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Extinction Thresholds: Insights from Simple Models

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Extinction Thresholds: Insights from Simple Models Ann. Zool. Fennici 40: 99–114 ISSN 0003-455X Helsinki 30 April 2003 © Finnish Zoological and Botanical Publishing Board 2003 Extinction thresholds: insights from simple models Jordi Bascompte Estación Biológica de Doñana, CSIC, Apdo. 1056, E-41080 Sevilla, Spain Received 12 Feb. 2003, revised version received 4 Mar. 2003, accepted 5 Mar. 2003 Bascompte, J. 2003: Extinction thresholds: insights from simple models. — Ann. Zool. Fennici 40: 99–114. There are two types of deterministic extinction thresholds: demographic thresholds such as the Allee effect, and parametric thresholds such as a critical effective coloniza- tion rate or a minimum amount of available habitat for metapopulation persistence. I introduce briefl y both types of thresholds. First, I discuss the Allee effect in the context of eradication strategies of alien species. Then, I consider an example of parametric threshold: the critical amount of suitable habitat below which a metapopulation goes deterministically extinct. I review how this spatial threshold changes in relation to the level of spatial detail and the complexity of the food web. Since classical metapopula- tion models assume an infi nite number of patches, I proceed by considering how the extinction threshold is affected by environmental variability acting on a small number of patches. Finally, I consider recent work suggesting that if the network of connectiv- ity among patches is not random but highly heterogeneous, the extinction threshold may disappear. “We donʼt record fl owers,” the geographer said. “Why not? Itʼs the prettiest thing!” “Because fl owers are ephemeral.” “What does ephemeral mean?” “It means, ‘which is threatened by imminent disappearance.ʼ” (A. de Saint-Exupéry 1943, “The Little Prince.”) Introduction invasions, climate change, pollution, and other types of environmental disturbance can only be Extinction has played a major role in the organi- guessed. zation of life on Earth. Almost all species which We face the big challenge of predicting under have existed at some point have already gone what circumstances extinction will occur. Cur- extinct. This unavoidable outcome, however, rently, there are two main schools of thought has been aggravated in the last few centuries when dealing with extinction. One deals with due to human activity. Since 1600, the extinc- stochastic processes, assumes that extinction is tion of more than 485 animal and 585 plant spe- unavoidable, and predicts the time to extinction. cies has been recorded (Lawton & May 1995), The other is deterministic and estimates the con- although the real number of species going ditions beyond which a population goes extinct. extinct due to habitat fragmentation, biological It predicts extinction thresholds, that is, critical 100 Bascompte • ANN. ZOOL. FENNICI Vol. 40 values of some variable of interest beyond which c is the threshold population size below which a population can no longer persist. F(n) < 0. In this paper I will review very simple In the absence of the Allee effect, F(n) is deterministic models which predict extinction always positive, as shown in Fig. 1. For this thresholds. In particular, I will focus on demo- situation there is a single solution, n* = k, which graphic thresholds, that is, critical population is stable. On the other hand, Eq. 1 has two solu- values below which the population goes extinct, tions, n* = k which is again stable, and n* = c, and thresholds derived from the reduction in the which is unstable (population either increases or fraction of available patches in a metapopulation decreases towards extinction, Fig. 1). context. These models are an oversimplifi cation A model like Eq. 1 may seem an oversimpli- of reality, but they can provide rules of thumb to fi cation of reality, multiple factors are missing. better understand extinction. I will explore how However, it defi nes a threshold in very simple these thresholds change with the level of spatial terms. One can plot the change in population resolution, the complexity of the food web, and size for different values of n in real data, and the effect of environmental variability acting on detect whether growth rate becomes negative at small networks of patches. One goal is to explore some critical density. One does not need further whether these simple rules of thumb can be used biological details. Also, although the previous regardless of the modelsʼ simplifying assump- model is deterministic, the Allee effect and tions. Finally, I will consider how the shape demographic stochasticity may interact to drive of the connectivity distribution of nodes (e.g., a population extinct (Lande 1998, Dennis 2002). patches) may affect the extinction threshold and Furthermore, demographic stochasticity by even make it disappear. itself can cause a population to decrease, which would constitute a type of stochastic