0.6 Second order model  (Page 2/9)

• B The Phase Plane

The pair of equations in [link] can be reduced to a single equation by eliminating the time variable $t$ . This can be done by simply dividing one by the other to give

The solution of this equation is examined inthe $p_1,p_2$ plane, which is called the phase plane .

As an example, consider the competition model in [link] in the phase plane

Solutions in the phase plane are

The trajectory starting at $A$ gives the value of $x$ and $y$ with time $t$ , an implicit variable, indicated by the values shown. If a different initial mixture of populations had been assumed, e.g., $B$ , then a different trajectory would result. Indeed, any initial mixture is a point on the phase plane, and thetrajectories indicate how they evolve in time. The more conventional time solutions are shown for the initialof $A$ by

and for $B$ by

Note the relation of the phase plane plots and the time plots. This particular problem will later be examined in greater detail.

One might wonder why this peculiar representation of the solutions is the form of one variable considered as a form of the other.This phase plane approach, although a bit unnatural at first, proves to be a very powerful tool.It is used by many in the literature [link] [link] [link] [link] and is a standard mathematical tool. [link] [link] It is worthwhile developing this concept before analyzing several physical systems.

Note that the phase plane contains all possible time plane plots for various mixtures.It can be shown that if the system has unique solutions, then the phase plane trajectories cannot cross.This means that a few key trajectories can be constructed which will make obvious what all other trajectorie will have to be.For example, in the above competition model, the initial mixture always determines who the eventual winner will be.Any initial mixture to the right of the line from the origin to $C$ results in $y$ increasing without bound and $x$ becoming extinct. Initial mixture to the left gives the opposite result.

There are several procedures that aid in the construction and interpretation of phase plane trajectories.There are special points on the plane known as equilibrium points or singular points that are important. If both $\frac{dx}{dt}$ and $\frac{dy}{dt}$ in [link] are zero, then $x$ and $y$ are constants and the system is in equilibrium.This means that at these points both the numerator and denominator of [link] are zero. For the competition model of [link] , there is a singular point at $x=0$ and $y=0$ , and another at $x=1$ and $y=1$ . Singular points may be stable or unstable depending on whether smallperturbations away from the point tend back to it or go away from it. Both points mentioned above are unstable.

A particular informative way of finding the singular or equilibrium points is to consider what are called partial equilibrium lines in the phase plane. The curve of all possible solutions of the equition

is called the partial equilibrium curve for population $p_1$ . This is understood by considering the first equation in [link] alone. The equation [link] implies $\frac{d{p}_1}{dt}=0$ , therefore, one side of this curve $\frac{d{p}_1}{dt}$ will be positive and on the other side it will be negative.If a particular $f=0$ curve was given by