When and why microbial-explicit soil organic carbon models can be unstable
Abstract. Microbial-explicit soil organic carbon (SOC) cycling models are increasingly recognized for their advantages over linear models in describing SOC dynamics. These models are known to exhibit oscillations, but it is not clear when they yield stable vs. unstable equilibrium points (EPs) – i.e. EPs that exist analytically, but are not stable to small perturbations and cannot be reached by transient simulations. Occurrence of such unstable EPs can lead to unexpected model behaviour in transient simulations or unrealistic predictions of steady state soil organic carbon (SOC) stocks. Here we ask when and why unstable EPs can occur in an archetypal microbial-explicit model (representing SOC, dissolved OC [DOC], microbial biomass, and extracellular enzymes) and some simplified versions of it. Further, if a model formulation allows for physically meaningful but unstable EPs, can we find constraints in the model parameters (i.e. environmental conditions and microbial traits) that ensure stability of the EPs? We use analytical, numerical and descriptive tools to answer these questions. We found that instability can occur when the resupply of a growth substrate (DOC) is (via a positive feedback loop) dependent on its abundance. We identified a conservative, sufficient condition on model parameters to ensure stability of EPs. Interactive effects of environmental conditions and parameters describing microbial physiology point to the relevance of basic ecological principles for avoidance of unrealistic (i.e. unstable) simulation outcomes. These insights can help to improve applicability of microbial-explicit models, aid our understanding of the dynamics of these models and highlight the relation between mathematical requirements and (in silico) microbial ecology.
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