Download Will Small Population Sizes Warn Us of Impending Extinctions?

vol. 160, no. 3
the american naturalist
september 2002
Will Small Population Sizes Warn Us
of Impending Extinctions?
Peter A. Abrams*
Department of Zoology, University of Toronto, 25 Harbord Street,
Toronto, Ontario M5S 3G5, Canada
Submitted March 2, 2001; Accepted February 15, 2002
abstract: Several models are used to show that population sizes
are often relatively insensitive to deteriorating environmental conditions over most of the range of environments that allow population
persistence. As conditions continue to worsen in these cases, equilibrium population sizes ultimately decline rapidly toward extinction
from sizes similar to or larger than those before environmental decline began. Consumer-resource models predict that equilibrium or
average population size can increase with deteriorating environmental conditions over a large part of the range of the environmental
parameter that allows persistence. The initial insensitivity or increase
in the population of the focal species occurs because changes in the
populations of other components of the food web compensate for
the decline in one or more fitness components of the focal population. However, the compensatory processes are generally nonlinear
and often approach their limits abruptly rather than gradually. When
there is steady directional change in the environment, populations
lag behind the equilibrium population size specified by current environmental conditions. The environmental variable can then decline
below the level required for population persistence while the population size is still close to or greater than its original value. Efficient
consumers and self-reproducing resources are especially likely to produce this outcome. More complex models with adaptive behavior,
additional consumers, or additional resources often exhibit similar
trajectories of population size under environmental deterioration.
Keywords: conservation, consumer-resource system, extinction, food
web, environmental deterioration, predation.
Conservation biologists need to understand how population sizes are expected to change as species experience
* E-mail: [email protected].
Am. Nat. 2002. Vol. 160, pp. 293–305. 䉷 2002 by The University of Chicago.
0003-0147/2002/16003-0002$15.00. All rights reserved.
deteriorating conditions (van Horne 1983; Doak 1994).
Such knowledge is important, given the key role of population size in current schemes for classifying species as
endangered (IUCN 2001) and because changes in numbers
are the key measure used to quantify population viability
in most situations (e.g., Gerber et al. 1999). This focus on
population size as a measure of demographic health can
be problematic for two reasons: first, equilibrium population size may be insensitive to, or increase with, environmental deterioration; and, second, there is a time lag
in every population’s response to altered environments
that may temporarily hide a decrease in the equilibrium
population size. The first problem has received very little
discussion in the conservation literature, although it is a
consequence of several well-known models in population
and community ecology. Predator and resource densities
change in ways that typically offset declines in the growth
parameters of a focal species. Previous work (e.g., Ives
1995) has generally not explored the nonlinearity of such
compensatory processes. The problems introduced by time
lags have primarily received attention from ecologists interested in metapopulations (Pulliam 1992; Tilman et al.
1994; Cowlishaw 1999) but may well be a greater problem
in more uniformly distributed populations. These assertions are the subject of the remainder of this article.
In the following analyses of simple models, I will explore
how actual and equilibrium populations are expected to
change in simple models in which a focal species experiences gradual deterioration of its environment. Deterioration will be defined as any change in the environment
that directly reduces the immediate per capita population
growth rate of individuals of a focal species. For example,
global warming may increase the death rate of a heatintolerant species or reduce the amount of time it can
forage actively. Toxins that interfere with activity can have
similar effects. Most of the analysis treats the case in which
a single consumer species in a food chain or food web
experiences deteriorating conditions; adverse effects on
prey species or in multispecies communities are treated
much more briefly.
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The American Naturalist
Population Size of Consumers Experiencing
Environmental Deterioration
Population Size and Environmental Deterioration
in Single-Species Models
Although single-species models are not the main focus of
this article, they do illustrate the two central features of
nonlinear responses and population lags. Assume that
population growth is described by the v-logistic model
(Gilpin and Ayala 1973):
[ ( )]
v
dN
N
p rN 1 ⫺
.
dt
k
(1)
When a consumer species experiences deteriorating conditions, subsequent increases in its resources will at least
partially compensate for those changes. In addition, the
actual consumer population lags behind changes in its
equilibrium density. I explore these phenomena using a
slight modification of the classical predator-prey models
explored by Rosenzweig (1971) and May (1972, 1973).
Consumers with population size N have a saturating functional response. Resources have v-logistic growth (Gilpin
and Ayala 1973). Thus, the form of the model is
[ ( )]
v
Here, v measures the extent to which density-dependent
effects are concentrated near the carrying capacity, k. Environmental decline could cause decreases in r, decreases
in v, or decreases in k. However, decreases in v have no
effect on equilibrium population size, and decreases in r
have no effect on the equilibrium population size until r
drops below 0 (at which point the equilibrium N becomes
0). Under a declining r, conditions already guarantee extinction at the time that the population first begins to
decrease. The equilibrium value of N, denoted N ∗, is defined by k in this model, so decreases in k are reflected
directly in N ∗. However, actual changes in N must lag
behind the changes in k, and the lag is more pronounced
when r is small. A small r implies considerable population
inertia. As a simple numerical illustration, assume that k
declines linearly to 0.1k over a 9-yr period. At the end of
that period, N p 0.350k if r p 1; N p 0.834k if r p
0.1, and N p 0.981k if r p 0.01. Whether r, k, or both
decline, population size changes in a sigmoid fashion over
time. The initial period of decline is slow because per
capita growth rate is only slightly negative, and the final
part of the decline is slow because the small population
size implies that the maximum decrease per unit time is
relatively small.
