Some Genetics, Demography, and Computer Simulation
Think about how to understand the consequences of a population that is growing at a rate of 5% per generation. In this case we can just write down an algebraic expression: call the initial population P0, the population at generation t Pt , then
P1 = P0 (1+.05)
P2 = P1(1+.2) = P0 (1+.05)2
and in general
Pt = P0(1+.05)t
giving us a general solution for any time t in the future. For example in 500 years or 25 generations an initial population of 100 people would grow to P25 = 100(1.05)25 or 339 people. Fortunately in this case the solution is easy and we can get the answer with a calculator.
But what if the average number of daughters per woman in this population were 1.2 but individual women varied at random in their actual output of daughters. If births of daughters occur to women as a process with a constant hazard per unit of time then the resulting number follows what is called a poisson distribution with a mean of 1.2. We can easily simulate such a scenario on a computer, for example with a spreadsheet. We start with 100 women, draw the number of daughters she has from a poisson distribution, and the total number of daughters now is our next generation. We simple repeat this process 25 times. Since we are interested in the variation from one trial to the next it is interesting to plot 20 such trials, 20 possible population histories, along with the result of the algebraic solution. The heavy black line in figure A1 is the algebraic solution of the model while the lighter lines are outcomes of our simulations with randomness incorporated. The algebraic result is called the deterministic model while the simulations the incorporate randomness are outcomes of the stochastic version of the model.
We learn that a small amount of randomness in births per woman leads to a lot of variation in the population after 500 years: the average achieved size is over 300 in agreement with the deterministic model but among the 20 worlds we simulated the population sometimes remained small, close to 100, and sometimes grew a lot, to nearly 600.
The growth rate in this example, 5% per generation, is glacially slow compared to what we see at any time in low technology human populations but extremely rapid compared to long term average growth rates in human history. Partly this difference reflects medicine and public health measures that have been made available to even the most remote humans, but it also reflects the boom and bust nature of most of human existence for most of our history. On short time scales, centuries for example, population histories have looked jagged as wars, famines, and epidemics took their toll, then populations grew at rates like 5% per generation or often much faster, then some other misfortune reduced numbers again.
Figure A1 Model outcomes of population growth of 5% per generation. The heavy line is the deterministic outcome while the others are various simulated histories with the number of daughters per woman random.
A Model with Carrying Capacity
Here we consider populations in a finite environment where growth is ultimately limited by resources. Initially the population is at such a low density that there is no competition for resources among people. Births and deaths occur at a constant per person rate. There is no age structure, no youth nor old people, so everyone is subject to the same rates—these assumptions make our algebraic models work and they reflect very well what happens in more realistic (but more complicated) model. We set the birth and death rates of the population to plausible generic values for low technology human populations of 50 births per thousand people per year and 30 deaths per thousand people per year. The difference, 20 per thousand per year, is the intrinsic growth rate, 20 per thousand or 2 per hundred, 2% per year. In the absence of any limitation the population grows according to this rate, exactly like money at compound interest. After a year an initial colony of 1000 people would grow to 1000 × (1.02) or 1020 people, after two years to 1020 × (1.02) or 1040 people (to the nearest whole person), and after a generation of 25 years the population size would be 1000 × (1.02)25 or 1641 people (again rounding off fractional people.) This population would double in about 35 years and would double slightly more than 14 times in 500 years to a size of nearly 20 million people. Compare the population in the initial section of the appendix that only tripled in 500 years.
There are two important points to be made about this example so far. First, demographers and economists conventionally describe birth and death rates as rates per year per thousand, hence our figures of 50 and 30. Second, geneticists and evolutionary biologists follow the convention of giving rates per generation rather than per year. In terms of a 25 year human generation an annual growth rate of 2% per year translates to a generational rate of 64.1%. The per-person per-generation growth, 1.641, is called the mean fitness of the population.
