An evolutionary maximum principle for density-dependent population dynamics in a fluctuating environment

Abstract

The evolution of population dynamics in a stochastic environment is analysed under a general form of density-dependence with genetic variation in r and K, the intrinsic rate of increase and carrying capacity in the average environment, and in s e 2 , the environmental variance of population growth rate. The continuous-time model assumes a large population size and a stationary distribution of environments with no autocorrelation. For a given population density, N, and genotype frequency, p, the expected selection gradient is always towards an increased population growth rate, and the expected fitness of a genotype is its Malthusian fitness in the average environment minus the covariance of its growth rate with that of the population. Long-term evolution maximizes the expected value of the density-dependence function, averaged over the stationary distribution of N. In the q-logistic model, where density dependence of population growth is a function of N q , long-term evolution maximizes E[N q ]Z[1Ks e 2 /(2r)]K q . While s e 2 is always selected to decrease, r and K are always selected to increase, implying a genetic trade-off among them. By contrast, given the other parameters, q has an intermediate optimum between 1.781 and 2 corresponding to the limits of high or low stochasticity.

Eco-evolutionary dynamics

“…Natural populations, however, frequently experience variable environments (Engen et al 2005) and density dependence (Bonenfant et al 2009). Lande et al (2009) apply diffusion theory to analyse a system of coupled ecological and evolutionary dynamics to derive an evolutionary maximum principle under these more realistic conditions. They show how 'the magnitude of environmental stochasticity and the form of density dependence govern the trade-off between r and K selection'.…”

Do Slower Life History Strategies Reduce Sociodemographic Sex Differences?

“…MacArthur and Wilson (1967) explained this phenomenon by describing the variation in organism's LH as the result of density-dependent selection. Density-dependent selection occurs when the genetic makeup of a population responds to changes in the total population size (Lande, Engen, & Saether, 2009 (Pianka,1970;Reznick et al, 2002). The theory of rand K-selection was prominent to the field of Life History evolution because it satisfied the desire to enumerate laws of nature (Reznick et al, 2002).…”

Correlative Changes in Life-History Variables in Response to Environmental Change in a Model Organism

“…Second, using a stochastic demographic model in which the temporal sequence of good and bad environments is driven by a Markovian process that governs the serial correlation of environment states, a perturbation analysis is conducted to investigate how the two types of environmental change affect the relative importance of demographic rates to the long-run stochastic population growth rate, l s . This will reveal if different stochastic environments select for fast life histories, which are characterized by early maturation, short life span, and high fecundity, or for slow life histories that have the opposite characteristics (Stearns 1983;Gaillard et al 1989;Heppell et al 2000; for examples of life-history speed in stochastic environments, see Lande et al 2009;Miller et al 2011). Note that the latter covariation between age at maturity, fecundity, and survival is captured in the lifehistory variable generation time (Stearns 1992).…”

“…They also forecast that the population growth rate of longer-lived species will be less sensitive to increases in climate variability (Morris et al 2008;Dalgleish et al 2010) but that the population consequences of a change in climate variability can be outweighed by the consequences of changes in the mean environment (van de Pol et al 2010;Coulson et al 2011). The theoretical studies, in turn, indicate that a change in the serial correlation of demographic rates may increase or decrease population growth rate depending on the structure of the life history (Tuljapurkar et al 2009) and that a change in the magnitude or type of environmental stochasticity can induce a shift from fast to slow life histories (Lande et al 2009) or even lead to extinction due to increased population fluctuations (Wilmers et al 2007). These are notable insights but the question of how common concurrent change in life-history variables such as the population growth rate, generation time, and lifetime reproductive success is in response to changing environmental change, is still open.…”

“…The first group uses data from field populations to decompose variation in demographic rates into different ecological and/or evolutionary contributions (Coulson et al 2001;van de Pol et al 2010), and it also uses data from field populations to assess how populations respond to perturbation of demographic rates (e.g., Forcada et al 2008;Morris et al 2008;Dalgleish et al 2010; and to recurrent disturbances (e.g., Tuljapurkar et al 2003;Barrows et al 2010;Vincenzi et al 2012). The second group consists of theoretical studies that investigate the demography of populations and stylized life histories in stochastic environments (e.g., Wilmers et al 2007;Lande et al 2009;Tuljapurkar et al 2009). Results from the first group of studies indicate that the relative importance of specific demographic rates to the growth rate of populations can depend on the frequency of disturbances (Tuljapurkar et al 2003).…”