GROWTH OF SINGLE PHYTOPLANKTON SPECIES WITH INTERNAL STORAGE IN A WATER COLUMN

In this paper, we analyze a system modeling the growth of single phytoplankton populations in a water column, where population growth increases monotonically with the nutrient quota stored within individuals. We establish a threshold result on the global extinction and persistence of phytoplankton. Condition for persistence is shown to depend on the principal eigenvalue of a boundary value problem, which is related to the physical transport properties of the water column (i.e. the diffusivity and the sinking speed), nutrient uptake rate, and growth rate.


1.
Introduction. In ecology, the understanding of competition between species for resources is a fundamental ecological issue. Much classical competition theory was developed for populations in well-mixed habitats, such as chemostats, in which a nutrient resource is supplied via an inflow, and a balancing outflow removes nutrient and organisms [26]. Classical competition models assume a direct relationship between the external concentration of nutrients and the population growth of organisms, without any intermediate steps of nutrient storage within cells. More precisely, those models ignore differences between individuals, using one ordinary differential equation to govern the growth of each species. These population growth are coupled to dynamics of one or more resources by assuming a constant quota of nutrient per individual, or equivalently, a constant yield of individuals from consumption of a unit of resource [6].

LINFENG MEI, SZE-BI HSU AND FENG-BIN WANG
In phytoplankton ecology, it has long been known that the quota is not a fixed constant. It can vary depending on the growth rate of population. This led to the following internal storage model [4,25,26]: (1.1) For i=1,2, Q i (t) represents the average amount of stored nutrient per cell of i-th population at time t, µ i (Q i ) is the growth rate of species i as a function of cell quota Q i , ρ i (R, Q i ) is the per capita nutrient uptake rate, per cell of species N i as a function of nutrient concentration R and cell quota Q i , Q min,i denotes the threshold cell quota below which no growth of species i occurs. The growth rate µ i (Q i ) takes the forms [2,3,4]: where (Q i − Q min,i ) + is the positive part of (Q i − Q min,i ) and µ i∞ is the maximal growth rate of the species. According to Grover [5], the uptake rate ρ i (R, Q i ) takes the form: 3) ρ max,i (Q i ) = ρ high max,i − (ρ high max,i − ρ low max,i ) where Q min,i ≤ Q i ≤ Q max,i . Cunningham and Nisbet [2,3] took ρ max,i (Q i ) to be a constant. Motivated by these examples, we assume that µ i (Q i ) is defined and continuously differentiable for Q i ≥ Q min,i > 0 and satisfies We assume that ρ i (R, Q i ) is continuously differentiable for R > 0 and Q i ≥ Q min,i and satisfies In particular, ρ i (R, Q i ) > 0 when R > 0. Next, we assume that U i = N i Q i , i = 1, 2, is the total amount of stored nutrient at time t for the species i. Dividing the quantity U i by N i , then yields the average quota per individual, which is identical to the quantity Q i under the assumption of a well mixed system. Then system (1.1) can be rewritten as follows: It is not hard to see that the following conservation properties hold (see, e.g., [25,26]): Thus, system (1.6) can be reduced into a limiting system which is a type-K monotone system (see, e.g., [25,26]). Smith and Waltman [25,26] used the theory of monotone dynamical system to prove only one species can survive for this internal storage model (1.1). Therefore, for models with constant quota [13] as well as for variable-internal-stores models [17,25], we have conclusions that two or more species cannot coexist in a well mixed habitat with only a single limiting resource, a result known as the Competitive Exclusion Principle [6]. For competition models with constant quota, several works have now