Many models assume that r is the primary factor affected by environmental deterioration; see, for example,
many population viability analysis programs (e.g., VORTEX [Miller and Lacy 1999]) and models of mutational
meltdown (Lande 1994; Lynch et al. 1995). In these models, there is little or no decrease in population size over
most of the period of decline in r. Unfortunately, parameters in simple models like equation (1) are difficult to
map onto observable biological quantities. However, both
of the potential features illustrated by equation (1)—
insensitivity of N ∗ to deterioration and changes in N lagging behind changes in N ∗—are predicted to be pervasive
features of the more realistic consumer-resource models
discussed in the remainder of this article.
dR
R
p rR 1 ⫺
dt
k
(
⫺
cRN
,
1 ⫹ chR
(2a)
)
(2b)
dN
bcR
pN
⫺d,
dt
1 ⫹ chR
where c is a per capita capture rate of resources by an
individual consumer when it is searching. Each resource
item caught takes a time, h, to handle and process, which
prevents search. The efficiency of conversion of resource
into new consumer individuals is given by b, and d is the
per capita death rate. Increasing d or h, or decreasing c
or b, are the only parameter changes that directly decrease
the per capita growth rate of the consumer. (Indirect deterioration via adverse changes in the prey is discussed
briefly in the section “Environmental Deterioration and
Population Size in Other Models.”) A sufficiently large
deterioration in any of these parameters causes extinction
of the consumer because it will no longer be able to meet
its resource requirements for replacement. It is explicit or
implicit in earlier analyses (e.g., Matessi and Gatto 1984;
Abrams 1992; Yodzis and Innis 1992; Case 1999) that equilibrium consumer density can increase with detrimental
changes in each of these parameters. However, some of
these population responses have never been illustrated, and
the transient dynamics of a system undergoing parameter
changes do not appear to have been explored. Table 1
summarizes the conditions under which equilibrium population density increases with detrimental changes in each
parameter. The table also gives the parameter value at
which extinction occurs. The parameters b and d have
qualitatively similar effects, as do h and c.
Figure 1, which assumes logistic resource growth
(v p 1), plots the equilibrium consumer population size
versus consumer death rate, d. Note that N ∗ increases with
a larger per capita death rate over most of the possible
range of death rates when the product chk is large relative
to 1. This product is the ratio of total time spent handling
Environmental Deterioration and Population Size
and processing resources to total search time, both measured when resources are at their carrying capacity. The
maximum equilibrium density exceeds the density at a
near-zero mortality by the factor (1 ⫹ chk)2/(4chk), which
can become arbitrarily large as chk becomes much larger
than 1. Several studies of functional responses in natural
or seminatural conditions have documented chk values on
the order of 10 or more (Abrams et al. 1990; Messier 1994;
Ruesink 1997). Because the equilibrium point is unstable
in systems where the equilibrium density increases with
d, the population will fluctuate around the equilibrium.
The average density can differ considerably from the equilibrium (Abrams and Roth 1994; Abrams et al. 1997).
However, it remains true that the mean predator density
either increases or remains approximately constant over
most of the range of death rates where the equilibrium
value increases (Abrams et al. 1997; P. A. Abrams, unpublished data).
The change in consumer density with consumer death
rate is determined by two relationships. First, higher consumer death rates always increase the equilibrium resource
density; more resources are required to balance the higher
mortality if the consumer is to attain equilibrium. Second,
equilibrium occurs where the number of consumers, N ∗,
is equal to the ratio of the per capita growth rate of the
resource to its per capita risk of death from one consumer.
If this were not true, the resource would not be at equilibrium (R ∗). From equation (2a), this ratio is given by
N∗ p
r [1 ⫺ (R ∗/k)v]
c/(1 ⫹ chR ∗)
.
(3)
Both the numerator and denominator of equation (3) decline with increasing R ∗, and R ∗ increases with increasing
d (see table 1). If the denominator declines faster than the
numerator, the ratio (N ∗) increases with an increased
death rate. When v is on the order of 1 or greater, h is
large and R ∗ is small, the hyperbolic decline of the denominator of N ∗ is larger than the linear decline in the
numerator, and a greater consumer death rate increases
Figure 1: Equilibrium consumer population size N∗ as a function of per
capita mortality relative to the maximum mortality that allows persistence, d/dmax in the system given by equations (2) when v p 1 . The death
rate has been scaled relative to b (equivalent to b p 1 ). The three lines
show N∗ for three values of the product, chk.
N ∗. The hyperbolic decline of the denominator becomes
smaller as R ∗ grows because of a higher consumer death
rate. This results in the eventual decline of the consumer.
When v k 1, the equilibrium and mean consumer population sizes increase with mortality over a larger fraction
of the permissible range of mortality rates because the
numerator of equation (3) is insensitive to R ∗ until R ∗
becomes large. Smaller values of v have the opposite effect;
population size increases with mortality over a smaller
portion of the permissible range. In both cases, the range
of mortality rates over which population size increases with
mortality corresponds to the range of mortality rates for
which there are limit cycles in population densities.
