Resolve parametrization degeneracy and absorbing boundaries in the reduced Hamiltonian framework

Develop a general and systematic method to resolve parametrization degeneracy and to handle absorbing boundary conditions within the effective one-dimensional reduction for multidimensional chemical reaction networks based on the Doi-Peliti mapping; specifically, construct criteria for selecting among multiple parametrizations Λ and deriving reduced Hamiltonians H⋆ that correctly recover all fixed points and phase transitions across parameter regimes.

Background

In the predator–prey example, the dimension reduction produced multiple parametrizations, Λ1 and Λ2, leading to different reduced Hamiltonians H⋆ and phase portraits. Only one parametrization captured all fixed points, and absorbing boundary conditions (e.g., predator extinction) necessitated switching parametrizations to recover the physical dynamics.

The authors emphasize that while they can resolve this case-specific degeneracy, a general methodological framework for treating parametrization degeneracy and systematically incorporating absorbing states into the reduced Hamiltonian analysis is lacking and needed for broader applicability.