considered populations and resources that are distributed in spatially variable habitats. A typical model is the unstirred chemostat model [13,20,21] that was introduced as a poorly mixed analog of the chemostat with transport of nutrient and organisms by diffusion. Similar systems were also constructed for competition in the flow reactor habitat, with transport of nutrient and organisms by both advection and diffusion [1,10]. The flow reactor and its modifications are very relevant because they can provide a simple model for riverine reservoirs [9,10]. In contrast to models in a well-mixed habitat, it is possible for two species with constant quota to coexist on a single limiting resource in a spatially variable habitat. As mentioned in the previous paragraphs, it was known that for many microorganisms the quota of nutrient per individual varies dynamically, so that nutrient is stored internally within individuals. Intuitively, quota variation in spatially variable habitats could allow individuals to obtain nutrient in a rich zone of a habitat, for later use to survive passage through a poor zone [12]. A full description of quota variation in a spatially variable habitat should represent the distribution of quotas over individuals, as well as that of individuals and nutrient over space. The existing models are mathematically difficult [11] and computationally challenging [7].
Hsu et al. [18] suggested another approach that arises from averaging over differences among individuals at a given location, in their amounts of stored nutrient. The authors in [12] extend the results obtained concerning competition with variable quota and nutrient storage in the unstirred chemostat [18] to the flow reactor habitat, using a similar approach. In [8], the author further proposes the following competition model with variable quota and nutrient storage in a water column: with boundary conditions (1.8) and initial conditions where the initial-value functions N 0 The spatial coordinate x represents depth of a water column, with x = 0 being the surface and x = L the bottom. Dissolved nutrient R(x, t) diffuses with diffusivity δ. The boundary conditions of R(x, t) are zero-flux condition at the surface (x = 0), and a periodically varying supply of the nutrient, R (0) (t), at the bottom of the habitat (x = L). Population density transports at the same diffusivity δ and moves by advection toward the bottom of the habitat at the sinking speed ν i . The nutrient taken up by individuals is carried within these individuals, so we assume that U i (x, t) follows the same transport processes as N i (x, t). The boundary conditions of U i (x, t) and N i (x, t) are zero-flux conditions at the surface of the habitat, and absorbing conditions at the bottom. The functions µ i (Q i ) and ρ i (R, Q i ) satisfy (1.4) and (1.5) respectively, i = 1, 2. Note that we can calculate the average quota at a location as Ni(x,t) , and apply the functions µ i and ρ i to this average. In this paper, we shall focus on the analysis of the single population model corresponding to system (1.7)-(1.9) with a constant supply of the nutrient, . In order to simplify notation, all subscripts are dropped in the remaining equations and we consider (1.11) and initial conditions (1.12) The organization of this paper is as follows. In section 2, we study the well-posedness and establish a threshold result on the global extinction and persistence for system (1.10)-(1.12) in terms of the principal eigenvalue of an eigenvalue problem by appealing to the theory of uniformly persistence. The loss of a conservative law in our system makes our analysis quite different from those in [18]. A brief discussion section completes the paper.
2. Mathematical analysis of system (1.10)-(1.12). This section is devoted to the analysis of system (1.10)-(1.12). Note that the boundary condition (1.11) at x = L is Dirichlet-type. Let the following transformation:

