Problem 12 - Calculus

Question

For the logistic differential equations (a) Give values for \(k\) and for \(L\) and interpret the meaning of each in terms of the growth of the quantity \(P\) (b) Give the value of \(P\) when the rate of change is at its peak. $$\frac{d P}{d t}=0.1 P-0.00008 P^{2}$$

Step-by-Step Solution

Answer

(a) \( k = 0.1 \), \( L = 1250 \). (b) \( P = 625 \).

Step 1: Identifying the Logistic Equation Form

The given differential equation \( \frac{dP}{dt} = 0.1P - 0.00008P^2 \) can be rewritten in the standard logistic form \( \frac{dP}{dt} = kP \left(1 - \frac{P}{L}\right) \). By matching terms, we identify \( k = 0.1 \) and \( L = \frac{0.1}{0.00008} \).

Step 2: Understanding Parameters

The parameter \( k = 0.1 \) represents the intrinsic growth rate of the population \( P \). The parameter \( L \) represents the carrying capacity of the environment, or the maximum population size that the environment can sustain.

Step 3: Calculating the Carrying Capacity

To find \( L \), we calculate \( L = \frac{0.1}{0.00008} = 1250 \). Thus, \( L = 1250 \) is the maximum population size (carrying capacity).

Step 4: Finding the Peak Rate of Change

The rate of change of \( P \) is at its peak when \( P = \frac{L}{2} \), which is the inflection point of logistic growth. Thus, \( P = \frac{1250}{2} = 625 \).

Key Concepts

Intrinsic Growth Rate

The intrinsic growth rate is a crucial concept in understanding logistic differential equations. It is denoted by the symbol \( k \) and represents the rate at which a population grows when there are no constraints such as limited resources. In simpler terms, it is how fast the population would increase under ideal conditions, where resources are unlimited and other factors that might limit growth are absent.

In the context of the logistic equation \( \frac{dP}{dt} = kP - \frac{kP^2}{L} \), the intrinsic growth rate is the constant \( k \), which can directly influence how rapidly the initial growth occurs. A higher \( k \) means that the population will try to grow more quickly, assuming no environmental limits. For our exercise, \( k \) is given as 0.1. This value indicates that under ideal conditions, without limits to growth, the population would expand by 10% per time unit.

• \( k = 0.1 \) implies a 10% growth rate.

It's important to understand that intrinsic growth rate is a theoretical speed of growth. In practice, populations are usually limited by available resources, which brings us to the concept of carrying capacity.

Carrying Capacity

The carrying capacity, denoted by \( L \), is the maximum population size that an environment can sustain indefinitely. It reflects the limits caused by the availability of resources like food, space, and other essentials necessary for population survival. In the logistic differential equation, carrying capacity is a key factor.

For the equation \( \frac{dP}{dt} = kP(1 - \frac{P}{L}) \), carrying capacity \( L \) is a threshold where population growth stabilizes and eventually stops when it is reached. In the solution we looked at, we calculated \( L \) as \( \frac{0.1}{0.00008} = 1250 \), which means the population will level off at 1250 when resources limit further growth.

• \( L = 1250 \) is the point where growth stops.
• Reflects resources available in the environment.

Carrying capacity is significant because it realistically captures the limits of growth through environmental constraints. Understanding this concept helps predict how populations interact with their surroundings, especially in real-world applications.

Population Dynamics

Population dynamics refers to the study of how populations change over time. It's an integral part of understanding the logistic differential equation, which models the growth of a population while considering limiting factors.

In the logistic growth model, population dynamics are manifested in three phases:

• Initial Exponential Growth: When \( P \) is small compared to \( L \), the population grows rapidly. This is driven by the intrinsic growth rate \( k \).
• Deceleration: As \( P \) begins to approach \( L \), the growth rate declines due to increased competition for resources.
• Equilibrium: At \( P = L \), population growth ceases. The rate of births equals the rate of deaths.

In our exercise, the population grows most rapidly when it is halfway to the carrying capacity. This occurs at \( P = \frac{L}{2} \), which simplifies to 625 in our example. This is where the rate of change of \( P \) is at its peak, and the population experiences the greatest growth.
Understanding these dynamics is essential for predicting population trends and for applications like resource management and conservation efforts.