An IPM that contains a seed bank is described using two equations. The first equation describes the number of seeds in the seed bank at time t + 1, i.e. seeds remaining in the seed bank + fresh seeds entering the seed bank, as
- ( eqn 1)
where ss is the constant seed survival in the seed bank, sr is recruitment from the seed bank and se is the establishment rate for fresh seeds. The fecundity function is described as fs(x) = fp(x)fn(x), where fp(x) is the probability of flowering and fn(x) is the number of seeds produced by plants of size x.
The second equation describes the density of individuals of size (y) at time t + 1 in the established population, including seedlings that germinate from the seed bank (first part), as
- ( eqn 2)
where the kernel k(y,x) describes all possible transitions from plant size x to plant size y, integrated over all sizes (Ω). Similar to other studies (Rees & Rose 2002; Rose et al. 2005), we used the integration of 0·9 times the minimum and 1·1 times the maximum rosette size observed, for evaluating the integrals see Table S1 (Supporting Information). The kernel consists of a survival-growth function, p(y,x), and a fecundity function, f(y,x), which both depend on plant size. For the monocarpic Cirsium where flowering is fatal, the survival-growth function is p(y,x) = s(x)[1 − fp(x)]g(y,x), where s(x) is the probability of survival for a plant size of x, fp(x) is the probability of flowering for a plant size of x and g(y,x) is the probability of a plant of size x growing to size y. For the iteroparous Primula, the survival-growth function is p(y,x) = s(x)g(y,x). For both species, the growth function, g(y,x), is a normal probability density function with mean and variance. The fecundity function is described as f(y,x) = fp(x)fn(x)se fd (y), where fp(x) is the probability of flowering and fn(x) is the number of seeds produced by plants of size x, and fd(y) is the probability distribution of seedling size with constant mean and variance. As a result of a lack of empirical data, we adopted the same procedure as others (Rees & Rose 2002; Childs et al. 2003; Rose et al. 2005; Williams & Crone 2006) and assumed that seedling size was independent of maternal plant size; matrix models make the same assumption. For the kernel parameters and equations, see Table S1 (Supporting Information).
To calculate the kernels from the data, we constructed regression models with plant size (rosette diameter for Cirsium and leaf length for Primula) at time t + 1 and seed production at time t as response variables and plant size at time t as an explanatory variable. Plant size and seed production were log-transformed in all the models. We estimated the dependence of plant survival and flowering probability on plant size using a generalized linear model with a logit link function (Table S1, Supporting Information). For each model, we included a quadratic size term and then selected the best model according to Akaike’s information criterion (Burnham & Anderson 1998). The selection of the best model for each sub-sample (termed the best model) allowed regression equations to vary from linear to quadratic depending on the sub-sample of data at hand. An exception was seed production, for which we always used a linear function to avoid drastic overestimates of seed production for small plants resulting from quadratic functions that sometimes fitted best for small data sets. The parameterization of the kernel from the data at hand is preferable for large data sets but it may not be the best solution for small data sets where additional information from other studies may be useful. In such a situation a priori knowledge of the species could be used to define the forms of the regression models. Therefore, we also used a constant model (termed the constant model), in which the forms of regression models were parameterized from the full data set and were kept fixed (Table S1, Supporting Information), while the parameters were estimated from the sub-samples.