An increase in the equilibrium consumer population
size with an increase in its per capita death rate is a general
feature of all ordinary differential equation models of consumer-resource systems in which consumer population
growth is determined by its intake of a self-reproducing
resource and the system has an unstable equilibrium point
(Rosenzweig 1971). If equations (2) are modified to in-
Table 1: Conditions for increasing population size and extinction as the result of adverse changes in parameter values
in equations (2) when v p 1
Parameter
d
b
c
h
Description
Per capita death rate
Efficiency of converting food to offspring
Resource capture rate while searching
Handling time for one resource item
295
N∗ increases with adverse changes
in parameter when
d ! b(chk ⫺ 1)/[h(chk ⫹1)]
b 1 dh(chk ⫹ 1)/(chk ⫺ 1)
c 1 2d/[k(b ⫺ dh)]
h ! (bck ⫺ 2d)/(dck)
N∗ p 0 when
d p bck/(1 ⫹ chk)
b p d(1 ⫹ chk)/(ck)
c p d/[k(b ⫺ dh)]
h p (bck ⫺ d)/(dck)
Note. Equilibrium point: R∗ p d/c(b ⫺ dh) ; N∗ p {rb[c vkv(b ⫺ dh)v ⫺ d v]}/[c v⫹1kv(b ⫺ dh)v⫹1]. Local stability when d 1 b(chk ⫺ 1)/
[h(chk ⫹ 1)]. The conditions for N∗ to increase with b or d cannot be solved for explicitly, given an arbitrary v. The condition for N∗ to
increase with a smaller c (or larger h) is c 1 [dv(1 ⫹ v)(1/v)]/[k(b ⫺ dh)v].
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The American Naturalist
Figure 2: The population size of a consumer undergoing a linear increase
in its death rate, with dynamics based on equations (2). In A, the initial
death rate is 0.1, and it increases by 0.0001 each time unit; the other
parameters are c p 1 , h p 5 , b p 1, r p 1, k p 1, and v p 1. In B, the
initial death rate is 1, and it increases by 0.05 each time unit; the other
parameters are c p 1, h p 1, b p 10, r p 0.1, k p 1, and v p 1.
clude an Allee effect in the resource population, both population cycles and an increasing N ∗ versus d relationship
can occur when d is low, even when handling time is 0.
Allee effects imply that the per capita growth rate of the
resource (the numerator of eq. [3]) increases with its population when that population is small, and this is what
produces the positive relationship between consumer
death rate and N ∗.
In the preceding examples, an increasing N ∗ versus d
relationship is associated with limit cycles in population
density, but cycles are not a necessary condition for such
a relationship. An increase in N ∗ with d may occur in
stable systems similar to equations (2) when there is consumer interference; that is, the per capita growth rate of
the consumer decreases as its own population increases.
Strong enough interference often leads to an increasing
consumer population with consumer per capita mortality
rate in systems with stable equilibria, especially those with
an Allee effect and a linear functional response (P. A.
Abrams, unpublished manuscript).
The preceding has considered equilibrium densities.
However, when consumer death rate increases steadily
over time, both consumer and resource populations lag
behind their equilibrium densities. The difference between
current density and the equilibrium density for current
conditions increases as the rate of environmental change
increases relative to the maximum rate of population response. Figure 2 illustrates the time course of consumer
population size for two cases in which consumer death
rate increases linearly over time. In figure 2A, at the time
that d exceeds the value where N ∗ p 0 (at t p 667; X on
the figure), the consumer population is still greater than
one-half its original mean size. This magnitude of lag is
common when the consumer population has a relatively
slow numerical response. The consumer population’s lag
is lengthened when its own dynamics are relatively slow
(i.e., the parameters b and d are small). Consumer population decline is also counteracted by a relatively rapid
resource response to altered consumer densities (large r).
Table 2 shows the consumer population size, N, at the
time when N ∗ p 0 for several sets of consumer and resource demographic rate parameters for an example otherwise identical to that in figure 2A. It confirms that slow
consumer dynamics and fast resource dynamics lead to
the largest consumer population size at the time it becomes
doomed to extinction by its increased death rate.
Another possible response to increasing death rates is
shown in figure 2B. Here, the consumer has a relatively
rapid numerical response and the resource has a slow response, which normally reduces both consumer lag and
resource compensation. The population is in fact close to
Table 2: Ratio of current population size to original equilibrium population size at the time when conditions imply eventual extinction
Resource growth
r p 10
rp1
r p 0.1
b p 10;
d(0) p 1
b p 1;
d(0) p 0.1
b p 0.1;
d(0) p 0.01
.267
.263
.2175
.588
.584
.547
.634
.743
.887
Note. This table compares how the relative speeds of consumer and
resource dynamics affect the trajectory of the consumer population under
linearly increasing consumer per capita mortality; b p birth efficiency;
d p death rate. Results are based on equations (2) in which the per
capita death rate, d, increases from an initial value of d(0) by an amount
equal to .001d(0) per year for 1,000 years. It reaches a value (1.667d(0)),
which implies an equilibrium N of 0, at time t p 667. The other parameters of the model are c p 1 , h p 5 , r p 1, and k p 1. The second
row, second column is based on parameters identical to those in figure
2A.
