More about Qualitative Methods for Differential Equations


What are qualitative methods ?

As we have seen on our last page, we can get a lot of information about the behaviour of solutions to a differential equation simply by looking carefully at what the equation is saying, without actually doing any calculations. These methods are called qualitative since here we emphasize what the solution "looks like" rather than its precise numerical values.

In the case of our simple Logistic Equation,
\[ \frac{dy}{dt} = r y (1 - y), ~~~y(0)=y_0 \]


we have discussed the following approaches:

(1) Full Solution:

\[ y(t) =\frac{y_o}{(1-y_o) e^{-r t} + y_o} \]

Getting this involved a lot of calculations, but it results in an exact description of each of the possible solutions. For each value of the time t, we can say exactly what the value of $ y(t) $ would be, according to the model.


(1) Qualitative Solution:

A plot of the RHS

A sketch of the flow


The RHS of the differential equation (plotted as a function of y) tells us a lot about what sorts of changes are taking place, and how the function $ y(t) $ evolves over time. The plot of the "flow" along the y axis that we can sketch from this information contains all of the important aspects of the qualitative behaviour of the solution.


(1) The Direction Field:

We discussed this last term, and showed that we can also get a qualitative description of the trajectories by plotting information on the yt plane directly. Here the quantity $ dy/dt $ acts as the slope of the tangents to the trajectories. See our summary about direction fields and the description of the direction field for the Logistic Equation from last term.




We are ready to apply our knowledge to a few new examples


A cubic curve


Consider the differential equation

\[ \frac{dy}{dt} = f(y)= y- y^3 \]

We have used the notation $ f(y) $ to represent the expression involving $ y $ on the RHS (right hand side) of the equation. We will not have a specific application in mind here, so that the unknown $ y(t) $ will not be restricted to positive values only. Here is a graph of the function $ f(y) $ :



We have already added arrows to the diagram, indicating the regions for which $ f(y)>0 $ (y increasing) and $ f(y)<0 $ (y decreasing), as well as points at the places where $ f(y)=0 $ (steady states). We could also summarize what this diagram is saying (this is optional) :

$ \frac{dy}{dt}=0 $ whenever

$ y- y^3 =0 $

i.e. $ y=0, 1, -1 $

at these values there is no change
$ \frac{dy}{dt}>0 $ whenever

$ y- y^3>0 $

i.e. $ 0<y<1 $ or $ y<-1 $

at these values y is increasing
$ \frac{dy}{dt}<0 $ whenever

$ y- y^3<0 $

i.e. $ y >1 $ or $ -1<y<0 $

at these values y is decreasing



For your consideration:




The effect of a parameter
We will look at a similar equation to the one we have already discussed, the differential equation

\[ \frac{dy}{dt} = f(y)=r y- y^3. \]

Here $ r $ is some constant (which need not necessarily be positive). In our previous example we used the value $ r=1 $ . But what will happen if $ r $ is changed? Its value will affect the shape of the graph of $ f(y) $ and thus also the behaviour of the flow and the trajectories describing the solutions to this differential equations. Here is a sequence of shapes, showing how the value of r affects the graph of $ f(y) $ .


r = 0.6

r = 0.3

r = -0.2




Clearly, the picture changes as the value of the parameter $ r $ is altered. In particular, this means that the number of zeros of the graph, i.e. the number of steady states also changes! When $ r $ is positive, there are three equilibria, but when $ r=0 $ only one, at the origin, is left. This is another example of the idea of a bifurcation .

The demonstration below allows you to vary the value of $ r $ smoothly and view the resulting changes in the shape of $ f(y) $ (only the positive values of y are shown here.)







For your consideration: