## INTRODUCTION

The optimum temperature for seed germination (*T*_{o}) can be defined as the soil temperature at which the highest germination percentage is achieved by a seed population in the shortest possible period of time (Mayer & Poljakoff-Mayber 1975). At supra-optimal temperatures (*T* > *T*_{o}), physiological changes occur in seeds, which inhibit both the rate of germination and the proportion of the seed population that will complete germination (Hills & van Staden 2003; Argyris *et al*. 2008). This progressive decline in germinability within a seed population as soil temperatures increase above *T*_{o} is known as thermoinhibition.

Surprisingly, the optimum germination temperatures for many species are quite low (20 °C or less) (Bradford 2002) so that small increases (<10 °C) above these optima will result in declining germination when temperatures are still favourable for subsequent growth of seedlings (Bloomberg *et al*. 2009).

Seed germination is a complex physiological process largely determined in non-dormant seeds by temperature and water potential of the seedbed. These two factors have been successfully combined in hydrothermal time (HTT) models to describe the time course of germination for a wide range of plant species, at both sub- and supra-optimal temperatures. The HTT model is a threshold model that simultaneously accounts for germination percentages and germination rates of a seed population (Gummerson 1986; Bradford 1995, 2002; Finch-Savage 2004).

In the HTT model, the order in which seeds germinate is specified by seed percentile (G), such that G = 1 is the first percentile of seeds to germinate, G = 2 is the second percentile to germinate and so on up to G = 100, which represents complete germination of a seed population. The time to complete germination (time from imbibition to radicle emergence) for a specific seed percentile *t*(G) is specified by

where *θ*_{HT} (Mpa °C d) is the HTT constant for the seed population; *T* (°C) is a constant temperature of the soil or other surrounding medium, which must be greater than *T*_{b} (°C), the base temperature, below which the radicle does not emerge to complete germination; *Ψ* (MPa) is a constant water potential of the soil or other surrounding medium; and *Ψ*_{b}(G) (MPa) is the base water potential, a threshold value analogous to base temperature.

Unlike *T*_{b}, which is relatively constant for all seeds in the population, *Ψ*_{b}(G) is variable, with the lowest (most negative) values corresponding to the lowest seed percentiles of seed germination order (G = 1, 2, 3 . . .) and with values increasing progressively so that the highest (closest to 0 MPa) values correspond to the highest seed percentiles. For seeds germinating at a constant *T* and *Ψ*, it is this variation in *Ψ*_{b}(G) that results in a distribution of times to germination, *t*(G), from the fastest seeds (G = 1, 2, 3) to the slowest seeds (G = 100). The original HTT model proposed by Gummerson (1986) assumes that the variation in *Ψ*_{b}(G) can be described by a normal frequency distribution so that

where *Ψ*_{b}(50) is the 50th percentile of the seed base water potential distribution; probit(G) is the probit function that calculates the standard normal deviate for a specified cumulative frequency (=G); and *σ*_{Ψb} is the standard deviation of *Ψ*_{b} values in the population.

Gummerson's HTT model does not account for thermoinhibition because it assumes that all parameters are independent of soil temperature and water potential. To account for thermoinhibition in the HTT model, an increase in seed base water potential at supra-optimal temperatures has been proposed (Alvarado & Bradford 2002; Rowse & Finch-Savage 2003). This increased seed base water potential reduces the rate of HTT accumulation at a specific soil water potential, and thus inhibits the rate of germination. This is consistent with physiological changes in the seed at supra-optimal temperatures that raise the threshold soil water potential at which seeds will germinate (Yoshioka *et al*. 2003; Finch-Savage & Leubner-Metzger 2006). The increase in seed base water potential at supra-optimal temperatures is modelled as a simple rightward shift in the location of the normal frequency distribution for *Ψ*_{b}(G) (Fig. 1). The magnitude of the shift scales with supra-optimal temperature and is specified by *k*(*T* − *T*_{o}), where *k* is a positive constant (Alvarado & Bradford 2002).

The assumption of normality underpinning the *Ψ*_{b}(G) distribution in the original HTT model (Eqns 1 and 2) may not be universally valid. Within the suboptimal temperature range, *Ψ*_{b}(G) for *Pinus radiata* D. Don and *Buddleja davidii* Franch. have been shown to have a skewed pattern best described by the Weibull distribution (Weibull 1951) that converges to a closed minimum value of *Ψ*_{b}(G), rather than an open left limit, as described by the normal distribution (Watt, Xu & Bloomberg 2010). Although research has not yet evaluated the utility of the Weibull distribution at describing *Ψ*_{b}(G) and germination in the supra-optimal temperature range, previous observations of germination do suggest that this approach could have merit. At supra-optimal temperatures, germination may be very rapid for the first few seed percentiles, with no evidence of thermoinhibition (Hills & van Staden 2003), but germination times for later percentiles usually scale positively with increasing temperatures. This behaviour implies that *Ψ*_{b}(G) for the initial seed percentiles is very similar for all supra-optimal temperatures, with divergence between *Ψ*_{b}(G) at contrasting supra-optimal temperatures occurring for higher percentiles of the population. This response can be accommodated by the Weibull distribution of *Ψ*_{b}(G), but not the normal distribution, which does not allow for divergence in *Ψ*_{b}(G) between contrasting supra-optimal temperatures across the seed percentile range.

Using data obtained from four unrelated plant species, we (1) fitted HTT models that use the Weibull and normal distribution to describe *Ψ*_{b}(G), and (2) compared the accuracy and bias of these two HTT models at predicting both *Ψ*_{b}(G) and seed germination. We discuss the results from an ecological and an adaptational perspective.