Bad Year:
\(N^{{}^{\prime}}=\par \begin{bmatrix}\left(1-d_{1\ }\left(1-c\right)\right)\left(S_{1}-\left(1-\left(\frac{1}{f}\right)\right)v\right)&((1-\mu)d_{2}S_{2})(1-c)\\ (\left(1-\mu\right)d_{1}\left(S_{1}-\left(1-\left(\frac{1}{f}\right)\right)v\right))(1-c)&\left(1-d_{2}\left(1-c\right)\right)S_{2}\\ \end{bmatrix}N\) (3)
This model contains two variables for dispersal,\(d_{1}\ \text{and}\ d_{2}\). This is to allow for the dispersal rates between to the two sub-habitats to be altered independently. The model was run for 1000 iterations in R, and the fitness (Q ) of the two sub-habitats was calculated at the end (4). Fitness was determined by the average growth rate of the population which was calculated by the change in population size divided by the number of simulations run.
\(Q=\ \frac{\Delta N_{x}}{\Delta t_{x}}\) (4)
The log of the average growth rate \(\operatorname{(log}Q)\) was then calculated. \(\operatorname{(log}Q)\) was then compared at different dispersal rates (d1 and d2), and the relationship between the two investigated. For a detailed description of the model see the supplementary material.