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.


1.
Introduction. Phytoplankton are microscopically small plants that drift in the water column of lakes and oceans, which form the base of the aquatic food chain. Phytoplankton require light for photosynthesis. Hence, phytoplankton populations should stay in the well-illuminated upper regions of the water column, since light availability decreases with depth. However, many phytoplankton species are heavier than water. They have a tendency to sink. Sinking phytoplankton plays an important role in several biogeochemical cycles as they can act as a carbon pump. For instance, sinking phytoplankton have the capability of directly affecting the global carbon cycle by exporting photosynthetic carbon from the atmosphere into the ocean interior [1,7,12]. Hence, a better understanding of the population dynamics of sinking phytoplankton species may contribute to a better understanding of the biogeochemical cycling of elements in aquatic ecosystems. On the other hand, some species, like the green algae Botryococcus, have a lower density than water and will float upwards, which is called buoyant. In freshwater lakes and rivers, phytoplankton communities can have a major impact on ecosystem dynamics. The appearance of algae blooms are often a signal of dangerous eutrophication and may result in serious water-quality problems.
The formation of phytoplankton blooms has attracted considerable attention from mathematical, experimental and numerical viewpoints. In the classic work, Riley et al. [24] proposed a linear partial differential equation with respect to phytoplankton concentration to describe the vertical structure of phytoplankton growth dynamics. Although they focused on the interplay between vertical turbulent diffusion and sinking velocity, their mathematical analysis neglected the light dependence of phytoplankton growth. Shigesada and Okubo [26] investigated the self-shading effect on algal vertical distribution in the water column by developing a nonlocal reaction-diffusion-advection model in which they incorporated light-dependent growth rate but neglected light absorption. In order to study the combined effect of vertical turbulent diffusion, sinking velocity, the light-dependent growth and death processes of phytoplankton, Huisman et al. proposed and analyzed a new reactiondiffusion-advection model of light-limited phytoplankton [11,15] in an eutrophic water column. It turned out that the conditions for phytoplankton bloom development can be captured by a critical depth and one or two critical threshold values for vertical turbulent diffusion rate. Moreover, their numerical simulations illustrated that phytoplankton as a whole can maintain a position in the well-illuminated zone near the surface of the water column at intermediate levels of turbulent diffusion. The rigorous mathematical analysis of the critical conditions for phytoplankton bloom has been established by Hsu and Lou in [14]. They showed that a critical water column depth, a critical sinking or buoyant velocity and a critical turbulent diffusion rate can exist for some intermediate range of phytoplankton death rate by means of the strict monotonicity of the critical death rate with respect to the water column depth, the sinking or buoyant velocity and the turbulent diffusion rate. Furthermore, their analysis indicated that the phytoplankton forms a thin layer at the surface of the water column for large buoyant rate, and forms a thin layer at the bottom of the water column for large sinking rate. In [9], Du and Mei extended the results in [8,14] to the model with variable diffusion and sinking rate. The effects of photo-inhibition on the growth of a single phytoplankton species are investigated in [10]. It turns out that the model with photo-inhibition possesses at least two positive steady states in certain parameter ranges. In [22], Peng and Zhao established a threshold type result on the global dynamics of the model of a single phytoplankton with time-periodic incident light intensity and time-periodic death rate in terms of the basic reproduction number. By analyzing various properties of the basic reproduction number with respect to the vertical turbulent diffusion rate, the sinking or buoyant rate and the water column depth, respectively, they revealed some new interesting effects of the modeling parameters and the time-periodic heterogeneous environment on persistence and extinction of the phytoplankton species.
