Homo neanderthalensis is the closest relative to Homo sapiens in the tree of human evolution. It is estimated that the Neanderthals became extinct around 30 thousand years ago. Several hypotheses have been put forward to explain this extinction. Larger brains hence larger heads were favoured by evolution among the Hominids for over two million years but stopped being an advantage due to trade-offs between bipedal walking, larger brains and birth. Our hypothesis states that H. neanderthalensis became extinct as a result of their larger head circumference, compared with H. sapiens, leading to birth complications. We use population dynamics to demonstrate that a small difference in the rate of mortality during birth can cause extinction of one population. We simulate two populations with an Agent Based Simulation technique to show that in the case of interbreeding between two populations, unexpected results can emerge depending on random events in simulations. We demonstrate that sharing the same habitat with capacity constraints over thousands of years, with a relatively higher death rate during birth could be the main cause of the slow decline in the H. neanderthalensis population, and its eventual extinction.