/LMAR=0/XLINE=3/USET=13␈↓ α∧␈↓␈↓ u1


␈↓ α∧␈↓α␈↓ αuBIOLOGICAL ADVANTAGES OF TERRITORIALITY AND EMIGRATION

␈↓ α∧␈↓α␈↓ ¬
John McCarthy, Stanford University

␈↓ α∧␈↓␈↓ αTIt␈αmay␈αsometimes␈αbe␈αadvantageous␈αfor␈αa␈αpopulation␈αto␈αput␈αmuch␈αof␈αits␈αefforts␈αinto␈αfinding
␈↓ α∧␈↓new␈αhabitats␈αat␈αthe␈αexpense␈αof␈αfully␈αpopulating␈αits␈αexisting␈αhabitats.␈α For␈αexample,␈αsuppose␈αfrog
␈↓ α∧␈↓ponds␈α⊂sometimes␈α⊂dry␈α⊂up␈α⊂and␈α⊂new␈α⊂frog␈α⊂ponds␈α⊂sometimes␈α⊂form.␈α⊂ A␈α⊂frog␈α⊂population␈α⊂that␈α⊂feels
␈↓ α∧␈↓crowded␈αas␈αsoon␈αas␈αit␈αhas␈αreasonably␈αpopulated␈αa␈αfrog␈αpond␈αand␈αexpels␈αits␈αsurplus␈αpopulation␈αto
␈↓ α∧␈↓hop␈α
randomly␈α
across␈α
dry␈α
land␈α
is␈α
likely␈α
to␈α
quickly␈α
occupy␈α
any␈α
new␈α
frog␈α
ponds␈α
that␈α
open␈α
up␈α
within
␈↓ α∧␈↓hopping␈α
distance.␈α
 This␈α
is␈α
advantageous␈α
for␈α
the␈α
population␈α
even␈α
though␈α
in␈α
most␈α
years␈α
no␈α
new
␈↓ α∧␈↓frog␈α∂ponds␈α∂arise␈α∂and␈α∂all␈α∂the␈α∂emigrants␈α∂die␈α∂when␈α∂they␈α∂have␈α∂hopped␈α∂as␈α∂far␈α∂as␈α∂possible.␈α∂ More
␈↓ α∧␈↓generally,␈α
any␈α
well-established␈α
population␈α
"should"␈α
send␈α
all␈α
the␈α
emigrants␈α
it␈α
can␈α
spare␈α
into␈α
nearby
␈↓ α∧␈↓potential␈α
ecological␈α
niches␈α
even␈α
at␈α
the␈α
cost␈α
of␈α
losing␈α
them␈α
almost␈α
all␈α
the␈α
time.␈α
 While␈α
emigration␈α
in
␈↓ α∧␈↓space␈αmay␈αexist␈αas␈αa␈αspecific␈αevolved␈αpattern␈αof␈αbehavior,␈αthe␈αconcept␈αalso␈αapplies␈αto␈αtrying␈αnew
␈↓ α∧␈↓food sources and other behavioral niches.

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

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

␈↓ α∧␈↓␈↓ αTThe␈α∩evolution␈α∩of␈α∩emigration␈α∩has␈α∩the␈α∩same␈α∩problems␈α∩as␈α∩other␈α∩hypothesized␈α∩"altruistic"
␈↓ α∧␈↓behaviors.␈α
 Several␈α
considerations␈α
apply.␈α
 First,␈α
the␈α
genes␈α
of␈α
the␈α
first␈α
emigrants␈α
to␈α
the␈α
new␈α
frog
␈↓ α∧␈↓pond␈αmay␈αrepresent␈αa␈αlarge␈αfraction␈αof␈αthe␈αgenes␈αof␈αits␈αultimate␈αoccupants.␈α Second,␈αif␈αfrog␈αponds
␈↓ α∧␈↓often␈αdry␈αup,␈αthen␈αthe␈αgenes␈αof␈αdetermined␈αstay-at-homes␈αare␈αlikely␈αto␈αbe␈αlost.␈α Third,␈αa␈αtrait␈αthat
␈↓ α∧␈↓is␈αusually␈αof␈αbenefit␈αto␈αthe␈αgroup␈αand␈αnot␈αto␈αits␈αposessor␈αmay␈αevolve␈αand␈αbecome␈αfixed␈αunder␈αthe
␈↓ α∧␈↓possibly␈αrare␈αconditions␈αunder␈αwhich␈αit␈αis␈αa␈αbenefit␈αto␈αthe␈αindividual␈αor␈αits␈αimmediate␈αkin.␈α The
␈↓ α∧␈↓subsequent␈α∂competition␈α∂between␈α∂groups␈α∂and␈α∂between␈α∂individuals␈α∂within␈α∂groups␈α∂may␈α∂affect␈α∂the
␈↓ α∧␈↓prevalence of the trait in quite complicated ways.