The modest growth rate of 2% per year is commonplace among human populations yet a growth of 2,000,000% over 500 years seems and is outlandish. Early in the process resources would become scarce and the rate of growth would slow. Assuming the initial colony occupied 100 square miles, the expanded population after five centuries would need to occupy nearly 2,000,000 (two million) square miles, about the area of Argentina or Kazakhstan. This is explosive growth in historical time but it corresponds to run of the mill population growth on the time scale of human lifetimes. We know that over the long period from the modern human diaspora out of Africa about 45,000 years ago to the industrial revolution about 200 years ago human numbers grew but at long term rates far below our modest 2% per year, far below even the 5% per generation of our initial example. On this long time scale they hardly grew at all. It is likely that most of the time populations were growing at rates like our 2%, perhaps slower, but that there were frequent catastrophic events like wars, famines, and plagues that cut populations back.
There is a convenient and standard way to make a model of population limited by resources called the logistic model. This may not be entirely accurate but it is simple and, given our poor understanding of detailed dynamics, more than good enough. The idea is that there is some carrying capacity K of the environment. Populations below the carrying capacity in size can grow while populations above the carrying capacity decline until they reach K. If we write Pt for population in some year t and Pt+1 for population the following year then simple population growth like compound interest, called geometric, follows this formula:
where R is just the intrinsic growth rate. Since we are concerned with evolution and its interaction with demography we will express rates in generational rather than annual units from now on. Thus we will write R=.641 as the intrinsic growth rate since a rate of 2% per year corresponds to growth of 64.1% per generation. In order to write a logistic model we imagine that the growth rate R is damped by the current ratio of population to carrying capacity:
In an empty environment without intraspecific competition population P is much less than carrying capacity K and population growth is almost the same as the simple geometric case. But as population increases the ratio P/K becomes significant, growth slows down, and eventually stops when population reaches carrying capacity, that is when P=K. If the carrying capacity of the environment into which our population moved was 10,000 people then the population would grow at a decreasing rate to reach 10,000. (Actually it would never quite reach 10,000 but such a mathematical nicety is irrelevant to thinking about a rough model of the world.)
What if the carrying capacity is not static but increases with the number of people? For example we might imagine that more people bring more farmland under cultivation so that K itself changes. It turns out (Cohen 1995) that nothing much changes if the increase in carrying capacity K is proportionally less than the increase in population P as would happen if the best land were cleared first while lower and lower quality land were subsequently brought under cultivation. The population still approaches a (new larger) carrying capacity so that as equilibrium is approached population P is equal to carrying capacity. The end result is that the standard of living, by which we mean the ratio of resources to people K/P, is still one. There are more people but they are not living any better than they did before the new land was cleared.
An interesting variant of this model, now, is to introduce a new source of mortality, perhaps disease or warfare. In areas of central Africa with high levels of falciparum malaria the cost to fitness of an individual may be around 25%: that means that with malaria an average individual will leave 25% fewer living descendants one generation later. With a growth rate of 20 per thousand per year an average individual has 1.64 daughters one generation later. If malaria now decreases fitness by 25% the average individual will only have 75% of 1.64 or 1.23 daughters one generation later. In terms of annual rates the malaria cuts population growth from 2% to 0.8% per year. (Notice that we count only daughters since our model is of a simple population that does not take into account sexual reproduction.)
Now we can consider the fixed carrying capacity K and examine the consequences for the population and for individual well-being. The algebraic model now becomes (writing M for the extra density-independent death rate, from malaria in our example but also likely to be from violence and local warfare):
This is starting to look messy, but we can easily find the equilibrium population, that is the population that would remain unchanging in this environment with the extra mortality. We simply set Pt+1 equal to Pt, rearrange some terms, and find that the new equilibrium is at
The meaning of this doesn’t immediately jump out at us, but if we substitute our assumed values, an intrinsic growth rate R of 0.64 and an extra mortality rate M of 0.25 we obtain
P/K = 0.39/0.64 ~ 0.61
The population now equilibrates at 61% of the old carrying capacity. A more interesting way to summarize what we have found by manipulating the logistic model is in terms of the standard of living, where a value of 1 means the bare subsistence minimum compatible with life and reproduction and a value of, say, 5, means that there is five times the subsistence minimum amount of resources available to the average person. In our model population the standard of living is the reciprocal of 0.61 or 1.6. There is more than half again as many resources per person now than there was before malaria appeared. What this means on the ground is that people do not have to work very hard to get enough to eat, that there is fruit on the trees for plucking, and that there are not great labor demands on anyone.
