On my first day of research in the BeeLab, my professor showed me this equation:
[1] equation of ant foraging model
I was scared! There are at least 8 parameters we can change and so many variables we want to manipulate. With the differential part? ewww! There is no simple way to summarize the equation or group the parameters together to make the equation looks nicer at the first glance. Is this what I’m going to deal with? I thought in my head.
To back up, this differential equation simulates ants’ behavior in a model that has one central nest and multiple food sources. Each food source has a given quality and distance to the nest, and ants in the colony forage to food sources based on the quality and distance. Pheromones are left behind and form trails that other ants can follow. Each equation describes the rate of ants foraging on the ith trail over time. A simpler version of this equation was first introduced in the paper “From nonlinearity to optimality: pheromone trial foraging by ants” by David Sumpter and Madeleine Beekman, where experiments were conducted on real ants with two food sources of different quality but the same distance to the nest. They summarized the results in a more general equation. Later, people before me in the Bee Lab (or Ant Lab) expanded this general equation to include food sources at different distances and try to understand whether (according to the ants) it’s worth going farther for a higher quality source. The result was summarized and published in the paper “From Foraging Trails to Transport Networks: How the Quality-Distance Trade-off Shapes Network Structure ''.
[2] color coded version of the equation
The color coded equation[2] groups the terms together and summaries the equation in a nicer way. The term in green is the rate at which one ant wanders around and finds a food source, which depends only on the distance from the nest. The term in blue is the recruitment rate at which an ant encounters and joins existing trails to food sources. We know that ants leave pheromones that other ants can track. Thus, there is a positive feedback loop where the more ants there are on the trail, the more pheromones and thus more ants will go on that trail. All these rates are multiplied by the ants available for foraging to get the rate at which the total number of ants join the trail. And last, we subtract the “loss” term where ants randomly wander off the trail.
After consolidating my understanding of what each term means, an idea from economics struck me. The problem we are concerned about is the trade off between distance and quality: how much farther are ants willing to travel to find a better resource?”. This exact question is asked in economics when dealing with trade-offs and utility. We could adapt the idea of an indifference curve, and consider the pairs of distance and quality that are “equivalent” to ants. An indifference curve in econ shows a combination of two goods in various quantities that provides equal satisfaction for a person.
[3] Example of an indifference curve
For example, consider the indifference curves on the right. Let X be the amount of new clothes and Y be the number of TV shows. Here a person is choosing between buying new clothes or watching TV shows to spend their time. Line I1 represents a set of pairs (#new clothes, # TV shows) that yield the same happiness, or utility. That is, the person buying 5 new clothes and watching 2 TV shows (point A) gains the same utility as that same person buying 2 clothes but watching 4.5 TV shows (point B). I2 is a separate indifference curve where points on I2 give more utility than points on I1. That is, buying 4 clothes and watching 4 TV shows (point C) yield a higher utility than A and B. A good thing about the indifference curve is that if we only care about utility, then it does not matter whether we consume at point A or B. All that matters is which indifference curve the point lies on and what utility it gives.
In our ant foraging model, we could define the term “equivalent” to be the set of (distance, quality) that yield the same number of ants on the trail. That is, we assume that a food source that is far away but good quality is the “same” as a food source close to the nest but bad quality because there are the same number of ants on each trail. The utility in this case would be represented by the number of ants on the trail. More ants on the trail would mean the food source gives the ants more utility. Constructing such a model would be very helpful since then we could understand what pairs of quality and distance are “same” to the ants.
But how do we get such pairs of distance and quality? After playing with the equation and giving multiple attempts to solve numerically, I decided to try shrinking the number of food sources to one. Luckily, this yields a cubic function and we were able to find a relationship between distance, quality, and the number of ants on the trail [4].
[4] The total number of ants is 10000. Graph of Q vs D where the lines represent the set of (Q,D) that give the same number of ants on the trail (x=number of ants)
With the excitement of thinking maybe we solved the problem, we tested our equivalence pair in a simulation. I picked two points on the same indifference curve and used the corresponding quality and distance to create a model with two food sources. Ideally, if the two food sources mean the “same” to the ants, which is what the indifference plot is supposed to represent, then the number of ants on the two trails should be the same. However, when running the simulation of the equation model using python, we figured out that all the ants eventually go to the closer food source instead of splitting evenly among the two. This means that the idea of equivalence can not be calculated assuming only one food source and expanding the equivalence relationship to multiple food sources. In other words, two food sources that are the same to ants when we place them one at a time does not mean that the food sources are still the same to ants when they are present at the same time. Attempting to create an indifference curve using two food sources also won’t work because this would yield a six order equation which has no analytic solution.
Although this is an ongoing problem and we haven’t quite figured out the exact solution, I am no longer afraid of this equation. The more I play with it the more I learn. Adapting another subject’s idea when there is a connection is a good approach to start with, and key insights can be found along the way. Even though this approach might not be the one we end up using in the end, I am now more confident in dealing with equations and am not afraid of them any more. More importantly, I learned how to approach this kind of problem by looking for connections between subjects.
Further reading:
David J.T Sumpter, Madeleine Beekman, “From nonlinearity to optimality: pheromone trail foraging by ants”, Animal Behaviour, Volume 66, Issue 2, 2003, Pages 273-280, ISSN 0003-3472, https://doi.org/10.1006/anbe.2003.2224. (https://www.sciencedirect.com/science/article/pii/S000334720392224X)
Lecheval, Valentin, Hannah Larson, Dominic D. R. Burns, Samuel Ellis, Scott Powell, Matina C. Donaldson-Matasci, and Elva J. H. Robinson. 2021. “From Foraging Trails to Transport Networks: How the Quality-Distance Trade-off Shapes Network Structure.” Proceedings of the Royal Society B: Biological Sciences 288 (1949): 20210430. https://doi.org/10.1098/rspb.2021.0430.
Banton, Caroline. “Indifference Curves in Economics: What Do They Explain?”. Investopedia. August 30, 2022. https://www.investopedia.com/terms/i/indifferencecurve.asp
Media credits:
[1]: from the paper “From Foraging Trails to Transport Networks: How the Quality-Distance Trade-off Shapes Network Structure.” https://doi.org/10.1098/rspb.2021.0430.
[2]: made by Fletcher Nickerson
[3]: Public domain image and modified by Yuki Yang. https://commons.wikimedia.org/wiki/File:Simple-indifference-curves.svg
[4]: generated by Yuki Yang using desmos
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