In the current paper, we study the following reaction-diffusion-advection model, which describes the population dynamics of a single phytoplankton species in a water column with crowding effect P t = DP xx − υP x + P [g(I(x, t)) − βP − d], x ∈ (0, L), t > 0, (1) with boundary conditions DP x (0, t) − υP (0, t) = 0, DP x (L, t) − υP (L, t) = 0, t > 0, SINGLE PHYTOPLANKTON SPECIES GROWTH 43 and initial condition where P = P (x, t) denotes the population density of the phytoplankton species. We assume that P 0 (x) ∈ C([0, L]) for simplicity. D > 0 is the vertical turbulent diffusion rate, υ is the sinking velocity (υ > 0) or the buoyant velocity (υ < 0), L > 0 is the depth of the water column, and d > 0 is the natural death rate. The positive parameter β gives rise to death rate βP which is due to crowding effect.
The assumption that the light gradient follows Lambert-Beer's law indicates that the light intensity I(x, t) can be given by where I 0 is the incident light intensity, k 0 is the total background turbidity due to all non-phytoplankton components, and k 1 is the specific light attenuation coefficient of the phytoplankton. g(I) is the specific growth rate of phytoplankton as a function of light intensity I(x, t). Here we assume that the specific growth rate g(I) ∈ C 1 ([0, ∞)) satisfies g(0) = 0, g (I) > 0 for I ≥ 0, and g(I) ≥ aI for I ∈ [0, where a > 0 and > 0. A typical example of g(I) takes the Michaelis−Menten form where m is the maximal growth rate and b is the half saturation constant. Du and Hsu investigated the special case β = 0, υ = 0 in [8]. They first established a special comparison lemma and a boundedness lemma (see Lemma 3.1 and Lemma 3.2 in [8]). Then using this two key lemmas, a complete description of the longtime dynamical behavior for the model with β = 0, υ = 0 is established. As mentioned before, Hsu and Lou studied the combined effect of the death rate, sinking or buoyant coefficient, water column depth, and vertical turbulent diffusion rate on the persistence of a single phytoplankton species when β = 0, υ = 0 in [14]. The asymptotic profiles of steady states for large advection rates are also investigated there. It turns out that the vertical distribution looks like a Dirac function at the bottom of the water column when the sinking velocity is sufficiently large, which indicates that the phytoplankton forms a thin layer at the bottom of the water column for large sinking rate. Similarly, the vertical distribution looks like a Dirac function at the surface of the water column when the buoyant coefficient is sufficiently large, which indicates that the phytoplankton forms a thin layer at the surface of the water column for large buoyant rate.
We focus on the longtime dynamical behavior for the general model (1)-(3) and the asymptotic profiles of steady states for large sinking or buoyant rates. Our results show that there is also 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, when the sinking velocity is sufficiently large, the phytoplankton species concentrates at the bottom of the water column with a finite population density. And the phytoplankton species concentrates at the surface of the water column with a finite population density when the buoyant velocity is large enough. Although the idea is motivated by [8,14], significant changes are needed in the detailed arguments due to the introduction of crowding effect (see Lemma 3.2 and Lemma 4.6).
The organization of this paper is as follows. In section 2, we establish the existence and uniqueness of positive steady states in terms of the death rate of the phytoplankton species. The longtime behavior of the solution is established by comparison principle and various analytical techniques in section 3. The aim of section 4 is to investigate the asymptotic profiles of positive equilibria for small crowding effect and large advection rate.
We end the introduction by mentioning some related mathematical research on the ecological models with crowding effect. Crowding effect is the phenomenon that population growth rate decreases with the increase of density in the process of population growth. If the environmental condition is infinite, the population may increase exponentially. But in the limited environment it follows the form of logistic growth, which is called environmental resistance or crowding effect. In [19,20], the well-stirred chemostat model with crowding effect is proposed. The mathematical analysis illustrates that coexistence occurs when crowding effect is large enough. In [18], the authors studied the unstirred chemostat with crowding effect. It turns out that crowding effect is sufficiently effective in the occurrence of coexisting, and overcrowding of a species has an inhibiting effect on itself. More works concerning crowding effect can be seen in [17,21,28] and references therein.
