Nonlinear dynamics and control of biological systems

Feedback, stability, and oscillations

The models describe how the frequencies of competing strategies evolve alongside environmental resources. My work asks when the coupled system settles to an equilibrium, when it develops persistent oscillations, and how the feedback mechanism changes these outcomes. Stability analysis and bifurcation theory explain the behavior observed in simulations and identify the conditions under which it changes.

From mathematical analysis to control

In our Automatica paper, we analyzed replicator–mutator dynamics with environmental feedback. We established conditions for Hopf and heteroclinic bifurcations that generate stable limit cycles and studied their persistence and stability. We also examined an incentive-based control policy, connecting the mathematical analysis to interventions that change the long-term behavior of the coupled system.

Related work investigates how resource dynamics and timescale separation affect these conclusions. Comparing self-renewing and externally supplied resources reveals that systems with similar equilibrium structure can exhibit different global oscillatory behavior. Earlier studies extend the analysis to two interacting populations or communities, where environmental feedback can support recurrent collective dynamics.

A foundation for modeling biological systems

This research established the mathematical foundation for my broader work on biological dynamics. Questions about feedback, stability, and separated timescales also motivate my models of neuron–astrocyte networks, where the focus shifts from population–environment interactions to the mechanisms supporting adaptive computation.