A Model for the Coexistence of Competing Mechanisms for \(\text {Ca}^{\text {2}+}\) Oscillations in T-lymphocytes

\(\text {Ca}^{\text {2}+}\) ${\text{Ca}}^{\text{2}+}$ is a ubiquitous signaling mechanism across different cell types. In T-cells, it is associated with cytokine production and immune function. Benson et al. have shown the coexistence of competing \(\text {Ca}^{\text {2}+}\) ${\text{Ca}}^{\text{2}+}$ oscillations during antigen stimulation of T-cell receptors, depending on the presence of extracellular \(\text {Ca}^{\text {2}+}\) ${\text{Ca}}^{\text{2}+}$ influx through the \(\text {Ca}^{\text {2}+}\) ${\text{Ca}}^{\text{2}+}$ release-activated \(\text {Ca}^{\text {2}+}\) ${\text{Ca}}^{\text{2}+}$ channel (Benson in J Biol Chem 29:105310, 2023). In this paper, we construct a mathematical model consisting of five ordinary differential equations and analyze the relationship between the competing oscillatory mechanisms.. We perform bifurcation analysis on two versions of our model, corresponding to the two oscillatory types, to find the defining characteristics of these two families.