## Introduction

One of the central issues that have informed life-history theory is the trade-off between size and number of offspring (Stearns 1992; Moles & Westoby 2006). For a given level of resource investment, a parent may choose to produce a few large-sized offspring (with higher individual probabilities of survival to reproductive maturity) or many small-sized offspring (with lower probabilities of survival). A standard mathematical model predicts the evolution of a single optimal offspring size that maximizes maternal fitness, in situations where offspring fitness increases with size according to a certain functional form (Smith & Fretwell 1974). Subsequent work has extended this model and related it to ecologically important phenomena including competition, dispersal, predation and host–parasite relationships (Parker & Begon 1986; Venable 1992; Fox, Thakar & Mousseau 1997; Marshall, Cook & Emlet 2006).

One such extension investigates whether the single-sized offspring strategy is still optimal when the environment varies unpredictably in time or in space. In this case, an alternative to a single-size strategy is for an individual to produce variably sized offspring. This may provide a buffer against environmental variability and constitutes a form of bet-hedging (Philippi & Seger 1989; Marshall, Bonduriansky & Bussière 2008; Olofsson, Ripa & Jonzén 2009). Since the 1970s, various authors have described offspring size variability in a large range of taxa, including plants, insects, fishes and birds, and suggested its possible adaptive value in variable habitats (for references, see the studies by McGinley, Temme & Geber 1987; Geritz 1995; Geritz, van der Meijden & Metz 1999; Westoby, Leishman & Lord 1996; Moles & Westoby 2006). However, variable offspring size within a species can arise from a number of genetic and environmental sources and can be partitioned in numerous ways: among populations, among individuals in the same population, within individuals, within a year or a clutch, or between years (Hangelbroek & Santamaria 2004). As a consequence, it is often difficult to determine whether the observed offspring size variation simply reflects developmental or resource constraints or whether it might also have adaptive value (Sakai 1995; Wolfe 1995; Simons & Johnston 1997, 2000; Vinuela 1997; Vaughton & Ramsey 1998; Forbes 1999; Einum & Fleming 2004). A second obstacle to establishing the adaptive value of offspring size variation is that the stipulation of specific functional forms relating fitness to offspring size and number is often difficult to establish (Einum & Fleming 2000; Torres-Vila & Rodriguez-Molina 2002; Moles, Warton & Westoby 2003; Gómez 2004; Moles & Westoby 2006). For example, it is generally admitted that the survival of seedlings increases with seed size; however, large seeds can also have higher risks of predation or reduced dispersal, resulting in a co-evolution of traits (Venable & Brown 1988; Moles, Warton & Westoby 2003). The relationship between offspring size and fitness may also vary between different stages of development or depend on parental behaviour (Hendry, Day & Cooper 2001). Hence, although there have been several empirical studies indicating that variation in offspring size can indeed be an adaptive response to unpredictable environmental variation (Crump 1981; Fox, Thakar & Mousseau 1997; Kudo 2001; Lips 2001; Koops, Hutchings & Adams 2003), empirical support for the hypothesis remains rare. Such obstacles also prevent the forging of more intimate links between theoretical and empirical approaches to the problem.

There have been a number of theoretical modelling studies of strategies for offspring size distribution and/or offspring numbers in the presence of temporally or spatially varying environments (Kaplan & Cooper 1984; Schultz 1991; Geritz 1995; Sasaki & Ellner 1995; Geritz, van der Meijden & Metz 1999; Kisdi 2007; Simons 2007; Olofsson, Ripa & Jonzén 2009). The process generally followed by mathematical models is to compare the fitness of a single-size strategy to a strategy of producing variably sized offspring under different conditions of habitat variability in space and/or time. Most of these theoretical approaches conclude that under certain assumptions, the optimal solution to the trade-off between size and number of offspring in a spatially and/or temporally variable environment is the production of variably sized offspring (Kaplan & Cooper 1984; Schultz 1991; Geritz 1995; Geritz, van der Meijden & Metz 1999). However, one modelling study concluded that variability in offspring size does not necessarily provide a fitness advantage over single-size strategies in a temporally varying environment that can be either “good” or “bad” (McGinley, Temme & Geber 1987). These studies have generally not parameterized their models with empirical data, aiming for generality instead.

Studies that parameterize theoretical models with empirical data to investigate the adaptive value of producing variably sized offspring in temporally varying environments remain relatively uncommon (see for example the study by Simons 2009). Here, we describe experiments carried out to investigate offspring size variation in a clonal plant species and use the experimental data to develop a species-specific mathematical model of growth. In particular, we experimentally determine the relationships between offspring size and fitness in the clonal plant *Scirpus maritimus* under different environmental conditions, and use these experiments to parameterize a mathematical (difference equation) model of biomass production over multiple seasons. We compare fitness under the observed offspring size distribution (i.e. lognormal) to fitness under two hypothetical alternatives: the *single*-*size* distribution, in which all offspring are of the same size, and the *uniform* distribution, in which the probability of producing tubers of a given size is a constant. The single-size distribution corresponds to the “optimal offspring size” theory and thus should be optimal in an environment without temporal variation. The uniform distribution represents an intermediate strategy between the single-size and lognormal strategies. These alternative strategies serve as theoretical controls for the lognormal strategy. We chose the route of “simulation as experiment” because of the difficulty in testing our hypothesis empirically (Peck 2004). We used longitudinal historical data on the temporally varying environment of *S*. *maritimus* to simulate continuous random temporal variability. To compare our predictions with previous models that predict a single optimal offspring size under a discrete number of possible states (good or bad) (Cohen 1966; Smith & Fretwell 1974; McGinley, Temme & Geber 1987), we investigate the impact of representing environmental variability through a continuous rather than a discrete distribution of possible states.