Environmental Deterioration and Population Size
297
extinction at t p 80, when N ∗ p 0. However, the increasing death rate initiates consumer-resource cycles in what
had been a stable system. The cycles hide the underlying
negative trend in N ∗ until close to t p 60, when the death
rate has already grown by approximately 75% of the increase required for ultimate consumer extinction.
Consequences of Reduction in the Consumer’s Capture
Rate for Consumer Population Size
If the consumer’s capture rate parameter, c, decreases,
equilibrium resource density must increase. This change
tends to decrease the denominator of equation (3),
which in turn tends to increase N ∗ . This increase occurs
because a reduced resource capture rate leads to a
greater total resource yield to the consumer when the
resource is overexploited. Figure 3 shows how the consumer’s equilibrium population size changes with c for
a consumer-resource system based on equations (2) with
zero handling time. The maximum consumer population
size occurs when c is only twice the minimum value that
allows the consumer to exist, and this result does not
depend on h. Regardless of the other parameters, equilibrium consumer population size increases as the capture
rate declines over most of the possible range of values of
c. Large values of c result in unstable equilibria when h is
nonzero. Numerical analysis of equations (2) suggests that
the mean consumer population size always increases as c
decreases within the unstable region (Abrams 1997; P. A.
Abrams, unpublished data), so the general form of the
relationship between the average N and c is similar to that
for N ∗ shown in figure 3. The system is always stable at
the point where c maximizes consumer population size
(see table 1).
The second parameter that alters the effective rate at
which a consumer can capture resources is the handling
time, h. As noted in table 1, N ∗ also has a unimodal
response to increasing h. If d is small relative to c or if v
is small relative to 1, the consumer equilibrium population
size increases with h over most of the potential range of
h and then rapidly decreases to 0, in a manner similar to
that shown in figure 3.
The time course of consumer population size under a
linear decline in a capture-related parameter (c or h)
largely mirrors the shape of the equilibrium relationship
when the decline is slow. Figure 4 shows three examples
of the population trajectory assuming an approximately
linear decrease in c from an initial value of 1 to a value
that approaches 0. Comparing figures 3 and 4 shows that
the shape of the equilibrium relationships is approximated
when environmental decline is relatively slow (fig. 4A).
This is not true of the more rapid decline in c assumed
in figure 4B, where N is still close to its maximum size at
Figure 3: The mean consumer population size as a function of its per
capita resource attack rate, c, in an examples based on equations (2) with
zero handling time. The other parameters are k p 1 , r p 1 , b p 1, and
d p 0.05.
t p 95, when c reaches a value where N ∗ p 0. Finally,
figure 4C shows the time course of N under the same rapid
decline in c that is assumed in figure 4B, but with a consumer that has a nonlinear numerical response (per capita
growth rate is b ⫺ d/[cR], rather than bcR ⫺ d). This nonlinear numerical response may provide a better model of
population decline under low food intake (Abrams and
Roth 1994) and has the same equilibrium as the original
model. In this case, the time course of N much more
closely mirrors the equilibrium curve because consumers
die rapidly when food intake drops.
Deteriorating conditions are likely to decrease the consumer’s capture rate of resource (reduced c or increased
h) and to decrease its efficiency of converting resources
to new consumer individuals (decreased b or increased d).
In some cases, these two types of effect will change population size in the same direction, and this is particularly
likely in the initial stages of the environmental decline,
when both changes often act to increase population size.
When the parameters have opposite effects, the net effect
can either increase or decrease N ∗. For example, assume
equations (2) with h p 1, b p 1, k p 1, v p 1, and initially c p 1 and d p 0.1 (which guarantees stability). If
c decreases by a factor x while d increases by (1/x), and x
decreases linearly from an initial value of 1, N ∗ does not
drop below its original value until x has changed by 88%
of the change that guarantees ultimate extinction
(N ∗ p 0). Unless x changes very slowly, the actual value
of N remains above its initial equilibrium density until
after the time at which N ∗ p 0.
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The American Naturalist
Environmental Deterioration and Population
Size in Other Models
This section analyzes a number of alternative models to
determine how several factors alter the above results: behavioral adaptation by the consumer, resources that do
not reproduce themselves, the presence of multiple resources, the presence of competing consumers, and focal
species that are resources rather than consumers. This set
of alternative models shows that the phenomena revealed
by simple one-consumer/one-resource systems are by no
means restricted to such systems.
Models with Behavioral Adaptation of Foraging
A growing body of literature suggests that adaptive behavior is a key component of food webs (e.g., Werner and
Anholt 1993; Abrams 1995; Peacor and Werner 1997;
Houston and McNamara 1999). Adaptive behavior tends
to couple the values of traits that are distinct in models
lacking behavioral flexibility. If there is an adaptive tradeoff between foraging and mortality, increasing mortality
factors will also usually decrease per capita consumption
rates. Because declining consumption rates are particularly
likely to increase population size, consumers may therefore
increase in numbers in response to adverse changes in
mortality risks. For example, if equations (2) are changed
so that the functional response is given by a model of
adaptive foraging presented in Abrams (1982, eqq. [5]),
consumer population density may increase with deterioration in three of the four parameters that affect consumer
population growth (appendix of Abrams 1984). For two
of these three parameters, increases in population size occur when the system has a stable equilibrium.
