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# Single phytoplankton species growth with light and crowding effect in a water column

• We investigate a nonlocal reaction-diffusion-advection model which describes the growth of a single phytoplankton species in a water column with crowding effect. The longtime dynamical behavior of this model and the asymptotic profiles of its positive steady states for small crowding effect and large advection rate are established. The results show that there is a critical death rate such that the phytoplankton species survives if and only if its death rate is less than the critical death rate. In contrast to the model without crowding effect, our results show that the density of the phytoplankton species will have a finite limit rather than go to infinity when the death rate disappears. Furthermore, for large sinking rate, the phytoplankton species concentrates at the bottom of the water column with a finite population density. For large buoyant rate, the phytoplankton species concentrates at the surface of the water column with a finite population density.

Mathematics Subject Classification: Primary: 35K57, 35B40; Secondary: 92D25.

 Citation: • • Figure 1.  The bifurcation diagrams of positive steady states of (1)-(2) versus the death rate with $\beta = 0$ in (a) and $\beta>0$ in (b).

Figure 2.  Vertical distributions of phytoplankton for large advection rates with crowding effect. Here we take a typical Michaelis-Menten form $g(I) = \frac{mI}{b+I}$ as the specific growth rate of phytoplankton, and choose the basic parameters of the species to be $D = 0.1, d = 0.2, I_0 = 1, k_0 = 1, k_1 = 0.1, m = 1, b = 1$. We further fix the parameter $\beta = 0.01$ in (a) and (b); $\beta = 0.05$ in (c) and (d); $\beta = 0.15$ in (e) and (f). The advection rates $\upsilon = 0.2, 0.5, 1, 5.$ in (a), (c) and (e) for the red, blue, green and black line respectively; the advection rates $\upsilon = -0.1, -0.5, -1, -3$ in (b), (d) and (f) for the red, blue, green and black line respectively. We observe that phytoplankton concentrates at the bottom or surface of water column with a finite population density for large advection rates.

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