Random dispersal speeds invasions
There's been a controversy in theoretical ecology over invasions for a number of years now. One question is how does the randomness (aka stochasticity) of dispersal events alter the speed of the invasion.
Consider a density-independent invasion process where the population density \(n_t(x)\) is a solution of the integrodifference equation
\[ n_{t+1}(x) = \mathscr{R}_0 \int_{-\infty}^{\infty} k(x-y) n_t(y) \mathrm{d} y\]
and the dispersal kernel is a shifted Gauss distribution
\[ k(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{- \frac{\left(x- u\right)^{2}}{2 \sigma^{2}}}.\]
This kernel converges to a delta-function as the standard deviation \(\sigma\) vanishes, which corresponds to the case of fixed deterministic dispersal. So we can use it to see how stochasticity alters invasion speed. The dispersal kernel's moment generating function
\[ M[k](s) = e^{ s^2 \sigma^{2}/2 + s u } \]
Using standard methods for density-independent branching processes, we can determine the wave-speed as a function of wave-number...
\[ c(s) = \frac{s \sigma^{2}}{2} + u + \frac{1}{s} \log{\left(\mathscr{R}_{0} \right)} \]
The minimum wave speed is then
\[ c^* = \sigma \sqrt{2 \log{\left (\mathscr{R}_{0} \right )}} + u \]
So increased dispersal stochasticity accelerates the waves compared to deterministic dispersal (which is the delta-function).