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

Step 1: List the data.
Starting population $N_0 = 10$ million cells, per-capita birth rate $b = 0.002$, per-capita death rate $d = 0.002$.
Step 2: Recall the growth idea.
Population change over time follows $\frac{dN}{dt} = rN$, where $r$ is the intrinsic rate of natural increase.
Step 3: Find $r$.
Here $r = b - d = 0.002 - 0.002 = 0$. The births exactly cancel the deaths.
Step 4: Apply the exponential formula.
Population size is $N_t = N_0 e^{rt}$. With $r = 0$, the exponent becomes zero.
Step 5: Simplify.
Since $e^0 = 1$, we get $N_t = N_0 \times 1 = N_0$. The number does not change no matter how many generations pass.
Step 6: Conclude.
The population stays at $10$ million cells, which is option (4). When birth rate equals death rate, the population is simply stable.
\[ \boxed{10\ \text{million}} \]