Singular perturbation approach to a plankton model generating harmful algal bloom

Figure 1.  1D blooming stationary solution $ u_1(x;\varepsilon), u_2(x;\varepsilon) $, and $ v(x;\varepsilon) $ of (9) with sharp internal transition layer at $ x = \ell(\varepsilon) $ for $ a = 0.95, b = 1.2, K = 2.9, R = 0.43, d = 2/3, \mu = 0.5, D = 5.0 $, and $ \varepsilon = 0.05 $

Figure 2.  Outer approximations $ U_1(x;\omega), U_2(x;\omega) $, and $ V(x;\omega) $. $ U_1(x;\omega) $ and $ U_2(x;\omega) $ have a discontinuous point at $ x = \ell(\omega) $

Figure 3.  Nullclines of two curves

Figure 4.  The solution $ V(x;\omega^*) \ ( x \in (0, 1)) $ of (13) with $ D = 5.0, \mu = 0.2, d = 1/(1+\mu) = 5/6, R = 0.43 $, and $ K = 2.9 $ (in red) uses both of the monotonically decreasing branches $ y = h_1(V) \ (V \leqq \omega^*) $ and $ y = h_2(V) \ ( V \geqq \omega^*) $, where $ \omega^* = 0.42731 $ and $ \ell^* = 0.324 $

Figure 5.  The solution $ V(x;\omega^*) \ (x \in (0, 1)) $ of (13) with $ D = 5.0, \mu = 0.5, d = 1/(1+\mu) = 2/3, R = 0.43 $, and $ K = 2.9 $ (in red) uses the monotonically decreasing branch $ y = h_1(V) \ (V \leqq \omega^*) $ but the monotonically increasing branch $ y = h_2(V) \ ( V \geqq \omega^*) $, where $ \omega^* = 0.272147 $ and $ \ell^* = 0.3390 $

Figure 6.  The solution $ \eta_2(x) $ of (18) on $ [\ell^*, 1] $ with $ D = 5.0, K = 2.9 $, and $ R = 0.43 $, where $ \omega^* = 0.272147 $ and $ \ell^* = 0.3390 $

Figure 7.  Spatial profiles of four cases of solutions $ z^{(i)}_N(x) $ (in red) and $ z^{(i)}_D(x) $ (in black) $ (i = 1, 2) $

Figure 8.  Profiles of function $ det^*(x) $ for cases (Ⅰ) and (Ⅱ). In case (Ⅰ), $ det^*(x) < 0 $ on $ I_1 $ and $ I_2 $. In case (Ⅱ), $ det^*(x) < 0 $ on $ I_1 $ and $ det^*(x) > 0 $ on $ I_2 $

Figure 9.  Profile of solution $ z_N(x) $ for boundary value problem of first three equations of (33) on $ I $ of case (Ⅱ)

Figure 10.  Solution trajectories of (a) (38) and (b) (39)

Figure 11.  The unstable manifold of $ (0, 0) $ for (38) meets the stable manifold of $ (1-\frac 1K) $ for (39) at $ V = V^* $

Figure 12.  Solution trajectories on composite phase plane at $ V = \omega $: (a) $ 0 < \omega \leqq V^* $ and (b) $ V^* < \omega < 1-\frac 1K $