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C00007 00003		A simple mathematical example is given by the Lotka-Volterra
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.require "memo.pub[let,jmc]" source
.cb BIOLOGICAL ADVANTAGES OF TERRITORIALITY AND EMIGRATION

.CB "John McCarthy, Stanford University"


	It may sometimes be advantageous for a population to
put much of its efforts into finding new habitats at the expense
of fully populating its existing habitats.  For example, suppose
frog ponds sometimes dry up and new frog ponds sometimes
form.  A frog population that feels crowded as soon as it has
reasonably populated a frog pond and expels its surplus population
to hop randomly across dry land is likely to quickly occupy
any new frog ponds that open up within hopping distance.  This is
advantageous for the population even though in most years no new
frog ponds arise and all the emigrants die when they have hopped
as far as possible.  More generally, any well-established
 population "should" send all the emigrants it can spare
into nearby potential ecological niches even
at the cost of losing them almost all the time.  While emigration
in space may exist as a specific evolved pattern of behavior, the
concept also applies to trying new food sources and other behavioral
niches.

	The object of this paper is give some mathematical
models that permit determining the
optimum fraction of a population's resources to be put into emigration.
Perhaps this will provide a teleological explanation of territoriality.
Perhaps we can regard regular bird migration as a development of
emigration behavior.

	A possible real example of this behavior is reported by xxx in
yyy.  It seems that mice confined to a field by fences reach stable population
levels several times those reached when young pregnant females are allowed
to emigrate.  Presumably they mostly emigrate to territory unsuitable for
mice or are prevented from surviving by predation or territoriality.  Otherwise,
immigrants would balance emigrants and the high population level would be
attained anyway.

	The evolution of emigration has the same problems as other
hypothesized "altruistic" behaviors.  Several considerations apply.
First, the genes of the first emigrants to the new frog pond may
represent a large fraction of the genes of its ultimate occupants.
Second, if frog ponds often dry up, then the genes of determined
stay-at-homes are likely to be lost.  Third, a trait that is usually
of benefit to the group and not to its posessor may evolve and become
fixed under the possibly rare conditions under which it is a benefit
to the individual or its immediate kin.  The subsequent competition
between groups and between individuals within groups may affect the
prevalence of the trait in quite complicated ways.
	A simple mathematical example is given by the Lotka-Volterra
predator-prey differential equations modified to model two
predator-prey regions with different coefficients and small
migration coefficients for both prey and predator between the
regions.  

	The original Lotka-Volterra system of differential equations
is 

	%2xq. = (a - b y)x%1

	%2yq. = (-c + d x)y%1,

where ⊗x is the population of prey and ⊗y is the population of
the predator.  The idea is that the prey has a net birth rate ⊗a and
a death rate ⊗b_y proportional to the population of predator,
and the prey has a net death rate ⊗c and a birth rate ⊗d_x proportional
to the population of prey.

The solution of this system of differential equations is an equilibrium
point characterized by

	%2x0 = c/d%1 and %2y0 = a/b%1

and a system of ovals (ellipses close to the equilibrium point) around
the equilibrium point.  For some initial conditions there are runaway
solutions in which either the predators or both die out.

	Now suppose there are two regions each with the same prey and
predators but with different coefficients on account of different
environment.  Suppose further that there is migration between the
two regions proportional to the populations of prey and predator
in the regions.  The equations become



	Intuitively what happens is that each region has its own
limit cycle, assuming the values of the coefficients are appropriate,
and each limit cycle has its own period.  Assuming that the
periods are not commensurate, the two systems will often be out
of phase, e.g. when one has a small number of prey, the death
rate of the predators will be mitigated by prey immigrating from
the other region.  The effect should be to reduce the size of
the limit cycles and for appropriate values of the coefficients
to stabilize the equilibrium points at the centers of the limit
cycles.  Actually the mathematical situation is complicated by the
fact that the system now has four dimensions instead of two.

Roughgarden
Bartlett Methuen stochastic models in biology