A dynamical approach to phytoplankton blooms

Figure 1.  The plane pictured above is $\{q=CP\}$, and the curve is $\gamma(L)$ defined by (28). This figure illustrates the existence and uniqueness of an initial condition $\alpha$ so that the solution $P(L;\alpha)$ satisfies the boundary condition $P'=CP$ at $z=0$ and $z=L$. For all $a\in (0, \alpha)$, we have $P' < CP$ when $z=L$, while for all $a>\alpha$, $P'>CP$. Parameter values are $A=10$, $B=0.5$, $C=1$ and $L=0.1$

Figure 2.  The phase portrait in the invariant plane $\{r=0\}$ of the linearized system (44). The dashed line is $\delta q = C\delta P$; the solid lines are the straight-line solutions determined by the eigenvectors $\lambda_+$ and $\lambda_-$. As shown in Lemma 3.4, in a small neighborhood of the origin, $(\delta P, \delta q)$ either tends to the origin on the stable manifold, or tends to the straight-line solution $\delta q = \lambda_+ \delta P$

Figure 3.  The top row is the projection of $\gamma(L)$ onto the $(P, q)$-plane for the same parameters $(A, B, C)=(10, 0.5, 1)$ at different depths $L$: $L=0.1$ (left), $L=0.3$ (center), and $L=3$ (right). As $L$ increases, the solid red curve $\gamma(L)$ passes above the lower dashed line $q=CP$ and tends toward the upper dashed line $q=\lambda_+ P$, as predicted by Lemma 3.4. The bottom row shows the corresponding pictures in $(P, q, r)$-space: the curve is $\gamma(L)$ for each $L$, and the plane is $\{q=CP\}$. In the first and second plot, $L < L^*$, so there is a nontrivial steady state solution to (1)-(3). In the third picture, $L>L^*$; consequently $\gamma(L)$ lies over the plane and there is no nontrivial solution

Figure 4.  The projection of $\gamma(L)$ onto the $(P, q)$-plane for the same parameters $(A, B, L)=(10, 0.5, 0.5)$ at two different values of the advection coefficient $C$: $C=0.5$ (left) and $C=-0.5$ (right). In each plot, the nontrivial intersection of the solution curve $\gamma(L)$ (solid) with the plane $\{q=CP\}$ (dashed) corresponds to a solution $P(z;a_0)$ to (4)-(6) and occurs so that $\zeta_{a_0}(L)>0$, satisfying Corollary 3. Parameter values are $(A, B, L)=(10, 0.5, 0.5)$ with $F(r)=r$

Figure 5.  Projections of the plane $\{q=CP\}$ and the curve $\gamma(z)$ onto $(p, q)$-space for (a) $z=L$ and (b) $z=z_1$, for $z_1 < L$ chosen in the proof of Lemma 4.3. Each point $P_a$ corresponds to the solution $(P(z;a), q(z;a), r(z;a))$ at the appropriate choice of $z$. (a) A configuration of $\gamma(L)$ with two solutions to (4)-(6), $P(z;A_1)$ and $P(z;A_2)$, both satisfying Corollary 3. By the same lemma, $P(z;\alpha_0)$ is not a solution to (4)-(6) as $P(z;\alpha_0)$ cannot be a nonnegative function on $[0, L]$. (b) As $P(z;A_1)$ and $P(z;A_2)$ must remain in the right-hand plane for all $z < L$, the assumed existence of $P(z;\alpha_0)$ in (a) gives rise to a subset of $\gamma(z_1)$ entirely contained in the left-hand plane. This configuration contradicts Lemma 4.2; as a result, any positive solution to (4)-(6) is unique