Question

Consider a population of 10 million cells. Given the per-capita birth rate of 0.002 (per unit time) and the per-capita death rate of 0.002 (per unit time), the expected number of cells after 10 generations is __________.

• A. 100 million
• B. 1 million
• C. 5 million
• D. 10 million

Hint:

• When the per-capita birth rate matches the per-capita death rate, the population achieves a state of dynamic equilibrium.

• Under these conditions, elapsed time and generations have no net effect on the total population size.

Answer 1

The Correct Option is D

Concept:

• Population growth rate over time can be mathematically expressed by the exponential growth equation: \[ \frac{dN}{dt} = rN \] where: itemize

• \(N\) is the population size.

• \(r\) is the intrinsic rate of natural increase, calculated as the difference between the per-capita birth rate (\(b\)) and the per-capita death rate (\(d\)): \[ r = b - d \]
If the growth rate \(r\) is zero, the population size does not undergo any net change over time, regardless of the time intervals or generations elapsed. itemize

Step 1: Calculate the intrinsic rate of natural increase (r)
Using the values provided:

• Per-capita birth rate (\(b\)) = \(0.002\)

• Per-capita death rate (\(d\)) = \(0.002\)
Calculate \(r\): \[ r = b - d = 0.002 - 0.002 = 0 \]

Step 2: Calculate the expected population size after 10 generations
The mathematical relationship for population size is: \[ N_t = N_0 e^{rt} \] Substituting \(r = 0\) into the equation: \[ N_t = N_0 e^{(0 \times t)} \] \[ N_t = N_0 e^0 \] \[ N_t = N_0 \times 1 \] \[ N_t = 10\ \text{million} \] Because the birth rate equals the death rate, the population remains stable at 10 million cells. This corresponds to option (D).