Problem 46 - Calculus

Question

Give an example of: A logistic differential equation for a quantity \(P\) such that the maximum rate of change of \(P\) occurs when \(P=75\)

Step-by-Step Solution

Answer

\(\frac{dP}{dt} = kP\left(1 - \frac{P}{150}\right)\) with \(P=75\) for maximum rate of change.

Step 1: Understand Logistic Differential Equation

A logistic differential equation models population growth with a limiting factor. The general form is \(\frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right)\), where \(k\) is the growth rate constant and \(L\) is the carrying capacity of the environment. The maximum rate of change occurs when \(P = \frac{L}{2}\).

Step 2: Set up the Condition for Maximum Rate of Change

The maximum rate happens at \(P = 75\). According to logistic growth, this is when \(P = \frac{L}{2}\). Therefore, \(\frac{L}{2} = 75\), which gives \(L = 150\).

Step 3: Write the Logistic Differential Equation

With \(L = 150\) and the general logistic form, the differential equation can be written as \(\frac{dP}{dt} = kP\left(1 - \frac{P}{150}\right)\), where \(k\) is the growth rate constant to be determined by specific environmental conditions or data.

Key Concepts

Population Growth

Population growth is an important biological process that can be modeled with different mathematical equations. One of the most significant models is the logistic differential equation, which describes how a population grows over time with a limiting factor.
Initially, populations tend to grow exponentially, meaning that the larger the population, the faster it grows. This is because each individual organism has the potential to reproduce multiple times, leading the population size to double, triple, or grow even more rapidly over time.
However, this exponential growth cannot continue indefinitely in a real-world environment, as resources such as food, space, and water become limiting factors. As a result, the growth rate slows down and eventually stabilizes, reaching a plateau. This is the essence of logistic growth.

• The logistic growth is represented by the equation: \[ \frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right) \], where:
  • \( P \) is the population size or quantity at a given time.
  • \( k \) is the intrinsic growth rate constant.
  • \( L \) is the carrying capacity or maximum population an environment can sustain.

Understanding population dynamics through logistic equations enables better prediction and management of biological resources.

Carrying Capacity

Carrying capacity is a crucial concept when analyzing how populations grow in nature. It refers to the maximum population size of a species that an environment can sustain indefinitely. In the logistic differential equation, it plays a pivotal role in predicting how and when a population will stabilize.
The carrying capacity is determined by various environmental factors, including available resources, habitat space, and competition among individuals. When the population reaches this capacity, the growth rate decreases until it approaches zero, as the resources are just enough to keep the population stable.

• In the logistic differential equation \( \frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right) \), the term \( \left(1 - \frac{P}{L}\right) \) indicates that as \( P \) approaches \( L \), the growth rate diminishes.

The environment can impose a natural limit through factors such as predation, disease, and resource availability. By understanding carrying capacities, ecologists can suggest sustainable practices for resource usage, ensuring that populations remain balanced over time.

Maximum Rate of Change

In the context of logistic growth, the maximum rate of change of a population occurs not at the beginning, but rather when the population is halfway to its carrying capacity. This is a key insight that distinguishes logistic growth from purely exponential growth.
Mathematically, this condition is realized when the population size \( P \) is equal to \( \frac{L}{2} \), where \( L \) is the carrying capacity. At this point, the force driving the growth is strongest, but it starts declining as the population continues to grow closer to \( L \).

• Using the logistic equation: \[ \frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right) \], when \( P = \frac{L}{2} \), the derivative \( \frac{dP}{dt} \) reaches its maximum value.
• For instance, if the carrying capacity \( L \) is 150, the maximum rate of population change occurs at \( P = 75 \).

This point of maximum change reflects how quickly a population can grow under optimal conditions—and how that growth rate slows as limits set by the environment become more significant.