Figure 4. Asymptotic evolution at constant Ag concentration.

(A) Analysis of the asymptotic deterministic evolution for the large-size limit of the model, at constant available concentration $C=30$. Top left: size of the population vs. number of maturation rounds, showing the exponential increase at rate $\varphi$. Bottom left: average binding energy of the B-cell population, decreasing linearly with speed $u$. Top right: evolution of the binding energy distribution, normalized to the number of cells in the GC, shows a travelling-wave behavior. Different times are represented with different colors, according to the color-scale on the right. Bottom right: distributions of binding energies, shifted by the time-dependent factor $-ut$ and rescaled by the exponential factor $\exp \{-\varphi t\}$. Notice the convergence to the invariant distribution ${\rho }^{*}$. (B) Values of the growth rate $\varphi$ (top) and maturation speed $u$ (bottom) as functions of the Ag concentration $C$. The points at which the two quantities are zeros define the two critical concentration $C^{*}$ and $C^{**}$ (red and purple vertical dashed lines). They split the asymptotic behavior of the system at constant Ag concentration in three different regimes: extinction for $C<C^{*}$ ($\varphi <0$), maturation for $C^{*}<C<C^{**}$ ($\varphi >0$ and $u<0$) and finally stagnation for $C>C^{**}$ ($\varphi >0$ but $u\gtrsim 0$). Results were obtained using parameter values reported in Table 1.