Problem 11 - Applied Calculus

Question

Let \(w\) be the number of worms (in millions) and \(r\) the number of robins (in thousands) living on an island. Suppose \(w\) and \(r\) satisfy the following differential equations, which correspond to the slope field in Figure \(10.42\). $$ \frac{d w}{d t}=w-w r, \quad \frac{d r}{d t}=-r+w r $$ Describe and explain the symmetry you observe in the slope field. What consequences does this symmetry have for the solution curves?

Step-by-Step Solution

Answer

The slope field is symmetric around \(w = r\), implying mirrored solution behaviors. This symmetry affects the dynamics and equilibrium nature of worm-robin interactions.

Step 1: Identify the Differential Equations

The given differential equations describe the change in number of worms and robins over time. These are: \( \frac{d w}{d t} = w - w r \) and \( \frac{d r}{d t} = -r + w r \). These represent the rate of change for worms \(w\) and robins \(r\) dependent on their current values and their interaction.

Step 2: Analyze System Properties

Both equations involve terms with \(w\) and \(r\). The term \(wr\) appears in both equations, indicating that the interaction between worms and robins is mutual and symmetrical. A change in one affects the other equivalently.

Step 3: Identifying Symmetry in Slope Field

The symmetry in the slope field is characterized by the interchangeability of the equations if we swap \(w\) and \(r\), leading to similar solution behaviors mirrored along the line \(w = r\). The slope field appears symmetric around this line, implying that if \((w(t), r(t))\) is a solution, \((r(t), w(t))\) might also exhibit solution-like behavior.

Step 4: Consequences of Symmetry for Solution Curves

The symmetry suggests that solution curves reflecting population dynamics will be mirrored: solutions starting below \(w = r\) will evolve similarly to those that start above it, leading to solution curves that are mirrored across \(w = r\). This affects their stability and equilibrium points, implying shared dynamics between worms and robins.

Key Concepts

Population Dynamics

Population dynamics investigates how populations of different species change and interact over time. In our case, we're exploring the relationship between worms and robins on an island. Specifically, the differential equations

• \( \frac{d w}{d t} = w - w r \)
• \( \frac{d r}{d t} = -r + w r \)

showhow each population affects the other.

Here, the term \(wr\) represents the interaction between the worms and robins. When robins encounter more worms, they tend to thrive due to increased food availability, which increases the term \(-r + wr\). Conversely, more robins might mean fewer worms as robins consume them, represented by \(w - wr\). These interactions lead to changes in population over time, creating complex dynamics.

Understanding these dynamics requires understanding how positive or negative feedback between species can lead to growth or decline in population sizes. Ultimately, the balance captured by these equations helps predict how the community structure may stabilize or disrupt over time.

Slope Field Symmetry

Slope fields help visualize solutions to differential equations by illustrating directions that a solution curve might take at any given point. For our case with worms and robins, examining the slope field reveals a notable symmetry.

• Interchanging \(w\) and \(r\) in the equations doesn't alter their fundamental behavior, as both rely similarly on the term \(wr\).
• This interchange results in a slope field that appears symmetric about the line \(w = r\), across which solution curves can be mirrored.

This symmetry implies that if you observe the relationship from the perspective of robins seeing worms as a resource and vice versa, you'll encounter a kind of mirror image.

This symmetry affects how we predict the solutions: solutions that are symmetric across a line like \(w = r\) are likely to show behaviors that are counterpart in nature on either side of this line. Therefore, solutions that begin in different initial conditions might still exhibit similar dynamics once mirrored across this axis.

Equilibrium Solutions

Equilibrium solutions occur where population levels remain constant over time. Under these conditions, the rates of change for both worms and robins turn to zero.

• For the worms, the equilibrium condition is \( \frac{d w}{d t} = 0 \).
• For the robins, it is \( \frac{d r}{d t} = 0 \).
• Hence, solving \( w - wr = 0 \) and \( -r + wr = 0 \) provides potential solutions for these steady states.

In this model, equilibrium emerge at points where populations cease to grow or decline, indicating a balanced interaction between species.

These equilibrium points are particularly essential as they hint toward stability or instability within an ecosystem.

• If small perturbations around these points result in convergence back to equilibrium, the system is stable.
• If small changes lead away from equilibrium, it indicates instability.

Thus, analyzing these points helps foresee whether the cohabitation of worms and robins will result in persistent sustainability or could lead to one group overshadowing the other over time.