Gregory Clark (Clark 2007) points out that the medieval Englishman had a higher standard than a medieval Japanese because there was much more sewage and filth in England and so a heavier burden of disease. This extra disease translated, as in our malaria example, to a lower population density and higher standard of living.
What are the social consequences of this new disease for our population? The most important immediate consequence is that there are plentiful resources for everyone and so, following the nature of the creature, males withdraw from subsistence work as they find that they can simply parasitize women for food. In much of central Africa the result is societies in which men don’t do anything very useful and women provision themselves, their children, and the men. The euphemism in economics for this kind of society is “female farming system.” Left free of the demands of subsistence we expect the men will start hanging out together, perhaps even all moving into a village men’s house (not so common in Africa). This may soon lead to local and regional raiding and warfare and an entrenched culture of local violence.
The warfare itself may cause enough excess death that a female farming system is sustainable even in the absence of a killer disease. Highland New Guinea, for example, seems to have had a classical cad system for millennia or more. (Ethnographic shorthand for societies where males put a lot of effort into competition with other males is that they are cad societies as opposed to dad societies where male effort is directed to provisioning a male’s own family.) Of course there are other contributing factors here like the extremely broken terrain that makes the establishment of a larger polity with an effective constabulary difficult or impossible.
Figure A2 Population size over time of a population of 100 introduced into an empty area with a carrying capacity of 1000. After 350 years falciparum malaria appears and the sickle cell gene appears after 750 years.
Adaptation to Malaria: Sickle Cell
We have so far looked at the demography of a typical low technology human population, first in an unrestricted environment, then in an environment in which there is a finite carrying capacity. Even when the number of humans itself changes the carrying capacity, as when more land is brought under cultivation, the qualitative result is the same: population grows to reach some carrying capacity and stays there. At this point the standard of living, a measure of resources per person, has declined to unity, i.e. the Malthusian limit. With fewer resources per person population would decline. We then introduced falciparum malaria and saw that the new higher death rate caused a decline in population to slightly more than half its former value along with a corresponding increase in resources per person, the standard of living. The new disease brought misery to many, especially parents faced with the death of a child. On the other hand there is more food and shelter and a better material life for the survivors.
Now we consider an introduced mutation in some members of the population so that they transmit the gene that produced sickle-cell hemoglobin HbS rather than the “normal” HbA version of the hemoglobin molecule. The sickle cell gene that is found in Africa and Asia actually represents several independent occurrences of the same mutation, but only several. Our population of 600 is so small that it is an unlikely target of one of the founding mutations: it is more likely that the new hemoglobin variant is introduced by migrants or visitors.
The new gene codes for a form of the hemoglobin molecule that barely does its job. People with two copies of HbS have the disease sickle cell anemia: they almost certainly die early in the central African environment while in modern industrial societies they suffer impaired and shortened lives. On the other hand people with one copy of HbS and one copy of the original version HbA are protected against malaria. This is called “heterozygote fitness superiority” since they have more successful offspring than either the HbA/HbA homozygote, who is more vulnerable to malaria, or the HbS/HbS homozygote, who likely suffers early death. In order to understand how evolution works in this case we consider individual genes. Individuals are formed by a process that is close enough to random mating. A very rare HbS in this population will be in a body with a random gene from the population at this locus, and since HbS is rare it will almost certainly find itself matched with HbA. The resulting heterozygote has a fitness of 1.64 on average if we assume that the protection against malaria is complete. Meanwhile most people are HbA/HbA homozygotes with an average fitness of 1.23. The fitness of HbS genes is higher when they are rare and they increase in frequency. But as they become more common their average fitness declines since more and more often an individual is formed with two copies of HbS chosen from the gene pool. Eventually equilibrium is reached in which 82% of the genes at this locus are HbA and 18% are HbS. Figure A2 show the effect of the new balanced polymorphism on the population—there is a slight increase but it amounts to very little. The population hardly benefits from the new malaria defense.
