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Chapter 7: Problem 13

A system of differential equations is given. (a) Construct the phase plane, plotting all nullclines, labeling all equilibria, and indicating the direction of motion. (b) Obtain an expression for each equilibrium. \(x^{\prime}=5-2 x-x y, \quad y^{\prime}=x y-y, \quad x, y \geqslant 0\)

Short Answer

Expert verified
Equilibrium is at \((1, 3)\).

Step by step solution

01

Identify Nullclines

The nullclines are where one of the derivatives is zero. For \(x' = 5 - 2x - xy = 0\), solve for \(y\) to get \(y = 5/x - 2\). For \(y' = xy - y = 0\), factor to get \(y(x - 1) = 0\). Thus, \(y = 0\) or \(x = 1\).
02

Plot Nullclines

Plot the nullcline \(y = 5/x - 2\) only for positive \(x\), where it is valid. Also plot horizontal line \(y = 0\) (x-axis) and vertical line \(x = 1\) on the phase plane.
03

Find Equilibrium Points

For equilibrium, both \(x'\) and \(y'\) must be zero. From \(x' = 0\), we get \(xy = 5 - 2x\). Substitute in \(y' = 0\) equations: \(y = 0\) does not provide new information, so set \(x = 1\) into \(xy = 5 - 2x\). We have \(y = 3\), thus equilibrium is \((1, 3)\).
04

Identify Directions on Nullclines

For each nullcline, determine the direction of the flow. If \(y = 5/x - 2\), for \(y > 5/x - 2\), then \(x' < 0\) and for \(y < 5/x - 2\), then \(x' > 0\). For \(y = 0\), if \(x > 1\), \(y' < 0\); for \(x < 1\), \(y' > 0\).
05

Construct Phase Plane

Create the plot with the equilibria, nullclines, and directions of motion. Indicate that solutions move right above the line \(y = 5/x - 2\) to approach equilibrium from left, and down past \(y = 0\) to approach equilibrium vertically down from \(\{x | x > 1\}\).
06

Expression for Equilibria

The only equilibrium is formed when both \(x'\) and \(y'\) are zero at \((x,y) = (1,3)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Phase Plane Analysis
Phase plane analysis is a graphical method to study systems of differential equations. It involves plotting the solutions of the system in a plane, called the phase plane, where each point represents a state of the system. For two dimensions, the variables are usually
  • one for the horizontal axis
  • one for the vertical axis
These axes represent the variables in the given system. In our example, we consider the variables as points (x, y) on the plane.

During phase plane analysis, you plot nullclines, equilibrium points, and direction fields. This helps in the visualization of how the state of the system changes over time and what behaviors are predicted by the differential equations. This process can tell us a lot about the stability of the system and what happens in different regions of the phase plane.
Equilibrium Points
Equilibrium points are crucial in understanding the behavior of differential equations. An equilibrium point occurs where both derivatives in the system of equations become zero, meaning that the system is in a state of constant rest or balance.

For the system given by the equations
  • x' = 5 - 2x - xy
  • y' = xy - y
Equilibrium points are found by setting both
  • x' = 0
  • y' = 0
Solve these equations simultaneously.

In our example, we solve these equations to find that the equilibrium point is (1, 3). This means at this point, both x' and y' will be zero, indicating no change in the system's state.
Nullclines
Nullclines are lines in the phase plane where one of the derivatives is zero. In other words, they help in identifying regions in the phase plane where the system does not move in one direction.

Identifying the nullclines helps in understanding the behavior of the system:
  • For x' = 5 - 2x - xy = 0, we solve for y and rearrange to find
    y = 5/x - 2.
  • For y' = xy - y = 0, factor to find that y = 0 or x = 1.
When plotted, these nullclines divide the phase plane into regions where the direction of motion differs, providing essential clues about the behavior of the system, especially near equilibrium points.
Direction of Motion
Understanding the direction of motion helps predict how the system evolves over time. It refers to the trajectory that the points follow as the system moves through different states.

For analyzing direction of motion:
  • Consider the regions defined by the nullclines. On the nullcline defined by
    y = 5/x - 2, above this line (y > 5/x - 2), the change in x is negative, so the motion is to the left. Below this line, the change in x is positive, so the motion is to the right.
  • On the axis y = 0 and line x = 1, examine the changes: When x > 1, y' < 0 indicates downward motion, and when x < 1, y' > 0 indicates upward motion.
Direction fields are often drawn on the phase plane to illustrate these movements, enhancing our understanding of the system's dynamics near equilibrium points and across the defined regions.

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Most popular questions from this chapter

Von Bertalanffy's equation states that the rate of growth in length of an individual fish is proportional to the difference between the current length \(L\) and the asymptotic length \(L_{\infty}(\) in \(\mathrm{cm}) .\) (a) Write a differential equation that expresses this idea. (b) Make a rough sketch of the graph of a solution to a typical initial-value problem for this differential equation.

Solve the differential equation. $$\frac { d u } { d t } = 2 + 2 u + t + t u$$

Competition and cooperation Each system of differential equations is a model for two species that either compete for the same resources or cooperate for mutual benefit (flowering plants and insect pollinators, for instance). Decide whether each system describes competition or cooperation and explain why it is a reasonable model. (Ask yourself what effect an increase in one species has on the growth rate of the other.) $$\begin{aligned} \text { (a) } \frac{d x}{d t} &=0.12 x-0.0006 x^{2}+0.00001 x y \\ \frac{d y}{d t} &=0.08 x+0.00004 x y \\ \text { (b) } \frac{d x}{d t} &=0.15 x-0.0002 x^{2}-0.0006 x y \\ \frac{d y}{d t} &=0.2 y-0.00008 y^{2}-0.0002 x y \end{aligned}$$

Metastasis of malignant tumors Metastasis is the process by which cancer cells spread throughout the body and initiate tumors in various organs. This sometimes happens via the bloodstream, where cancer cells become lodged in capillaries of organs and then move across the capillary wall into the organ. Using \(\mathrm{C}\) to denote the number of cells lodged in a capillary and \(I\) for the number that have invaded the organ, we can model this as \(C^{\prime}=-\alpha C-\beta C \quad I^{\prime}=\alpha C-\delta I+\rho I\) where all constants are positive, \(\alpha\) is the rate of movement across the capillary wall, \(\beta\) is the rate of dislodgment from the capillary, \(\delta\) is the rate at which cancer cells in the organ die, and \(\rho\) is their growth rate. (a) Suppose \(\rho<\delta .\) Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane. (b) Suppose \(\rho<\delta .\) Construct the phase plane, including all nullclines, equilibria, and arrows indicating the direction of movement in the plane (c) What is the difference in the predicted dynamics between part (a) and part (b)?

(a) Program a calculator or computer to use Euler's method to compute \(y(1),\) where \(y(x)\) is the solution of the initial-value problem $$\frac{d y}{d x}+3 x^{2} y=6 x^{2} \quad y(0)=3$$ for each of the given step sizes. \(\begin{aligned} \text { (i) } h=1 & \text { (ii) } h=0.1 \\ \text { (iii) } h=0.01 & \text { (iv) } h=0.001 \end{aligned}\) (b) Verify that \(y=2+e^{-x^{3}}\) is the exact solution of the differential equation. (c) Find the errors in using Euler's method to compute \(y(1)\) with the step sizes in part (a). What happens to theerror when the step size is divided by 10\(?\)
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