Discuss how a limiting factor leads to logistic growth of a population. Derive the logistic growth equation and explain the ecological meaning of the carrying capacity (K).
Answer
Limiting factor reduces the per‑capita growth rate, giving the logistic equation dN/dt = rN(1 - N/K); K represents the maximum sustainable population size.
Step-by-step solution
- State that in the absence of limiting factors, population growth is exponential: dN/dt = rN, where r is the intrinsic rate of increase.
- Introduce a limiting factor (e.g., food scarcity) that causes the growth rate to decline as population size N approaches a maximum sustainable value.
- Assume the per‑capita growth rate decreases linearly with N: effective rate = r (1 - N/K), where K is the population size at which growth stops.
- Substitute the effective rate into the exponential model: dN/dt = rN (1 - N/K).
- Explain that K is called the carrying capacity, the largest number of individuals that the environment can maintain indefinitely without degradation.
- Summarize the derived logistic equation and the role of K.
Common mistake: Confusing carrying capacity (K) with the initial population size or treating K as a fixed number regardless of resource changes.
Concept tested: Logistic growth & carrying capacity
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