A Reaction-Diffusion-Advection Glucose-Insulin-Epinephrine Model: Well-Posedness, Dissipativity, and Equilibrium Structure

ABSTRACT

In this paper, we propose and analyze a spatial glucose–insulin–epinephrine model formulated as a coupled
reaction–diffusion–advection system. The model extends classical glucose–insulin formulations by incorporating
epinephrine as a counter-regulatory hormone, so that insulin-mediated glucose removal and epinephrine-mediated glucose production are represented within a unified spatial framework.
Methods: For nonnegative initial data, positive physiological parameters, and homogeneous Neumann boundary conditions,we establish local well-posedness, positivity preservation, uniform-in-time bounds, and global existence of classical solutions. We further construct positively invariant and absorbing regions, showing that the associated semiflow is dissipative in the biologically relevant phase space. For spatially homogeneous equilibria, the steady-state system is reduced to a single scalar equation for the glucose concentration.
Results: We prove that the corresponding scalar balance function is strictly decreasing, which implies the existence
and uniqueness of a positive homogeneous equilibrium for every Hill exponent (m\geq1). In addition, explicit bounds for the equilibrium are derived, and the monotone dependence of the equilibrium glucose level on several
physiologically relevant parameters is characterized. These results show that homogeneous multistability does not
arise at the homogeneous equilibrium level in the present feedback structure.
Conclusion: The model provides a well-posed, positive, and dissipative spatial framework for subsequent studies of
linear stability, transport effects, and bifurcation phenomena in glucose regulation.
Keywords: reaction-diffusion-advection system; glucose-insulin-epinephrine model; well-posedness; dissipativity;
invariant region; positive equilibrium; parameter dependence