Two-pool calcium model ($\dot{c}=r-kc-{\dot{c}}_s$)

A two-pool calcium model couples cytosolic calcium and an intracellular store. With compatible reference volumes, uptake $J_{+}$ moves calcium into the store, release $J_{-}$ and leakage $k_s{c}_s$ return it, and ${\dot{c}}_s=J_{+}-J_{-}-k_s{c}_s$. External supply $r$ and export $kc$ give $d(c+c_s)/dt=r-kc$. Thus an equilibrium has $c=r/k$ independently of the internal exchange law. Distinct physical reference volumes require their appropriate volume factors before adding the balances.

The unique equilibrium has $u=\mu$ and $f=0$. Its determinant is $-f_v/ϵ>0$ and trace is $-1+(f_v-\gamma f_u)/ϵ$. Thresholds are obtained from the two displayed equations, and a positive-trace interval exists only if the fast normal derivative rises above $ϵ$. At a nondegenerate trace crossing, the linear oscillation frequency is $\sqrt{-f_v/ϵ}$. The sign of a nonlinear Hopf coefficient is needed to classify the local periodic branch; in a bounded positive planar flow, a unique repelling equilibrium also allows a Poincaré-Bendixson existence argument for a periodic orbit.

For $J=\beta u^2/(1+u^2)$ and $K=u^4/(1+u^4)$ with small leak $\delta$, the positive nullcline is unique at each $u$ because its derivative with respect to $v$ is strictly negative. Its asymptotic regions are a high-store balance $v\simeq (J-K)/\delta$, a crossing balance near $J(u_c)=K(u_c)$, and a negligible-leak balance $v^2/(1+v^2)\simeq J/K$. The crossing relation is $(K_c^{\prime }-J_c^{\prime })(u-u_c)\simeq K_c/v^2-\delta v$. These describe one continuous curve, not three distinct equilibria at a fixed cytoplasmic concentration.

For $\dot{u}=\mu -u-\gamma f/ϵ$ and $\dot{v}=f/ϵ$, total scaled calcium $w$ changes only on the external-input/extrusion time scale. Fast motion therefore follows lines $u+\gamma v=\mathrm{constant}$. The fast normal eigenvalue on $f=0$ is $f_v-\gamma f_u$, so $f_v<0$ alone does not establish attraction. If $v=V(u)$, the slow equation is $\dot{u}=(\mu -u)/(1+\gamma V^{\prime }(u))$. Folds of this reduced description satisfy $V^{\prime }=-1/\gamma$.