Appendix F. Calculating median dispersal distance.

Calculating median dispersal distance

Appendix C. Scaling collapse.

Scaling collapse

Appendix B. Generating psuedorandom dispersal vectors.

Generating psuedorandom dispersal vectors

Appendix D. Small speciation results and z.

Small speciation results and z

Are Anomalous Invasion Speeds Robust to Demographic Stochasticity?

<div><p>Two important issues for conservation are the range expansion of species as a result of climate change and the invasion of exotic species. Being able to predict the rate at which species spread is key for successful management. In deterministic models, the invasion speed of a polymorphic population can be faster than that of any of the component phenotypes, and these “anomalous” invasion speeds persist even when the mutation rate between phenotypes is vanishingly small. Here we investigate whether the same phenomenon is observed in a model with demographic stochasticity. The model that we use is discrete in time and space and we carry out numerical simulations to determine the invasion speed of a population that has two morphs which differ in their dispersal abilities. We find that anomalous speeds are observed in the stochastic model, but only when the carrying capacity of the population is large or the mutation rate between morphs is high enough. These results suggest that only species with large population sizes, such as many insect species, may be able to invade faster if they are polymorphic than if there is only a single morph present in the population.</p></div

Comparison of analytical and numerical predictions of the invasion speed.

<p>This is an example of the case when polymorphism results in faster invasions than either single morph. The curve represents the analytical predictions of the invasion speed in the limit given by Eqn. (13). The crosses represent numerical predictions calculated numerically using Eqn. (10), and the circles the numerically integrated predictions, when using a space increment of 0.1. Parameter values used were , , and . For the analytical prediction was varied and for the numerical simulations the values of used were 0.6, 1, 1.4, 1.8 and 2.2.</p

Anomalous invasion speeds when individual morph speeds are similar.

<p>The triangles represents the establisher morphs speed, the crosses the disperser morphs speed and the circles the invasion speed when both morphs are present. Parameter values used were , , , , and ranges from to .</p

Parameter regions where each invasion speed occurs.

<p>The area between each of the curves (given by (14) and (15)) and the axes is where the polymorphic invasion occurs at approximately the same monomorphic speed as one of the phenotypes. The area above the curves is where the polymorphic invasion occurs faster than either monomorphic invasion, with the shading from white to grey representing the extent to which the polymorphic invasion is faster. (a), (b) and (c) show the parameters used in Fig. 3. The area where the ratio of the net growth rates and dispersal rates is less than one is not shown in this figure because that region of parameter space violates our assumption of net growth rate of establisher net growth rate of disperser and dispersal rate of disperser dispersal rate of establisher. If there were a tradoff in only one of the traits, for example, if both morphs have the same net growth rate but different dispersal rates then the ratio of growth rates would be 1 and so we can see that the invasion would follow the speed of the disperser. Similarly if the morphs only differed in their net growth rate then we can see that the invasion would follow the speed of the establisher.</p

Comparison of invasion speeds with different values of .

<p>The parameter values are the same as in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0067871#pone-0067871-g003" target="_blank">Fig 3(c)</a> with the value of varied from (crosses), (diamonds), (circles), (plus) and (triangles point down). The filled circle represents the polymorphic deterministic speed which is the same for all mutation rates. The triangles point up represent the fastest single morphs speed, which here is the establisher, with the filled triangle the deterministic speed.</p

Invasion profiles with different parameter values.

<p>These show that invasion speeds are the same when (i) morphs have asymmetrical mutation rates, (ii) there is non-neutral competition and (iii) morphs have asymmetrical mutation, non-neutral competition and different carrying capacities. The simulations were initiated with the first 100 cells occupied by each phenotype at its equilibrium population density and the remaining cells unoccupied. The simulations were run on a lattice consisting of 8000 cells, using a space increment of 0.1. For all graphs each line represents the density profiles at a different time point, shown by the different shades of grey, with each time point 500 units apart. In column (A) the polymorphic invasion speed is the same as the monomorphic establisher speed; in column (B) the polymorphic invasion speed is the same as the monomorphic disperser speed, and in (C) the polymorphic invasion speed is faster than either monomorphic invasion. For all the simulations the parameters used are the same as in Fig. 3 apart from in (i) have that  = 0.001 and  = 0.00025, in (ii)  = 0.8,  = 0.7, where is the competition coefficient, and in (iii) have both the parameters used in (i) and (ii) and additionally  = 1.2 and  = 1.</p