2. Existence and uniqueness of positive steady states. The purpose of this section is to study the existence and uniqueness of positive steady states of the single population growth model in a water column where D > 0, υ ∈ R, g(I) satisfies (5), and With this in mind, we consider the linear eigenvalue problem Lemma 2.1. [3,14] All eigenvalues of (8) are real, and the smallest eigenvalue λ 1 (q) can be characterized as which corresponds to a positive eigenfunction ψ 1 , and λ 1 (q) is the only eigenvalue whose corresponding eigenfunction does not change sign. Moreover, (i) q 1 (x) ≥ q 2 (x) implies λ 1 (q 1 (x)) ≥ λ 1 (q 2 (x)), and the equality holds only if For every υ ∈ R, L > 0 and D > 0, we define Obviously, d * (υ, L, D) > 0. At first, we derive the priori estimates for positive equilibria of (6).
At first, if x 0 ∈ (0, L), then Q x (x 0 ) = 0 and Q xx (x 0 ) ≤ 0. By using the equation (10), we get that (11) holds. If x 0 = 0, we argue by contradiction. Suppose that Q(0)[g(I 0 ) − βQ(0) − d] < 0. Then by the continuity of g and Q, there exists a small interval [0, δ) such that e (υ/D) It follows from (10) that (e (υ/D)x Q x ) x > 0 for x ∈ (0, δ). Since x 0 = 0 is the maximum point of Q on [0, δ], it follows from the Hopf boundary lemma that Q x (0) < 0, which contradicts the boundary condition in (10). Hence, (11) holds if x 0 = 0. Similarly, we can show that (11) holds if x 0 = L. In summary, (11) For any positive solution P of (6), let Q = e −(υ/D)x P . Then Q satisfies (10), which can be rewritten as 46 DANFENG PANG, HUA NIE AND JIANHUA WU It follows from (12) and Lemma 2.1 that To investigate the existence of positive steady states of (6), we need first establish the existence and uniqueness of positive solutions to the following auxiliary system where I(x) takes the form (7). This auxiliary system plays an important role in determining the profiles of positive solutions to (6), and makes the difference between bifurcation diagrams Fig.1(a) and Fig.1(b).
Proof. We first show the existence of positive solutions of (13). Let U = e −(υ/D)x P . Then U satisfies whereÎ(x) = I 0 exp −k 0 x − k 1 x 0 e (υ/D)s U (s)ds . By similar arguments as in Lemma 2.2, we can show that any positive solution U of (14) satisfying Introduce the following spaces Define a differentiable operator T τ : where K is the solution operator ξ = K(m(x)) for the problem  We claim that T 0 has a unique fixed point 0 in Ω. To this end, let T 0 (U ) = U . Then Integrating this equation over [0, L], we obtain that β L 0 e 2(υ/D)x U 2 dx = 0, which implies that U (x) ≡ 0. Hence, 0 is the unique fixed point of T 0 in Ω, and index(T 0 , Ω, W ) = index(T 0 , 0, W ).
Next, we show that index(T 0 , 0, W ) = 1 by using Lemma A.2. Let T 0 (0) be the Fréchet derivative of T 0 with respect to U at 0. For ϕ ∈ W , let T 0 (0)ϕ = λϕ and ϕ ≡ 0. Then Multiplying this equation by ϕ and integrating over [0, L] by parts, we get Thus, λ < 1 and the spectral radius r(T 0 (0)) < 1. It follows from Lemma A.2 that index(T 0 , 0, W ) = 1. Hence, we deduce that Next, we show that index(T, 0, W ) = 0 by Lemma A.2 again. To this end, let It is easy to see that U ≡ 0, that is, 1 is not an eigenvalue of T (0) in W . Hence, 0 is an isolated fixed point of T in W . Let T (0)U = λU and U ≡ 0. Then Consider the eigenvalue problem Let µ 1 −e (υ/D)x g(I 0 e −k0x ) be the smallest eigenvalue of (16), and let µ 1 (0) be the smallest eigenvalue of Clearly, µ 1 (0) = 0. In view of e (υ/D)x g(I 0 e −k0x ) > 0, it follows from monotonicity of the smallest eigenvalue with respect to the weight function that has an eigenvalue less than 1, denoted by η 1 . Hence, it is easy to see that 1 η1 > 1 is an eigenvalue of T (0). That is, T (0) has an eigenvalue greater than 1. It follows from Lemma A.2 that index(T, 0, W ) = 0.
