Problem 7 - Calculus for Biology and Medicine

Question

In Problems \(7-12\), find all equilibria of each system of differential equations and use the analytical approach to determine the stability of each equilibrium. \(\frac{d x_{1}}{d t}=-x_{1}+2 x_{1}\left(1-x_{1}\right)\) \(\frac{d x_{2}}{d t}=-x_{2}+5 x_{2}\left(1-x_{1}-x_{2}\right)\)

Step-by-Step Solution

Answer

Equilibria are (0,0) and \((\frac{2}{3}, \frac{2}{3})\).

Step 1: Setting the Differential Equations to Zero

To find the equilibrium points, we need to set the right-hand side of each differential equation to zero. Thus, we have:- For \( \frac{dx_1}{dt} \), set \( -x_1 + 2x_1(1-x_1) = 0 \).- For \( \frac{dx_2}{dt} \), set \( -x_2 + 5x_2(1-x_1-x_2) = 0 \).

Step 2: Solving for Equilibrium in \(x_1\) Equation

First, solve \( -x_1 + 2x_1 (1-x_1) = 0 \). Rewriting, we have \( x_1 (2 - 3x_1) = 0 \).This gives \( x_1 = 0 \) or \( 2 - 3x_1 = 0 \) which solves to \( x_1 = \frac{2}{3} \).

Step 3: Solving for Equilibrium in \(x_2\) Equation

Using the equation \( -x_2 + 5x_2(1-x_1-x_2) = 0 \), rewrite it as \( x_2[5(1-x_1-x_2) - 1] = 0 \).So, \( x_2 = 0 \) or \( 5 - 5x_1 - 5x_2 - 1 = 0 \), which simplifies to \( 5x_1 + 5x_2 = 4 \).

Step 4: Finding Equilibrium Points

From previous steps, combine conditions:1. \( (x_1, x_2) = (0,0) \),2. \( x_1 = \frac{2}{3} \) and plugging into \( 5(\frac{2}{3}) + 5x_2 = 4 \) leading to \( 5x_2 = \frac{2}{3} \) and \( x_2 = \frac{2}{3} \), leading to \( (x_1, x_2) = (\frac{2}{3}, \frac{2}{3}) \).

Key Concepts

Equilibrium Points

Equilibrium points in a differential equation system occur where the rate of change is zero; that is, where the system is at rest or unchanging. To find these points, we need to solve for when the derivatives with respect to time equal zero. Let's consider the equations:

• For \( x_1 \): \( \frac{dx_1}{dt} = -x_1 + 2x_1(1-x_1) \)
• For \( x_2 \): \( \frac{dx_2}{dt} = -x_2 + 5x_2(1-x_1-x_2) \)

To find equilibrium points, we set these derivatives to zero, leading us to:
\(-x_1 + 2x_1(1-x_1) = 0 \) and \(-x_2 + 5x_2(1-x_1-x_2) = 0 \).

By solving these equations, we discern potential equilibrium points where the behaviors of \( x_1 \) and \( x_2 \) are constant. Thus, the system's state is identified by finding the intersections of these solutions, such as \((x_1, x_2) = (0, 0)\) and \((x_1, x_2) = \left(\frac{2}{3}, \frac{2}{3}\right)\).
This understanding forms the base for further analyzing how the system behaves near these points.

Stability Analysis

To decide the nature of equilibrium points, we perform a stability analysis. This analysis helps us determine if a system will remain at equilibrium or diverge when slightly disturbed. The stability of a point informs us whether it is "stable" (the system returns to equilibrium), "unstable" (the system deviates further from equilibrium) or "neutral" (the system stays at new close states).

To perform stability analysis:

• Compute the Jacobian matrix, which consists of partial derivatives of the right-hand side of each differential equation with respect to each variable.
• Evaluate the Jacobian at each equilibrium point.
• Analyze the eigenvalues obtained from the Jacobian.

If the real parts of all eigenvalues are negative, the equilibrium is stable; if positive, it is unstable. For example, a Jacobian matrix derived from our system indicates the nature of stability of \((x_1, x_2) = (0, 0)\) in contrast to \((x_1, x_2) = \left(\frac{2}{3}, \frac{2}{3}\right)\), aiding in predicting long-term system behaviors.

Analytical Methods

Analytical methods offer precise ways to solve and understand differential equations without resorting to numerical approximations. These methods, crucial for finding exact solutions, simplify the process of determining behaviors in complex systems like our given problem.

For this differential equation system, solutions came by systematically algebraically manipulating:

• Setting the derivatives to zero to identify equilibrium points, where changes cease.
• Simplifying equations to solve for individual \(x_1\) and \(x_2\) to deduce rest states.
• Analyzing the system’s behavior around these equilibria through linearization and examining the Jacobian matrix.

These methodological steps ensure a thorough understanding of dynamics, providing insights into equilibrium states and the nature of stability for systemic behavior predictions. In real-world applications, such approaches demystify complex, interacting phenomena, leading to better control and prediction capabilities.