Glossary
Carrying Capacity (M)
The maximum population size that a particular environment can sustain indefinitely, representing the upper limit of logistic growth.
Example:
A small pond can only support a certain number of fish; this maximum number is its carrying capacity.
Differential Equation
An equation that relates a function with its derivatives, used in logistic models to describe the rate of change of a population over time.
Example:
The equation dP/dt = kP(M - P) is a differential equation that models how a population P changes over time t.
Growth Rate (k)
A positive constant in the logistic differential equation that represents the intrinsic growth rate of the population when it is far from the carrying capacity.
Example:
In the equation dP/dt = 0.5P(1000 - P), the growth rate constant is 0.5, indicating how quickly the population would grow initially.
Logistic Model
A differential equation that describes how a population grows over time, initially exponentially, but then slowing down as it approaches a maximum limit.
Example:
The spread of a new viral trend on social media often follows a logistic model, starting fast but eventually leveling off as most users adopt it.
Point of Fastest Change
The specific population size at which the rate of growth in a logistic model is at its maximum, occurring when the population is half of the carrying capacity (M/2).
Example:
A bacterial colony growing in a petri dish will experience its point of fastest change when its population reaches half of the dish's maximum sustainable capacity.