In view of index(T, Ω, W ) = index(T, 0, W ), it follows from Leray-Schauder degree theory that T has at least one nonzero fixed point in Ω. Namely, (14) (thus (13)) has at least one non-trivial nonnegative solution. By similar arguments as in Lemma 2.2, we can show that the non-trivial nonnegative solution of (13) is a positive solution of (13) by the strong maximum principle and Hopf boundary lemma.
Next, we verify the uniqueness of positive solutions to (13). The idea is motivated by the arguments in [8]. Firstly, it follows from the strong maximum principle that all nontrivial nonnegative solutions of (14) must be strictly positive on [0, L]. Suppose (14) has two positive solutions By the uniqueness of the ODE, we obtain that ( Without loss of generality, we may assume Multiplying (17) with i = 1 by U 2 and (17) with i = 2 by U 1 , integrating over (0, x 0 ) by parts, and subtracting each other, we obtain that

SINGLE PHYTOPLANKTON SPECIES GROWTH 49
The right-hand side of (18) is positive, while its left-hand side is non-positive, a contradiction. Therefore the problem (14)(hence (13)) has a unique positive solution P 0 (β; x).
The following results show that d * is the critical death rate, that is, the phytoplankton species survives if and only if its death rate is less than d * .
, then zero is the only nonnegative steady state of (1)-(2); (ii) If 0 < d < d * (υ, L, D), then (1)-(2) has a unique positive steady state, denoted by P d (β; x), which satisfies lim Proof. It follows from Lemma 2.2 that part (i) holds. Now, we focus our attention on the proof of (ii).
Step 1. Local bifurcation. Let Q = e −(υ/D)x P . Then (6) is equivalent to (10). Let Clearly, K is a strongly positive compact operator. By standard elliptic regularity theory we know that F is a compact differential operator on X.
x correspond to the positive solutions of (10).
Now we begin to construct a positive solution branch (d, Q) ⊂ R + ×X bifurcating from the trivial solution branch (d, 0) by bifurcation theory (see [2,25] or Theorems A.4 and A.5 in the appendix). Let L(d * , 0) be the Fréchet derivative of G(d * , Q) with respect to Q at 0. Then L(d * , 0)Φ = 0 gives By the definition of d * , we know that Φ = cψ 1 , where c is a constant, and ψ 1 is the corresponding eigenfunction of λ 1 (−g(I 0 e −k0x )). Hence, we conclude that the kernel N (L(d * , 0)) = span{ψ 1 }.
Next we determine the range of Multiplying the first equation of (19) by ψ 1 , and integrating over (0, L) by parts, we obtain and codimR(L(d * , 0)) = 1. Thus L(d * , 0) is a Fredholm operator with index zero (see Definition A.3 in the appendix). Moreover, by the strong maximum principle, it is easy to see that It is easy to see that Z ⊕ span{ψ 1 } = X. By the application of the standard bifurcation theorem from a simple eigenvalue (see Theorem A.4), (d * , 0) is a bifurcation point, and there exists δ > 0 and Then the bifurcation branch Γ = {(d( ), P ( )) : 0 < < δ} is exactly the positive solution of (6).
Step 2. Global bifurcation. We show that the local solution branch Γ can be extended to a global one by the application of the global bifurcation results for Fredholm operators (see Theorems A.4 and A.5 in the appendix.) Noting that F : R + ×X → X is C 1 smooth and compact, we can conclude that the Fréchet derivative Then Γ ⊂ C . Let Let C * = C ∩ (R + × X 0 ). Then C * consists of the local positive solution branch Γ near the bifurcation point (d * , 0). That is C * ⊂ R + × X 0 in a small neighborhood of (d * , 0).
