What causes exponential growth of a population?

Question

Isaiah Allen · Accepted Answer

The rate of growth of a population in people per year is the births minus the deaths for that year. But what determines those numbers? Births are caused by women having babies, and the number of pregnant women is proportional to the size of the population. The deaths are also proportional to the population P(t) at time t, so the rate of change of P(t) is about P'(t) = (births - deaths) in year t. The equation #P'(t) = k P(t)# has the family of solutions 
#P(t) = A e^(kt)#, where A is the initial population (when t=0) and k is the growth rate as a decimal. (For example #P(t) = 200 e^(0.03t)# would model a population initially 200 people growing at a rate of 3% per year.) You can prove that P'(t) = k P(t) by taking the derivative of 
#P(t) = A e^(kt)#  getting  #P'(t) = A k e^(kt) = k P(t)#. To prove these are the only solutions, let g(t) = #(P(t))/e^(kt)#, then you can use the quotient rule to show that g'(t) = 0 meaning 
g(t) = constant, say A, so #P(t) = A e^(kt).# \ Populate the earth, with awesome answers from dansmath… /

Christopher Agresta · Answer

undefined Exponential growth of a population occurs when the birth rate exceeds the death rate consistently over time, leading to an increasing population size. This phenomenon is often observed in environments where resources are abundant and there are minimal limiting factors such as disease, predation, or competition for resources. Exponential growth can also occur in situations where individuals have high reproductive rates and the offspring themselves begin reproducing early, contributing to a rapid increase in population size.