by Jess Behrens
© 2005-2018 Jess Behrens, All Rights Reserved
By now it should be obvious that I believe the field of ecology has a lot to contribute to our understanding of human, societal phenomenon that don't involve adorably fuzzy bunnies, iridescent butterflies, cute furry dogs, or stomach churning millipedes; the birds & bees. In other words, not just nature 'outside' of us. For those of you reading this wondering why in the hell anyone would spend so much time on something like this, there you go. One of the common tools used in making predictions about populations of species in 'nature' that I haven't yet included in my work here are the Lotka-Volterra predator/prey equations. I've left them out purposely; because predator-prey interactions involve issues of 'death' which is different from my primary hypothesis that competition in the NCAA Basketball Tournament is structured by shared experience beyond just winning and losing. And the Hawk/Dove/Owl game is one that is based exclusively on competition over shared resources. However, it's not hard to see how being eliminated from the tournament is a type of death. As such, it's possible that Lotka-Volterra may provide interesting insights into the evolutionary game theory simulations I've been describing in my other posts.
The Lotka-Volterra predator/prey equations were first developed in the 1920s. They are first order, non-linear differential equations that are based around a species population's carrying capacity. While they've been used in many different ways, the idea here is to place two species from the tournament data set in Cartesian space relative to one another along with isoclines (a line) that represent thresholds in the species' carrying capacity. The method is well outlined in two ecology textbooks, cited here:
Begon, M., J. L. Harper, and C. R. Townsend. 1996. Ecology: Individuals, Populations, and Communities,
3rd edition. Blackwell Science Ltd. Cambridge, MA. Gotelli, N. J. 1998. A Primer of Ecology, 2nd edition. Sinauer Associates, Inc. Sunderland, MA.
However, a very good, illustrated, brief description of the process is outlined on this University of Tennessee course site. All of the plots I present here will reference this site.
The idea is to place one species on y-axis and the other on the x-axis, similar to what was done in the species plots I posted in other chapters, but with different elements and purpose. Traditionally, these two axes reference the species' population size. But, given that this project is based on simulation results, the x & y axes will reference total species fitness, which is determined by the evolutionary game theory math as defined in Chapter 4. Figure 1 shows an example of one of these bi-species plots. It compares
Figure 1. Lotka-Volterra, Inter-species Competition Plot, All Tournament Years
the relative fitness of Hawks & Owls across all 14 tournament years. I will use it to describe the basic elements of the plot as described in the referenced text books & the website.
As you can see, the plot has two lines, one orange and the other blue, & 14 points that have been labeled by tournament year. The yellow points represent years in which a Hawk won the Tournament and the blue years where an Owl won. The y-axis references Owl fitness totals derived from the evolutionary game theory simulations and the x-axis is the same, but for Hawk fitness. There are 4 additional, important points on this plot that do not correspond to the gray Tournament Year Average Fitness points:
Where the blue line (k1) meets the x-axis
Where the blue line (k1) meets the y-axis
Where the orange line (k2) meets the x-axis
Where the orange line (k2) meets the y-axis
The two lines, K1 (blue) & K2 (orange), in Figure 1 are called isoclines, which are commonly used in Cartesian math. Isoclines represent a consistent threshold that crosses the Cartesian space defined by the x & y axes. Because this project uses the Lotka-Volterra equations to construct the Cartesian plot, these isoclines represent each species' (K1 & K2) total fitness carrying capacity.
Since this is a competitive system, one end of each line represents the species fitness carrying capacity and the other is the same carrying capacity, but mitigated by the impact of the other species on it (i.e. K1's impact on K2 & K2's impact on K1). Thus, the point where the orange line (K2) intersects the y-axis, which references Owls (species 2 - K2), corresponds to the Owl's fitness carrying capacity independent of their interaction with Hawks. At the other end of the orange line, where K2 intersects the x-axis, represents this same fitness carrying capacity, but as it is dependent Owl & Hawk competition. Likewise, these same points exist for the blue line (K1), but they are switched. So, the point where the blue line intersects the x-axis is fitness carrying capacity for Hawks independent of Owls & the y-axis intersect is the Hawk fitness carrying capacity as dependent on competition with Owls. Methodologically speaking, the axis intersects for each line are derived from the best fit regression line for Hawks & Owls independently. The yellow & blue dots represent the average simulated fitness for each species in a given tournament year.
Determining the impact of Hawks on Owls & vice versa is a bit trickier because they require something known as a competition coefficient. However, since we have the game theory simulation results as a surrogate to use in estimating the competition coefficients, we can use the slope of the best fit regression line instead. These simulations, as I discussed in Chapter 4, break the tournament structure and simulate all probable interactions using the percentage of species by type as a simulation weight for determining who plays whom. The game theory math described in Chapter 4 is then used to calculate the total population fitness that results from these repeated, random, simulated interactions. Thus, the resulting data set is effectively a simulated measurement of the species impact on one another through competition. As noted above, the slope of a best fit regression that is based on those simulations is a good estimate of the effect that each species has the other and is what we use here.
There 4 main scenario's that derive from plots like you see in Figure 1, and all of them involve considering the position of the yellow & blue dots relative to K1 & K2. Rather than spell each of them out generally here, I'll point you to the University of Tennessee website. Specifically, Figure 1 corresponds to scenario 3 on that site, which is an unstable equilibrium between Hawks & Owls. This is because K2, or Owl fitness, is higher than Hawk fitness to the left of their intersection point & K1 is higher than K2 to the right of it. What this means is that the carrying capacity for Owls when Hawks fitness is zero (on the Y axis) is higher than the competition mitigated fitness of Hawks at the same point. That relationship switches at values of K1 (x-axis) higher than their intersection point, meaning that Hawks are dominant in these areas. What this means is blue & yellow dots located in between K1 & K2 to the left of the intersection point favor Owls, while Hawks are favored in areas between the two lines and to the right of the intersection point. Points that fall above or below the line do not favor either Hawks or Owls, and represent, respectively, areas that are below or above both carrying capacities. Again, for more explanation, check out the Univ. of Tennessee site.
In Figure 1, five of the points fall in the area between the K1 & K2 (2011, 2014, 2008, 2009, & 2018). The remaining points fall either above or below the lines. In following from the description above, 2011, 2014, & 2018 should all favor Owls, and indeed Owls won in those years. However, 2009 & 2018 fall in the region that should favor Hawks. As such, this portion of Figure 1 is 50% correct (2009 - Hawk; 2018 -Owl). The remaining years include 4 Owl champions (2010, 2017, 2006, & 2016) & 5 Hawk Champions (2012, 2013, 2005, 2007, & 2015), which is nearly 50/50 and is what one would expect given that these points are either above or below total carrying capacity.
Traditional, biological interpretation of the results seen in Figure 1 are based on a different set of assumptions, some of which are physically impossible for basketball teams to replicate. For example, basketball teams are not able to reproduce, which means the entire concept of tournament population 'moving' across the Cartesian space depicted in Figure 1, in iterative reproductive time steps as is normally the case for a Lotka-Volterra plot, is also impossible. However, the position of a static average fitness point (the blue & yellow points) relative to the respective fitness isoclines should still make sense if Lotka-Volterra applies. In general, this seems to be the case as the area between the lines is 75% correct while the areas above and below K1 & K2 are 50/50. However, it is likely that modifications to the simulation may be necessary & will, perhaps, improve the accuracy of predictions made using Figure 1. In fact, the model will definitely be updated in future posts & will include the use of Nash Mixed Method equilibriums. But, this should help introduce the concepts and logic.
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