Adaptive investment of defensive effort by the prey may
also lead to increases in the predator’s density as its own
mortality rate increases. The mechanism decreases in optimal prey defense with increasing predator mortality. This
is demonstrated by Abrams and Matsuda (1997, p. 1744,
eq. [3a]).
Figure 4: Three examples of the time course of decline in a consumer
population when there is a linear decline in the consumption-rate constant, c. All of the figures are based on equations (2) with parameters
k p 1, r p 1, b p 1, d p 0.05, and h p 0. In all cases, consumer and
resource populations are initially at their equilibrium values. The rate of
decline in c is given by ⫺dexp(⫺g/c), where d is 1 divided by the period
of decline, and g p .0001. The exponential function has the effect of
drastically slowing the decline in c when it approaches 0, which prevents
negative values. For most of the range of c, its rate of decline is approximately constant. In A, the parameter c declines from 1 to almost 0
over the course of 1,000 time units; in B and C, the decline occurs over
100 time units. In C, the consumer has a nonlinear numerical response,
given in the text. In B and C, c falls below the value for which N∗ p
0 at t p 95.
Models in Which Resource Growth Is
Not via Self-Reproduction
All of the models considered above have assumed that the
resources are biotic entities that reproduce themselves. The
class of abiotic (non-self-reproducing) resources provides
less scope for increased resource population growth to
compensate for greater consumer mortality rates, decreased consumption rates, or other adverse changes. This
case can be modeled by equations (2) with the v-logistic
resource growth term being replaced by a function of the
following form: dR/dt p b ⫺ g(R), where b is a resource
Environmental Deterioration and Population Size
input rate and g is a monotonically increasing function
describing resource loss from the system. The per capita
resource growth rate is then [b ⫺ g(R)]/R, which typically
decreases rapidly as R increases. Because the per capita
resource growth rate is the numerator of the expression
for N ∗ (eq. [3]), and deterioration of the consumer’s environment increases R ∗, this means that the consumption
rate per resource (the denominator of eq. [3]) must decrease very rapidly with R ∗ for N ∗ to increase as conditions
deteriorate. It can be shown that a decrease in c or b or
an increase in d or h in equation (2a) cannot increase the
equilibrium population size of the consumer under the
framework of equations (2). However, under some forms
of adaptive change in c, it is possible for population size
to increase as c decreases in models with abiotic resources
(Abrams 1989; Richards and Wilson 2000).
Models with Two or More Resource Species
Relatively few consumers are complete specialists. The basic consumer-resource model with a type 2 functional response (eqq. [2]) can be modified by adding a second
resource, potentially having a different vulnerability to the
consumer. The appendix presents the two-resource model
considered here and in the next section. The resource having the lower vulnerability and/or a larger maximum per
capita growth rate can exclude the more vulnerable or
slower-growing resource via apparent competition, when
the consumer is efficient (Holt 1977). Figure 5 provides
an example of the effects of two resources having different
vulnerabilities for the relationship between consumer
death rate and mean consumer population size. Increasing
consumer death rate can allow coexistence of the more
vulnerable resource species, which is excluded by apparent
competition at low consumer death rates. This, and the
fact that there are two transitions between stability and
limit cycles as d is increased, leads to a multimodal relationship between death rate and mean consumer population. More complicated shapes are possible with more
resources. In all of these cases, the quality (as well as the
quantity) of resources increases as the consumer death rate
is increased. This occurs because the more vulnerable resources tend to be excluded or overexploited under low
consumer mortality or high consumer exploitation rates.
However, the decline to zero-equilibrium consumer population size occurs just as abruptly with increasing death
rate as in comparable systems with one resource. In multiresource systems, the changes in equilibrium and mean
population size are also often multimodal when deterioration affects consumption-rate constants rather than
mortality rates.
If the two-resource species themselves compete, the consumer population increases with a higher consumer death
299
Figure 5: The average consumer population size as a function of its per
capita death rate when the consumer uses two resources differing in
vulnerabilities but requiring equal handling times. Results are based on
the one-consumer version of the equations in the appendix, with a small
immigration rate (0.0001) added to each resource to prevent very low
densities. The dynamics are cyclic for most values of d but stable for a
small range of d near d p 0.3 and stable at the upper end of the range,
where the mean population size decreases. The parameter values are
r p 1, k p 1, b p 1, c1 p 4, c2 p 1, h1 p h2 p 2.
rate whenever the two midlevel resources are able to coexist and there is a finite handling time (eq. [A-1] in
Abrams 1999). Here, higher mortality of the consumer
means that the more vulnerable resource increases at the
expense of the less vulnerable resource, which is what allows the consumer population to increase. This increase
occurs in stable as well as unstable systems. Proportional
decreases in both its consumption rates also increase the
consumer’s population size for the same reason. These
changes are examples of compensatory increases in resource quality. When resources compete, however, a large
enough increase in consumer mortality or decrease in consumption will often result in elimination of the less vulnerable resource (Abrams 1999). After this, further worsening of conditions for the consumer will begin to decrease
its population size.
