Our model simulates a population of annual plants living in an arid environment. The model uses empirically based and realistic assumptions about the environment and its interaction with the three seed traits. Thus, in our model, environmental favorability is determined by the total seasonal precipitation; which, given the complete lack of precipitation during the dry season, is also the annual precipitation. Inter-annual variation in total annual precipitation represents temporal heterogeneity. Environments that are highly heterogeneous in time are less predictable, because in highly variable environments the information about precipitation that is available to the annual plant population in the current year is less indicative of the precipitation in the next year.
We chose an approach of optimization and sensitivity analysis. Under this approach, the mean population size after many generations of a population composed exclusively of individuals of a given trait or trait combination value(s) (i.e. of a single genotype) is considered as an indication of the genotype fitness. Thus, our model cannot simultaneously estimate the fitness of more than one genotype in a common environment, as is done in an alternative evolutionary stable strategy (ESS) approach. The ESS was previously used by Ellner (1985a,b), Rees (1994), Tielbörger & Valleriani (2005) and Satterthwaite (2010) to study the evolutionary effects of seed dormancy, and by Kobayashi & Yamamura (2000) to study those of seed dormancy and dispersal. However, the ESS approach is impractical when several axes of continuous traits are considered simultaneously. Here, we conducted a sensitivity analysis of the fitness effect of two interacting continuous traits: seed mass and seed germination fraction (i.e. dormancy). The analysis was conducted using all combinations of these two traits within the prescribed ranges under a broad range of environmental conditions. The environmental conditions were defined by the two axes – environmental favorability and its predictability. In our simulated semiarid climate conditions of the Mediterranean, these two axes corresponded to mean annual precipitation and inter-annual variation in precipitation, respectively.
For simplicity, we assume that the reproductive biomass produced by a plant is fixed (i.e. differences in yield among the plants are solely determined by seedling survival) and is converted to seed number through the inverse function of seed mass. Following Weiner et al. (2001), we assign conventional units (0–1) to measures of mass for ease of interpretation; however, these units are essentially arbitrary and should not be taken literally. This relationship between seed mass and number is independent of environmental conditions. We assume that seeds are identical in shape and therefore seed mass is equivalent to seed size. Several mechanisms of plant adaptation through seed traits, seed mass and dormancy are well accepted in the literature and are listed below as model assumptions.
Our model is based on the approach of Cohen (1966), subsequently used by Brown & Venable (1986) and Venable & Brown (1988). The model describes the population fitness of an annual plant species in terms of the geometric mean of its population size over the last 9000 yr of a 10 000-yr simulation experiment. At any annual time-step, t, the model calculates the total number of seeds in a population just before the start of the next growing season, Nt+1, that is, the number of seeds that will be available after seed production, distribution, and mortality before germination.
- (Eqn 1)
where the evolutionary traits being tested are G, the fraction of the seed bank that germinates each year, and M, the seed mass. The variable environmental forcing is represented by the mean annual precipitation, μ [P], and its standard deviation, σ[P]. We ran simulations for each combination of G and M values within a relevant range under each of the ranges of mean annual precipitation regimes with each of the standard deviation levels. The values we used for all parameters in the simulations described here, following the formulation detailed below (equations 2-5), are presented in Table 1. Nt is the current seed population size, V is the survival fraction in the soil, S is seedling survival after germination and Y is fecundity. These three variables are functions of G, M, Nt and the actual annual precipitation in each year, Pt, as follows:
Table 1. Variable symbols and parameter values used in simulations
| a s ||Shape parameter for precipitation and size-dependent seedling survival||0.27||3|
| a v ||Shape parameter for size–survival in soil relationship||5||2|
| a y ||Maximal fecundity in seed mass units||10||4|
| b s ||Shape parameter for precipitation and size-dependent seedling survival||12.5||3|
| b v ||Shape parameter for size–survival in soil relationship||10||2|
| b y ||Shape parameter for density dependence of survival to maturity||4E-5||4|
| c s ||Scale parameter for precipitation and size-dependent seedling survival||4||3|
| c v ||Proportionality coefficient for the effect of precipitation (soil moisture) on survival in soil||0 – no effect 0.003 – intermediate 0.004 – strong||2|
| c y ||Shape parameter for density dependence of survival to maturity||5.3E-5||4|
| d s ||Shape parameter for precipitation and size-dependent seedling survival||7||3|
| G ||Annual germination fraction||0.02 : 1||1|
| I PN ||Sign of survival-in-soil and seed mass relationship||−1 – positive relationship +1 – negative relationship||2|
| L ||Autocorrelation time-scale of random precipitation time series||2 [yr]||5|
