## 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

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