Immediately after the flood there would have been a great increase in population due to Gods command to Noah to repopulate the earth. If we consider the six children of Noahs (3 couples) and assume they had a moderate number of children, six, of course this would lead to 18 births. If we maintain this formula and take into consideration a generation is 25 years, well, by the third generation the population would have grown to 81 couples, having six children each, thus, 486 people. Now lets keep in mind that six children per couple is an extremely conservative number in Old Testament times. But if this pace were maintained, then by just the eighth generation, or 200 years after the flood, there would be 118,098 people on the earth. Now simple math will tell you that after 4,450 years the population would easily attain six billion persons without question and Im not aware of a one secular scientist that refutes this possibility.
Let's look at the growth rate another way using the exponential growth model. Let's first recognize that population growth has never been constant. Starting from eight people after the Flood, the population would only have to double 30 times to reach 8.6 billion in 4,450 years. Now there is a well known 'Rule of 72', which says divide 72 by the percentage growth rate to get the time required for doubling. E.g. if inflation is 8% p.a., then in 72/8 = 9 years, the cost of living will have doubled.
If we actually use the population growth rate from the last 2000 years, then again, we have absolutely no problem in reaching the six billion people. So what is a realistic growth rate? The Encyclopedia Britannica claims that by the time of Christ, the world's population was about 300 million. It apparently didn't increase much up to the year AD 1000. In addition, it was up and down in the Middle Ages due to plagues and various other impacts, but may have reached 800 million by the beginning of the Industrial Revolution in 1750 (an average growth rate of just 0.13% in the 750 years from 10001750).
By 1800, the population was one billion while the second billion was reached by 1930 an average growth rate of 0.53%. This period of population growth cannot be due to improved medicine, because antibiotics and vaccination campaigns did not impact till after WWII. From 1930 to 1960, when the population reached three billion, the growth rate was 1.36 %. By 1974, the fourth billion was reached, so the average growth rate was 2.1% from 1960 to 1974. From 1974 to 1990, when the mark hit five billion, the growth rate had slowed to 1.4%. The increase in population growth this century is due to fewer deaths in infancy and through disease.
If we figure the average growth rate was a mere 0.4 %, then the doubling time would be 180 years. Then after only 30 doublings or 4,450 years, the population could have reached eight billion.
If you want something more rigorous, there are standard mathematical formulas that can be used to calculate population growth. They must include birth and death rates as well as generation time. However, the simplest formula involves just a constant growth rate, and Ive seen many scientists use this method while never hearing any one of them discredit its relative accuracy.
Evolutionary anthropologists say that the Stone Age lasted for at least 100,000 years, during which time the world population of Neanderthal and Cro-Magnon men was roughly constant, between one and 10 million. Think about that, if the population of earth was even 10,000 of these supposed "one to 10 million", the earth population today would be well into the double-digit billions.