From Wikipedia:
"In pre-industrial society, death rates and birth rates were both high and fluctuated rapidly according to natural events, such as drought and disease, to produce a relatively constant and young population. Family planning and contraception were virtually nonexistent; therefore, birth rates were essentially only limited by the ability of women to bear children. Emigration depressed death rates in some special cases (for example, Europe and particularly the Eastern United States during the 19th century), but, overall, death rates tended to match birth rates, often exceeding 40 per 1000 per year. Children contributed to the economy of the household from an early age by carrying water, firewood, and messages, caring for younger siblings, sweeping, washing dishes, preparing food, and working in the fields.[5] Raising a child cost little more than feeding him; there were no education or entertainment expenses and, in equatorial Africa, there were no clothing expenses either. Thus, the total cost of raising children barely exceeded their contribution to the household. In addition, as they became adults they become a major input to the family business, mainly farming, and were the primary form of insurance for adults in old age. In India, an adult son was all that prevented a widow from falling into destitution. While death rates remained high there was no question as to the need for children, even if the means to prevent them had existed.[6]
During this stage, the society evolves in accordance with Malthusian paradigm, with population essentially determined by the food supply. Any fluctuations in food supply (either positive, for example, due to technology improvements, or negative, due to droughts and pest invasions) tend to translate directly into population fluctuations. Famines resulting in significant mortality are frequent. Overall, the population dynamics during stage one is highly reminiscent of that commonly observed in animals."
Couldn't have put it better myself.
To be honest, I was a little surprised to see a creationist argument on a civ forum. Not that the game overtly pushes the idea of evolution, but because (to me, anyway) it seems to have an aura of history, science and education, both in its content and among its players.
I feel a little qualified to comment on Kruelgor's original post, as I have studied demographics at the postgrad level. But I won't bore you or me with descriptions of demographic transition or how the study of osteology can impact our understanding of generation cohort mortality in medieval Britain. Simple commonsense can tell us what is wrong with the OP's graph: all its assumptions are wrong. Doesn't matter how many children women are having or not having. If the number of births equals the number of deaths, then the population will be stable. In pre-industrial society the birth rate tended to be high (much higher than 3, Kruelgor), but so was infant mortality and mortality in general.
The truth is, relying on pure mathematics does not tell us that the population 6,000 years ago was 2, or 200,000, or 6.5 billion, or 20 trillion for that matter. Population extrapolations into the past require all-important assumptions about past birth and death rates. And if you want to assume that the global population has doubled every 100 years for the last 6,000 years (a massive and extraordinary falsehood) then you must provide massive evidence for this.
By the way a global population of 2 in 4,000 BC must have come as quite a surprise to the citizens of the Sumerian civilization, who were living in established cities of tens of thousands each, and had well-developed trade, writing, and religion. The ancient Egyptians might also have been surprised by the claim made by Adam and Eve to be the first humans on Earth, as they had been living in the valley of the Nile in well developed large communities for 1500 years, with copper-working, agriculture and animal husbandry.
Incidentally, as part of my master's thesis I was involved in paleodemographic modelling of the aboriginal population of Australia, in an attempt to estimate the population before European contact. Suffice it to say it is an extremely difficult thing to do, and pretty much impossible if you want to come up with an accurate population figure. The smallest variation in inititial assumptions can result in huge outcome differences. Did the 1836 smallpox plague result in 34 or 42 % casualty rate? This can make a difference of tens of thousands of people, and that is just in 1788, when contact with Europeans was first made.
The truth is that the OP is a serious Christian struggling with evolution. That's fine, but you just have to find your own way to make it work. Most other Christians are able to do it.