Problem 25 - Calculus for Biology and Medicine

Question

Let $$ \begin{array}{l} \frac{d x_{1}}{d t}=x_{1}\left(2-x_{1}\right)-x_{1} x_{2} \\ \frac{d x_{2}}{d t}=x_{1} x_{2}-x_{2} \end{array} $$ (a) Graph the zero isoclines. (b) Find all equilibria and classify them, by linearizing the system near each equilibrium. (c) Draw the directions of the vector field on the zero isoclines, and in the regions between the zero isoclines.

Step-by-Step Solution

Answer

Graph the zero isoclines, find equilibria at \( (0,0), (2,0), (1,1) \), classify by linearizing, and analyze vector field directions in regions between isoclines.

Step 1: Identify Zero Isoclines

The zero isoclines for a system of differential equations are found by setting each differential equation equal to zero separately. For \( \frac{d x_1}{dt} = 0 \), solve \[ x_1(2-x_1) - x_1 x_2 = 0. \]This gives us two isoclines: 1. \( x_1 = 0 \) (the x-axis) 2. \( x_2 = 2-x_1 \) (a downward sloping line with intercept at 2).For \( \frac{d x_2}{dt} = 0 \), solve \[ x_1 x_2 - x_2 = 0. \]This gives two isoclines: 1. \( x_2 = 0 \) (the y-axis) 2. \( x_1 = 1 \) (a vertical line at \( x = 1 \)).

Step 2: Find Equilibria

Set both differential equations to zero and solve them simultaneously.1. From \( x_1 = 0 \) and \( x_2 = 0 \), one equilibrium is at \( (0, 0) \).2. From \( x_2 = 0 \) and \( x_1(2 - x_1) = 0 \), equilibria at \( (2, 0) \).3. From \( x_1 = 1 \) and \( 1x_2 - x_2 = 0 \), equilibria at \( (1, 1) \).

Step 3: Classify Equilibria

Linearize the system around each equilibrium point by finding the Jacobian matrix,\[J = \begin{bmatrix}\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \\frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \\end{bmatrix}\]where\( f_1 = x_1(2 - x_1) - x_1x_2 \) and \( f_2 = x_1x_2 - x_2 \).Calculate the Jacobian:- At \( (0, 0) \), determine stability using eigenvalues.- At \( (2, 0) \), determine stability (often a sink or source).- At \( (1, 1) \), linearize, find eigenvalues to determine saddle or center or spiral.

Step 4: Analyze the Vector Field

Graph the zero isoclines and determine the direction of the vector field in-between. Examine the signs of the derivatives \( \frac{dx_1}{dt} \) and \( \frac{dx_2}{dt} \) over different regions between the intersection of isoclines.Draw arrows indicating increasing or decreasing \( x_1 \) and \( x_2 \) based on the sign of derivatives in the regions. Examine where the vector field points towards or away from equilibria, additionally noting any boundary behavior.

Key Concepts

Equilibrium Points

Equilibrium points in differential equations are essential because they indicate where the system does not change over time. In a two-dimensional system like ours, equilibrium points occur where both derivatives \( \frac{dx_1}{dt} \) and \( \frac{dx_2}{dt} \) equal zero simultaneously. This means the rates of change in both variables stop, leading to a stable or neutral state.
To identify these points, solve the system of equations by setting each differential equation to zero. The solutions are the equilibrium points. For example, in our system, we found equilibria at:\[(0, 0), (2, 0), \text{and} (1, 1).\]
Once identified, classify these points by determining their nature, such as stable, unstable, or saddle points. This classification helps predict the behavior of the system near these points, indicating how nearby trajectories might move. Knowing equilibrium points helps understand the long-term behavior of a dynamic system.

Jacobian Matrix

The Jacobian matrix presents a powerful tool for analyzing the behavior of dynamical systems around equilibrium points. It is a matrix of all first-order partial derivatives for a system of equations. The matrix helps understand how small changes in the system's variables influence changes in the system itself.
The Jacobian matrix for our system is computed by deriving:\[J = \begin{bmatrix}\frac{\partial f_1}{\partial x_1} & \frac{\partial f_1}{\partial x_2} \frac{\partial f_2}{\partial x_1} & \frac{\partial f_2}{\partial x_2} \end{bmatrix}\]
Where \( f_1 \) and \( f_2 \) are the original differential equations. Evaluating this matrix at each equilibrium point provides insight into the system's local stability. Eigenvalues of the Jacobian tell us whether the point is a sink (stable), source (unstable), or saddle (unstable).
The Jacobian gives a linear approximation of the system near equilibrium, making it easier to predict system behavior around these crucial points.

Vector Fields

Vector fields visually represent the direction and magnitude of the rates of change of a dynamical system at various points in the space. They provide insights into how the system evolves over time. In a 2D system, for any point \((x_1, x_2)\), the vector \(( \frac{dx_1}{dt}, \frac{dx_2}{dt} )\) represents the direction in which the system is moving at that location.
To analyze the vector field, graph the zero isoclines (lines where either \( \frac{dx_1}{dt}=0 \) or \( \frac{dx_2}{dt}=0 \)) and regions between them. By observing the system behavior in these regions, you can determine where the system is heading in terms of increasing or decreasing values of \( x_1 \) and \( x_2 \).
This graphical analysis lets us visualize the attractors and repellers, paths or orbits in phase space, and the overall flow of the system. Examining a vector field can help understand potential paths the system might take starting from any given point in space, thus making it a valuable tool for dynamical analysis.