## Introduction

Seed germination is greatly influenced by both temperature (Roberts, 1988; Probert, 2000) and water potential (Bradford, 1990, 1995) and these factors largely determine the timing of onion and carrot seed germination in the field (Finch-Savage & Phelps, 1993; Finch-Savage *et al*., 1998). The timing and spread of time to germination within the seed population have a major impact on the efficiency of production in these two crops (Finch-Savage, 1995). Accurate models are therefore required to describe seed responses to these two variables for effective field predictions. Two different population-based threshold modelling approaches are considered in the present work to describe seed responses to the hydrothermal environment. In both approaches, it is assumed that seeds germinate in a set order unaffected by germination conditions, so that each seed can be assigned a value of *G*, which is the percentage of the population at which it germinates. Gummerson (1986) developed a theory in which germination time *t*(*G*) is a function of the extent to which the constant water potential (Ψ) and constant suboptimal temperature (*T*) of each seed (*G*) exceed thresholds (bases; Ψ_{b}, *T*_{b}) below which germination will not occur.

_{HT}= (Ψ − Ψ

*(*

_{b}*G*))(

*T − T*

_{b})

*t*(

*G*)(Eqn 1)

Hydrothermal time (θ_{HT}) and the base temperature (*T*_{b}) are assumed constant. Only the base water potential (Ψ_{b}) varies with (*G*) and so the distribution of the germination times of individual seeds within the population are determined by the distribution of this parameter. This approach has been shown to adequately describe germination curves produced in a wide range of suboptimal constant temperatures and water potentials (Gummerson, 1986; Dahal & Bradford, 1994; Finch-Savage *et al*., 1998; Roman *et al*., 1999; Shrestha *et al*., 1999; Allen *et al*., 2000). Accepting these assumptions it is possible to describe seed response of the whole population in a single equation by incorporating a suitable distribution (usually a normal distribution) of base water potentials within the population (Gummerson, 1986; Dahal *et al*., 1993; Dahal & Bradford, 1994; Bradford, 1995). In the case of a normal distribution:

*G*) = [(Ψ − θ

_{HT}/(

*T − T*

*)*

_{b}*t*(

*G*)) − Ψ

_{b}(50)]/σ

_{Ψb}(Eqn 2)

where Ψ_{b}(50) is the base water potential of the 50th percentile and σ_{Ψb} is the standard deviation of Ψ_{b} within the population. The best fit to the model can be obtained by repeated probit regressions varying the values of θ_{HT} (e.g. Bradford, 1995). In a range of tomato seed lots this model accounted for 73–93% of the variation in radicle emergence timing across a range of temperatures and water potentials (Cheng & Bradford, 1999).

However, to be useful for field predictions the hydrothermal time model must also be able to describe the reduction in germination rate and nongermination that occurs at supra-optimal temperatures. This factor can be accommodated in the model by employing a shift in the distribution of Ψ_{b} with temperature. For example, during thermoinhibition of lettuce (*Lactuca sativa*) water potential thresholds were found to increase as the temperature approached the upper limit for germination (Bradford & Somasco, 1994). Germination was then prevented as thresholds shifted above ambient water potential. Subsequently, other studies have shown that as temperature increases above the optimum (*T*_{o}) and approaches the ceiling temperature (*T*_{c}), the Ψ_{b} distribution shifts progressively towards and above 0 MPa (Kebreab & Murdoch, 1999; Meyer *et al*., 2000; Bradford, 2002). As the distribution shifts above 0 MPa, germination is progressively prevented in the population accounting for the distribution of ceiling temperatures shown by Ellis *et al*. (1986).

Bradford (2002) modified eqn 1 above to describe the response of seed germination to supra-optimal temperatures when the shift in distribution of Ψ_{b}(*G*) with temperature is linear thus:

_{HT}= [Ψ − Ψ

*(*

_{b}*G*)

_{o}+

*k*

_{T}(

*T − T*

_{o})](

*T*

_{o}−

*T*

*)*

_{b}*t*(

*G*)(Eqn 3)

where *k*_{T} is a constant (the slope of the Ψ* _{b}*(

*G*) vs

*T*line when

*T*> (

*T*

_{o}) and Ψ

*(*

_{b}*G*)

