Question

The time-dependent growth of a bacterial population is governed by the equation \[ \frac{dx}{dt}=x\left(1-\frac{x}{200}\right), \] where \(x\) is the population size at time \(t\). The initial population size is \(x_0=100\) at \(t=0\). As \(t\to\infty\), the population size of bacteria asymptotically approaches ________________________.

• A. 150
• B. 200
• C. 300
• D. 500

Hint:

- Logistic growth has two equilibria: \(x=0\) (unstable) and \(x=K\) (stable). - The parameter \(K\) is the carrying capacity—the long-term population level.

Answer 1

The Correct Option is B

Step 1: The differential equation is of logistic form \[ \frac{dx}{dt}=rx\left(1-\frac{x}{K}\right), \] with growth rate \(r=1\) and carrying capacity \(K=200\). Step 2: In a logistic model, regardless of the initial value \(x_0>0\), the solution \(x(t)\) approaches the carrying capacity \(K\) as \(t\to\infty\). Hence, \[ \lim_{t\to\infty} x(t)=K=200. \] Therefore, the bacterial population asymptotically approaches \(\boxed{200}\).