Let C + be the connected component of C \{(d( ), P ( )) : −δ < < 0}. Then C * ⊂ C + . It follows from Theorem A.5 that C + satisfies one of the following alternatives: whereĨ n (x) = I 0 exp −k 0 x − k 1 x 0 e (υ/D)s Q n (s)ds . Integrating the first equation of (20) from 0 to x, we have As g(Ĩ n (x)), Q n (x) andQ n (x) are uniformly bounded in (0, L), we conclude that Q n,x is uniformly bounded in (0, L). By (20),Q n,xx is uniformly bounded in (0, L).
Notice that any positive solution of (6) satisfies 0 IfP ≥ 0 andP (x 0 ) = 0, then it follows from the strong maximum principle that where Integrating the first equation of (22) from 0 to x, we have As g(Ĩ m (x)), Q m (x) andQ m (x) are uniformly bounded in (0, L), we conclude that Q m,x is uniformly bounded in (0, L). It follows from (22) thatQ m,xx is uniformly bounded in (0, L). Hence, for any p > 1,Q m ∈ W 2 p (0, L). By Sobolev embedding theorems, there exists a convergent subsequence ofQ m , which we still denote bỹ Q m for simplicity, such thatQ m →Q ≥ 0( ≡ 0) in C 1 ([0, L]) as n → ∞. Taking the limit in (22) as m → ∞, we get It follows from the strong maximum principle thatQ > 0 on [0, L], which implies thatd = d * , a contradiction.
The remaining possibility is thatd = 0,P > 0, namely, (d m , P m ) → (0,P ). Integrating the first equation of (21) from 0 to x, we have Since g(I m ) and P m are uniformly bounded, one concludes that P m,x is uniformly bounded. Thus P m,xx is uniformly bounded by (21) again. Passing to a subsequence if necessary, we may assume that x 0P (s)ds . By virtue ofP > 0, it follows from Lemma 2.3 thatP = P 0 (β; x). Thus the global bifurcation branch C * must meet the branch {(0, P ) : P > 0} at the point (0, P 0 (β; x)) as d → 0 due to the continuity of P d (β; x). That is, lim Step 3. Uniqueness. The proof of uniqueness is exactly similar to that of uniqueness of (13), see Lemma 2.3.
3. Longtime behavior. In this section we study the longtime dynamical behavior of the reaction-diffusion-advection model (1)-(3) satisfying (4) and (5), which describes the population dynamics of a single phytoplankton species in a water column with crowding effect. To this end, let Q = e −(υ/D)x P . Then Q satisfies where I(x, t) = I 0 exp −k 0 x − k 1 x 0 e (υ/D)s Q(s, t)ds , and Q 0 (x) = e −(υ/D)x P 0 (x). Clearly, Q 0 (x) 0 on [0, L], and Q 0 (x) ∈ C([0, L]) by the assumption on P 0 (x). Hence, we only need to investigate the longtime dynamical behavior of the solution Q(x, t) to (23).
By standard arguments, it is not hard to prove the uniqueness and global existence of the solution Q(x, t) of (23). Moreover, Q(x, t) > 0 for t > 0, x ∈ [0, L] by the strong maximum principle. The main result of this section is as follows, which indicates that d * is the critical death rate, that is, the phytoplankton species survives if and only if its death rate is less than d * .
Remark 1. The estimation (28) is crucial for us to extend the comparison lemma in [8] to the case of β > 0. In view of I(x, t) = I 0 exp −k 0 x − k 1 x 0 e (υ/D)y Q(y, t)dy , we have It follows from the equation (23) that Namely, Integrating for x from 0 to L, we get . Q(x, s). Clearly W(t) is nondecreasing. Suppose for contradiction that W(t) → ∞ as t → ∞. Then we can find t n → ∞ such that W(t n ) = max x∈[0,L] Q(x, t n ) and W(t n ) → ∞. We may assume that t n > 1 for all n ≥ 1. Define z n (x, t) = Q(x, t + t n − 1) W(t n ) .
