Growth-fragmentation model for a population presenting heterogeneity in growth rate: Malthus parameter and long-time behavior

Figure 1.  Scheme of the growth mechanism of a cell with growth rate $ \tau = \tau(x) $

Figure 2.  Time evolution of the size distribution per feature, either the whole distribution at a few times (bottom) or at many times but only at size $ x = 1 $ (top). Obtained with coefficients $ \tau (v, x) = vx $, $ \gamma(v, x) = x^2 \tau(v, x) $ and the two variability kernels $ \kappa^{red} $ (2a) and $ \kappa^{irr} $ (2b)

Figure 3.  Time evolution of the estimates of the instantaneous population growth rate obtained with coefficients $ \tau (v, x) = vx $, $ \gamma(v, x) = x^2 \tau(v, x) $ and the two variability kernels $ \kappa^{red} $ (left) and $ \kappa^{irr} $ (middle). For any $ t>0 $, $ \lambda_n(t) $ is obtained by linear regression of the log of the total number, $ s \mapsto \ln \big( \int \! \! \! \int_{ \mathcal{S}} n(s, v, x) \, {\text{d}} v {\text{d}}x \big) $ (right), on the time interval $ \big[ \frac{t}{2}, t \big] $.

Figure 4.  Time evolution of the size distribution per feature at size $ x = 1 $, for the variability kernels $ \kappa^{_{F t S}}(p_0 \pm \varepsilon) $, with $ p_0 $ defined by (22) and $ \varepsilon = 0.05 $ to show that $ p_0 $ appears as a critical value for $ p $ in the case Fastest to Slowest. Obtained with the coefficients $ \tau (v, x) = vx $, $ \gamma(v, x) = x^2 \tau(v, x) $

Figure 5.  Time evolution of the size distribution per feature, either the whole distribution at a few times (bottom) or at many times but only at size $ x = 1 $ (top). Obtained with coefficients $ \tau (v, x) = vx $, $ \gamma(v, x) = x^2 \tau(v, x) $ and the variability kernels $ \kappa^{_{S t F}}(0.5) $, $ \kappa^{_{F t S}}(0.2) $ and $ \kappa^{_{F t S}}(0.8) $. The corresponding Malthus parameters were estimated to be (up to $ 10^{-3} $ precision) $ v_2 = 2 $, $ v_1 = 1 $, and $ \lambda_{2, 0.8} \approx 1.356 $, respectively, in accordance with the conjecture (see Table 2)