Models in Which the Consumer Suffers Competition
In the previous models, a consumer could make use of
resources that increased when its own mortality rate increased. Interspecific competitors could potentially prevent such compensation by utilizing the extra resources.
Because competitive coexistence at stable population sizes
requires two or more resources, it is most useful to examine this question in the context of a model with two
consumers and two or more resources. The appendix outlines the model and its analysis. Here, I will reexamine
two cases in which initial insensitivity to environmental
decline is followed by a rapid population collapse in the
single-resource models examined above. The first is a de-
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The American Naturalist
cline in the two consumption rate parameters, ci1, and ci2,
of focal consumer species i (where, for simplicity, I will
assume no handling time). The second is an increase in
di when (ci1 ⫹ ci2)hk is large and the system is initially
unstable. Figure 6 shows the change in equilibrium population size with declining c11 and c12 in a model with two
consumer species whose relative consumption rates of two
resources are mirror images of each other (see appendix).
Each line within a panel represents a different degree of
similarity in relative consumption rates of the two consumer species. The two panels represent systems with
moderate (A) or low (B) resource requirements. Each
panel contains examples in which N ∗ of the focal species
increases as its consumption-rate constants decrease. However, strong enough competition (i.e., sufficient similarity
in resource use) prevents this increase. The presence of a
sufficiently similar competitor will always prevent compensatory increases in the equilibrium density of a focal
species as its conditions decline. However, this is not the
case when overlap is moderate, which is more likely to be
the case among coexisting species. Figure 6 shows that
increasing population size with decreasing c is most likely
when consumers are efficient and there is a significant
difference between consumer species in their relative capture rates of different resources. Another phenomenon that
occurs when consumers are efficient is a discontinuous
drop in equilibrium population from a finite size to 0; this
is indicated by the lines in figure 6B that terminate before
reaching 0. The actual time course of change in population
size in these cases shows a rapid, but not discontinuous,
decrease.
Environmental deterioration may also affect the per
capita death rate of one of two competing consumer
species (see appendix). Numerical analysis of this situation leads to patterns similar to those shown in figure
6. If the consumers are very efficient, the equilibrium
Figure 6: Population sizes of a consumer in a two-consumer/two-resource system (eqq. [A1]) as a function of reduction in the focal consumer’s
resource consumption rates. The X-axis gives the proportional reduction in each of the two consumption rates. A and B assume initial consumer
efficiencies (R∗/k) of 0.2 and 0.05, respectively. Each line in a panel corresponds to a different degree of resource overlap, defined as 2(1 ⫺ q) , where
q p c11/(c11 ⫹ c12) p c22/(c21 ⫹ c22). Overlap is given to the left of each line.
Environmental Deterioration and Population Size
population size increases with increasing death rate of a
focal consumer, followed by a discontinuous jump to a
zero equilibrium. As in the single-resource models, increases in population size with an increasing death rate
are associated with a relatively large handling time
(chk k 1), as shown in the appendix. A very similar competitor will prevent any increase in N ∗ in response to
increased mortality.
Models in Which the Focal Species Is a Resource (Prey)
It has long been known (e.g., Volterra [1926] 1931) that
the equilibrium resource population in a consumerresource system is independent of its own death rate, provided that the only variable affecting the consumer’s
growth rate is resource population size. With a sufficiently
high death rate of the resource, the consumer will go extinct, and, thereafter, the resource population declines as
its death rate continues to increase. If the consumer is
efficient (can subsist on a small amount of resource), then
the range of resource death rates over which its equilibrium population is constant is much larger than the range
over which its population declines. If dR is the resource
death rate and dN is the consumer death rate, the fraction
of the possible increase in dR over which R ∗ remains constant is
v
{ [
]}
dN
r1⫺
kc(b ⫺ d Nh)
r ⫺ dR
⫺ dR
.
(4)
This means that if {d N/[kc(b ⫺ d Nh)]}v is small relative to
1, the equilibrium R will remain constant over most of
the potential range of added mortality, dR, that will allow
it to persist. Furthermore, R ∗ will reach 0 after a small
additional increase in mortality, once the consumer is absent. An efficient consumer, high resource carrying capacity, or resource density dependence concentrated near
its carrying capacity (v k 1) will produce this pattern of
decline. It should be noted that, if the consumer is a generalist, the focal resource declines in a less abrupt fashion
with decreases in its own growth parameters. Direct effects
of consumer population size on its own population growth
also produce more gradual declines in resource density
with increases in its own mortality. Finally, consumer population tends to remain higher than its equilibrium value
as resource death rates increase; the resource will therefore
often decline slightly during the period when its equilibrium remains constant.
301
Models in Which Environmental Decline
Affects Two or More Species
There are too many possible combinations of parameters
to treat this topic adequately here. However, results from
the simple consumer-resource model of table 1 can be
used to illustrate the range of possibilities. On the one
hand, declines in the resource’s r or k reduce the consumer’s equilibrium population size, either offsetting potential increases or enhancing decreases caused by adverse
changes in predator parameters. On the other hand, adverse changes in the predator’s parameters lead to larger
resource populations, which offsets potential declines in
R ∗ because of a direct decline in resource growth parameters. The capture rate C may either increase or decrease
if both predator and prey are directly and adversely affected
by environmental change. Increases in C would decrease
R ∗ but could increase or decrease N ∗. Clearly, many more
possibilities exist in larger food webs.
