Residence time and turnover times in a changing environment
In many ecological systems, there are pools or classes through which materials or individuals flow. For example, in a stage structured population individuals change stages through time, moving between different classes. Individuals also enter a system through birth or immigration and leave through death or emigration. In other examples, such as following carbon, the carbon moves among different pools, as in a two pool model between terrestrial vegetation and litter, and can leave or enter the system as well. In these examples, questions of interest are the mean time an individual spends within the system, or within a single class or pool. For the case where all rates are constant, independent of time and density, these questions are easily answered using linear models. For more complex cases with density dependence and changing rates, such as in a model of carbon cycling where rates are changing due to global change, questions of how to define and compute residence time lead to difficult questions.
We develop an approach to both defining and computing residence times in a model with multiple pools cases with changing parameters by beginning with a demographic approach with time varying parameters based on the McKendrick – von Foerster approach. We provide an explicit solution for cases where there are no loops (once a carbon atom leaves a pool it cannot return), and general for a two pool model. We develop implicit solutions for general models, and relate all the dynamics to a non-autonomous set of equations which can then provide a general formula for residence time. The results are discussed in light of multi-pool models for carbon cycling.