Fig. 3: Optimising dispersal strategy with changing frequency of bad years. Figures 3a and 3b differ in frequency of good years. 3a has a frequency of \(\frac{1}{3}\) and 3b has a frequency of \(\frac{1}{5}\). The heat maps demonstrate the fitness (growth rate) of the population at different dispersal ratios. indicates the optimal dispersal ratio to maximise fitness. It is mostly assumed in dispersal models that sub-habitats have used the same dispersal rate for all habitats, and so this is indicated by the black line where the dispersal rate from sub-habitat 1 is equal to the dispersal rate from sub-habitat 2. indicates the optimal dispersal ratio along this line. In both, there is additional fitness when you move away from the black line and have a dispersal rate dependent on the sub-habitat and the frequency of variability. Variables used to generate graphs: v = 20, S1 = 18, S2 = 14, f in 3a = 3, f in 3b = 5.
By how much the two sub-habitats differ in fecundity is critical for contingent dispersal strategies to evolve. Consider two sub-habitats. In sub-habitat 1, the environment is highly variable, with fecundity being high in intermittently good years, and low in bad years. Sub-habitat 2 has a constant environment. In Figure 4, the fecundity in sub-habitat 2 is put on a sliding scale. If the average in sub-habitat 2 is higher than that of sub-habitat 1, then the dispersal rate from sub-habitat 2 will be 0 and sub-habitat 1 will be left empty. The optimal dispersal rate from sub-habitat 1 can then take any value (Fig. 4, region 3, see supplementary material). If the average fecundity in sub-habitat 1 is much higher than sub-habitat 2 the reverse happens and the dispersal rate from sub-habitat 1 is zero and sub-habitat 2 is left empty. When, however, fecundity in the sub-habitats are roughly the same, the optimum strategy is to have contingent dispersal rates so the that the offspring is distributed over the two sub-habitats: one with high variability and is very productive in good years, and one with low variability but is lowly productive, causing both to be of similar fitness. There is no obvious best location, and in some years one sub-habitat will be better, in other years the other. In environments such as this, site-specific, non-zero dispersal rates will evolve. In order to demonstrate this, we completed a sensitivity analysis which modulated the model parameters. Figure 5 shows the result of modulating the parameter ยต. Further results can be found in the supplementary material.