Pattern formation on a growing oblate spheroid. an application to adult sea urchin development

Figure 1.  Case of no growth: $ r = 0 $. Bifurcation diagram in the $ (c_1, v_e) $ plane. The branches of the spatially homogeneous equilibria $ (u_e, v_e) $ for model (16) are shown as functions of the parameter $ c_1 $. The parameter $ c_2 $ is fixed as $ c_2 = 0.04 $. In this specific case, the trivial equilibrium $ P_0 $ is an equilibrium for all the values of $ c_1 $ whereas the two equilibria $ P_1 $ and $ P_2 $ exist only in the range $ 0.01 < c_1 < 0.16 $

Figure 2.  Case of growth: $ r \neq 0 $. In the parameter space $ (c_1, r) $, the region below the curve is where existence of the two non trivial equilibria $ P_1^{(r)} $ and $ P_2^{(r)} $ is ensured. The parameter $ c_2 $ is fixed as $ c_2 = 0.04 $. (left) The case $ \omega = 1 $ (right) The case $ \omega = 10 $

Figure 3.  Case of growth: $ r \neq 0 $. The branches of the spatially homogeneous equilibria $ (u_e, v_e) $ for model (16) as functions of the growth parameter $ r $. The parameters $ c_1 $ and $ c_2 $ are fixed at: $ c_1 = 0.03 $, $ c_2 = 0.04 $. The trivial equilibrium $ P_0 $ is an equilibrium for all the values of $ r $ whereas the non trivial equilibria $ P_1^{(r)} $ and $ P_2^{(r)} $ exist only within a particular range of the parameter $ r $ and disappear because of a saddle-node bifurcation. (Left) The case $ \omega = 1 $, (right) The case $ \omega = 10 $

Figure 4.  Effects of the growth parameter $ r $ on the Turing region of the equilibrium $ P_{2} $. Turing regions in the parameter space $ (c_1, d) $ are shown for different values of the parameter $ r $. The other parameters are fixed as: $ c_2 = 0.04 $ and $ \omega = 1 $. Turing regions are bounded by the solid curve, the dashed vertical line and the dash-dot vertical line. Top row: (left) $ r = 0 $; (right) $ r = 0.001 $. Bottom: $ r = 0.002 $

Figure 5.  The stationary case: $ r = 0 $. LSFEM numerical solutions $ v({\bf{x}}, T) $ of the model (16)-(2) on the oblate spheroid with $ f_0 = 1.9645 $ and $ \xi = 4.1134 $ attained at the corresponding integration times $ T = [0, 20, 100, 500, 1500] $. The model parameters are chosen so that $ (c_1, c_2) = (0.03, 0.04) $, $ d = 0.05 $. The size parameter is $ \omega = 10 $

Figure 6.  The slow exponential evolving case: $ r = 0.001 $. LSFEM numerical solutions $ v({\bf{x}}, T) $ of the model (16)-(2) on the oblate spheroid with $ f_0 = 1.9645 $ and $ \xi = 4.1134 $ attained at the corresponding integration times $ T = [0, 20, 100, 500, 1500] $. The model parameters are chosen so that $ (c_1, c_2) = (0.03, 0.04) $, $ d = 0.05 $. The size parameter is $ \omega = 10 $

Figure 7.  Spatial pattern in the biological plot related to the variable $ v({\bf{x}}, T) $ of model (16)-(2) at $ T = 1500 $. The oblate spheroid parameters and the model parameters are fixed as in Fig. 6. (Left) stationary case, $ r = 0 $. (Right) slow exponential evolving case, $ r = 0.001 $