By Lemma 3.3, h(x, t) is bounded for all x ∈ [0, L] and t > 0. Hence lim t→∞ h(x, t) = h * (x) exists. Meanwhile, it follows from Lemma 3.3 again that Q(·, t) ∞ is also bounded. Hence, we can apply the standard parabolic regularity theory to (23) to conclude that, for any sequence t n → ∞, there is a subsequence of {Q(·, t n )} which converges in C 1 ([0, L]), say Q(·, t n k ) → Q * . Since h(·, t n ) → h * (x), we necessarily have h * (x) = we know that lim t→∞ P (x, t) = h * (x) and thus h * (x) must be a nonnegative steady state of (1)-(3). Since h * (0) = 0 and h * (x) is the limit of an increasing sequence, we have h * (x) > 0 for x ∈ (0, L] and h * (x) ≡ 0. Therefore h * (x) is a nontrivial nonnegative steady state of (1)- (3). By the strong maximum principle h * (x) is positive, and hence we can use Theorem 2.4 to conclude that h * (x) ≡ P d (x).
Let Ψ A (x) = −g(I 0 e −(k0+k1A)x ), and Φ A (x) be the positive eigenfunction corresponding to λ 1 (Ψ A ) with Φ A ∞ = 1. It is easy to see that λ 1 (Ψ A ) → 0 and Φ A → 1 in C 1 ([0, L]) as A → ∞ by a regularity and compactness argument. Therefore we can find Let Q(x, t) be the solution of (23) with initial condition Q(x, 0) = 2BΦ A (x). Then we can findδ > 0 small so that Q 0 (x) < Q(x, t), A < Q(x, t)e (υ/D)x for t ∈ (0,δ] and x ∈ [0, L]. Hence for t ∈ (0,δ], we have Thus for ω(x, t) By the strong maximum principle we deduce ω = Q − 2BΦ A (x) < 0 for t ∈ (0,δ] and x ∈ [0, L]. It follows that Q(x, τ ) < Q(x, 0) for 0 < τ ≤δ. By the same argument as before, we deduce that h(x, t) = x 0 P d (s)ds. Repeating the above arguments, we can conclude that lim It remains to consider the case d = d * . The proof is exactly similar to that of the case d < d * with some simple modification. Let Q(x, t) be defined exactly as in the proof above. Then we know that h(x, t) := x 0 e (υ/D)s Q(s, t)ds > 0 is strictly decreasing in t. Hence lim t→∞ h(x, t) = h * (x) ≥ 0 exists. By the same consideration we can show that P (x, t) = e (υ/D)x Q(x, t) → (h * ) (x) as t → ∞ in C 1 ([0, L]), which indicates that (h * ) (x) is a nonnegative steady state of (1)-(3). However, by Theorem 2.4, the only nonnegative steady state of (1)-(3) is the trivial solution 0 when d = d * . Hence P (x, t) → 0 as t → ∞ uniformly for x ∈ [0, L], thus h(x, t) → 0 as t → ∞.
By Lemma 3.2 we deduce 0 < h(x, t) < h(x, t), which implies that h(x, t) → 0 as t → ∞. By the application of this fact and parabolic regularity, as before, we deduce lim t→∞ e (υ/D)x Q(x, t) exists in the C 1 ([0, L]) norm, and the limit is a nonnegative steady state of (1)-(3). By virtue of d = d * , this limit must be 0. This complete the proof. 4. Asymptotic profiles of positive steady states.

4.1.
Asymptotic profiles for small crowding effect. In this subsection, we focus on the asymptotic profiles of positive steady states when the crowding effect disappears. The motivation comes from the following observation. It follows from Theorem 2.4 and Theorem 3.1 that d * is the critical death rate, and the phytoplankton species survives if and only if its death rate is less than the critical death rate. Moreover, the results in [14] indicate that the bifurcation diagram of positive steady states for the model without crowding effect with respect to the death rate looks like Fig.1(a). That is, when the death rate goes to zero, the population density of the phytoplankton goes to infinity at the bottom or surface of the water column. In contrast to the model without crowding effect, Theorem 2.4 shows that the density of the phytoplankton species will have a finite limit rather than go to infinity when the death rate disappears (see Fig.1(b)), which is due to the crowding effect. Theorem 4.1. The unique positive solution P 0 (β; x) of equation (13) satisfies P 0 ∞ → ∞ as β → 0.
