Generalized Lotka-Volterra Model with Species Turnover in a Variable-Basis State Space

The state space is a fundamental concept for describing the trajectory of a dynamic system. Depending on its form, it can highlight certain changes over time while ignoring others. This is particularly the case for the spaces associated with theoretical ecology models, notably the generalized Lotka-Volterra (gLV) model, which allows the modeling of interacting populations. The fixed-dimension state space classically used in gLV models does not account for the effective renewal of species through addition, removal, or mutation. To address this limitation, we propose a new variable-base state space, introduced in a previous study. This framework leads to a reformulation of the gLV model within the context of hybrid dynamical systems. To illustrate the approach, we apply the proposed model to the gut microbiota, particularly in the context of bacteriotherapy following antibiotic treatment.