(a)
In a population primarily consisting of A types with only a small proportion (
x
) of
invading T types, the A-type fitness is
F
(A) = 864(1 –
x
) + 936
x
= 864 + 72
x
and the T-type fitness is
F
(T) = 792(1 –
x
) + 972
x
= 792 + 180
x.
F
(A) >
F
(T) as long as 864 + 72
x
> 792 + 180
x,
or 72 > 108
x,
or
x
< 72/108 = 2/3. An all-A population can’t be invaded by T types unless the T types are more than two-
thirds of the population, so a small number of mutant Ts cannot successfully invade. Similarly for a small
proportion (
x
) of invading Ns:
F
(A) = 864(1 –
x
) + 1,080
x
= 864 + 216
x
and
F
(N) = 648(1 –
x
) + 972
x
=
648 + 324
x.
F
(A) >
F
(N) as long as 864 + 216
x
> 648 + 324
x,
or 216 > 108
x,
or
x
< 216/108 = 2. This is
always true because
x
must be between 0 and 1. Therefore, the A types are always fitter than the N types,
so an all-A population can’t be invaded by Ns.
(b)
In a primarily N population, mutant Ts have fitness
F
(T) = 972(1 –
x
) + 972
x
= 972
and
Ns have fitness
F
(N) = 972(1 –
x
) + 972
x
= 972. The fitnesses are equal, so Ts and Ns do equally well in
the population and Ns cannot prevent Ts from invading. A population of Ns invaded by Ts thus exhibits
neutral stability, where both the primary and secondary criteria for an ESS give ties. Since neither type is
more fit than the other, their proportions in the population will persist, only slightly adjusting as
mutations occur.
Against a group of mutant As, the N types have fitness
F
(N) = 972(1 –
x
) + 648
x
= 972 – 324
x
and the A types have fitness
F
(A) = 1,080(1 –
x
) + 864
x
= 1,080 – 216
x.
F
(N) >
F
(A) when 972 – 324
x
>
1,080 - 216
x,
or 108 + 108
x
< 0. This condition never holds, so As can invade an all-N population. An all-
N population is unstable when an A mutation is possible.
(c)
In a primarily T population, mutant As have fitness
F
(A) = 936(1 –
x
) + 864
x
= 936 –
72
x
,
and the T-type fitness is
F
(T) = 972(1 –
x
) + 792
x
= 972 – 180
x.
F
(T) >
F
(A) when 972 – 180
x
> 936
– 72
x,
or 36 > 108
x,
or
x
< 36/108 = 1/3. An all-T population can’t be invaded by A types unless the A
types are more than one-third of the population, so a small number of mutant As cannot successfully
invade. When the mutants are type N,
F
(T) = 972(1 –
x
) + 972
x
= 972,
and the mutant Ns also have
fitness
F
(N) = 972(1 –
x
) + 972
x
= 972. Again, the fitnesses are equal, so Ts and Ns do equally well in the
population, and Ts cannot prevent Ns from invading.
S3.
(a)
The payoff table is at right.
Solutions to Chapter 13 Solved Exercises
2 of 10
Column
A
T
Row
A
20, 20
11, 35
T
35, 11
6, 6