Remark 2. It follows from Theorem 3.1 and Theorem 4.1 that lim d→0 P d (β; x) = P 0 (β; x), and lim The limits illustrate that the bifurcation diagram Fig.1(b) of positive steady states for the model (1)-(3) will gradually evolve into Fig.1(a) when the crowding effect disappears.

4.2.
Asymptotic profiles for large advection rates. This subsection is devoted to investigate the asymptotic profiles of the unique positive steady state P (x; υ) of (1)-(3) when the advection coefficient is large enough. For simplicity of notation and clarity of the presentation, we may assume D = 1, β = 1, L = 1 by some scaling. Thus we consider the following steady state system Let P (x; υ) denote the unique positive solution of (35). Note that g(I 0 e −k0 ) < −λ 1 (−g(I 0 e −k0x )) = d * by Lemma 2.1. It follows from Theorem 2.4 that P (x; υ) exists for any υ ∈ R if 0 < d < g(I 0 e −k0 ). The following results describe the asymptotic profiles of P (x; υ) for large sinking rate (υ > 0) and large buoyant rate (υ < 0).  (ii) As υ → −∞, P (x; υ) → 0 uniformly in any compact subset of (0,1], Remark 3. Theorem 4.2 indicates that the sinking species is monotone increasingly distributed in the water column, and the phytoplankton species concentrates at the bottom of the water column with a finite population density when the sinking velocity is sufficiently large. It follows from Theorem 4.3 that the buoyant species is monotone decreasingly distributed in the water column and the phytoplankton form a thin layer at the surface of the water column with a finite population density when the buoyant coefficient is sufficiently large. The proof of Theorems 4.2 and 4.3 is very complicated and lengthy, we divide it into the following nine lemmas. Lemma 4.4. Suppose P (x; υ) is the unique positive solution of (35). Then (i) Proof. Set ω(x) = e −υηx P (x; υ), where η is some constant which will be chosen differently for different purposes. Then ω satisfies Let η = 1/2. Then ω satisfies (i) If υ > 2 g(I 0 ) − d, then υ 2 4 − g(I(x)) + e υx 2 ω + d > 0 in (0, 1). Namely, ω xx > 0 in (0, 1). Since ω x (0) > 0, we have ω x > 0 on [0, 1]. This implies that P x = e υηx (υηω + ω x ) > 0 on [0, 1].
Lemma 4.11. There exist positive constants C 5 , C 6 , both independent of υ, such that Proof. Here we only prove (i), (ii) can be shown similarly and we omit it. It follows from Lemma 4.5(i) and Lemma 4.7(i) that where g i (x; υ)(i = 1, 2) are given by It is easy to check that For large υ, the only critical point (denoted by x 1 ) of g 1 on [0,1] is determined by which implies that (1)) for large υ. Hence, υ 2 for some positive constant C 7 independent of υ. As g 1 attains the global minimum at x = x 1 on [0,1], we see that For g 2 we have For large υ, the only critical point (denoted by x 2 ) of g 2 on [0,1] is determined by which implies that x 2 = 1 − (1/υ)(1 + o(1)) for large υ. Hence, where C 8 is some positive constant independent of υ. As g 2 attains the global maximum at x = x 2 on [0,1], we see that for every x ∈ [0, 1]. Let C 6 = max{C 7 , C 8 }. Then (i) holds. The proof of part (ii) is similar to that of part (i), and we omit it here. we know that there exist some positive constants C 9 , C 10 , both independent of υ, such that for υ ≤ −C 9 , |P (x; υ) − P (0; υ)e υx | ≤ C 10 υ 2 on [0, 1]. Letting υ → −∞, we immediately get that lim υ→−∞ P (x; υ) − P (0; υ)e υx L ∞ (0,1) = 0, which implies that P (x; υ) → 0 uniformly in any compact subset of (0,1] as υ → −∞. The proof is finished.

