BIOL 4120 Effective Population Size

BIOL 4120

Principles

Ecology

Opuntia cactus on a Canaveral sand dune

When there are unequal numbers of each sex, random error will be more influential in the rarer sex. If you report only the population size, it is usually assumed that the population is 50% male and 50% female. This is the most common situation. Consider, however, two populations: one with 100 of each sex and the other with 20 males and 180 females. The next generation will receive 50% of its genes from each sex, no matter what the sex ratio is. In the first population, 100 males contribute and random error in the transmission of alleles to the next generation happen at a rate appropriate for a population of that size. In the other population, there are only 10 males, and random events are much more likely to influence which alleles get passed to the next generation. You might argue that this effect is countered by the larger female population -- 180 versus 100. However, the effects are not symmetric. The male ratio (10 males to 100 males) is not the reciprocal of the female ratio (180 females to 100) and the influence of the change in male population is not completely offset by the greater number of females.

This means that we can't compare the effect of genetic drift in different populations unless they both have the same sex ratio. This is OK, because there is a way to adjust for the effect of an excess of males or females. What we do is to calculate the size of a population that has a 1 to 1 sex ratio, but is equally prone to the influence of genetic drift as is the population with a unequal sex ratio. This population size is called the effective population size, and we can compare any two effective population sizes because they all are assumed to have a 1 to 1 sex ratio.

We will calculate the effects in the problems below. If you want to check your answers, send me an email, come into the office, or call. I'll give you my solutions and discuss any problems you might have.

As an exercise in calculating the effect, you should fill in this table (remember how computer savvy you are now) and construct a graph of N_e versus the sex ratio. The sex ratio can be either males divided by females or females divided by males. I used the first ratio when I did this problem.

Number of Males	Number of Females	sex ratio m/f	4NmNf	Nm+Nf	Ne

100	100
90	110
80	120
70	130
60	140
50	150
40	160
30	170
20	180
10	190
5	195
1	199

A second instance in which one can't simply calculate the effect of genetic drift is when a population is fluctuating in size. In the years when the population is small, the effect is large and when the population gets large, the influence of genetic drift is small. So, how is one to judge the overall effect. This involves and averaging of population sized. However, we have that same asymmetric effect as above. Little numbers count more. When you calculate the average (called the arithmetic mean), big and little sizes have the same influence. This means that we have to look for a mean that gives more weight to small than large size. This mean is called the Harmonic Mean and is calculated as the reciprocal. I give you the formula here from the notes and then an easier formula (a simple algebraic rearrangement of the previous formula).

Sample (t)	Pop. Size (Nt)	1/Nt
1	500	0.002
2	500	0.002
3	500	0.002
4	500	0.002
5	500	0.002
6	500	0.002
7	500	0.002
8	500	0.002
9	500	0.002
10	500	0.002

Total	5000	0.02
	Arithmetic Mean	Effective Population Size (Ne)
	500	500.000

Let's calculate it when there is one bad year, when only 50 are in the population:

Sample (t)	Pop. Size (Nt)	1/Nt
1	50	0.02
2	500	0.002
3	500	0.002
4	500	0.002
5	500	0.002
6	500	0.002
7	500	0.002
8	500	0.002
9	500	0.002
10	500	0.002

Total	4550	0.038
	Arithmetic Mean	Effective Population Size (Ne)
	455	263.158

You should try to calculate both the arithmetic mean and Ne to make sure you know how to do it. Notice the big gap between the two population sizes. The average greatly overestimates the size and will greatly underestimate the effect of genetic drift. Look at what happens when the population gets large for a year:

Sample (t)	Pop. Size (Nt)	1/Nt
1	5000	0.0002
2	500	0.002
3	500	0.002
4	500	0.002
5	500	0.002
6	500	0.002
7	500	0.002
8	500	0.002
9	500	0.002
10	500	0.002

Total	9500	0.0182
	Arithmetic Mean	Effective Population Size (Ne)
	950	549.451

As you can see, the arithmetic mean once again over estimates the effective population size over this time period and under estimates the importance of genetic drift. Finally, what is the effect of more variation than a single bad year. You can calculate this:

Sample (t)	Pop. Size (Nt)	1/Nt
1	50
2	50
3	500
4	500
5	500
6	500
7	500
8	500
9	500
10	500

Total
	Arithmetic Mean	Effective Population Size (Ne)

Now, a last point. I have calculated the arithmetic mean and effective population size when you systematically add bad years. The results are in the table below. The first column is the number of bad years (with a value of 50) out of 10 total years. The years not bad have a population size of 500, as above. You should graph the arithmetic mean and effective population size versus the number of bad years to get a feel for how these two things relate to one another.

Number of years with 50	Arithmetic Mean	Ne
1	455	263.158
2	410	178.571
3	365	135.135
4	320	108.696
5	275	90.909
6	230	78.125
7	185	68.493
8	140	60.976
9	95	54.945
10	50	50.000

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