Will be talking about limited dispersal– get traveling waves, etc.
We are assuming that the new allele is established. How quickly does it take over the population? Spread of advantageous gene, spread of disease, etc.
Example of plague: slow, only 300-600km/yr. SARS: very fast. 6 months becomes global. Broad range dispersal of a dollar bill– long tail.
Noisy traveling waves with local dispersal. Spread of advantageous allele follows a differential equation (Fisher and Kolmogorov): diffusion + growth. Speed of this wave: can derive form by dimensional analysis. If we consider discrete population, introduces a noise term. Affects the wave significantly. Leads to a significant correction of the wave speed. (Brunet and Derrida 1998)
Long range dispersal. Rather than a diffusion term, integrate WRT a kernel. So far, mostly mean field approaches.
Simulations. Start with one seed on a $d={1,2}$ lattice. Draw new seeds from an exponential distribution centered on previous seeds.
Three parameter regimes:
Determine the form of the radius function by a beautiful self-consistency argument. This comes from reflecting the shape of the funnel to determine the boundary. Saddle point approximation gives recurrence relation -> diff equation.
Helps us understand what happens when I matters. These three parameters give three different type of infection trees.