Blooming in a Nonlocal, Coupled Phytoplankton-Nutrient Model

Zagaris, Antonios; Doelman, Arjen; Thi, N. N. Pham; Sommeijer, B. P.

doi:10.1137/070693692

Cited by 18 publications

(58 citation statements)

References 17 publications

(45 reference statements)

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“…(See [25] for a short discussion on the nature and specificity of P (L, N).) The light intensity L at depth z and time t is determined by the total amount of planktonic and non-planktonic components in the column [ Hence, the system is non-local-a typical feature of most realistic phytoplankton models.…”

Section: Introductionmentioning

confidence: 99%

“…Effectively, ε 1/4 characterizes the extent of the zone where DCMs appear relative to the depth of the ocean. In this paper, we follow [25] and treat the parameter ε as an asymptotically small parameter, i.e., we assume that 0 < ε ≪ 1 so that (1.5) has, indeed, a singularly perturbed character. The nonlinearity p in (1.5) is given by p(ω + , η, x) = 1 − η (η H + 1 − η) (1 + j H /j(ω + , x)) , (…”

Section: Introductionmentioning

confidence: 99%

“…The associated spectral problem has been investigated in full asymptotic detail in [25], where we worked with the linearization of (1.10) around u + = (0, 0) T , 11) in which f (x) = ν 1 + j H e κx and ν = 1 1 + η H ∈ (0, 1).…”

Section: Introductionmentioning

confidence: 99%

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Emergence of steady and oscillatory localized structures in a phytoplankton–nutrient model

Zagaris

Doelman

2011

Nonlinearity

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Abstract. Co-limitation of marine phytoplankton growth by light and nutrient, both of which are essential for phytoplankton, leads to complex dynamic behavior and a wide array of coherent patterns. The building blocks of this array can be considered to be deep chlorophyll maxima, or DCMs, which are structures localized in a finite depth interior to the water column. From an ecological point of view, DCMs are evocative of a balance between the inflow of light from the water surface and of nutrients from the sediment. From a (linear) bifurcational point of view, they appear through a transcritical bifurcation in which the trivial, no-plankton steady state is destabilized. This article is devoted to the analytic investigation of the weakly nonlinear dynamics of these DCM patterns, and it has two overarching themes. The first of these concerns the fate of the destabilizing stationary DCM mode beyond the center manifold regime. Exploiting the natural singularly perturbed nature of the model, we derive an explicit reduced model of asymptotically high dimension which fully captures these dynamics. Our subsequent and fully detailed study of this model-which involves a subtle asymptotic analysis necessarily transgressing the boundaries of a local center manifold reduction-establishes that a stable DCM pattern indeed appears from a transcritical bifurcation. However, we also deduce that asymptotically close to the original destabilization, the DCM looses its stability in a secondary bifurcation of Hopf type. This is in agreement with indications from numerical simulations available in the literature. Employing the same methods, we also identify a much larger DCM pattern. The development of the method underpinning this work-which, we expect, shall prove useful for a larger class of models-forms the second theme of this article.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Emergence of steady and oscillatory localized structures in a phytoplankton–nutrient model

Zagaris

Doelman

2011

Nonlinearity

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“…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).…”

Section: P (S T)ds mentioning

confidence: 99%

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

Hsu¹,

Lou²

2010

SIAM J. Appl. Math.

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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.

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“…The model of [16] is one among many mathematical models of phytoplankton proposed and investigated in recent years; see [20,9,14,10,15,21,26] and the references therein for related study on the formation of phytoplankton blooms from mathematical, experimental, and numerical viewpoints. It is our hope that some of the mathematical theory and techniques developed here for treating the model of [16] can also find applications in the study of other phytoplankton models.…”

mentioning

confidence: 99%

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

Du¹,

Hsu²

2010

SIAM J. Math. Anal.

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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.

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Blooming in a Nonlocal, Coupled Phytoplankton-Nutrient Model

Cited by 18 publications

References 17 publications

Emergence of steady and oscillatory localized structures in a phytoplankton–nutrient model

Emergence of steady and oscillatory localized structures in a phytoplankton–nutrient model

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

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

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