## Hybrid Zones, I: Inferior Heterozygotes

In a number of locations one finds two regions of space which contain relatively homogeneous populations, that differ considerably from each other, and are separated by a narrow zone in which hybrids are found. A textbook example is the Northern flicker which is red in the Western part of the United States and yellow in the East, with the transition occurring sharply in the western half of South Dakota, Nebraska, and Kansas. See Harrison (1990) for other examples.

There are a number of possible explanations for hybrid zones. An obvious possibility, is that each type has a selective advantage in the region it occupies. A second possibility we will explore here is that hybrids are less fit.

To start with the simplest possible situation, we will suppose that there is a single locus with two possible alleles: A and a. Since our individuals are diploid (have two copies of their chromosones) each site can be in one of four states AA, Aa, aA, or aa. To formulate the dynamics we introduce the relative fitnesses f(A,A) = f(a,a) = 1 and f(A,a) = f(a,A) = delta in (0,1].

At rate 1, each individual is "replaced". To make the new individual we first choose one parent from the four nearest neighbor sites, pick one of its alleles, and call the result u (which will be either A or a). Then choose a second parent independently, pick one of its alleles and call the result v. We accept the new individual and change the state of x to uv with probability f(u,v), otherwise x remains unchanged.

s3 Exercise. Set delta=0.9 (which represents very strong selection) and start from the random initial condition. Quite quickly blobs of the pure types develop and their sizes grow in time. The interfaces (or hybrid zones) seem to obey the rules of motion by mean curvature, i.e., curves tend to become straight and sharper curves do so at a higher rate.

Another system in which boundaries seem to follow this dynamic is the majority vote model. In that system and here we

Conjecture. Clustering occurs, i.e., for any x and y the probability state_t[x] is not equal to state_t[y] converges to 0 as t tends to infinity.

Harrison, R.G. (1990) Hybrid zones: windows on evolutionary processes. In D. Futuyama and J. Antonovics (eds) Oxford Surveys in Evolutionary Biology. Oxford U. Press