Discussion
The above analysis has shown that density dependence
and interspecific interactions can compensate for environmental deterioration. As a result, the equilibrium, average, or actual population size of a species may change
very little or even increase during the initial period of
environmental deterioration. However, this compensation
is generally a nonlinear process, and a sufficient decline
in environmental quality will eventually lead to a rapid
decline in population size, often ending in extinction. This
pattern is certainly not universal, but it is common enough
in simple models to argue against the general use of population size as an adequate measure of population health.
Nonlinearity of compensatory processes may underlie
some of the many examples of fish stocks that have
abruptly collapsed from what were estimated to be high
population sizes (Hilborn and Walters 1992; Quinn and
DeRiso 1998).
The second theme of the preceding analysis has been
that changes in population density lag behind changes in
environmental conditions, and these lags can interact with
delayed compensatory factors in a complex manner. Thus,
it is often difficult to detect the deteriorating conditions
from the record of population density over time until conditions are close to or past those that will guarantee ultimate extinction. In fact, most of the populations illustrated in the figures of this article would not have been
classified as endangered under current IUCN criteria
(IUCN 2001) until after their equilibrium population size
had already dropped to 0.
Neither increasing populations nor significant lags are
always expected when there is environmental deteriora-
302
The American Naturalist
tion. For example, abiotic resources tend to lead to significant reductions in equilibrium population size with
deleterious changes in any parameter affecting consumer
growth directly. Lags in the population response may be
relatively insignificant when the turnover rate of the population is sufficiently high. Cycling consumer species or
stable but efficient consumers of self-reproducing resources seem especially likely to undergo counterintuitive
changes in population size. Surveys by Ellner and Turchin
(1995) and by Kendall et al. (1998) suggest that consumerresource cycles are reasonably common. As figure 3 illustrates, increases in consumer population size with decreasing capture rate characterize most of the potential range
of capture rates. Coexistence of competing species of vulnerable and well-defended resources is especially likely to
lead to consumer populations increasing as their own per
capita death rate increases, and this type of food web seems
to be common in aquatic systems (Leibold 1996). Unfortunately, we lack the survey data on demographic rates
and functional responses that would be required to quantify the likelihood of the counterintuitive population trajectories described here.
Even if the phenomena discussed here occur in a minority of species experiencing declining environments, the
results should lead to a reevaluation of the status of some
species whose current populations are relatively large. Current global changes in both physical and biological conditions in many cases cannot be reversed until many years
after the source of the deterioration is removed or altered.
Human-caused environmental declines will often occur at
an accelerating rather than linear rate, contrary to the
numerical results presented here. If so, the period of population decline will be compressed in time. Nevertheless,
the conservation implications of these results are not entirely pessimistic. When environmental decline is slow,
population monitoring may be able to detect the population decline soon enough to reverse the underlying environmental decline. Even if environmental decline is
rapid, there may be persistence of a small remnant population for a considerable time beyond when the equilibrium size becomes 0. If conditions can be reversed during
this period, the species can be restored from these survivors. The models predicting a population crash with deterioration also suggest that it may often be possible to
restore species to close to their original abundances with
only a partial reversal of the negative environmental trend.
In addition, it is important to note that the spatially homogeneous models considered here are unlikely to describe the entire geographic range of any moderately widespread species. In such species, population declines in one
part of the range are likely to warn of conditions that may
be deteriorating at slower rates in other parts of the range.
Most models in resource management and in conser-
vation biology are based on descriptions of the dynamics
of a single species. This is reflected in textbooks on fisheries
management (e.g., Hilborn and Walters 1992; Quinn and
DeRiso 1998), which are almost entirely based on singlespecies models (but see Walters et al. 1997, 2000). The
focus on single species is also reflected in many computer
programs to perform population viability analysis for conservation purposes (e.g., VORTEX; Miller and Lacy 1999).
This focus has remained true in spite of many previous
calls for considering interacting species or other food-web
components when making conservation or management
decisions (e.g., Pimm 1991). The current article has shown
that, in continuous models with two or more species, deteriorating conditions affecting a focal species often increase its population size, unlike continuous-time singlespecies models. In addition, food web models suggest that
patterns of rapid declines in population size at the end of
a long steady period of environmental decline are much
more common than might have been thought on the basis
of single-species models. This discrepancy is largely a consequence of the inability to fully mirror compensatory
food web responses in single-species models.
The two major messages of this work are not entirely
novel. For example, van Horne (1983) argued against using
population size as an indicator of environmental quality.
However, her work stressed the possibility of large numbers of subdominant animals being forced into suboptimal
habitats, while this work is concerned with entire populations. Ives and Gilchrist (1993) presented a general
framework for calculating the indirect effects of density
dependence and other community members in predicting
the consequences of climate change for population densities. There has also been some discussion of lags in the
response of the population to a change in the environment.