5.
Discussion. Phytoplankton are microscopically small plants that drift in oceans and lakes, which form the base of the aquatic food chain. Since they transport significant amounts of atmospheric carbon dioxide into the deep oceans, they may play a crucial role in the climate dynamics. In freshwater lakes and rivers, phytoplankton communities can have a major impact on ecosystem dynamics. The appearance of algae blooms are often a signal of dangerous eutrophication and may result in serious water-quality problems. Hence, the formation of phytoplankton blooms has recently attracted considerable attention from mathematical, experimental and numerical viewpoints. Under the assumption that phytoplankton transport is governed by turbulent diffusion, Du and Hsu [8] studied the global dynamics of a nonlocal reaction-diffusion model proposed by Huisman [16] (i.e. (1)-(3) with β = 0, υ = 0), which describes the evolution of a single phytoplankton species in an eutrophic vertical water column where the species relies solely on light for its metabolism. It turns out that there exists a critical death rate d * > 0 such that if d ∈ (0, d * ), the model admits a unique positive steady state which is a global attractor of it, whereas it has no positive steady state and zero is a global attractor if d ∈ [d * , ∞). Considering the effect of sinking or buoyant motion, Hsu and Lou [14] studied a nonlocal reactiondiffusion-advection model of light-limited phytoplankton [11,15] in an eutrophic water column. The model is a special case of (1)-(3) with β = 0, υ = 0. Their results show that there also exists a critical death rate d * > 0 so that for d ∈ (0, d * ), the model has a unique positive steady state and for d ∈ [d * , ∞) it has no positive steady state. Moreover, the bifurcation diagram of positive steady states with respect to the death rate looks like Fig.1(a). That is, when the death rate goes to zero, the population density of the phytoplankton goes to infinity at the bottom or surface of the water column. By means of the strict monotonicity of the critical death rate with respect to the water column depth, the sinking or buoyant velocity and the turbulent diffusion rate, they showed that a critical water column depth, a critical sinking or buoyant velocity and a critical turbulent diffusion rate can exist for some intermediate range of phytoplankton death rate. Furthermore, their analysis on the asymptotic profiles of steady states for large advection rates indicates that the vertical distribution looks like a Dirac function at the bottom of the water column when the sinking velocity is sufficiently large, which indicates that the phytoplankton forms a thin layer at the bottom of the water column for large sinking rate. Similarly, the vertical distribution looks like a Dirac function at the surface of the water column when the buoyant coefficient is sufficiently large, which indicates that the phytoplankton forms a thin layer at the surface of the water column for large buoyant rate.
The purpose of this paper is to incorporate the crowding effect into the population dynamics of a single phytoplankton species in a water column, and to study the longtime dynamical behavior of the nonlocal reaction-diffusion-advection model (1)-(3) and the asymptotic profiles of its positive steady states for large sinking or buoyant rates. Our results show that there is also a critical death rate d * such that the phytoplankton species survives if and only if its death rate is less than the critical death rate d * . In contrast to the model without crowding effect, Theorem 2.4 shows that the density of the phytoplankton species will have a finite limit rather than go to infinity when the death rate disappears (see Fig.1(b)), which is due to the crowding effect. Furthermore, the limits lim d→0 P d (β; x) = P 0 (β; x), and lim β→0 P 0 (β; x) ∞ = ∞ illustrate that the bifurcation diagram Fig.1(b) of positive steady states for the model (1)-(3) will gradually evolve into Fig.1(a) when the crowding effect disappears. Furthermore, Theorems 4.2 and 4.3 indicate that the sinking phytoplankton species concentrates at the bottom of the water column with a finite population density, and the buoyant phytoplankton species concentrates at the surface of the water column with a finite population density when phytoplankton transport is governed by the advection motion (see Fig.2).