Problem 45 - Calculus for Biology and Medicine

Question

Assume that \(N(t)\) denotes the size of a population at time \(t\) and that \(N(t)\) satisfies the differential equation $$ \frac{d N}{d t}=3 N\left(1-\frac{N}{20}\right) $$ Let \(f(N)=3 N\left(1-\frac{N}{20}\right)\) for \(N \geq 0\). Graph \(f(N)\) as a function of \(N\) and identify all equilibria (i.e., all points where \(\frac{d N}{d t}=0\) ).

Step-by-Step Solution

Answer

Equilibria occur at \( N = 0 \) (unstable) and \( N = 20 \) (stable).

Step 1: Define the Function

The function given by the differential equation is \( f(N) = 3N\left(1 - \frac{N}{20}\right) \). We need to analyze this function to understand the population dynamics.

Step 2: Find the Equilibria

Equilibria occur where the rate of change is zero, i.e., where \( \frac{dN}{dt} = f(N) = 0 \). Solving \( 3N\left(1 - \frac{N}{20}\right) = 0 \), we find:1. \( N = 0 \)2. \( 1 - \frac{N}{20} = 0 \rightarrow N = 20 \)So, the equilibria are at \( N = 0 \) and \( N = 20 \).

Step 3: Analyze Stability of Equilibria

To analyze stability, consider the sign of \( f(N) \) around the equilibrium points:- For \( N < 0 \), \( f(N) < 0 \), but this is not biologically meaningful as population can't be negative.- For \( 0 < N < 20 \), \( f(N) > 0 \) indicating growth.- For \( N > 20 \), \( f(N) < 0 \) indicating a decrease in population.Hence, \( N = 0 \) is a unstable equilibrium and \( N = 20 \) is a stable equilibrium.

Step 4: Graph the Function

Plot \( f(N) \) against \( N \). This involves drawing the parabolic curve, showing intercepts at \( N = 0 \) and \( N = 20 \), peaking at \( N = 10 \) (since it's a symmetric parabola with a maximum in this range). The graph illustrates population growth accelerating until \( N = 10 \) and declining afterwards, showing stability at \( N = 20 \).

Key Concepts

Population Dynamics

Population dynamics is the study of how and why populations change in size and structure over time. In the given differential equation, \[ \frac{d N}{d t}=3 N\left(1-\frac{N}{20}\right) \]we observe how the rate at which the population size, \( N(t) \), changes over time is dependent on its current size.

• The term \( 3N \) suggests that the population grows at a rate proportional to its current size—a common assumption called exponential growth.
• The factor \( 1-\frac{N}{20} \) introduces a limiting factor, reflecting how population growth slows as \( N \) approaches 20, suggesting limited resources, such as food or space.

This combination of growth regulation is typical in models like the logistic growth model, which describes many real-world populations. Such equations help us understand scenarios where a population might grow rapidly when small but stabilize or even decline as resources become scarce.

Equilibrium Analysis

Equilibrium analysis focuses on identifying points where the system does not change over time. In our context, these are the values of \( N \) where the rate of change \( \frac{dN}{dt} \) is zero. By setting \( f(N) = 0 \), we find these points.

• Calculating \[ 3N \left(1 - \frac{N}{20}\right) = 0 \] shows that equilibrium exists where either \( 3N = 0 \) or \( 1 - \frac{N}{20} = 0 \).
• This gives us \( N = 0 \) and \( N = 20 \) as equilibrium points.

The equilibrium points tell us crucial information:

• \( N = 0 \) is the scenario where there is no population.
• \( N = 20 \) represents the carrying capacity, a state where the population can sustain itself long-term.

These points help indicate how populations will behave over time, predicting when they might stabilize or face risk of decline due to overpopulation.

Stability of Equilibria

The stability of equilibria explains whether a system returns to equilibrium after a disturbance.In our exercise, we look at the behavior around each equilibrium.

• By examining \( f(N) \), if it's positive, the population is increasing; if negative, it's decreasing.
• Near \( N = 0 \): For values just above zero, \( f(N) > 0 \), indicating that any small population grows, making 0 an unstable equilibrium.
• Between \( 0 < N < 20 \): \( f(N) > 0 \) shows population growing up until carrying capacity.
• Near \( N = 20 \): If \( N > 20 \), \( f(N) < 0 \), causing the population to decrease back to 20, making it a stable equilibrium.

Stability analysis can reveal whether a firm perturbation will return the system to equilibrium. In our case, if populations stray from \( N = 20 \), the system tends to correct itself, reflecting the real-world resilience of ecological systems.