Evolution of size–dependent flowering in a variable environment: construction and analysis of a stochastic integral projection model

Abstract

Understanding why individuals delay reproduction is a classic problem in evolutionary biology. In plants, the study of reproductive delays is complicated because growth and survival can be size and age dependent, individuals of the same size can grow by different amounts and there is temporal variation in the environment. We extend the recently developed integral projection approach to include size-and age-dependent demography and temporal variation. The technique is then applied to a long-term individually structured dataset for Carlina vulgaris, a monocarpic thistle. The parameterized model has excellent descriptive properties in terms of both the population size and the distributions of sizes within each age class. In Carlina, the probability of flowering depends on both plant size and age. We use the parameterized model to predict this relationship, using the evolutionarily stable strategy approach. Considering each year separately, we show that both the direction and the magnitude of selection on the flowering strategy vary from year to year. Provided the flowering strategy is constrained, so it cannot be a step function, the model accurately predicts the average size at flowering. Elasticity analysis is used to partition the size-and agespecific contributions to the stochastic growth rate, s . We use s to construct fitness landscapes and show how different forms of stochasticity influence its topography. We prove the existence of a unique stochastic growth rate, s , which is independent of the initial population vector, and show that Tuljapurkar's perturbation analysis for log( s ) can be used to calculate elasticities.

Evolutionary demography of iteroparous plants: incorporating non-lethal costs of reproduction into integral projection models

“…For example, it has been shown for semelparous plants that the direction and magnitude of selection on flowering size can vary from year to year [8] and that covariance between demographic rates could amplify the effects of stochasticity [14]. Our qualitative conclusions regarding the importance of non-lethal costs in selecting for delayed reproduction are probably robust to our assumption of environmental constancy, as stochasticity alone does not favour reproductive delay when adult survival exceeds juvenile survival [22], as was the case for O. purpurea (figure 2).…”

“…limitation of regeneration sites and not antagonistic interactions among established individuals) [45]. We tested the assumption that density dependence operates at the establishment stages by examining the relationship between population-level seed production and seedling recruitment [8,16]. Because the first 3 years of development occur below ground, we tested for correlations between total seed production and seedling recruitment 3 years later, separately, for each of the two study sites in each light environment.…”

“…The development of demographic tools, especially continuously size-structured integral projection models (IPMs) [6,7], has allowed for the estimation of this optimum and quantitative comparisons of observed and optimal reproductive strategies in semelparous plants. We have now accumulated a wealth of such studies [8][9][10][11][12][13][14][15][16][17][18][19]. The popularity of semelparous plants for the study of life-history evolution and reproductive delay is due in part to the ease and elegance with which lethal costs of reproduction can be incorporated into demographic models: the probability of survival is simply conditioned on the probability of not flowering.…”

Integral Projection Models for Species with Complex Demography

“…The observed size dependence of flowering is gradual, representing possibly a constraint or else a decision that depends on plant size at some time between censuses (Rees et al 1999). We therefore imposed gradual size dependence by holding the size slope fixed at its estimated value and optimizing the intercept and age slope (see Childs et al 2003). Rees et al (1999) compared estimated and ESS flowering strategies and found that plants following the ESS strategy would flower later in life and at larger sizes (on average) than real plants.…”

“…Under similar assumptions to matrix models, the integral projection model (IPM) predicts a population growth rate l with associated eigenvectors and statedependent sensitivity and elasticity functions (Easterling 1998). An IPM is implemented on the computer as a matrix iteration, but this is just a technique for computing integrals and not a discretization of the life cycle.Several empirical studies have illustrated how a kernel can be estimated from the same data as a matrix model (Easterling et al 2000;Rees and Rose 2002;Childs et al 2003Childs et al , 2004Rose et al 2005). The functions making up the kernel can often be estimated by regression; for example, size-dependent survival and fecundity can be fitted by generalized linear or additive models and growth by parametric or nonparametric regression (Metcalf et al 2003).…”