Journal of Applied Blackwell Publishing Ltd. Ecology 2003 The effect of density-dependent catastrophes on population 40, 859–871 persistence time CHRIS WILCOX* and BRET ELDERD† The Ecology Centre, University of Queensland, St. Lucia, QLD 4068, Australia, and †Center for Integrating Statistical and Environmental Science, Henry Hinds Laboratory for Geophysical Sciences, 5734 South Ellis Avenue, Chicago, IL 60637 Summary 1. There has been increasing recognition that catastrophes are an important factor in modelling threatened populations. However, density dependence has generally been omitted from models of threatened populations on the assumption that this omission yields conservative predictions. We explore the significance of including density- dependent catastrophes in models of threatened populations. 2. Using an analytical model, we show that density-dependent catastrophes have a sig- nificant effect on population persistence, decreasing mean persistence time at large population sizes and causing a relative increase at intermediate sizes. 3. We illustrate our results with empirical data from a disease outbreak in crabeater seals Lobodon carcinophagus and show that intermediate population sizes have the longest predicted persistence times. 4. The pattern we found is qualitatively different from previous results on persistence time based on density-independent models, in which persistence time increases with population size to an asymptote. 5. Synthesis and applications. This study has important implications for the conserva- tion of species that may experience density-dependent catastrophes, such as disease out- breaks or starvation. Our results indicate that small and intermediate sized populations may contribute disproportionately to species persistence. Thus populations that have been dismissed as ‘marginal’ may actually be important for conservation. In addition, culling may increase the persistence of populations that experience density-dependent catastrophes. Key-words: birth-death Markov process, disease, extinction, PVA, starvation. Journal of Applied Ecology (2003) 40, 859–871 lation size, reaching exceedingly long periods once a Introduction population reached a relatively small size (Shaffer 1987). Predicting persistence times and extinction rates for These results stimulated interest in the effect of infre- threatened populations is a central focus in conserva- quent, large increases in death rates, or ‘catastrophes’, tion biology and has been one of the main tools for on persistence times (Ewens et al. 1987). Mathematical making important species and habitat management work on similar problems concerning extinction decisions (Boyce 1992; Burgman, Ferson & Akcakaya and loss of alleles due to extreme mortality events or 1993; Beissinger & Westphal 1998; Reed et al. 2002). emigration was also in progress, although it was not gen- Early studies used simple exponential growth models erally applied directly to endangered species (Brockwell, with variation in the birth or death rates due to varying Gani & Resnick 1982; Brockwell 1985; Ewens 1989). environmental conditions (Goodman 1987; MacArthur Catastrophes can cause large decreases in mean persist- & Wilson 1967). These initial investigations predicted ence time, which can be described as a power function that persistence time increased very rapidly with popu- of the population ceiling (Lande 1993). Depending on the relative size of the catastrophes and the popula- tion growth rate, persistence times can either increase © 2003 British Correspondence: Chris Wilcox. Tel.: +61 7 3365 1686. Fax: at an increasing or a decreasing rate with the popula- Ecological Society +61 7 3365 1655. E-mail: [email protected]. tion ceiling (Lande 1993; Mangel & Tier 1993). These 860 results have led to widespread recommendations to examined modelled catastrophes in this way and even C. Wilcox & increase the population sizes deemed necessary to this analysis did not examine its influence directly B. Elderd maximize persistence (Mace & Lande 1991; Mangel & (Table 1). In fact, several analyses modelled factors Tier 1993; Meffe & Carroll 1994). that are likely to be density-dependent, such as disease However, these general results are based on a simple mortality, as density-independent (e.g. Marmontel treatment of catastrophes. In fact, a lack of attention et al. 1997). This contradicts empirical evidence that to the effects of catastrophic mortality has been cited both disease transmission and mortality are density- as one of the primary weakness of models of declin- dependent in some wildlife populations (Hochachka & ing species (Coulson et al. 2001). Catastrophes are Dhondt 2000). generally included in these models as either extreme In light of their prevalence in natural