Solving the Predator and Prey Equation
At Florida Tech, math is a large part of many student’s curriculum and many students make a career out of it.
Applied math major Greta Polo’s student design project deals with understanding and modeling a prey and predator and a control population to find their ideal number of meetings while simultaneously maximizing their total numbers.
According to Polo, the project provides a solution to model the meetings of potential populations.
“Say you have an invasive bacteria in the ocean,” Polo said. “You can model how it would affect the fish and what control, like fishing, you would use to solve this problem and at what periods of time you would allow this to happen to help the overall population.”
For example, take a rabbit, a fox and a hunter. Without the fox, the rabbits would multiply infinitely many times; without rabbits, the foxes would die out as they do not have anything to eat, and without hunters, the foxes would eat all the rabbits.
However, most populations cannot be modeled in a continuous manner, as a real-life setting is discontinuous and non-linear. This is what makes the process more complicated and where the math comes in. Polo used a space called L 2, which is the space of discontinuous functions to help model this relationship.
Additionally, when something is not continuous, it is hard to know whether it has a derivative in order to proceed. That is why Polo used the method of Frechet Differentiation.
“When we do this, we prove uniqueness and the existence of a solution which allows us to proceed within our process,” Polo said. “Solving our adjoint system backwards, from a terminal time, allowed us to solve the original system in the normal forward method to proceed to formulate a gradient.”
The gradient method is a method that shows the direction to move in, in order to find the optimal solution. Iterating this and using numerical approximations, Polo modeled how certain conditions would affect the population and when the best hunting seasons would be in order to maximize the numbers of foxes and rabbits.
Polo was inspired to pursue this project because math at a higher level can become very abstract and many people fail to see the applications it has to something in the real world.
“I think this project was very applicable to a real world situation and I was also happy to see it myself as sometimes when doing theoretical math, I find it myself harder to see and understand it,” Polo said.
Polo was very surprised to see how math in infinite dimensions could apply to fully understandable real-life situations.
“This project helped me see how the math you do in theory is fully applicable to something in the real world and without it, it would be a lot harder to properly understand and find a solution,” Polo said. “It will definitely help me in my career if I wanted to go into modeling and controlling certain variables.”
Polo advises students to choose a research topic they like and have the ability to explain it to people.
“Its important to love what you are doing or else you will never get it done in the best way you can,” she said. “This is definitely an important part of your college experience as it allows you to find a way to put together all the things you have learned throughout the years into a project which ties it all together.”
Polo received “Best in Show for Mathematical Sciences” at the 2016 Northrop Grumman Engineering and Science Student Design Showcase.