Analyzing the Edwards- Buckmire 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 G(0)=0 S’ = -(S-A) S(0) = S 0 A’ = (S / (S+ ) + G)A A(0) = A 0

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) = = 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 = 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

Graph of System Near the Equilibrium Point When G*>0

Graph of System Near the Equilibrium Point When G*= 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 = 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’ =

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.