48. Growth of a microbe in a test tube is modeled as
dX/dt = rX (1 − X/K), where X is the biomass, r is the growth rate and K is the carrying capacity of the environment (r ≠ 0; K ≠ 0).
If the value of starting biomass is K/100, which one of the following graphs qualitatively represents the growth dynamics?
Microbial growth in a closed test tube does not increase indefinitely. As nutrients deplete and waste accumulates, cell growth slows and eventually stops. This behavior is accurately modeled by the logistic growth equation:
dX/dt = rX(1 − X/K)
Where:
- X = biomass
- r = intrinsic growth rate
- K = carrying capacity (maximum sustainable biomass)
Given an initial biomass of K/100, the growth follows the classic S-shaped sigmoid curve.
Understanding Logistic Growth
Growth phases:
- Lag/low population phase – growth slow when X ≪ K
- Exponential phase – nearly exponential rise when X approaches mid-range
- Deceleration – growth slows due to nutrient limitation
- Steady state – biomass approaches K asymptotically and levels off
This yields an S-shaped curve approaching a plateau at K.
Final Answer
🎯 (A) represents true logistic microbial growth.
Conclusion
The logistic model captures realistic microbial population limits in batch culture. From a very small initial biomass of K/100, the growth begins slowly, accelerates, then stabilizes at K.
Understanding logistic dynamics forms the basis of:
- bioprocess scaling
- fermentation engineering
- ecological population modeling
