Problem 12 - 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 \(10.42\). $$ \frac{d w}{d t}=w-w r, \quad \frac{d r}{d t}=-r+w r $$ Assume \(w=2\) and \(r=2\) when \(t=0 .\) Do the numbers of robins and worms increase or decrease at first? What happens in the long run?

Step-by-Step Solution

Answer

Initially, worms decrease while robins increase. In the long run, neither settles at equilibrium due to instability.

Step 1: Analyze Initial Conditions

Given the initial conditions where \(w = 2\) and \(r = 2\) at \(t = 0\), we substitute these values into the differential equations to determine the initial changes in the populations of worms and robins.

Step 2: Substitute into Worms' Equation

Substitute \(w = 2\) and \(r = 2\) into the differential equation for worms:\[\frac{d w}{d t} = w - w r = 2 - 2 \times 2 = 2 - 4 = -2\]This indicates that initially, the number of worms decreases.

Step 3: Substitute into Robins' Equation

Substitute \(w = 2\) and \(r = 2\) into the differential equation for robins:\[\frac{d r}{d t} = -r + w r = -2 + 2 \times 2 = -2 + 4 = 2\]This indicates that initially, the number of robins increases.

Step 4: Analyze Equilibrium Points

The equilibrium points occur where both \(\frac{d w}{d t} = 0\) and \(\frac{d r}{d t} = 0\). Solve the system of equations:1. \(w - wr = 0\) implies \(w(1 - r) = 0\) giving \(w = 0\) or \(r = 1\).2. \(-r + wr = 0\) implies \(r(w-1) = 0\) giving \(r = 0\) or \(w = 1\).The equilibrium points are \((0, 0)\) and \((1, 1)\).

Step 5: Long-term Behavior Analysis

The long-term behavior depends on the stability of these equilibrium points. Linearize the system near each equilibrium point using Jacobian matrices, showing the stability characteristics:1. At \((0, 0)\), the system is unstable.2. At \((1, 1)\), analyze eigenvalues to find it is a saddle point, indicating unstable behavior.Thus, the system oscillates around the equilibrium but doesn't settle down at one due to inherent instability.

Key Concepts

Population Dynamics

Population dynamics is a branch of life sciences that studies the size and age composition of populations as dynamic systems. It explores how these populations change over time and what factors drive these changes. In our scenario involving worms and robins, population dynamics helps understand how quantities of each species interact and fluctuate.
For example, individuals in one population might affect the birth and death rates of another. Here, we have differential equations capturing these interactions, where the number of worms and robins influence each other reciprocally.

• The worms' equation, \( \frac{dw}{dt} = w - wr \), implies worms proliferate unless limited by robins.
• The robins' equation, \( \frac{dr}{dt} = -r + wr \), suggests that robins decline unless they benefit from eating worms.

The dynamic interactions captured here are a simple representation of predator-prey relationships, demonstrating the underlying principles of how species affect each other's growth through competing or symbiotic relationships.

Equilibrium Points

Equilibrium points in a differential system refer to the states where populations remain constant over time. Mathematically, these are points where the rates of change are zero.
Solving our given equations for equilibrium, we seek points where:

• \( \frac{dw}{dt} = 0 \), meaning worms neither increase nor decrease.
• \( \frac{dr}{dt} = 0 \), meaning robins neither increase nor decrease.

Solving \( w - wr = 0 \) gives us \( w = 0 \) or \( r = 1 \), and solving \( -r + wr = 0 \) gives \( r = 0 \) or \( w = 1 \). The intersection provides equilibrium points at \((0, 0)\) and \((1, 1)\).
Analyzing these points shows us scenarios where both populations might "balance" out; however, it's essential to study their stability to understand the system's long-term behavior fully.

Jacobian Matrix

The Jacobian matrix is a mathematical tool used to study the local behavior of dynamical systems near equilibrium points. It provides a way to linearize a system of differential equations, offering insights into the system's stability.
Given our model equations:

• \( \frac{dw}{dt} = w - wr \)
• \( \frac{dr}{dt} = -r + wr \)

we can form a Jacobian matrix by taking partial derivatives of these expressions with respect to \(w\) and \(r\).
The resulting Jacobian is:
\[\begin{bmatrix}\frac{\partial (w - wr)}{\partial w} & \frac{\partial (w - wr)}{\partial r} \\frac{\partial (-r + wr)}{\partial w} & \frac{\partial (-r + wr)}{\partial r} \\end{bmatrix} = \begin{bmatrix}1 - r & -w \ r & w - 1 \\end{bmatrix}\]
Evaluating this matrix at equilibrium points like \((0, 0)\) and \((1, 1)\) helps determine system stability. For example, calculations might reveal that \((1, 1)\) is a saddle point, indicating unstable behavior. This information is pivotal for predicting whether populations will stabilize, oscillate, or diverge over time in the real biological context.