_{o}is the threshold distribution at

*T*

_{o}. This equation adjusts Ψ

*(*

_{b}*G*)

_{o}to higher values as

*T*increases above

*T*

_{o}and stops the accumulation of thermal time at the value equivalent to that accumulated at

*T*

_{o}. Thus, temperatures above

*T*

_{o}do not contribute additional thermal time in the supra-optimal range. Instead, effects on germination are accounted for by the change in Ψ

*(*

_{b}*G*). Thus a germination rate response to temperature is described that has a clearly defined optimum at the convergence of two straight lines, as shown in a number of studies (e.g. Garcia-Huidobro

*et al*., 1982; Covell

*et al*., 1986). Germination response can then be described for the full range of temperatures and water potentials using a combination of equations 1 and 3. However, eqn 3 cannot predict the plateau or curved relationship between germination rate and temperature that has been observed around

*T*

_{o}in a number of species (e.g. Labouriau & Osborn, 1984; Orozco-Segovia

*et al*., 1996). To achieve this a different relationship between Ψ

*(*

_{b}*G*) and temperature is needed.

A second population-based threshold modelling approach known as the Virtual Osmotic Potential (VOP) model is being developed to assist simulation of germination response under variable seedbed conditions (Rowse *et al*., 1999). To initially develop the model, germination of an imbibed seed was considered to result from further water uptake and growth of the radicle. It was assumed that the growth was driven by an increase in turgor pressure exceeding the yield threshold pressure (*Y*) due to an accumulation of solutes and a consequent decrease in osmotic potential (Ψ_{π}). The model used changes in Ψ_{π} to integrate the history of water potential experienced by the seed. For simplicity, *Y* was considered as a constant for every seed in the population. However, growth could also result from a decrease in *Y*. This necessarily means that values of Ψ_{π} used in the model are empirical, and to avoid confusion with real measurable osmotic potentials the term VOP with the symbol Ψ_{πv} was used.

The VOP model utilises the concepts of Ψ* _{b}* and

*T*

_{b}, but also incorporates progress towards germination at water potentials below Ψ

*and above the minimum for metabolic advancement Ψ*

_{b}_{min}(Tarquis & Bradford, 1992) that is not described by the basic hydrothermal time model. It is at these water potentials that priming occurs. As originally derived (Rowse

*et al*., 1999) the VOP model described the effects of water potential on germination time at a single temperature (15°C). In this paper the scope of the VOP model has been extended by assuming a linear relationship between germination rate and suboptimal temperature and therefore replacing the term

*k*

_{0}(T)(1 – Ψ/Ψ

_{min}) of eqn 9 in Rowse

*et al*. (1999) by

*R*(1 – Ψ/Ψ

_{min})(

*T*/

*T*

_{b}– 1). This gives an equation that describes germination time of a seed fraction as:

where Ψ, Ψ* _{b}*, Ψ

_{min},

*T*and

*T*

_{b}are as described above and

*k*and

*Y*are constants determined by fitting. A linear relationship between germination rate and suboptimal temperature has been reported both for carrot (Finch-Savage

*et al*., 1998) and onions (Ellis & Butcher, 1988) and is therefore justified in the present work. A problem can arise with this formulation because when

*T*

_{b}= 0,

*T/T*

_{b}= infinity and

*t*(

*G*) = 0, but this situation can be avoided by the use of absolute temperature. However, for most practical purposes this problem can be ignored.

To simulate the effects of a changing environment, advancement towards germination can be determined by integrating changes in Ψ_{πv} that are proportional to the history of temperature and water potential experienced by the seed relative to *T*_{b}, Ψ_{min} and Ψ* _{b}*. The increase in Ψ

_{πv}during each finite step of the simulation at a fixed temperature of 15°C is given by Rowse

*et al*. (1999). Modification of this equation to include the effects of suboptimal temperatures yields:

*d*Ψ

_{πv}(

*G*)/

*dt*=

*k*(1 – Ψ/Ψ

_{min})(

*T*/

*T*

_{b}– 1)(Ψ

_{b}(

*G*) –

*Y*– Ψ

_{π}

*(*

_{v}*G*))(Eqn 5)

In the present work the VOP model is further extended to include response to temperature across both sub- and supra-optimal ranges and the same approach is also adopted to describe the later response within the hydrothermal time model.