General applications of nonautonomous system theory to ecological research

Background/Question/Methods

Mathematical models have been widely used to describe ecological systems from population, community to ecosystem levels. The typical approach is based on using ordinary differential equations to model the dynamics of population density, productivity, or biogeochemical cycles. Almost all these models are phrased as autonomous differential equations, which ignores any explicit time dependence in external forcing or parameters.  Thus, rates of change of dependent variables (i.e. population density of a species) depend on constant parameters and other dependent variables (i.e. population density of other species), but do not directly depend on time. However, in many ecological systems the parameters describing external influences are not constant and vary explicitly with time in ways that are not well described by stochastic models. In mathematics, this kind of behavior is described by a nonautonomous system, as opposed to the more commonly used autonomous systems. However, the concept of nonautonomous system has seldom been applied to ecological research. We compared different approaches to the description of nonautonomous system versus autonomous systems and developed a conceptual framework to foster applications of nonautonomous systems in ecological research.

Results/Conclusions

The most striking example of nonautonomous system are periodic systems such as seasonality. For example, seasonality can drive phytoplankton community composition in lakes.  At the ecosystem level, seasonal changes in soil temperature and moisture can significantly change microbial processes and thus affect carbon and nutrient cycling. Such seasonal systems can be described by periodic differential equations and much of the intuition from the behavior of autonomous systems, such as ideas of stability, carries over to this case. However, many important ecological systems are subject to forcing that is nonautonomous and not periodic.  The most striking example is global change. Here, even when ignoring the role of density dependence, a linear system, notions of steady states and stability which have been at the heart of ecological thinking, must be changed.