␈↓ α∧␈↓␈↓ αTA␈α
simple␈α
mathematical␈α
example␈α
is␈α
given␈α
by␈α
the␈α
Lotka-Volterra␈α
predator-prey␈α
differential
␈↓ α∧␈↓equations␈α∩modified␈α∩to␈α∩model␈α∩two␈α∩predator-prey␈α∩regions␈α⊃with␈α∩different␈α∩coefficients␈α∩and␈α∩small
␈↓ α∧␈↓migration coefficients for both prey and predator between the regions.

␈↓ α∧␈↓␈↓ αTThe original Lotka-Volterra system of differential equations is

␈↓ α∧␈↓␈↓ αT␈↓↓xq. = (a - b y)x␈↓

␈↓ α∧␈↓␈↓ αT␈↓↓yq. = (-c + d x)y␈↓,

␈↓ α∧␈↓␈↓ αTwhere␈α
␈↓↓x␈↓␈α
is␈α
the␈α
population␈α
of␈α
prey␈α
and␈α
␈↓↓y␈↓␈α
is␈α
the␈α
population␈α
of␈α
the␈α
predator.␈α
 The␈α
idea␈α
is␈α
that
␈↓ α∧␈↓the␈αprey␈αhas␈αa␈αnet␈αbirth␈αrate␈α␈↓↓a␈↓␈αand␈αa␈αdeath␈αrate␈α␈↓↓b y␈↓␈αproportional␈αto␈αthe␈αpopulation␈αof␈αpredator,
␈↓ α∧␈↓and the prey has a net death rate ␈↓↓c␈↓ and a birth rate ␈↓↓d x␈↓ proportional to the population of prey.
␈↓ α∧␈↓␈↓ u2


␈↓ α∧␈↓␈↓ αTThe␈αsolution␈α
of␈α
this␈α
system␈α
of␈α
differential␈α
equations␈α
is␈αan␈α
equilibrium␈α
point␈α
characterized
␈↓ α∧␈↓by

␈↓ α∧␈↓␈↓ αT␈↓↓x0 = c/d␈↓ and ␈↓↓y0 = a/b␈↓

␈↓ α∧␈↓␈↓ αTand␈α
a␈αsystem␈αof␈αovals␈α(ellipses␈αclose␈αto␈αthe␈αequilibrium␈αpoint)␈α
around␈αthe␈αequilibrium␈αpoint.
␈↓ α∧␈↓For␈αsome␈αinitial␈αconditions␈αthere␈αare␈αrunaway␈αsolutions␈αin␈αwhich␈αeither␈αthe␈αpredators␈αor␈αboth␈αdie
␈↓ α∧␈↓out.

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

␈↓ α∧␈↓␈↓ αTIntuitively␈αwhat␈αhappens␈αis␈αthat␈αeach␈αregion␈αhas␈αits␈αown␈αlimit␈αcycle,␈αassuming␈αthe␈αvalues␈αof
␈↓ α∧␈↓the␈α⊂coefficients␈α⊂are␈α⊂appropriate,␈α⊂and␈α⊂each␈α⊂limit␈α⊂cycle␈α⊂has␈α⊂its␈α⊂own␈α⊂period.␈α⊂ Assuming␈α⊂that␈α⊂the
␈↓ α∧␈↓periods␈α
are␈α∞not␈α∞commensurate,␈α∞the␈α∞two␈α∞systems␈α∞will␈α∞often␈α
be␈α∞out␈α∞of␈α∞phase,␈α∞e.g.␈α∞when␈α∞one␈α∞has␈α∞a
␈↓ α∧␈↓small␈α
number␈α
of␈α
prey,␈α
the␈α
death␈α
rate␈α
of␈α
the␈α
predators␈α
will␈α
be␈α
mitigated␈α
by␈α
prey␈α
immigrating␈α
from
␈↓ α∧␈↓the␈αother␈αregion.␈α The␈αeffect␈αshould␈αbe␈αto␈αreduce␈αthe␈αsize␈αof␈αthe␈αlimit␈αcycles␈αand␈αfor␈αappropriate
␈↓ α∧␈↓values␈α⊂of␈α⊂the␈α⊂coefficients␈α⊂to␈α⊂stabilize␈α⊂the␈α⊂equilibrium␈α⊂points␈α⊂at␈α⊂the␈α⊂centers␈α⊂of␈α⊂the␈α⊂limit␈α⊂cycles.
␈↓ α∧␈↓Actually␈α⊂the␈α⊂mathematical␈α⊂situation␈α⊂is␈α⊂complicated␈α⊂by␈α⊂the␈α⊂fact␈α⊂that␈α⊂the␈α⊂system␈α⊂now␈α⊂has␈α⊂four
␈↓ α∧␈↓dimensions instead of two.

␈↓ α∧␈↓␈↓ αTRoughgarden Bartlett Methuen stochastic models in biology
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