Subtopic: Fundamentals of Microbial growth kinetics
Question Statement:
Growth of a microbe in a test tube is modeled as dX/dt = rX(1 - X/K), where r is the specific growth rate and K is the carrying capacity. The phase-plane trajectory of growth rate (dX/dt) versus population X is:
Options:
(A) A parabolic curve concave downwards with peak at X = K/2
(B) A straight line with negative slope passing through origin
(C) An exponential curve increasing indefinitely
(D) A horizontal line of constant growth
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Step-by-Step Derivation (GATE BT 2017 Q.48)**
**1. Governing Concept & Question Classification**:
- **Section**: BT
- **Question Type**: **MCQ** (2 Marks)
- **Topic**: Bioprocess Kinetics & Population Ecology (Logistic Growth Dynamics)
**2. Detailed Step-by-Step Derivation**:
- The microbial growth in the closed test tube is governed by the differential logistic equation:
$\frac{dX}{dt} = rX \left(1 - \frac{X}{K}\right)$
where:
- $X$ is the biomass concentration.
- $r > 0$ is the intrinsic specific growth rate.
- $K > 0$ is the carrying capacity of the environment.
- Given that the initial biomass is very small relative to the carrying capacity ($X_0 = K / 100 \ll K$):
1. When $X \ll K$: The term $(1 - X/K) \approx 1$, so $\frac{dX}{dt} \approx rX$. The population undergoes an initial **exponential acceleration phase**.
2. As $X$ approaches $K/2$: The absolute growth rate $\frac{dX}{dt}$ attains its maximum value at the inflection point ($X = K/2$).
3. When $X > K/2$: Nutrients deplete and growth slows down (deceleration phase).
4. As $X \to K$: The term $(1 - X/K) \to 0$, so $\frac{dX}{dt} \to 0$. Biomass asymptotically approaches the carrying capacity $K$, reaching a flat, stable stationary phase plateau.
- This produces the classic **sigmoidal (S-shaped) logistic growth curve** (Graph A).
- **Option-by-Option Evaluation**:
- **Option (A) [CORRECT]**: Displays the classic sigmoidal logistic growth curve starting from $X_0 \ll K$ and asymptoting at carrying capacity $K$.
- **Option (B) [INCORRECT]**: Shows linear growth, ignoring both exponential growth and saturation.
- **Option (C) [INCORRECT]**: Shows exponential growth without reaching an environmental carrying capacity plateau.
- **Option (D) [INCORRECT]**: Shows an inverted decay profile.
**3. Final Answer / Verification Summary**:
- **Official GATE Answer Key**: **A**
- **Scoring Criteria**: +2.0 Marks (Negative marking: -0.67 marks).
Practice This in Real CBT Exam Interface
Test yourself with the official TCS-iON palette, virtual calculator, and immediate performance analytics.