Question

For the following problems, consider the logistic equation in the form \(P^{\prime}=C P-P^{2} .\) Draw the directional field and find the stability of the equilibria.[T] Rabbits in a park have an initial population of 10 and grow at a rate of \(4 \%\) per year. If the carrying capacity is 500, at what time does the population reach 100 rabbits?

Short answer

It takes approximately 39 years for the rabbit population to reach 100.

Step-by-step solution

Understanding the Logistic Equation

The given logistic equation is \( P^{\prime} = CP - P^2 \). Where \( C \) is the intrinsic growth rate, \( P \) is the population size, and \( P^{\prime} \) is the rate of change of the population. We need to identify the values of \( C \) and the carrying capacity.

Determine Parameters

From the problem, the growth rate \( C \) is \( 4\% \) or \( 0.04 \) and the carrying capacity is 500. Therefore, the differential equation becomes: \[ P^{\prime} = 0.04 P - \frac{1}{500} P^2 \].

Find Equilibria and Their Stability

To find equilibrium points, set \( P^{\prime} = 0 \): \( 0 = 0.04P - \frac{1}{500}P^2 \). Factor out \( P \): \( P(0.04 - \frac{1}{500}P) = 0 \). Solutions are \( P = 0 \) and \( P = 500 \). \( P = 0 \) is unstable (since growth is positive for small \( P \)) and \( P = 500 \) is stable (since growth slows as \( P \) approaches 500).

Draw the Directional Field

The directional field can be drawn by plotting several values of \( P \) versus \( P' = 0.04P - \frac{1}{500}P^2 \) and observing the direction of the population size over time. Around \( P = 0 \), the slope is positive, implying instability, and flattens as it approaches \( P = 500 \), where it stabilizes.

Solve for Time to Reach 100 Rabbits

Use the logistic growth solution formula \( P(t) = \frac{K}{1 + \left(\frac{K-P_0}{P_0}\right)e^{-Ct}} \), where \( K = 500 \), \( P_0 = 10 \), and \( C = 0.04 \). Plug in \( P(t) = 100 \) to solve for \( t \): \[ 100 = \frac{500}{1 + \left(\frac{500-10}{10}\right)e^{-0.04t}} \], simplify to find \( t \).

Compute and Simplify

From the equation \( 100 = \frac{500}{1 + 49e^{-0.04t}} \), find \( 1 + 49e^{-0.04t} = 5 \), and solve \( 49e^{-0.04t} = 4 \). Therefore, \( e^{-0.04t} = \frac{4}{49} \). Take the natural logarithm of both sides: \( -0.04t = \ln\left(\frac{4}{49}\right) \) and calculate \( t \).

Conclude the Calculation

Solving \( t = -\frac{\ln\left(\frac{4}{49}\right)}{0.04} \) gives \( t \approx 39.09 \). Thus, it takes approximately 39 years for the population to reach 100 rabbits.

Key concepts

Population Ecology

Population ecology is a fascinating field of biology that examines how populations of organisms, such as rabbits in a park, interact with their environment and change over time. At the heart of population ecology is the study of growth patterns, including logistic growth, which describes how a population grows rapidly when it's small, slows down as resources become limited, and ultimately levels off at a stable size called the carrying capacity.

The logistic growth model is especially useful in population ecology because it accounts for the fact that resources like food, space, and mates are limited in any real-world setting. These limitations lead populations to stabilize at an equilibrium point, rather than growing indefinitely. Understanding these dynamics helps ecologists predict changes and manage wildlife effectively.

Given the initial population, growth rate, and carrying capacity, the logistic growth model can provide insights into when a population will reach a certain size, which is crucial for conservation and resource management.

Stability Analysis

Stability analysis is a key concept in studying population models, particularly logistic growth equations. This process involves determining whether the population size will return to an equilibrium after small disturbances.

In our rabbit population example, we found two equilibrium points: one at zero rabbits, and another at 500 rabbits (the carrying capacity). The stability of these points is determined by examining how small changes affect the population growth rate.

To perform stability analysis, we consider the sign of the derivative of the population with respect to time at these points:

• If small changes cause the population to move away from the equilibrium, the point is unstable.
• If small changes cause the population to return to the equilibrium, the point is stable.

In this case, a small increase from zero results in a growing population, marking zero as unstable, while the population nearing 500 leads to little change, suggesting stability.

Differential Equations

Differential equations are mathematical tools that play a central role in modeling population dynamics through the logistic growth model. A differential equation expresses the rate of change of a quantity, in this case, population size, in terms of the population itself.

The logistic growth model is represented by a differential equation: \[P' = CP - P^2\] where \(P'\) is the change in population, \(C\) is the growth rate, and \(P\) is the current population size. This equation reflects how the growth rate declines as the population approaches the carrying capacity.

Understanding how to solve these equations is crucial for predicting population trends. Techniques involve setting \(P' = 0\) to find equilibrium points, and using integration for more precise predictions of population sizes over time.

Equilibrium Points

Equilibrium points in a logistic growth model represent population sizes where the growth rate is zero. Identifying these points provides valuable insights into population stability and potential for change. In logistic growth, equilibrium points are determined by solving the differential equation when the rate of change, \(P'\), equals zero.

For the rabbit population exercise, solving \(0.04P - \frac{1}{500}P^2 = 0\) yields two equilibrium points: 0 and 500.

• The point \(P = 0\) (extinction point) shows instability because adding even a few rabbits increases population growth rate.
• The point \(P = 500\) (carrying capacity) is stable since any increase or decrease corrects back to 500 over time.

Understanding these points helps predict long-term behavior and the sustainability of populations under given environmental conditions.