1. 2: Population genetics
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2. Thomas Malthus ( )
Malthus observed that in nature plants and animals produce far more offsprings than can survive, and that Man too is capable of overproducing if left unchecked.
Malthus was a political economist who was concerned about, what he saw as, the decline of living conditions in 19th century England.

3. He blamed this decline on three elements: The overproduction of young.
Thomas Malthus ( )
He blamed this decline on three elements:
The overproduction of young.
The inability of resources to keep up with the rising human population.
The irresponsibility of the lower classes.
To combat this, Malthus suggested that the family size of the lower class ought to be regulated such that poor families do not produce more children than they can support…

4. Thomas Malthus ( )
Darwin and Wallace read Malthus, and extended his logic further than Malthus himself could ever take.
They realized that producing more offspring than can survive establishes a competitive environment among siblings, and that the variation among siblings would produce some individuals with a slightly greater chance of survival.

5. “A simple model of population growth”
Assumptions:
Time is measured in generations, i.e., the model is discrete.
Generations are non-overlapping.
: the average progeny per individual.
: the number of individuals in time t.
Then

6. “A simple model of population growth”
The concept of “average fitness”:
Let n1,n2,…,nk be the number of k types in a population at time t-1, and the fitness of these types be w1,…,wk.
Assume N(t) = n1+n2+…nk. What is the ratio between N(t+1) and N(t)? The average fitness will be:
This value changes with time.

7. “A simple model of population growth”
Assume that the average fitness from generation to generation is not constant. Let w(1) be the average fitness in generation 1, w(2) in generation 2, etc... Now:
G(w) is the geometric mean of the w’s.

8. “A less simple model of population growth”
In this model N(t) is large, and in a time interval Δt, a fraction bΔt of the population produce an offspring, and a fraction dΔt die. Thus,
Where m is (b-d). As Δt->0, this becomes
Comparing the 2 models we obtain:

9. “A less simple model of population growth”
Thus, m=ln(w), and a w of 2 (population doubles each year), is the same as m=ln(2)=0.693.
m is called the Malthusian parameter.

10. The ln(1+x) approximation
Taylor series of ln(1+x):
Hence, for very small x we can approximate ln(1+x) by x.

11. “When growth rate is small”
w=1 indicates no growth at all. Assume that the population grow, but very slowly, w=1+s, s is small. Now,
Comparing the 2 models we obtain that for population with a small growth rate:

12. Population without birth
In this standard model
Where m is (b-d). Now assume that b=0.
What is the probability that an individual lives more than t years? Since all individuals have the same chance to die, the answer is N(t)/N(0)=e-dt.
Let X be a random variable denoting the life span of an individual, then
Hence, the life span in such a case is exponentially distributed.

13. Mixed Continuous Population
Let n1(t), n2(t),,…, nk(t), be the number of k types in a population at time t, and their Malthusian parameters be m1,…,mk. Assume the population are independent.
The m parameter of each population type is fixed. But we will study the average m parameter over the entire population, which is not fixed:

14. Mixed Continuous Population
Let n1(t), n2(t),,…, nk(t), be the number of k types in a population at time t, and their Malthusian parameters be m1,…,mk. Assume the population are independent.
The m parameter of each population type is fixed. But we will study the average m parameter over the entire population, which is not fixed:

15. Mixed Continuous Population
Conclusion: at each time point, the population size increases as a “standard” continuous population, with m equal to the weighted mean.

16. Mixed Continuous Population
We next ask how change with time. It is reasonable to assume that it increases with time (those type of population with high m values will contribute more to m each generation). The question is what is the rate at which increases.

17. Mixed Continuous Population

18. Mixed Continuous Population
This equation is Fisher’s “Fundamental Theorem of Natural Selection” (1930).
It states that the increase in average fitness (weighted Malthusian parameters) at each time t, equals the variance of these parameters, at that time.
Note, with time the variance decreases, till in practice the population is dominated by the type with the highest fitness.