COEXISTENCE SOLUTIONS OF A COMPETITION MODEL WITH TWO SPECIES IN A WATER COLUMN

. Competition between species for resources is a fundamental eco-logical process, which can be modeled by the mathematical models in the chemostat culture or in the water column. The chemostat-type models for resource competition have been extensively analyzed. However, the study on the competition for resources in the water column has been relatively neglected as a result of some technical diﬃculties. We consider a resource competition model with two species in the water column. Firstly, the global existence and L ∞ boundedness of solutions to the model are established by inequality estimates. Secondly, the uniqueness of positive steady state solutions and some dynamical behavior of the single population model are attained by degree theory and uniform persistence theory. Finally, the structure of the coexistence solutions of the two-species system is investigated by the global bifurcation theory.

1. Introduction. Competition between species for resources is a fundamental ecological process [7,20]. The chemostat-type models of resource competition have been extensively analyzed (see, e.g., [10,14,15,16,22,23,24]). However, the study on resource competition in the water column has been relatively neglected as a result of some technical difficulties. Firstly, for the competition models in a water column, the usual reduction of the system to a competitive system of one order lower through the "conservation of nutrient" principle is lost. Thus the system with predation and competition is non-monotone, and the single population model can't be reduced to a scalar system. Hence, it is much more difficult to study the uniqueness and stability of the semitrivial nonnegative equilibria. Secondly, by virtue of the complex 2692 HUA NIE, SZE-BI HSU AND JIANHUA WU boundary conditions, it is hard to establish the global existence of the solutions and a priori estimates of the positive steady state solutions.
Motivated by the biological significance, the study of the models in a water column began to be a problem of considerable interest recently. In [25], a mathematical model describing the vertical distribution of phytoplankton and two resources in a water column was proposed. Numerical results show a catastrophic transition between a surface maximum pattern and a subsurface maximum pattern of phytoplankton. The authors analyzed a model of competition between two phytoplankton species in a stratified water column in [26]. Multiple regions of alternative stable states are possible in parameter space by numerical simulations. In [13], the authors developed a model to explore how phytoplankton respond through growth and movement to opposing resource gradients and different mixing conditions. Numerical computation indicates that the model is able to replicate the diverse vertical distributions observed in nature and explain what underlying mechanisms drive these distributions. The mathematical analysis on competition for resources in a water column can be found in [8], which discussed the existence and uniqueness of steady-state solutions of the system with one resource and one species. Hsu and Lou [9] investigated 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. 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 was analyzed. Du and Mei [6] studied a general reaction-diffusion-advection equation that models the dynamics of a single phytoplankton species in a eutrophic vertical water column. The asymptotic profiles of the positive steady-state solution for small diffusion, large diffusion and deep water column are given in [6], respectively. However, the dynamical behavior of the resource competition model with two species in a water column is unclear until now.
This paper deals with a general competition model with one resource and two species in a water column with boundary conditions and initial conditions S(x, 0) = S 0 (x) ≥ 0, u(x, 0) = u 0 (x) ≥ 0, ≡ 0, x ∈ [0, L], v(x, 0) = v 0 (x) ≥ 0, ≡ 0, Here S(x, t), u(x, t), v(x, t) are the concentrations of the nutrient and the two species respectively. D 0 (x) is the vertical eddy diffusion coefficient, and D(x) is the diffusion rate of species across the thermocline. ν i (x) denotes the velocity of cells, α i is the yield coefficient, and d i > 0 is the death rate of species i(i = 1, 2). L is the depth of the water column. S 0 > 0 is the nutrient concentration at the sediment. β > 0 is the relative transfer velocity of nutrients at the sediment interface. f i (S) = miS ai+S with i = 1, 2, which is the nutrient-limited growth rate of species i. m i > 0 is the maximum growth rate, and a i > 0 is the half-saturation constant. The initial concentrations of the nutrient and the two species are all assumed to be nonnegative continuous functions on the water column. The detailed biological explanation for this model can be found in [13,25,26].
By suitable scaling, we may take S 0 = 1 and L = 1. Then the original system (1)-(3) becomes with boundary conditions and initial conditions We concentrate on positive solutions of the following steady state system with boundary conditions Throughout this paper, we assume the diffusion rates and velocity of species satisfy the following hypotheses where γ ∈ (0, 1), i = 1, 2. Moreover, we can extend the response functions f i : [16,23]). We will denotef i (S) by f i (S) for the sake of simplicity.
As mentioned before, the conservation principle is invalid, and the system with predation and competition is non-monotone. Moreover, the single population model can't be reduced to a scalar system. Hence, it is hard to study the uniqueness and stability of the semitrivial nonnegative equilibria. The organization of the paper is as follows: In Section 2, by Gronwall inequality and an indirect argument, we establish the global existence and L ∞ boundedness of solutions to the parabolic system (4)- (6). In Section 3, by the general maximum principle and a crucial spectral analysis, we show any positive solution of the single population model is nondegenerative, which produce the uniqueness of semitrivial nonnegative equilibria (see Lemma 3.3 and Theorem 3.4). Some dynamical behaviors of the single population model are attained by uniform persistence theory. The structure of the coexistence solutions of the system (4)-(6) is investigated in Section 4 by bifurcation theory.
The nondegeneracy of any positive solution of the single population model also plays a key role in proving the existence of the local and global bifurcation. Finally, some numerical results on the coexistence region are given, which complement the analytic results.
2. Preliminaries. The goal of this section is twofold. One is to provide some wellknown lemmas related to our study. The other is to establish the global existence and L ∞ boundedness of solutions of the parabolic system (4)- (6).
By the maximum principle of the parabolic equation, the solution (S(x, t), U (x, t), V (x, t)) of (11) satisfies S(x, t) > 0, U (x, t) > 0, V (x, t) > 0 for all x ∈ [0, 1], t > 0. Moreover, it is easy to see that lim sup t→∞ S(x, t) ≤ 1, which implies for > 0 small there exists T 1 > 0 such that S(x, t) ≤ 1 + for all x ∈ [0, 1], t ≥ T 1 . Hence, there exists positive constant ρ 0 depending only on the initial data S 0 (x), such that 0 < S(x, t) ≤ ρ 0 for all x ∈ [0, 1], t > 0, and we only need to show the boundedness of U (x, t), V (x, t). Let d = min{d 1 , d 2 } and By integrating each equation in (11) and summing together, we obtain By Gronwall inequality we get the L 1 estimates Next, we show U (x, t) is bounded for all x ∈ [0, 1] and t > 0. Let φ(t) = max U (x, τ ). Clearly, φ(t) is nondecreasing. Suppose for contradiction that φ(t) → ∞ as t → ∞. Then we can find t n → ∞ such that φ(t n ) = max We may assume that t n > 1 for all n ≥ 1. DefineŨ n (x, t) = U (x,t+tn−1) . Theñ Noting that |f 1 (S(x, t + t n − 1)) − d 1 | ≤ |f 1 (ρ 0 ) − d 1 | := Λ 0 , the comparison principle for parabolic system leads to 0 ≤Ũ n (x, t) ≤ e Λ0t for x ∈ [0, 1] and t ≥ 0. Hence by the application of standard parabolic regularity, we can conclude that ) for any γ ∈ (0, 1). Hence, by passing to a subsequence if necessary we getŨ n (x, t) →Ũ in 2]) by passing to a further subsequence if necessary. Moreover, |g(x, t)| ≤ Λ 0 , andŨ is a weak solution to It follows that and t > 0, we obtain that there exists and t > 0. Repeating the same arguments as before, we assert that there exists In order to figure out the non-trivial nonnegative solutions of (7)-(8), we derive some estimates for the nonnegative solutions of (7)-(8). 1), and for any given Proof. (i) At first, for any nonnegative solution (S, u, v) of (7)-(8), we have S(0) > 0. Indeed, if S(0) = 0, then it follows from the existence and uniqueness of the solution to the ordinary differential equation that S ≡ 0, which is a contradiction to the boundary condition S It follows from the strong maximum principle that U > 0 on [0, 1], and hence u > 0 (ii) From the equation for u and the positivity of u, we obtain −d (iii) Integrating the equation for u, we get Noting that S(x) is strictly monotone increasing in (0, 1), one can assert that f 1 (S(x)) − d 1 is strictly monotone increasing with respect to x in (0, 1). Hence, there exists some . Repeating the similar arguments as above, we obtain Next, we establish a priori estimates for u and v by an indirect argument. To this end, for any δ 0 > 0, suppose there exists a sequence (d Without loss of generality, we assume Passing to a sequence if necessary, we may assume by passing to a subsequence d It follows from the strong maximum principle thatŨ > 0 on [0, 1]. From the equation for S i , we obtain 1], and integrating by parts, we obtain

