Case Study: Differential Equations (MATH4827)

Developed and taught by John S Butler (TU Dublin)

Description of Module

This is an introduction to the mathematics differential equations motivated by Leaky Integration and the Predator-Prey model using python notebooks.

Leaky Integration

The Leaky Intergration differential equation describes how infromation is leaked over time. Here I use the example of multisensory distance estimation to illustrate the model [1].

1. Distance estimation
2. Multisensory distance estimation

Predator Prey

The Predator-Prey model is a differential equation that describes the interaction between two or more species [2, 3, 4, 5].

3. Linear population differential equation
4. Logistic population non-linear differential equation INTERACTIVE

• Numerical Assignment 1: Dodos

5. Logistic population non-linear differential equation with oscillation INTERACTIVE

• Numerical Assignment 2: Dodos and Humans

6. Predator-Prey differential equations INTERACTIVE
7. Predator-Prey differential equations Phase Plane

• Numerical Assignment 3: Rabbits, Foxes and Myxomatosis

8. Predator-Prey differnetial equations Phase Plane Jacobian

Example Applications of the Predator Prey Models

• The re-introduction of wolves into Ireland
• Modelling the impact of mirco-plastics on fish and humans
• Modelling the spread of a disease using an SIR model INTERACTIVE

References

[1] Campos, J. L., Butler, J. S., & Bülthoff, H. H. (2014). Contributions of visual and proprioceptive information to travelled distance estimation during changing sensory congruencies. Experimental brain research, 232(10), 3277-3289.

[2] Stover, Christopher and Weisstein, Eric W. "Population Growth." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/PopulationGrowth.html

[3] Vito Volterra. Fluctuations in the abundance of a species considered mathematically. Nature,118:558–560,1926.

[4] Alfred J Lotka. Analytical note on certain rhythmic relations inorganic systems.Proceedings of the National Academy of Sciences,6(7):410–415,1920.

[5] Strogatz, S. Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (studies in nonlinearity), Westview Press; 2 edition (29 July 2014)

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Supplemental Reading

Brooks H, Kanjanasaratool U, Kureh Y and Porter M (2021) Disease Detectives: Using Mathematics to Forecast the Spread of Infectious Diseases. Front. Young Minds. 9:577741. doi: 10.3389/frym.2020.577741

Brady RM and Butler JS (2021) The Circle of Life: The Mathematics of Predator-Prey Relationships. Front. Young Minds. 9:651131. doi: 10.3389/frym.2021.651131

Campos JL, Pandi M and Butler JS (2020) “Feeling” Ourselves Move: A Team Effort by Our Senses. Front. Young Minds. 8:9. doi: 10.3389/frym.2020.00009

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Supplementary Video Lectures

Steven Strogatz. (2021, March 1). Nonlinear Dynamics and Chaos - Steven Strogatz, Cornell University [Video]. YouTube. https://www.youtube.com/playlist?list=PLbN57C5Zdl6j_qJA-pARJnKsmROzPnO9V

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Popular Videos

The Relationship Equation - Numberphile. (2015, April 3). [Video]. YouTube. https://www.youtube.com/watch?v=BkOIw7vAZCQ

Monbiot, G., How Wolves Change Rivers TED TALK. (2014, February 13). [Video]. YouTube. https://www.ted.com/talks/george_monbiot_for_more_wonder_rewild_the_world?language=en

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Popular Press Reading

Tree, I. (2018). Wilding: The return of nature to a British farm. Pan Macmillan.

Strogatz, S. (2004). Sync: The emerging science of spontaneous order. Penguin UK.

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Podcasts

Strogatz, S. (2019-2021). Joy of X. Quanta Magazine. https://www.quantamagazine.org/tag/the-joy-of-x