Single Phytoplankton Species Growth with Light and Advection in a Water Column

Abstract

Abstract. We investigate a nonlocal reaction-diffusion-advection equation which models the growth of a single phytoplankton species in a water column where the species depends solely on light for its metabolism. We study the combined effect of death rate, sinking or buoyant coefficient, water column depth, and vertical turbulent diffusion rate on the persistence of a single phytoplankton species. Under a general reproductive rate which is an increasing function of light intensity, we establish the existence of a critical death rate; i.e., the phytoplankton survives if and only if its death rate is less than the critical death rate. The critical death rate is a strictly monotone decreasing function of the sinking or buoyant coefficient and water column depth, and it is also a strictly monotone decreasing function of the turbulent diffusion rate for buoyant species. In contrast to the critical death rate, a critical sinking or buoyant velocity, a critical water column depth, and a critical turbulent diffusion rate may or may not exist. For instance, if the death rate is suitably small with respect to the water column depth, the phytoplankton can persist for any sinking or buoyant velocity; i.e., there is no critical sinking or buoyant velocity under such a situation. We further show that a critical water column depth, a critical sinking or buoyant velocity, and a critical turbulent diffusion rate for both buoyant species and species with large sinking rates can exist for some intermediate range of phytoplankton death rates and, whenever they exist, are always unique. In strong contrast, there may exist two critical turbulent diffusion rates for species with small sinking rates. The phytoplankton forms a thin layer at the surface of the water column for large buoyant rates, and it forms a thin layer at the bottom of the water column for large sinking rates. Precise characterizations of these thin layers are also given.

Effects of diffusion and advection on the principal eigenvalue of a periodic-parabolic problem with applications

“…Before presenting our monotonicity result, we should point out that, in general, λ N 1 (α, D, L) does not enjoy any monotonicity property on either D, α or L. Indeed, Huston et al in [39] showed that, unlike the autonomous case, λ N 1 (α, D, L) is not monotone in D > 0 even when h ≡ 0. In [34], Hsu and Lou proved …”

“…4 to show that the assertions (ii) and (iii) of Theorem 1.5 remain true. We also mention that though the proof of Theorem 1.5 mainly follows [34], considerable modifications are needed.…”

“…Taking into account the biological meaning, g(I ) is usually assumed to satisfy g(0) = 0 and g is strictly increasing in [0, ∞). Problem (5.1) has received extensive studies numerically as well as theoretically; see, e.g., [22][23][24][25][26]29,31,[34][35][36]41,42,53,54] and the references therein. It should be mentioned that in all those papers, a common assumption is that the incident light intensity I 0 (t) = I 0 is a positive constant.…”

“…As in [25,34], it is easy to show that, given the initial data u 0 ∈ C([0, L]) with u 0 ≥ 0, ≡ 0, (5.5) admits a unique solution, which exists globally in time and is positive for all x ∈ [0, L] and t > 0.…”

Analytical solution of the nitracline with the evolution of subsurface chlorophyll maximum in stratified water columns

“…Major theoretical results include photoacclimation (increase in chlorophyll per cell) (Steele, 1964;Fennel and Boss, 2003), bistability (Yoshiyama and Nakajima, 2002;Ryabov et al, 2010), oscillating SCM (Huisman et al, 2006), hysteresis conditions (Kiefer and Kremer, 1981;Navarro and Ruiz, 2013) and the ESS (evolutionary stable strategy) depth obtained by gametheory approach (Klausmeier and Litchman, 2001;Mellard et al, 2011). Recent mathematical studies solved the persistence and uniqueness of the steady-state solution (Du and Hsu, 2010;Hsu and Yuan, 2010;Du and Mei, 2011) and gave rigorous proofs for the abovementioned ESS depth and the game-theory approach (Du and Hsu, 2008a, b). Additionally, several modeling studies have been conducted to quantitatively assess the importance of different physical-biological processes leading to the SCML (Jamart et al, 1977(Jamart et al, , 1979Varela et al, 1994;Klausmeier and Litchman, 2001;Hodges and Rudnick, 2004;Beckmann and Hense, 2007).…”