systems and the variation in vital rates, driven by environmental stocha- potential effect on extinction risk, it is important to sticity, or as mortality events occurring at random examine the impact that density-dependent catastrophes intervals that remove a large portion of the population have on persistence time. Understanding the role that (e.g. Lande 1993; Doak, Kareiva & Klepetka 1994; density-dependent catastrophes may play is especially Marmontel, Humphrey & Oshea 1997). Empirical evid- pressing, given that they may qualitatively change the ence suggests that these representations are too simple. results of models that are often used to make decisions Catastrophes are often more complex, and in particular among management alternatives. may be caused by factors that have density-dependent Catastrophes have two general characteristics: prob- feedbacks. The two factors identified by Young (1994) ability and intensity. The probability of a catastrophe is as the most common causes of massive die-offs, or the chance that it will occur in an interval of time, for catastrophes, in his review of large mammal die-offs instance the chance of an extreme drought or a pest were disease and starvation, both of which are likely to outbreak in a given year. Independently, one can also be density-dependent. Inclusion of density dependence in characterize the intensity of the catastrophe, for exam- catastrophic mortality may radically change the extinc- ple how much does the 50-year drought increase the per tion time predictions made by population models. capita death rate over the average rate. Each of the two Density dependence has often been ignored in models parameters, probability of occurrence and intensity, of extinction risk (e.g. Inchausti & Weimerskirch 2001), can either be density-dependent or density-independent. based on some early modelling results demonstrat- This gives four possible ways for catastrophes to enter ing that this was a conservative assumption relat- into population dynamics. ive to extinction risk (Ginzburg, Ferson & Akcakaya It is not difficult to imagine real examples of each 1990). However, recent results using a very well under- of these types of feedback in natural systems. For stood population of feral sheep have shown that extinc- instance, coastal dune systems in the south-eastern tion risk is quite sensitive to density dependence, at USA are heavily impacted by hurricanes, and small least for density dependence in the population growth mammal populations in this habitat decline precipi- rate (Chapman et al. 2001). Thus, it is unclear whether tously the year after hurricanes due to starvation ignoring density dependence is a conservative assump- (Swilling et al. 1998). While the probability of a catas- tion, and it probably depends on the particular system trophe, i.e. a hurricane, in this case is density-independent, in question. To determine if density-dependent cata- the resulting population declines due to starvation are strophes were included in population models when probably density-dependent phenomena. appropriate, we searched for papers on the viability of Disease outbreaks may provide an example of species that had potentially density-dependent cata- catastrophic events that are density-dependent in both strophes in Young’s (1994) review. While the importance their probability and intensity. For instance, the rate of of including density dependence may seem an obvious spread of Mycoplasma gallisepticum, a poultry pathogen, point, only one of the population viability analyses we among house finch Carpodacus mexicanus populations Table 1. Catastrophes in population models for species where there is evidence for density dependence based on Young (1994) Density Evidence Included dependent for density Authors Species Population model catastrophes Source in model dependence Kokko, Lindstrom & Ranta (1997) Seals Difference equation, No Not applicable Not applicable Yes matrix Song (1996) Deer Matrix Yes Drought (starvation) No Yes Swart, Lawes & Perrin (1993) Monkey Differential equations Yes Not specified No Yes Caughley & Gunn (1993) Kangaroo Difference equations Yes Drought (starvation) No Yes © 2003 British Pascual, Kareiva & Hilborn (1997) Wildebeest Matrix, nonlinear Yes Drought (starvation) Yes Yes Ecological Society, difference equations Journal of Applied Pascual & Hilborn (1995) Wildebeest Difference equation Yes Drought (starvation) Yes Yes EcologyArmbruster, 40, & Lande (1993) Elephant Leslie matrix Yes Drought (starvation) No Yes 859–871 861 throughout the eastern USA depends on density, with dynamics, including density-dependence (Mangel & Effect of density- larger populations experiencing the epizootic earlier Tier 1993). In most cases the matrix inversion will have to