Not necessarily. In many species the birth (actually reproduction) rate is very high, but the mortality rate is also high. If you look at modern species, most are in equilibrium, with birth matching death and populations remaining somewhat stable. There are local exceptions of course; some species die out (e.g., mammoth and mastodon) while other species thrive for a while (us).
So what year did the modern man age begin?
Not sure what you mean by this.
In fossils, modern man has been around somewhere between 100,000 and 200,000 years according to current evidence (see below). It depends on where you draw the line. Some scientists would go older.
If you mean modern civilization, you are looking at closer to 10,000 years.
Herto skulls (Homo sapiens idaltu)Some new fossils from Herto in Ethiopia, are the oldest known modern human fossils, at 160,000 yrs. The discoverers have assigned them to a new subspecies, Homo sapiens idaltu, and say that they are anatomically and chronologically intermediate between older archaic humans and more recent fully modern humans. Their age and anatomy is cited as strong evidence for the emergence of modern humans from Africa, and against the multiregional theory which argues that modern humans evolved in many places around the world.