Question

which of the following best characterizes a population that is growing exponentially with time?
a equal increases in number over equal increments of time
b controlled growth over a finite time
c doubling in size over equal increments of time
d reaching a maximum then experiencing a rapid decline
e growing at a rate proportional to the square root of time

Explanation:

• For exponential growth, the population size \(N(t)\) follows the formula \(N(t)=N_0\times2^{\frac{t}{T}}\), where \(N_0\) is the initial population, \(t\) is time, and \(T\) is the doubling - time.
• In option A, equal increases (\(\Delta N\)) over equal time increments (\(\Delta t\)) is characteristic of linear growth (\(N(t)=N_0 + kt\), where \(k\) is a constant).
• Option B, controlled growth over a finite time, is more related to logistic growth (\(\frac{dN}{dt}=rN(1-\frac{N}{K})\), where \(r\) is the intrinsic growth rate and \(K\) is the carrying capacity) which is not exponential.
• Option D, reaching a maximum then declining, is also a feature of logistic - like or other non - exponential (e.g., density - dependent with overshoot) population models.
• Option E, growth proportional to \(\sqrt{t}\) (\(N(t)=N_0 + k\sqrt{t}\)) is a non - exponential (power - law, but not exponential) relationship.
• For exponential growth \(N(t)\), if \(t = T\) (doubling - time), \(N(T)=2N_0\); if \(t = 2T\), \(N(2T)=2N(T) = 4N_0\); if \(t=3T\), \(N(3T)=2N(2T)=8N_0\). So the population doubles over equal increments of time.

Answer:

C. Doubling in size over equal increments of time