Randomization method to test for spatial variation
Principles - To test for differences in between populations, we first calculated the asymptotic growth rate of each of the six populations, by pooling data over years within each population (See Table 2). The principle of the test is to assess if the observed standard deviation of over the six populations reflects environmental stochasticity as a result of spatial variation in the environment. The null hypothesis is that the life history of an individual is independent of which population it belongs to. By randomly permuting individuals between populations, the values of the standard deviation of calculated from the permuted data set estimate the distribution of the standard deviations of under the null hypothesis that populations have no effect (i.e., that the life history of an individual is independent of its population).
Permutations of transition data - To obtain transition data, we generated 10000 new projection matrices per population by randomly permuting individuals, i.e. their life history, among populations. For each population, the permutation process kept the total number of life histories identical to the one observed in the real population. For each permutation, a transition matrix T (excluding the fecundity term) was built per population, according to the number of individuals observed in stage i at time 1 over the total number of individuals observed in stage j at time 0.
Permutations of fecundity data - The fecundity matrix F characterized by a single non-zero term (a13) was obtained as follows. As our demographic survey allows only the estimation of the mean number of seedlings produced per flowering plant (a13) for each population and each pair of years, we used additional individual reproductive data collected between 1994 and 1996 in natural populations (Colas et al. 2001) to draw the individual number of seedlings produced by each flowering plant under demographic stochasticity or sampling error. Colas et al. (in prep) showed that the distribution of the number of seeds per plant fitted a log-normal distribution with the mean equal to the standard deviation. As the parameter f measured in the demographic survey is the mean number of seedlings per plant, f is the product of the mean number of seeds per plant by the emergence rate. The emergence rate was set to 0.015 as suggested by field data (Colas et al. in prep) . This mean value was used for every population and every year, as no additional data were available. For any flowering plant in a population P and a year Y, the number of seeds was first drawn from a log-normal distribution which mean was equal to the observed f value of the population P in year Y, divided by the emergence rate (0.015). The number of just-emerged seedlings per flowering plant fi was then drawn from a binomial distribution with mean equal to the emergence rate 0.015, times the number of seeds. Finally, the number of seedlings produced per flowering plant and having survived until June, was drawn in a binomial distribution with a mean equal to s0 as observed in pop P and year Y, times the number of just-emerged seedlings. Individual fecundity values were then corrected by multiplying them by the observed mean value fs0 and by dividing by the mean value calculated over the trials. Thus, the mean number of seedlings calculated over the individual trials was equal to the observed mean number of seedlings produced per flowering plant for each population and year, fs0.
The result of this procedure was a set of seedlings per individual flowering plant drawn from our best guess at the appropriate distribution and with the mean constrained to match the observed data. Now these plants, with their associated seedling production, were randomly permuted among the populations, maintaining the number of flowering plants in each population at the value used in calculating the actual fertility values.
It should be noted that the permuted life-histories for the transition calculations are completely independent of the permuted seedling production values. Thus changes in the number of flowering plants in a permuted set of transitions have no effect on the estimation of fertility. By doing so, our aim was to maintain the sample sizes of the permuted data equal to the ones in the real data sets, in order to follow as closely as possible the structure of the data as suggested by Caswell (2001) .
Statistical test - For each permutation, we could define a matrix transition T and a fecundity matrix F for each population. The asymptotic growth rate of each population was calculated as the highest eigenvalue of the matrix A defined as A = T + F. We then calculated the standard deviation of lambda over the six populations for each permutation and we computed its distribution. A one-tailed test was used to assess the probability that the observed standard deviation of lambda over the six populations could be drawn from the distribution of standard deviations of lambda over the six populations under the null hypothesis of no population effect.
Principles - To test for temporal variation in our data set, we developed a new test using an extension of the randomization procedure over groups. We first obtained a single projection matrix for each pair of year by pooling data over populations (see Table 2). Similarly to the analysis of spatial variation, the principle of this test was to assess if the observed standard deviation of over the six pairs of years (19951996, 19961997, 19971998, 19981999, 19992000 and 20002001) could be drawn from a distribution of standard deviations of lambda over the six pairs of years under the null hypothesis of no year effect.
Permutations of transition data - Random permutations of individual data were performed as follows: any time interval (for instance, 19951996) was represented by the transition of each individual during that period, that is a pair of states (i, j) with i the state of the plant at time t+1 and j its state at time t. We then generated 10000 new transition matrices T for each pair of years, by randomly permuting individual pair of states (i, j), among pairs of years. For each permutation, a transition matrix T was built per pair of years, according to the number of individuals observed in stage i at time t+1 over the total number of individuals in stage j at time t.
Permutations of fecundity data - Individual fecundity were generated by simulating demographic stochasticity and sampling error as described above. Flowering plants with their associated seedling production, were randomly permuted among pairs of years, maintaining the number of flowering plants in each year at the value used in calculating the actual fertility values.
Statistical test - For each permutation, the asymptotic growth rate of each pair of years was calculated as the highest eigenvalue of the matrix A defined as A = T + F. The standard deviation of over the six pairs of years was calculated for each of the 10000 permutations. A one-tailed test was used to assess the significance of the observed standard deviation of .
Caswell, H. 2001. Matrix population models: construction, analysis and interpretation, Second edition. Sinauer Associates, Inc., Sunderland, Massachusetts, USA.
Colas, B., I. Olivieri, and M. Riba. 2001. Spatio-temporal variation of reproductive success and conservation of the narrow-endemic Centaurea corymbosa (Asteraceae). Biological Conservation 99:375-386.