Suppose a population is growing according to the logistic...

Suppose a population is growing according to the logistic equation, $$ \frac{d P}{d t}=r P\left(1-\frac{P}{K}\right) $$ Prove that the rate at which the population is increasing is at its greatest when the population is at one-half of its carrying capacity. Hint: Consider the second derivative of $P$.

Suppose a population is growing according to the logistic equation,
$$
\frac{d P}{d t}=r P\left(1-\frac{P}{K}\right)
$$
Prove that the rate at which the population is increasing is at its greatest when the population is at one-half of its carrying capacity. Hint: Consider the second derivative of $P$.

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Key Concepts

Logistic Differential Equation

The logistic differential equation is a mathematical model used to describe population growth in environments where resources are limited. It captures both the natural growth of the population at a rate proportional to its current size and the slowing growth as the population nears a maximum sustainable size.

Carrying Capacity

Carrying capacity is the maximum population that an environment can support indefinitely. In logistic models, it represents the upper limit on growth due to finite resources, and it is a key parameter that determines how the growth rate changes over time.

Maximum Growth Rate and Inflection Point

In logistic growth, the point of maximum growth rate occurs at the inflection point of the population curve, where the curvature changes from concave upward to concave downward. This point is crucial because it marks the transition from acceleration to deceleration in population growth.

Second Derivative Test

The second derivative test is a method in calculus used to determine the concavity of a function. In the context of the logistic equation, calculating the second derivative and setting it to zero helps identify the inflection point, confirming that the maximum rate of increase in population occurs when the population is exactly half of the carrying capacity.

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