On a Nonlocal Reaction-Diffusion Problem Arising from the Modeling of Phytoplankton Growth

Abstract

Abstract.In this paper we analyze a nonlocal reaction-diffusion model which arises from the modeling of competition of phytoplankton species with incomplete mixing in a water column. The nonlocal nonlinearity in the model describes the light limitation for the growth of the phytoplankton species. We first consider the single-species case and obtain a complete description of the longtime dynamical behavior of the model. Then we study the two-species competition model and obtain sufficient conditions for the existence of positive steady states and uniform persistence of the dynamical system. Our approach is based on a new modified comparison principle, fixed point index theory, global bifurcation arguments, elliptic and parabolic estimates, and various analytical techniques.Key words. phytoplankton, competition for light, uniform persistence, reaction-diffusion equation, steady state AMS subject classifications. 35J55, 35J65, 92D25DOI. 10.1137/090775105 Introduction.In this paper we analyze a reaction-diffusion model which describes the growth of phytoplankton species in a eutrophic environment. In such environments there are ample nutrients and the phytoplankton species typically compete for light. In [17,18,25] Huisman and Weissing developed a theory of interspecific competition for light that assumes complete mixing of phytoplankton species. This theory is based on a system of ordinary differential equations (ODEs) and predicts that complete mixing leads to competitive exclusion similar to that in [13,12,1,23]; namely, the species with the lowest "critical light intensity" wins the competition.However, in many aquatic environments, phytoplankton species are not thoroughly mixed. To understand the effect of incomplete mixing on the growth of phytoplankton species in a eutrophic environment, Huisman, van Oostveen, and Weissing [16] introduced a reaction-diffusion model and analyzed the model through numerical simulations. But a thorough mathematical treatment of the model has been lacking. The purpose of this paper is to prove some basic mathematical facts for this model which provide a basis for further rigorous mathematical analysis of the involved reaction-diffusion system. Our uniform persistence result indicates that with incomplete mixing of the phytoplankton species, competitive exclusion does not always happen, and coexistence can occur in some parameter ranges.

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

“…The proof of Theorem 3.1 is similar to that of case v = 0, which was studied in [8], with some modifications. For the sake of completeness we give the proof here in detail.…”

“…They also analyzed in depth the phase transition curve for the case g(I) = aI γ , 0 < γ ≤ 1, by means of reducing the equation to a Bessel equation. In [25] the authors studied the asymptotic behaviors of the eigenvalues and eigenfunctions associated with the linearized operator of (2.1) when D is small and v > 0 is of the order √ D. In [8] the authors study both single species and two species competing for light in a eutrophic ecosystem with no advection, and the dynamics of single species growth is also completely analyzed in [8]. In this paper, we will use several critical rates to give a complete classification of the phase transition from bloom to no bloom for the general single phytoplankton species model (2.1)-(2.5).…”

“…The proof of the uniqueness of a positive solution of (4.1) basically follows from the argument in [8] applying to (4.10). By the strong maximum principle [21], all nonnegative and not identically zero solutions of (4.8) must be strictly positive in [0, L].…”

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).…”

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…[3], [5], [6], [7], [21], [22], [25], [26], [29], [31], [35], [42], [50], [56], [57], [60], [67], [68]. We also recall the related problems faced in [23] and [24] also for higher order operators, and in [19] for p = 2 and N = 1.…”