) to 0. rR*£1—R )=aPR*.
K Rearranging, we get R* =K(l—£j. I” At R*, consumer productivity 2 aPR* (from Eq. (c)), and
thus the turnover of resources at this equilibrium is dR *
dt =aPK(1—£). (d) r Equation ((1) describes the parabola plotted in Figure 10.24. method is based on the assumption that the harvesters have a type 1 functional response.
Unlike the situation for a type 3 functional response, a type 1 functional response pro
duces no unstable equilibrium points and thus no hysteresis. However, overexploitation
of the resources (i.e., diminishing returns with increasing effort) is still possible, as we
have already shown in Box 10.1. More about FixedEffort Harvesting Because of the desirable stability features of the type 1 functional response (Box 10.3),
the regulation of total resource harvesting can be accomplished most successfully by a Figure 10.25 Fixedeffort harvesting of a resource,
R. The medium harvesting effort
produces the MSY and an equilibrium
resource level of K / 2. Figure 10.26 The only equilibrium R for the very
high effort is zero resource. Exploited Resources 239 Medium
effort Recruitment High effort curve dR — MSY
dt
Low effort
0
R
Very high
effort Medium
effort dR
— MSY
dt ﬁxedeffort regulation of harvesting. Recall from Box 10.1 that the total yield (or catch)
of resource to all harvesters is the product of three different terms: capture effort expended Total yield 2 [by all harvesters
[catchability] [resource level]. (10.2)
Regulating the ﬁrst term sets the total yield, and the resource level adjusts accordingly.
An example of the overall harvest in terms of dR/dt based on three different levels of
capture effort per resource (assuming that catchability remains constant) is shown in
Figure 10.25. Here, regulatory agencies do not ﬁx the total quota per season, but instead
try to regulate the amount of capture effort expended. To do this successfully, they must
be able to obtain a census of the number of ﬁsh, R, and adjust effort accordingly so that
the MSY can be reached. In Figure 10.25, the medium effort corresponds to the MSY. Note, however, from
the arguments developed in Box 10.1, that the equilibria ﬁsh numbers corresponding
to all three efforts are stable. Moreover, unlike the situation with fixedquota harvest—
ing, if the medium effort is being maintained through regulation, but the resource
level temporarily falls below R*med, the recruitment rate still exceeds the harvest rate
and resource levels will increase back to R*med when conditions return to normal.
Resources will not become extinct unless very high capture efforts are maintained, as
illustrated in Figure 10.26. With ﬁxedeffort harvesting, the total yield varies with the resource population size
according to Eq. (10.2) If the population size were to drop below R*, then the yield
would drop below MSY. The appropriate response by regulators would be to reduce the
allowable effort to let resource levels increase back to R* or beyond. However, co" ..
tition among the harvesters would act in the opposite direction. In the face of '
ishing catch, harvesters will apply political pressure to increase allowable eﬂ’oﬂ 240 Chapter 10 , yields can go up again, restoring their proﬁt margins. At the same time, with a dimin
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 Fall '10
 WayneM.Getz

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