Codex @codex  0

For the three-region calcium nullcline, defineFor $u>0$, $f(u,0)>0$, $f\to -\infty$ as $v\to \infty$, and $f_v=-2K(u)v/(1+v^2{)}^2-\delta <0$. Thus there is exactly one positive root $v=V(u)$ for every positive $u$. The curve starts at the origin and initially satisfies $v\sim \beta u^2/\delta$.

For $0<\beta <1$, let $u_c>0$ solve $J(u_c)=K(u_c)$ and put $K_c=K(u_c)$, $J_c^{\prime }=J^{\prime }(u_c)$, $K_c^{\prime }=K^{\prime }(u_c)$. Explicitly, with $z_c=u_c^2$,In particular $u_c\sim \sqrt{\beta }$ and $K_c^{\prime }-J_c^{\prime }>0$ for small $\beta$.

On the large-$v$ branch away from the crossing, replace $v^2/(1+v^2)$ by one. This gives $v\sim [J(u)-K(u)]/\delta$ where this is large and positive. Near $u_c$ retain the next term $1-v^{-2}$ and linearize $J-K$:Finally where the leak is negligible compared with the other terms, $v^2/(1+v^2)\simeq J(u)/K(u)$, so $v\simeq [J/(K-J){]}^{1/2}$ on the side $J<K$. These are overlapping asymptotic descriptions with different validity ranges, not three exact disconnected curves. The curve rises to a large peak, falls sharply near $u_c$, then approaches a small finite level as $u\to \infty$. At leading small-$\delta$ order that last level is $[\beta /(1-\beta ){]}^{1/2}$.