Understanding Logistic Population Growth

The logistic growth model describes how a population's size changes over time when limited by resources. The rate of population growth is given by the equation:

$ \frac{dN}{dt} = rN \left( 1 - \frac{N}{k} \right) $

where:

• N is the population size
• t is time
• r is the intrinsic rate of increase
• k is the carrying capacity (maximum population size the environment can sustain)

This model reflects that population growth is initially slow, then accelerates, and finally slows down as it reaches the carrying capacity.

Phases of Logistic Growth

The logistic growth curve typically consists of three distinct phases:

• Lag phase: This is the initial phase where the population size is small. The population is adapting to the new environment, and the growth rate is slow.
• Exponential phase (Log phase): As the population adapts and resources are plentiful relative to population size, the growth rate increases rapidly. The population size increases exponentially. Mathematically, when N is much smaller than k, the term $(1 - N/k)$ is close to 1, and the equation approximates exponential growth ($\frac{dN}{dt} \approx rN$).
• Stationary phase (Plateau phase): As the population size (N) approaches the carrying capacity (k), resources become limited, and environmental resistance increases. The growth rate slows down significantly, eventually becoming zero when $N = k$. The population size stabilizes.

Correct Sequence

Based on the characteristics of the logistic growth model and the phases described, the correct sequence is:

Lag phase $\rightarrow$ Exponential phase $\rightarrow$ Stationary phase

This sequence reflects the progression from initial adaptation and slow growth, through a period of rapid increase, to stabilization at the carrying capacity.