the matrix model
The mathematical theory behind matrix population models is extensive (see Caswell 2001 for a review). First, a column vector representing the number of individuals in each size class is defined. In this model, this meant four classes for C. montagui (C1,1, C2,1, C3,1 and C4,1) and three classes for S. balanoides (S5,1, S6,1 and S7,1), producing the following population vector (V) (see Hyder et al. 2001 and Svensson et al. 2004 for details):
By projecting a transition matrix on to this vector, the result is a new vector (of the same size) that displays the number of individuals for each species and in each size class one time-step later. In a deterministic model there is only one matrix describing the environment, eventually producing constant asymptotic estimates of population variables. By adding variability to the matrix or by using several matrices, the variability in model outcome increases. Further nonlinearity can be added by introducing dependent transition elements (i.e. replacing transition elements with functions).
In this analysis, the transition matrix (A) consisted of two submatrices. The survivals transitions for C. montagui were represented in the upper left corner and the survival transitions for S. balanoides were represented in the lower right corner, producing a matrix with seven rows and seven columns. All other elements were set at 0:
where S represents individuals that survive but stay within a size-class and G stands for individuals that survive and grow into another size-class.
As recruitment for both species was dependent on available free space, recruitment was not included in the transition matrix. Instead, the recruitment procedure was separated from the rest of the projection and, in model simulations, recruitment at time t (Rt) was calculated independently as:
- Rt = Rft × (A − OCt − OSt)(eqn 1)
where Rft is the linear recruitment function at time t, A is total area (25 cm2) and OCt and OSt, respectively, are the space occupied by individuals of C. montagui and S. balanoides at time t. Hence, free space was defined as space without barnacles and other potential occupiers of space were not included in the model. In nature, C. montagui settles and recruits during summer and autumn while S. balanoides settles and recruits during spring. Therefore, in model simulations, C. montagui recruited during period 1 (summer–autumn) and S. balanoides during period 2 (winter–spring).
Based on the measured variability in survival and recruitment, represented by the transition matrices and recruitment functions, a periodic and stochastic two-species matrix model was constructed. The model was constructed on certain biologically derived and tested assumptions (Hyder et al. 2001; Svensson et al. 2004). The assumptions were:
Individuals could either survive in their current size-class, increase in size to any of the other size-classes or die.
The population was considered open, with all recruitment into the first size-class.
Mortality was size-specific and calculated from data and independent of vital rates.
There was no spatial covariation between mortality and recruitment.
Recruitment was partly limited by space.
In the model, the simulated time-step equalled 1 year and was divided into two periods. Measured variability was included in the model by, for each time-step and period, randomly drawing each transition element and recruitment factor for each species from a normal distribution based on their mean and standard deviation and introducing them into the transition matrix. As the survival of individuals cannot exceed 1, the sum of the elements in a column cannot be larger than 1. Therefore, if the stochastic process produced a column that summed to more than 1, each element in that column was replaced with its relative proportion. The formula for this was:
- (eqn 2)
where atij is the transformed element, aij is the original element and i is the row number. The process was repeated for all j columns.
The initial population vector v(0) was set to zero and a stochastic sample path representing a total of 10 000 years was produced. A sample path of this length is not used to predict population size after 10 000 years, but as a means of estimating response variables and their variation at the stochastic steady state. In simulations, the generated sequence of transition matrices together with the recruitment functions was applied to v(0) to produce a sequence of population vectors (v(1), v(2), … , v(10 000)). The stochastic growth rate (λs), that is the geometric mean of the growth rate, was then calculated numerically by averaging a number of one-step estimates of log λs over 10 000 time units and then calculating its natural exponent (see Caswell 2001 for details).
To exclude transient effects and allow the population to reach stochastic steady state, the first 1000 years was disregarded from the analysis. For all simulations, the average population size (N), population growth rate (λs), population structure, recruitment and occupied space and the variation were calculated for each period and species. To compare the variation in population size (N) and recruitment between species with different means, the variation in population size (N) and recruitment were converted to coefficients of variation.
A perturbation analysis was designed to investigate the relative importance of different vital rates (i.e. survival, growth and recruitment) to barnacle densities and free space. As the model includes nonlinear properties (i.e. stochastic variability and space-limited recruitment) classic growth rate elasticity was not applicable (Caswell 2001). An alternative perturbation analysis was constructed, which comprised making a small positive proportional change to each transition element and recruitment separately and then calculate the change in mean population size (NCm and NSb) and available free space (F). This produced three different perturbation estimates (for NCm NSb and F) for each of the transitions in the life cycle. An identical Markov chain was used for all simulations, eluding differences in population variables resulting from variability in the stochastic sample path. Perturbation estimates (Pij) were calculated accordingly:
- Pij = ((Xnij − Xo)/Xo)/(p − 1)(eqn 3)
where Xnij is the value of the studied variable, resulting from a proportional change p to element eij, Xo is the original value of the same variable and (p − 1) is a scaling factor correcting for the size of factor p. The analysis was performed with a set of different p-values ranging on a logarithmic scale from 1·1 to 1·0001 (1·1, 1·01, … , 1·0001). Perturbation estimates converged at p = 1·01 and was determined at an accuracy of 0·01.
simulations on altered recruitment
Three scenarios were designed to simulate the effects of changes in recruitment on barnacle densities and free space. First, decreasing recruitment for S. balanoides was simulated (scenario 1) and, secondly, increasing recruitment for C. montagui was simulated (scenario 2). Thirdly, scenario 3 comprised simulating scenario 1 and 2 concurrently. Data on high recruitment for C. montagui were taken from Lisbon, Portugal (Hyder et al. 2001), and estimates of low recruitment for S. balanoides were taken from the Isle of Man (Svensson et al. 2005).
For each species there were two normal distributions of recruitment, one around a low mean and one around a high mean (Table 1). In the original model, C. montagui recruitment values (eqn 1) were drawn from their low mean distribution and S. balanoides recruitment values (eqn 1) from their high mean distribution. For scenario 1, the low mean distribution for S. balanoides was included in the recruitment process at 100 different discrete frequencies, spanning in between 0 and 1 (0, 0·01, 0·02, … 1). In scenario 2, the same procedure was repeated for C. montagui, by including the high mean distribution at 100 discrete frequencies. In scenario 3, scenario 1 and 2 were simulated simultaneously. An identical Markov chain was used in all simulations. Barnacle population sizes and amount of occupied space were recorded for each frequency in all three scenarios.