SINGLE PHYTOPLANKTON SPECIES GROWTH IN A WATER COLUMN 611
Then system (1.10)-(1.12) becomes and initial conditions Let Ω = (0, L) ⊆ R, and p ∈ (1, ∞) be fixed. For each β ∈ ( 1 2 + 1 2p , 1), let X β be the fractional power space of L p (Ω) with respect to the operator − ∂ 2 ∂x 2 and the boundary condition ∂W ∂x (0, t) = 0, W (L, t) = 0 (see, e.g., [14]). Then X β is an ordered Banach space with the order cone X + β consisting of all non-negative functions in X β , and X + β has non-empty interior Int(X + β ). Moreover, [16]). Similarly, we assume that Y β is the fractional power space of L p (Ω) with respect to the operator − ∂ 2 ∂x 2 + ν ∂ ∂x and the boundary condition Then E is an ordered Banach space with the order cone P consisting of all non-negative functions in E, and P has non-empty interior Int(P). Denote the norm on E by · β . Then, there exists a constant k β > 0 such that The biologically relevant domain for the system (2.1)-(2.3) is given by It is easy to see that
Next, we consider a special case where the growth and uptake functions are chosen specifically [8]. In this case, we can further obtain an improved relevant domain for the system (2.1)-(2.3).

Consider the following system
It is easy to see that there exists Q c such that Q c is the unique equilibria of system (2.17), and hence, Then we consider the following eigenvalue problem: Suppose λ 0 is the principal eigenvalue corresponding to the positive eigenfunction φ 0 (x) which is uniquely determined by the normalization max [0,L] φ 0 (x) = 1.
Claim 2. (0, 0, 0) is a uniform weak repeller for the system (2.1)-(2.3) in the sense that Then, there exists t 1 > 0 such that Thus, there exists t 1 > 0 such that This implies that From the last two equations of (2.1) and (2.24), it follows that (2.25) Consider the following auxiliary system (2.26) Note that (2.26) is a cooperative system. Given m > 0, we set (2.18), it is easy to see that (N (x, t),Û (x, t)) satisfies system (2.26) and Thus, we may choose m > 0 such that Then the Comparison Principle implies that Since λ 0 0 > 0, it follows that lim t→∞N (x, t) = ∞, and hence, lim t→∞ N (x, t, v 0 ) = ∞. This contradiction proves claim 2.
3. Discussion. In this paper, we analyze a PDE system (1.10)-(1.12), or equivalently (2.1)-(2.3) that models the growth of single phytoplankton species consuming nutrients in a water column, which are capable of storing the nutrient within their cells. Due to the loss of a conservative law, our system can not be reduced to a monotone system, and the arguments used in [18] can not be applied to our model. We first show that the biologically relevant domain for system (2.1)-(2.3) is positively invariant (see Lemma 2.1), and solutions of our system are eventually bounded (see Lemma 2.2). We can also obtain another biologically relevant domain for system (2.1)-(2.3) if the growth and uptake functions are specifically chosen as in [8] (see Proposition 2.1 and Remark 2.1). Next, we use theory of uniform persistence to prove that the extinction/persistence of phytoplankton species is determined by the principal eigenvalue of (2.19) (see Theorem 2.1). The principal eigenvalue involves the diffusivity, the sinking speed, nutrient uptake rate, and growth rate (see (2.18) and (2.19) ). It is worth noting that the crucial observations in the proof of Theorem 2.1 are two auxiliary systems, (2.21) and (2.26). Since systems (2.21) and (2.26) are both cooperative, the standard comparison principle can be used in our analysis.
For sufficiently large sinking rate ν, we have shown that λ 0 (ν) must be negative, and hence, the phytoplankton species goes to extinction. The larger ν is, the greater the tendency is for the species to sink, that is, the phytoplankton species will tend towards the bottom of water columns. Due to the absorbing boundary conditions at the bottom (i.e., N (L) = 0), this makes it easier for the phytoplankton species to die out.
• If we consider λ 0 as a function of δ, that is, λ 0 = λ 0 (δ), then for any ν ≥ 0 and L > 0, we have lim δ→∞ λ 0 (δ) = −∞; lim Although we are unable to investigate the monotonicity of the principal eigenvalue λ 0 (δ) on the turbulent diffusion rate δ, we determine the asymptotic behaviors of λ 0 (δ) for small turbulent diffusion rate δ, and large turbulent diffusion rate δ under the case where the advection ν ≥ 0. For sufficiently large turbulent diffusion rate δ, we have shown that λ 0 (δ) must be negative, and hence, the phytoplankton species cannot bloom. In such situation, the habitat is a well-mixing water column and the density of the phytoplankton species is the same at any position of the habitat. Then the absorbing boundary conditions at the bottom (i.e., N (L) = 0) ensures that the phytoplankton species cannot bloom. For sufficiently small turbulent diffusion rate δ (poorly mixing water columns), we showed that the phytoplankton species will always persist if advection ν = 0, while the phytoplankton species will always go to extinction if advection ν > 0. We point out that the asymptotic behaviors of λ 0 (ν) or λ 0 (δ) remains open when advection ν is negative. Although it will be of interest to understand the asymptotic behaviors of λ 0 (L) for large/small water column depth L, it remains unclear to us. We have discussed the extinction/persistence of phytoplankton species, and established the existence of steady-state solutions of the single population model (2.1)-(2.3) under suitable conditions (see Theorem 2.1). However, the uniqueness of steady-state solutions of system (2.1)-(2. 3) remains open. To make the analysis more tractable, we also assumed that the supply of the nutrient from the bottom of the habitat is a constant, and we only investigated the single population model in this study. In the near future, we shall study the two-species model (1.7)-(1.9) with variable quota and nutrient storage in a water column, where the supply of the nutrient is a time-periodic function.