One of the standard population dynamics models that I learned in my undergrad mathematical modelling units was the Lotka-Volterra equations. These represent a very simple set of assumptions about populations, and while they don’t necessarily give physically realistic population trajectories they’re an interesting introduction to the idea that differential equations systems don’t necessarily have an explicit solution.
The assumptions are essentially: prey grow exponentially in the absence of predators, predation happens at a rate proportional to the product of the predator and prey populations, birth of predators is dependent on the product of predator and prey populations, predators die off exponentially in the absence of prey. In SEB113 we cover non-linear regressions, the mathematical models that lead to them, and then show that mathematical models don’t always yield a nice function. We look at equilibrium solutions and then show that we orbit around it rather than tending towards (or away from) it. We also look at what happens to the trajectories as we change the relative size of the rate parameters.
Last time we did the topic, I posted about using the logistic growth model for our Problem Solving Task and it was pointed out to me that the model has a closed form solution, so we don’t explicitly need to use a numerical solution method. This time around I’ve been playing with using Euler’s method inside JAGS to fit the Lotka-Volterra system to some simulated data from sinusoidal functions (with the same period). I’ve put a bit more effort into the predictive side of the model, though. After obtaining posterior distributions for the parameters (and initial values) I generate simulations with lsode in R, where the parameter values are sampled from the posteriors. The figure below shows the median and 95% CI for the posterior predictive populations as well as points showing the simulated data.
The predictions get more variable as time goes on, as the uncertainty in the parameter values changes the period of the cycles that the Lotka-Volterra system exhibits. This reminds me of a chat I was having with a statistics PhD student earlier this week about sensitivity of models to data. The student’s context is clustering of data using overfitted mixtures, but I ended up digressing and talking about Edward Lorenz’s discovery of chaos theory through a meteorological model that was very sensitive to small changes in parameter values. The variability in the parameter values in the posterior give rise to the same behaviour, as both Lorenz’s work and my little example in JAGS involve variation in input values for deterministic modelling. Mine was deliberate, though, so isn’t as exciting or groundbreaking a discovery as Lorenz but we both come to the same conclusion: forecasting is of limited use when your model is sensitive to small variations in parameters. As time goes on, my credible intervals will likely end up being centred on the equilibrium solution and the uncertainty in the period of the solution (due to changing coefficient ratios) will result in very wide credible intervals.
It’s been a fun little experiment again, and I’m getting more and more interested in combining statistics and differential equations, as it’s a blend of pretty much all of my prior study. The next step would be to use something like MATLAB with a custom Gibbs/Metropolis-Hastings scheme to bring in more of the computational mathematics I took. It’d be interesting to see if there’s space for this sort of modelling in the Mathematical Sciences School’s teaching programs as it combines some topics that aren’t typically taught together. I’ve heard murmurings of further computational statistics classes but haven’t been involved with any planning.