Best practices for incorporating noise into stochastic differential equation population models

Background/Question/Methods

Stochastic differential equations (SDE’s) are an important tool used in population modeling. They represent time and population density as continuous variables, and hence are the stochastic counterparts of deterministic ordinary differential equations. They can accommodate random influences (i.e., noise) from demographic and environmental phenomena. Unfortunately, there is little consistency in the ecological literature about how SDE models should be constructed. Although mechanistic approaches for deriving SDE’s from first principles exist (e.g., the work of Linda and Edward Allen), most ecological SDE models incorporate noise in a phenomenological, ad hoc manner. Whether these simpler phenomenological approaches adequately capture the mechanisms they are meant to represent remains largely untested.

We translated a suite of important deterministic population models into SDE’s, using the mechanistic approach and the more common phenomenological approach. We compared the predictions of the two approaches for each of the models in the suite to assess whether and when alternative representations of stochasticity give conflicting results.

Results/Conclusions

For many models in our suite, the predictions generated by the mechanistic SDE’s were meaningfully different from those generated by the phenomenological SDE’s. We also found important interactions between environmental and demographic stochasticity in the mechanistic SDE models that were absent from the phenomenological models. Because phenomenological SDE models contain simplistic assumptions about environmental noise that are unlikely to be met in natural systems, our results indicate that researchers should use care when converting differential equations into their stochastic counterparts, and should use a mechanistic approach.