| M ||Seed mass||0.02 : 1 [arbitrary units]||1|
| N t ||Number of seeds in the current population||[number of seeds]||1|
| N t+1 ||Number of seeds after annual seed production||[number of seeds]||1|
|[P]||Vector of autocorrelated actual annual precipitation rates||[mm]||2, 3, 5|
| P t ||Actual annual precipitation||[mm]||2, 3|
|[R]||Vector uniform random numbers||0–1 [unitless]||5|
| S ||Seedling survival||Fraction [unitless]||3|
|[T]||Time-span vector||1 : 10 000 [yr]||5|
| t ||Current time-step||[yr]||1–5|
| V ||Survival fraction of seeds in the soil||Fraction [unitless]||2|
| V max ||Maximal annual survival in soil||0.9||2|
| Y ||Number of seeds produced by a germinated seed (fecundity)||Number of seeds||4|
|[λ]||Vector of random autocorrelated values||Arbitrary [unitless]||5|
|δPN||Kronecker delta for size-dependent or -independent survival in soil||0 – size-independent 1 – size-dependent||2|
|μ[P]||Mean annual precipitation||100 : 280 [mm]||5|
|μ[λ]||Mean of random vector (for normalization)||Arbitrary [unitless]||5|
|σ[P]||Inter-annual standard deviation of precipitation||0 : 40 [mm]||5|
|σ[λ]||Standard deviation of random vector (for normalization)||Arbitrary [unitless]||5|
V, the survival fraction of seeds in the soil:
- (Eqn 2)
V is a sigmoid-shaped survival function which is dependent on seed mass, M, with two empirical shape parameters, av and bv. Vmax represents the maximal mean survival-in-soil rate under favorable conditions; av and bv were parameterized such that the inflection point of the sigmoid will be around the median seed mass (Fig. 1). cv is a parameter describing the sensitivity of survival-in-soil to annual precipitation, Pt. cv was set to 0.004 or 0.003 in the set of tests that assumed a negative effect of soil moisture on survival in soil, and 0 in the set of tests that did not include this effect. Setting cv to values larger than 0.004 led to almost complete mortality of the seeds in the soil under the ranges of precipitation amounts in our model. δPN is a Kronecker delta and used as a switch: δPN = 1 when the modeled relationship between seed survival in the soil and seed mass is either positive or negative (as detailed in assumption 5) and δPN = 0 for an independent relationship, in which case the function (Eqn (Eqn 2)) is reduced to V = Vmax − cvPt. In these independent cases, we tested three different fixed mean survival rates in the soil: Vmax = 0.1, 0.5 and 0.9. The formulation is further reduced to V = Vmax when there is no dependence on soil moisture (i.e. cv = 0). We used Vmax = 0.9 in that case. IPV is a sign function that corresponds to the type of relationship between mass and survival-in-soil: it is +1 for the case of a negative relationship and −1 for a positive relationship (Fig. 1).
Figure 1. Three types of modeled hypothetical relationships between seed mass (M) and the probability of survival in the soil seed bank for 1 yr: negative (bold solid line), positive (gray solid line), and constant, that is, independent of seed mass (dashed lines). The simulated size-independent survival range included low, intermediate and high survival probabilities (0.1, 0.5 and 0.9, shown as light, gray and black dashed lines, respectively).
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S, seedling survival:
- (Eqn 3)
The survival function describes a precipitation, Pt (subscript t indicates a specific year), and a seed mass, M, dependent Weibull function with four empirical shape and scale parameters, aS, bS, cS, and dS, which were parameterized such that the shape of the survival curve will correspond to observations on seedling survival of two annual grasses, Avena sterilis and Hordeum spontaneum, in semi-arid and desert conditions (S. Volis, unpublished) and the range will fit the arbitrary mass units. This parameterization allows spanning of the full range of possible values by changing only M while keeping the shape parameters constant (Fig. 2a).
Figure 2. Survival sof seedlings as a function of seed mass (M) under different amounts of precipitation (Precip., gray-scale) (upper panel), and fecundity (seed yield, i.e. number of seeds produced per germinated seed) as a function of seed size (M) and population density (number of individuals, Nt, gray-scale) (lower panel).
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Y, the number of seeds produced by a germinated seed (fecundity):
- (Eqn 4)
Y describes a density-dependent and seed size-dependent yield function of seedlings, which includes both the probability of the seedling survival to maturity and their seed production, where M is seed mass. ay is an empirical coefficient for the maximal fecundity (per unit seed mass) and by and cy are shape parameters for the density and size dependence of young plants’ survival to reproductive maturity (Fig. 2b).
- (Eqn 5)
[P] is a random vector of actual annual precipitation rates in all years composed of elements, Pt, of the precipitation in each particular year, t. [T] is a vector of all years (1:10 000). The precipitation is calculated as an autocorrelated time series with a random phase, prescribed autocorrelation time, L (2 yr in this case), and prescribed mean and standard deviation. [R] is a vector of 10 000 uniform random numbers. [λ] is a resulting vector of random autocorrelated values (Bohrer et al., 2007). [λ] (in arbitrary units) is converted to precipitation with a prescribed mean and standard deviation (μ[P] and σ[P], respectively, in mm yr−1) by normalizing it against its own mean and standard deviation (μ[λ] and σ[λ], respectively).