1. Systems of two differential equations

The general form for a system of two differential equations is as follows

dx

dt= f(t, x, y)

dy

dt= g(t, x, y)

We will focus on systems of autonomous equations, where the right-hand sides do not depend ontime t and are functions of only x and y.

dx

dt= f(x, y)

dy

dt= g(x, y)

As a specific example, consider

dx

dt= (2− x)y

dy

dt= x(y − 1)

What it means to solve this system of two differential equations is to find functions x(t) and y(t)that satisfy the equations. Each of the equations would contain both of the unknown functions xand y, so the equations cannot be solved separately but must be investigated together. x and y arecalled state variables because their values at any time t describe the state of the system.

Just as with one equation, we are often looking to find a particular solution for a system ofdifferential equations, that is we are seeking functions x and y which satisfy a certain initialcondition. The initial condition specifies values for x and y at some specific time, often t = 0:x(0) = x0 and y(0) = y0.

Of main interest in studying a system of differential equations is finding constant solutions(equilibria) and exploring the long-term behavior of the system.

There are three types of plots which are very helpful in visualizing the behavior of solutions of asystem of differential equations:

• a direction field plot• a phase plane plot and• component plots.

The direction field is a plot similar to the slope field plot for a single differential equation. However,rather than having time on the horizontal axis as in the slope field, the direction field plot has thesecond state variable on the vertical axis and the first state variable on the horizontal axis. (Forour example system, we will have y(t) on the the vertical axis and x(t) on the horizontal axis.) So,the solution curves we can visualize on the direction field consist of points (x(t), y(t)).

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Systems of two differential equations 2

If we know values that x and y take on at some specific time (most often time t = 0), we can traceout a curve on the direction field plot. Different initial conditions (x(0), y(0)) would lead to differentsolutions of the system of differential equations, and therefore different curves on the direction field.

Because the direction field plot does not explicitly show time, the direction and speed of motion fora solution curve are indicated with an arrow. Such an arrow is called a vector. The direction ofthe arrow allows to determine the direction in which the values for x and y move. The length of thearrow can be used to determine how fast we are moving along the solution curves x(t) and y(t). Alonger arrow indicates faster motion, and a shorter arrow indicates slower motion. However, somecomputer tools for generating direction fields, like the one we are using, draw the arrows with thesame length. For the analysis we will do in this course, just knowing the direction of motion willbe sufficient. (A computer tool called pplane draws the arrows with different lengths. If you wouldlike to download and use it, visit: https://www.cs.unm.edu/∼joel/dfield/).

The direction field plot helps us to figure out how solution curves behave. The plot is constructedby taking a grid of points (x(t), y(t)) and drawing arrows at all points. The arrow (vector) at eachpoint is tangent to a curve that passes through that point.

Shown to the right is the direction field for

dx

dt= (2− x)y

dy

dt= x(y − 1)

We can visualize a curve in the direction field by locating a specific initial condition and followingthe flow of the field. Visualizing a representative sample of curves in the direction field, we obtaina phase plane plot for the system of differential equations.

Shown to the right is a phase plane plot for

dx

dt= (2− x)y

dy

dt= x(y − 1)

Systems of two differential equations 3

The graphs of x(t) and y(t) versus time t are called component plots. Each visualized curve inthe direction field has corresponding component plots for x(t) and y(t).

Below are some examples fordx

dt= (2− x)y

dy

dt= x(y − 1)

Component plots corresponding to orange curve in the direction field. The horizontal axis is time.

Component plots corresponding to purple curve in the direction field. The horizontal axis is time.

Component plots corresponding to gray curve in the direction field. The horizontal axis is time.

Systems of two differential equations 4

Constant solutions (equilibria)

For a system of two differential equations, we have a constant solution (equilibrium) when bothdx

dt= 0 and

dy

dt= 0. An equilibrium appears as a point (x(t), y(t)) = (c, d) in the direction field.

The values x(t) = c and y(t) = d make bothdx

dt= 0 and

dy

dt= 0. A system of differential equations

can have more than one equilibrium.

If solution curves move toward an equilibrium, the equilibrium is called stable. If solution curvesmove away from an equilibrium, the equilibrium is called unstable. If solution curves loop aroundan equilibrium, that equilibrium is neither stable nor unstable.

As we can see in the phase plane plot for

dx

dt= (2− x)y

dy

dt= x(y − 1)

that we have two equilibria. We find them algebraically as follows:

dx

dt= (2− x)y = 0⇒ x = 2, y = 0

dy

dt= x(y − 1) = 0⇒, x = 0, y = 1

dx

dt= 0 and

dy

dt= 0 are both zero when (x, y) = (2, 1) or (x, y) = (0, 0).

So, the two equilibria are (x, y) = (2, 1) and (x, y) = (0, 0).

(x, y) = (2, 1) is unstable.

(x, y) = (0, 0) is neither stable nor unstable.

Nullclines

Curves where eitherdx

dt= 0 or

dy

dt= 0 are called nullclines of the system of differential equations.

Systems of two differential equations 5

For our example system,dx

dt= (2− x)y = 0 along the vertical line x = 2 and the horizontal line

y = 0. These lines are called x−nullclines sincedx

dt= 0 along them.

dy

dt= x(y − 1) = 0 along the vertical line x = 0 and the horizontal line y = 1. These lines are called

y−nullclines sincedy

dt= 0 along them.

Note that the equilibrium (x, y) = (0, 0) is the intersection of the x−nullcline y = 0 and they−nullcline x = 0.

The equilibrium (x, y) = (2, 1) is the intersection of the x−nullcline x = 2 and the y−nullcliney = 1.

The nullclines divide the phase plane plot into regions where arrows share a common orientation.

Arrows on nullclines are either vertical or horizontal.

Here is again the phase plane plot for

dx

dt= (2− x)y

dy

dt= x(y − 1)

where you can see this exemplified.

Systems of two differential equations 6

Interpretation of arrows in the direction field

The arrows in the direction field have a horizontal and a vertical component. The horizontal compo-nent corresponds to the variable on the horizontal axis, and the vertical component corresponds tothe variable on the vertical axis. Arrows in different directions tell us if the variables are increasing,decreasing or staying constant.

Below is the list of different cases, considering x as the variable on the horizontal axis and y as thevariable on the vertical axis.

For our example system

dx

dt= (2− x)y

dy

dt= x(y − 1)

the nullclines divide the phase plane plot intonine regions. In each region we know what valuesx and y take on, and that allows us to determine

ifdx

dtand

dy

dtare positive or negative.

Systems of two differential equations 7

Then, in each region we draw the direction of the horizontal and vertical arrow components, and thisdetermines the orientation for all the arrows in the region. We also draw arrows on the nullclines,which show if we cross a nullcline or move along it, and in what direction this happens.

The sketch below shows that we cross the nullclines x = 0 (y-axis) and y = 0 (x-axis). On the otherhand, we move along the nullclines x = 2 and y = 1. On the nullcline y = 1, we move toward theequilibrium (x, y) = (2, 1), while on the nullcline x = 2, we move away from it.

Application example: Interacting populations

The videos will present an example of a system of differential equations in the context of twointeracting populations, more specifically a predator-pray model.