This adaptation to malaria with heterozygote advantage at the Hb locus is the kind of quick and dirty adaptation that appears under strong selection in a new environment. It damages the human but it causes more damage to the parasite and so is favored by selection. After a long time we expect that evolution will come up with something better and so over long evolutionary time the sickle cell polymorphism is certainly transient, it will be replaced by a better fix for the problem.
Figure A2 makes it clear that this adaptation to malaria really doesn’t do much for the population. Worse yet the system persists for many generations after the elimination of the malaria. Sickle cell anemia is an enduring source of family tragedy and impaired lives in the American Black community, an enduring scar of evolution, a response to malaria many centuries in the past.
Effect of a New Favorable Mutant
The mutation that leads to lactase persistence in Europe and neighboring areas apparently happened only once in history. Other identical or equivalent mutations may have occurred but they were quickly lost because of random drift. We can understand this by simulating the evolution of population on a computer, that is with a stochastic model like the one we used at the beginning of this appendix.
Figure A3 shows the results of simulating the appearance of a mutation with a 4% selective advantage in a population of 50,000 people. The mutation is introduced at generation 50. Without the mutation the carrying capacity of this population is 200,000, while the carrying capacity of the same population with everyone carrying the mutation is nearly a million people. Fifty population histories are shown in the figure: we can think of these as 50 possible histories of the same population or as outcomes of evolution in 50 parallel universes.
Theory says that the probability that a new advantageous mutation with selective advantage s persists is about 2s so we expect persistence in 4 of 50 trials. In this case we observe 3. In the 3 successful introductions the mutation hung around at low frequency for about 100 generations or 2500 years and then increased rapidly to become nearly fixed in 300 generations or 7,500 years.
The probability of persistence of a new mutation is described well by the 2s approximation, but in the simulations the later progress toward fixation of the mutant is governed both by individual selective advantage and by differences between the two variants in carrying capacity. We set the carrying capacity of the advantageous mutant to five times the carrying capacity of the ancestral type because, with lactase in mind, food production from dairying is on the order of five to ten times greater than using a herd only for meat production.
We know that in a sample of European skeletons from about 8,000 years ago the common European lactase persistence mutant was not present. From another skeletal sample the frequency was around 0.5 at 4,000 years ago. In the simulations the frequency was significantly lower at 160 generations before the present, suggesting that the selective advantage of lactase persistence may have been even higher than 4% in agreement with estimates that have appeared in the literature.
Another interesting pattern in the simulations is that advantageous genes that will eventually spread to everyone spend a long time, several thousand years, hanging around at low frequency. There must be hundred or even thousands of advantageous genes present in our species that are rare but that will eventually spread and replace the current alleles at their loci.
As the advantageous mutant increases in the simulations it causes population increase: the numbers we use are not out of the range of what lactase persistence probably caused since some estimate that dairying and using dairy products without fermenting the lactose away may yield up to ten times as much food as simply raising cattle for beef. One new gene in our simulation leads to a massive increase in population over several thousand years. In the world such an increase would be accompanied by a spatial expansion of population just as we see in human history. We know some details of the expansion of Indo-Europeans, Bantu speakers in Africa, Han Chinese, and so on. Human history sometimes looks like a kaleidoscope of population booms and busts, and most of these were almost certainly driven by genetic changes of the kind we have simulated.
Figure A3 Simulations of 50 population histories in which a new mutant with a 4% selective advantage was introduced at generation 50. The top panel show the frequency of the mutant over time in the five populations in which it was not lost due to drift, the bottom panel shows population size change in each population.
Clark, G. (2007). A Farewell to Alms: A Brief Economic History of the World. Princeton, NJ, Princeton. Cohen, J. E. (1995). How many people can the earth support? New York, W.W. Norton.