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# A dynamical approach to phytoplankton blooms

Both authors supported by the National Science Foundation under grant DMS-0940363.
• Algae in the ocean absorb carbon dioxide from the atmosphere and thus play an important role in the carbon cycle. An algal bloom occurs when there is a rapid increase in an algae population. We consider a reaction-advection-diffusion model for algal bloom density and present new proofs of existence and uniqueness results for the steady state solutions using techniques from dynamical systems. On the question of stability of the bloom profiles, we show that the only possible bifurcation would be due to an oscillatory instability.

Mathematics Subject Classification: Primary:34B15, 92D25;Secondary:35B35, 35J65.

 Citation: • • 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

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