As early as 1967, MacArthur and Wilson (1967) pointed
out that recently fragmented populations might persist
long after colonization rate dropped below extinction rate,
ensuring extinction of the entire set of populations. This
idea was termed an “extinction debt” by Tilman et al.
(1994). Recent examples include Brooks et al. (1997) and
Cowlishaw (1999). Ironically, metapopulations that are decreasing because of habitat loss may be more likely to
provide adequate warning of impending extinction than
are homogeneous populations that are experiencing declining conditions. Metapopulations or recently fragmented populations will generally decline at least as rapidly
as inhabited areas are lost, and the process of habitat loss
is seldom invisible. Species whose demographic parameters
are declining may themselves increase, and the parameters
are seldom measured. Extinction debts are not confined
to metapopulations; all populations experiencing changed
conditions exhibit some lag in their population-level response. Other theoretical examples of such lags are pro-
Environmental Deterioration and Population Size
vided by Doak (1994), Cantrell et al. (1998), and Fagan
et al. (1999). Byers and Goldwasser (2001) provide an
empirical example where low demographic rates of a native
snail species hid its eventual exclusion of an invasive exotic
species for over 25 yr.
Both theoretical and empirical work are required to better understand the conditions under which environmental
deterioration will fail to be reflected in significant declines
in population size. This work has considered only differential equation models of homogeneous populations
within communities having few species. It has also assumed no stochastic variation in the environmental parameters and has largely ignored the question of how negative changes in the environmental parameters affecting
the per capita growth rate of one species affect the population size of other species that depend on that one. All
of these theoretical gaps need to be addressed. Empirically,
we clearly need more information on such topics as the
form of functional responses and density dependence, as
well as more quantitative descriptions of interspecific interactions in communities.
303
c 22 p q; c 21 p 1 ⫺ q, with q 1 0.5; and before any environmental deterioration occurs, c 11 p c 22, c 12 p c 21, and
d 1 p d 2. These simplifications allow equations (A1) to be
solved for equilibrium population sizes. The necessary
condition for a discontinuous drop in N1∗ to 0 as d1 is
raised is that
d2 !
1 ⫺ 3q ⫹ 2q 2
.
h ⫺ q ⫺ 3hq ⫹ 2hq 2
(A2)
If this condition is satisfied, then extinction of both N1
and R2 occur when
d1 p
d 2q
.
1 ⫺ d 2h ⫺ q ⫹ 2d 2hq
(A3)
A small increase in d1 at the initial symmetric equilibrium
increases the equilibrium population size of consumer 1
if
d1 p d2 ≤
⫺1 ⫹ h ⫹ 2q ⫺ 4hq ⫺ 2q 2 ⫹ 4hq 2
.
h(1 ⫹ h)(2q ⫺ 1)2
(A4)
Acknowledgments
This article was supported by an operating grant from
National Sciences and Engineering Research Council of
Canada to the author. T. Day, K. Schertzer, W. W. Murdoch, D. Doak, C. Brassil, and several anonymous reviewers provided useful comments on one or more earlier
drafts.
APPENDIX
Consumer-Resource Models with Two Resource Species
The two-consumer/two-resource extension of equations
(2) in the text is
[ ( )] 冘
vj
dR j
R
p rj R j 1 ⫺ j
dt
kj
dNi
p Ni
dt
(冘
jp1, 2
1⫹
⫺
ip1, 2
1⫹
cij R j Ni
冘 cij hij R j ,
jp1, 2
)
bicij R j
冘 cij hij R j ⫺ d i .
jp1, 2
(A1)
Figures 5 and 6 are based on this system. The oneconsumer/two-resource system (fig. 5) is given by these
equations with only a single consumer species, i. The calculations for the two-consumer case below make the simplifying assumptions that r1 p r2 p 1; k 1 p k 2 p 1;
v1 p v2 p 1; b1 p b2 p 1; hij is identical for all i, j;
The basic implication of all of these results on death rates
is that efficient consumers (that produce a low equilibrium
R/k) are most likely to exhibit both initial increases in
population size with increasing death rate and abrupt extinction as their own per capita death rate increases when
they undergo competition. In addition, high overlap in
resource use (q close to 0.5) makes both of these phenomena less likely.
The effects of declines in the consumption rates of one
consumer on its population size can be analyzed in a similar fashion, but the algebra becomes much more complicated. Adopting the same parameter simplifications as
above and the assumption that h p 0 allows some reasonably simple results; these are the basis of figure 6. Here
we reduce c11 and c12 by a factor f, while leaving d 1 p
d 2. This causes an abrupt extinction of species 1 when
f p (1 ⫺ q)/q, provided that d ! (3q ⫺ 1 ⫺ 2q 2)/q. Figure
6A does not show any cases of abrupt extinction because
d is too large to meet this criterion for the values of q
illustrated. The condition required for the population size
of consumer 1 to increase when the factor f initially starts
to decrease from f p 1 is
d!
(2q ⫺ 1)2
.
2(1 ⫺ 3q ⫹ 3q 2)
(A5)
When q is very close to 1/2, d must be extremely small to
satisfy this condition, which is why the lines for q p
0.53 (overlap p 0.94) in both panels in figure 6 decline
monotonically.
304
The American Naturalist
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Associate Editor: Daniel F. Doak