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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. 294 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]. 296 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. 298 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- 300 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! 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