Qualitative Analysis

Very often it is almost impossible to find explicitly of implicitly the solutions of a system (specially nonlinear ones). The qualitative approach as well as numerical one are important since they allow us to make conclusions regardless whether we know or not the solutions.

Recall what we did for autonomous equations. First we looked for the equilibrium points and then, in conjunction with the existence and uniqueness theorem, we concluded that non-equilibrium solutions are either increasing or decreasing. This is the result of looking at the sign of the derivative. So what happened for autonomous systems? First recall that the components of the velocity vectors are tex2html_wrap_inline123 and tex2html_wrap_inline125 . These vectors give the direction of the motion along the trajectories. We have the four natural directions (left-down, left-up, right-down, and right-up) and the other four directions (left, right, up, and down). These directions are obtained by looking at the signs of tex2html_wrap_inline123 and tex2html_wrap_inline125 and whether they are equal to 0. If both are zero, then we have an equilibrium point.

Example. Consider the model describing two species competing for the same prey

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Let us only focus on the first quadrant tex2html_wrap_inline133 and tex2html_wrap_inline135 . First, we look for the equilibrium points. We must have

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Algebraic manipulations imply

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and

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The equilibrium points are (0,0), (0,2), (1,0), and tex2html_wrap_inline149 .
Consider the region R delimited by the x-axis, the y-axis, the line 1-x-y=0, and the line 2-3x-y=0.

Clearly inside this region neither tex2html_wrap_inline123 or tex2html_wrap_inline125 are equal to 0. Therefore, they must have constant sign (they are both negative). Hence the direction of the motion is the same (that is left-down) as long as the trajectory lives inside this region.

In fact, looking at the first-quadrant, we have three more regions to add to the above one. The direction of the motion depends on what region we are in (see the picture below)

The boundaries of these regions are very important in determining the direction of the motion along the trajectories. In fact, it helps to visualize the trajectories as slope-field did for autonomous equations. These boundaries are called nullclines.

Nullclines.

Consider the autonomous system

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The x-nullcline is the set of points where tex2html_wrap_inline161 and y-nullcline is the set of points where tex2html_wrap_inline163 . Clearly the points of intersection between x-nullcline and y-nullcline are exactly the equilibrium points. Note that along the x-nullcline the velocity vectors are vertical while along the y-nullcline the velocity vectors are horizontal. Note that as long as we are traveling along a nullcline without crossing an equilibrium point, then the direction of the velocity vector must be the same. Once we cross an equilibrium point, then we may have a change in the direction (from up to down, or right to left, and vice-versa).

Example. Draw the nullclines for the autonomous system and the velocity vectors along them.

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The x-nullcline are given by

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which is equivalent to

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while the y-nullcline are given by

displaymath171

which is equivalent to

displaymath141

In order to find the direction of the velocity vectors along the nullclines, we pick a point on the nullcline and find the direction of the velocity vector at that point. The velocity vector along the segment of the nullcline delimited by equilibrium points which contains the given point will have the same direction. For example, consider the point (2,0). The velocity vector at this point is (-1,0). Therefore the velocity vector at any point (x,0), with x > 1, is horizontal (we are on the y-nullcline) and points to the left. The picture below gives the nullclines and the velocity vectors along them.

In this example, the nullclines are lines. In general we may have any kind of curves.

Example. Draw the nullclines for the autonomous system

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The x-nullcline are given by

displaymath167

which is equivalent to

displaymath139

while the y-nullcline are given by

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which is equivalent to

displaymath191

Hence the y-nullcline is the union of a line with the ellipse

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Information from the nullclines

For most of the nonlinear autonomous systems, it is impossible to find explicitly the solutions. We may use numerical techniques to have an idea about the solutions, but qualitative analysis may be able to answer some questions with a low cost and faster than the numerical technique will do. For example, questions related to the long term behavior of solutions. The nullclines plays a central role in the qualitative approach. Let us illustrate this on the following example.

Example. Discuss the behavior of the solutions of the autonomous system

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We have already found the nullclines and the direction of the velocity vectors along these nullclines.

These nullclines give the birth to four regions in which the direction of the motion is constant. Let us discuss the region bordered by the x-axis, the y-axis, the line 1-x-y=0, and the line 2-3x-y=0. Then the direction of the motion is left-down. So a moving object starting at a position in this region, will follow a path going left-down. We have three choices

tex2html_wrap_inline201
First choice: the trajectory dies at the equilibrium point tex2html_wrap_inline149 .
tex2html_wrap_inline201
Second choice: the starting point is above the trajectory which dies at the equilibrium point tex2html_wrap_inline149 . Then the trajectory will hit the triangle defined by the points tex2html_wrap_inline149 , (0,1), and (0,2). Then it will go up-left and dies at the equilibrium point (0,2).
tex2html_wrap_inline201
Third choice: the starting point is below the trajectory which dies at the equilibrium point tex2html_wrap_inline149 . Then the trajectory will hit the triangle defined by the points tex2html_wrap_inline149 , (1,0), and tex2html_wrap_inline225 . Then it will go down-right and dies at the equilibrium point (1,0).

For the other regions, look at the picture below. We included some solutions for every region.

Remarks. We see from this example that the trajectories which dye at the equilibrium point tex2html_wrap_inline149 are crucial to predicting the behavior of the solutions. These two trajectories are called separatrix because they separate the regions into different subregions with a specific behavior. To find them is a very difficult problem. Notice also that the equilibrium points (0,2) and (1,0) behave like sinks. The classification of equilibrium points will be discussed using the approximation by linear systems.

If you would like more practice, click on Example.

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Author: Mohamed Amine Khamsi

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