Prediction and Predictability in Population Biology: Noise and Chaos

To determine best predictors and quantify prediction insecurities we investigate an analytically solvable stochastic system from epidemiology in which the time dependent solution of the model, the likelihood function and Bayesian posterior among other quantities can be calculated explicitly as function of given data. We give analytical expressions for the prediction probability conditioned on best estimators of parameters versus conditioned on data only, and marginalized over the parameters, and observe that the prediction insecurity is wider in the second case of conditioning on the data only, as it should be done in empirical studies. Though the concept becomes clear in the analytical study the differences between prediction based on data directly and prediction based on best estimates of parameters is small due to the simplicity of the model which allowed the analytic treatment. In an only slightly more complex model which however already cannot be treated analytically we clearly observe the expected large differences between the two predictions. Finally, we discuss additional aspects in more extended stochastic population biological systems which do not only exhibit fixed point dynamics but also bifurcations into deterministic chaos with short term predictability and long term unpredictability as quantified by the characteristic exponent, the largest Lyapunov exponent. All is developed in a unified framework of time dependent and state discrete stochastic processes typical for many population biological systems, in ecology and in epidemiology, and can also incorporate non-exponential waiting times by simply including more intermediate classes between transitions.