Qualitative analysis of two systems of nonlinear first-order ordinary differential equations for biological systems

Abstract

We consider two systems of nonlinear first-order ordinary differential equations proposed to describe Ca2+-levels in renal vascular smooth muscle cells and in liver cells. Initially, we present the models and its assumptions. We next investigate an approach to local solvability by Picard–Lindelöf 's Theorem. Further, we prove nonnegativity of the systems' possible solutions and we especially conclude global unique existence of the models' solutions by Gronwall-type arguments and the concept of trapping regions. After finishing our theoretical part with some aspects of stability analysis, we provide evidence of our findings by some numerical experiments.