Allee effect (Lande 1998). Demographic thresholds: the The importance of the Allee dynamics in Allee effect extinction has been widely explored in the context of conservation biology (Lande 1988, The Allee effect is a mechanism fi rst described Groom 1998). Endangered species tend to have by Allee et al. (1949) and describes any process small densities, and under this situation, they in which any component of fi tness is correlated may go extinct due to the Allee effect, demo- with population size (Courcham et al. 1999, graphic stochasticity, or a combination of both. Stephens & Sutherland 1999, Stephens et al. This interaction between stochastic processes 1999). In a population dynamics context, the and the Allee effect is also important in determin- Allee effect depicts a situation in which popula- ing the likelihood of successful establishment by tion growth rate decreases below some critical alien species (Hacou & Iwasa 1996, Fagan et al. minimum density. In some circumstances this 2002, Petrovskii et al. 2002), and the extent of growth rate may even be negative, originating the range expansion during invasions (Lewis & an extinction threshold. Different mechanisms Kareiva 1993, Keitt et al. 2001). responsible for this demographic threshold are However, almost no study has incorporated failure to locate males, failure to satiate preda- the Allee effect in the context of biological con- tors, and inbreeding depression. trol. Large efforts to eradicate alien animal and The Allee effect can be incorporated into plant species have increased with the accelera- a logistic growth model in the following way tion in the arrival of alien species due to global (Dennis 1989, Amarasekare 1998, Keitt et al. travel and commerce (Simberloff 2001). The 2001): introduction of alien species is now believed to be one of the major causes of biodiversity loss (1) worldwide. It is usually accepted in studies of pest eradication, that eradication can only be where n is population size, g is the intrinsic rate achieved by the elimination of all the individuals. of natural increase, k is the carrying capacity, and Liebhold and Bascompte (2003) have studied the ANN. ZOOL. FENNICI Vol. 40 • Models of extinction thresholds 101 0.6 1.0 0.1 ) n 0.8 ( 0.4 extinction F 0.3 0.6 0.2 c 0.5 0.4 0 Per capita growth rate, 0.7 0.2 –0.2 establishment 0 200 400 600 800 1000 0.0 Number of individuals, n 020406080100 120 140 160 180 200 x0 Fig. 1. Per capita growth rate as a function of popula- tion size according to Eq. 1. Continuous line depicts a Fig. 2. The simulated probabilities of gypsy moth estab- situation with no Allee effect (c = 0), and broken line lishment are plotted as a function of the population level at the time of treatment (x ), and eradication rate corresponds to a case with an Allee effect (c = 100, 0 c indicated by the solid dot). In the fi rst case, there is a (fraction of individuals killed, ). Based on Liebhold and single solution n* = k (empty dot), which is stable. In Bascompte (2003). the last case, there is an additional, unstable solution indicated by the solid dot. Population sizes larger than c have a positive per capita growth rate and will grow (3) until reaching their carrying capacity k. Population sizes below c have a negative per capita growth rate, and so, will go extinct (indicated by the arrows). Other param- In this way, at the very low densities observed eters are: g = 2, k = 1000. at the initial stages of a biological invasion, the Allee effect can be represented simply by ln(rt) g as a linear function of nt, with intercept –c /k combined role of stochasticity and Allee effect and slope g /k (Liebhold & Bascompte 2003). on the extinction of isolated populations follow- By fi tting the data on gypsy moth populations to ing eradication treatments. As a case study, they Eq. 3, Liebhold and Bascompte (2003) estimated used historical data on the dynamics of isolated the Allee extinction threshold as the negative populations of gypsy moth (Lymantria dispar) intercept (–cg /k) divided by the slope (g /k), in North America. This insect species, a native which yield c ~ 107 males trapped per colony. to most of the temperate forest in Eurasia, was Thus, when the number of males is lower than accidentally introduced in the east coast of the this threshold, the population will go determin- USA in 1869. Since then, it has been expand- istically extinct. Note, however, that the Allee ing through northeastern USA and southeastern effect interacts with environmental stochasticity, Canada, where it has defoliated large forest areas which would transform the extinction threshold (Liebhold et al. 1992). into a probability function (Lande 1998, Dennis Liebhold and Bascompte (2003) used tempo- 2002, Liebhold & Bascompte 2003). Thus, Lie- ral data on the total number of males of gypsy bhold and Bascompte (2003) modifi ed Eq. 3 to moth caught in pheromone traps to fi t parameters add an additive environmental noise term, which of a discrete time version of Eq.
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