Journal of Animal Ecology 2013, 82, 1117–1119
doi: 10.1111/1365-2656.12146
IN FOCUS
Climate change heats matrix population models
Temperature responses of life-history traits and per capita growth rates in a constant thermal environment.
In Focus: Amarasekare, P. & Coutinho, R. M. (2013) The intrinsic growth rate as a predictor of population viability under climate warming. Journal of Animal Ecology, 82, 1240–1253. Metabolic theory predicts that demographic rates can be expressed as a function of environmental temperature. Amarasekare & Coutinho (2013) build a novel matrix model where demographic rates (fertility, mortality, development) vary according to expected rates of climate warming. They challenge recent studies that claim low population viability of tropical species based on rmax estimated from the Euler–Lotka equation, because the latter assumes a constant stage distribution that is unrealistic under fast rates of warming and for organisms with long development. In those cases, the measurement of the temperature responses of life-history traits could be based in niche theory.
Correspondence author. E-mails:
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Ecology is about identifying the causes and consequences of population change across species in space and time. Conservative estimates of approximately
© 2013 The Author. Journal of Animal Ecology © 2013 British Ecological Society
1118 S. Herrando-P erez 5 3 million species on Earth (Costello, May & Stork 2013) illustrate the dimension of the task the ecological research community faces, but also indicates that it is unrealistic to obtain detailed demographic data for every single taxon – let alone their different populations. For that reason, ecological disciplines such as demography, biogeography and macroecology must resort to proxies for unknown life-history information, which are biologically meaningful and ideally comparable across organisms and locations. The ‘intrinsic [population] growth rate’, ‘Malthusian parameter’, ‘Darwinian fitness’ or ‘maximum [population] growth rate’ (rmax) is one of those proxies. rmax represents the maximum theoretical rate of multiplication of a population in a perfectly benign world; that is, one with unlimited supply of resources (food, space, etc.) and free of enemies (competitors, predators, etc.) and environmental stochasticity (fires, storms, etc.). Thus, shrews or daisies have higher rmax, and hence can grow faster from low to high abundance, than cervids or sequoias. rmax scales inversely with traits such as body size (Fenchel 1974) and age at maturity (Hone, Duncan & Forsyth 2010). It can be estimated from demographic models, through allometric relationships (but see Duncan, Forsyth & Hone 2007), and from experiments by driving populations to low abundance and estimating growth rate as recovery proceeds. The robust estimation of rmax is relevant to a full range of situations, such as modelling long-term population trajectories (Delean, Brook & Bradshaw 2013), calculating sustainable harvest (Hone 1999) or predicting the ecological effects of climate change (Tewksbury, Huey & Deutsch 2008). In this issue, Amarasekare & Coutinho (2013) caution against applying the results from recent studies that estimate rmax from the Euler–Lotka equation and that claim that the depression of such estimates of population fitness in response to global warming will be more accentuated for tropical insects (Huey et al. 2009) and lizards (Deutsch et al. 2008) than for their temperate counterparts. Amarasekare & Coutinho (2013) do not question whether such pattern exists, but set out to test whether the assumption of a constant stage (or age) distribution of the Euler–Lotka equation (hence the biological meaning of associated parameters like rmax) holds as rapid climate change progresses. Succinctly, the Euler–Lotka equation accounts for the proportion of survivors to stage x (lx), and the mean number of offspring produced per surviving individual (bx), providing that there is the same, fixed proportion of the population in the period between any two given stages (x + dx). The equation can be expressed in polynomial (only solvable by iteration, Eqn 1) and integral (Cole 1954; Lewontin 1965; Eqn 2) forms. rmax is often calculated by resolving the integral equation or through mathematical approximations (e.g. Caswell & Hastings 1980; Begon, Townsend & Harper 2006).
Euler–Lotka polynomial: Xn x¼1
ermax x lx bx ¼ 1
eqn 1
Euler–Lotka integral: Z
1
ermax x lx bx dx ¼ 1
eqn 2
x¼1
Over 100 years, Amarasekare & Coutinho (2013) projected population shifts under three scenarios of temperature fluctuation: seasonal across years, seasonal with associated warming (by 003, 005 and 01 °C annually) and seasonal with increased amplitude of fluctuation (i.e., following scenarios predicted by IPCC 2007). Based on metabolic theory (see Brown et al. 2004; Savage et al. 2004), they used two contrasting modelling frameworks that account for the temperature response of rmax based on the thermal responses of fertility, development (from juvenile to adult) and mortality rates: (i) a modified version of the Euler–Lotka equation (Amarasekare & Savage 2012) and (ii) a life-history table of fertility and mortality profiles (Caswell 2001) allowing for a variable stage distribution and tailored to an insect with two stages (juvenile and adult) in a temperate climate (Mediterranean). In the two frameworks, population size grows exponentially across years, because compensatory density feedbacks (Herrando-Perez et al. 2012) are intentionally excluded. The matrix model reveals that under predictable temperature fluctuation, juvenile-to-adult ratios and the population growth rate remain stable around their mean values. In contrast, juvenile-to-adult and population growth rate rise for a short period with a warming climate, then plummet at a rate that correlates negatively with increasing warming and most notably, with increasing amplitude of thermal fluctuations. Three key outcomes emerge: (i) the stage distribution of the population is not constant under warming; (ii) the Euler–Lotka equation overestimates the initial rise of rmax when warming is slow and the time to extinction when warming is fast; and (iii) population responses based on rmax will be reliable under predictable thermal environments or under slow rates of warming (2 stages or age classes) and nonlinear increases in environmental temperature. I concur with Amarasekare & Coutinho (2013) in that ‘…developing a mathematical framework based on the temperature responses of life-history traits to estimate population viability is an important research priority’. In that effort, it is critical to acknowledge that population responses to current warming are framed within the evolutionary history of species. So studies of rmax (and other measures of population fitness) across multiple species, such as those of Huey et al. (2009) and Deutsch et al. (2008), need to include phylogenetic controls (Duncan, Forsyth & Hone 2007). Amarasekare & Coutinho (2013) further suggest that whenever reliable estimates of rmax are unavailable, one could instead estimate the upper and lower thresholds of temperature beyond which a population cannot grow from low abundance – that alternative has merit, yet those thresholds are unknown for most species (like rmax!) and might require estimation through experimentation, and should not neglect evolutionary potential. Mounting evidence reveals that the capacity to evolve tolerance to thermal stress is low in response to heat stress and high in response to cold stress (Ara ujo et al. 2013). Mechanistic models must capture such ‘thermal niche asymmetry’ – otherwise, we will be bound to overestimate extinction risk from climate change for coldadapted species and underestimate it for warm-adapted species that are unable to disperse to suitable thermal environments. Salvador Herrando-Pe´rez1,2 Department of Biogeography and Global Change, National Museum of Natural Sciences (CSIC), Madrid 28006, Spain 2 School of Earth and Environmental Sciences, University of Adelaide, Adelaide, SA 5005, Australia
1
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