Analyzing the EdwardsBuckmire Model for Movie Sales
Mike Lopez & P.J. Maresca
What is the Edwards-Buckmire Model?
A three dimensional system of non-linear,
coupled, ordinary differential equations
which predict the rate of change in the
amount of total money grossed (G’ (t)) by
a movie over a period of several weeks.
• G’ = AS
• S’ = -(S-A)
• A’ = (S / (S+ ) + G)A
G(0)=0
S(0) = S0
A(0) = A0
What Do we Want to Accomplish?
- Classify the behavior of the system near the
equilibrium point and use this analysis to
illuminate the practical significance of the model
- linearize the system so as to isolate the
relationship between S’ and A’
- Both seek to simplify the model and work with
parts of it by tweaking the variables and
dimensions of the model.
How to Define Gross?
The formula for the change in gross
revenue made by a movie is given by
the equation:
G’ (t) = SA
In this equation, S represents the
amount of screenings over the time
period in which we are measuring
changes in gross and A represents
the money made from each
screening.
Equation Governing the Number of Movie Screenings
The formula for the change of screenings of a specific
movie over time is given by:
S’ (t)=-(S-A)
Looking at the equation, if the number of screens is
greater than the revenue (i.e. the number of people
attending the movie) the equation predicts that the
number of showings of the movie decreases.
Equation Governing Revenue Made By A Screening
The equation describing the revenue from a particular movie holds all the
parameters of the model:
A’ (t) =  ( S /( S



  )  G ) A
= decay parameter
= the effectiveness of advertisement
= the people who dislike the film (based on a percentage)
Finding The Equilibrium Point
We found the equilibrium point of the system to be (0, 0, G*)by first
setting each of the equations in the system equal to zero as follows:
0 = AS
0 = -(S-A)
0 =  ( S /( S   )  G ) A
One can then observe from the first equation that either A or S must
equal zero. By rearranging variables, the second equation indicates
that S=A. Therefore, plugging these values into the third equation, we
find that G can be any value, which we designate G*.
The Significance of the Equilibrium Point
The value (0, 0, G*) seems to indicate a
state in which the change in the number of
screens and the revenue earned from
each screen remains constant while the
movie’s gross still changes. However, it is
not physically possible for the gross to
change if the two variables which it
depends on, notably S and A, do not
change.
How then can we interpret the
model near this equilibrium point
and will such analysis help us
understand the model from a
practical standpoint?
Graph of System Near the Equilibrium Point When G*= 0
In order to perform an qualitative analysis of our
equilibrium point we allowed the value of G* to
vary. We predicted, from a practical standpoint,
that no matter the values of G*, the system
should reveal the same 3-D behavior because
G* cannot effectively vary because the two
variables on which it depends do not vary.
The following graphs, generated by
Mathematica, display the 3-D behaviors of the
system for different values of G*.
Graph of System Near the Equilibrium Point When G*<0
1.0
0.5
x
0.0
0.5
1.0
1.0
0.0
0.5
0.5
0.0
1.0
y
z
0.5
1.5
1.0
2.0
Graph of System Near the Equilibrium Point When G*>0
1.0
0.5
x
0.0
0.5
1.0
1.0
0.0
0.5
0.5
0.0
1.0
y
z
0.5
1.5
1.0
2.0
Graph of System Near the Equilibrium Point When G*= 0
1.0
0.5
x
0.0
0.5
1.0
1.0
0.0
0.5
0.5
0.0
1.0
y
z
0.5
1.5
1.0
2.0
3-D Analysis of the Model Near Equilibrium Point
Indeed, our 3-D graphs support our idea that no
matter the values of G*, as long as S’ and A’ are
zero, the system will not bifurcate.
This observation raises a practical consideration
of the model: as long as the revenue from each
screening as well as the number of screenings is
not increasing, the film should not be grossing at
different rates!
Linearizing the System
When we linearized the system about the
equilibrium point we obtain the following
Jacobian matrix :
J=
1
0
 1


 0 G * 0 
0

0
0


Notice that when we linearized the system, the entire bottom row of coefficients
(corresponding to the coefficients of the G’ equation) becomes zero. This
Jacobian contains the 2-D system given by:
S’ = -S + A
A’ = ( G*) A
Graph of the 2-D Linearized System
Analysis of the Linearized 2-D System
Effectively eliminates the equation for G’, allowing us to
isolate the relationship between A’ and S’.
Our graphs of the two variables suggest that attendance
will exponentially increase if there is a positive initial
attendance. In other words, if there already an audience
attending the screen, A is bound to increase over time, t,
and as a result, so will the number of screens.
However, this implication is unreasonable because it
implies that attendance will increase indefinitely but there
are only a certain population that can watch the film.
Conclusions
- Near the equilibrium point, we discovered
that the behavior of the system does not
change with G*, yielding a stiff system.
Our analysis indicates that G* cannot vary
unless A’ and S’ are varying leading us to
the practical implication that in this system
changes in gross are dependent on
changes in revenue from a particular
movie which essentially governs the
number of screenings of a movie.
Conclusions (Cont.)
- Analysis of the linearized system reveals
that more information is required to draw
generalizations about G. The elimination
of G in the linearization effectively
highlights the relationship between S’ and
A’ through A and S where a positive initial
amount of movie goers leads to an
exponential growth in the release of
screens per week and money earned.