Multiplying this equation by a smooth function
3. Dynamical behavior of single population model. In order to investigate positive solutions of the two-species system (4)-(6), we first study the following single population model where f (S), α, ν and d are exactly the simplification of the associated parameters or variables with subscript i = 1 or 2. Moreover, the vertical diffusion rates D 0 (x), D(x) and the velocity of species ν(x) still satisfy the hypothesis (H). The first step is to work out the properties of solutions to the steady state system It follows from Lemma 2.5 that the following lemma holds, which establishes a priori estimates for nonnegative solutions of (15).  1), and for any given δ 0 > 0, there exists a positive constant (1)).
Next, we show the uniqueness of positive equilibrium of (15) by degree theory.
To this end, let Then (15) is equivalent to It follows from Lemma 3.1 that any nonnegative solution of (16) with U ≡ 0 satisfies We introduce the spaces: where K 0 , K are the solution operators φ = K 0 (h 1 (x)) and ψ = K(h 2 (x)) for the problems respectively Proof. (i) It follows from similar arguments as in Lemma 3.1 that A τ has no fixed point on ∂Ω. By the homotopic invariance of the degree, we obtain (ii) Let A (0, 0) be the Fréchet derivative of A at (0, 0) with respect to (χ, U ).
(i) The case of φ > 0 in (x p , 1). By the above arguments, we have Ψ(x p ) > 0. Note that The general maximum principle implies Ψ/U 0 cannot reach its non-positive minimum in (x p , 1). By virtue of Ψ(x p ) > 0, one can conclude that Ψ/U 0 cannot reach its non-positive minimum at x = x p . Then min (ii) The case of φ < 0 in (x p , 1). By the above arguments, we have Ψ(x p ) < 0. Note that L 2 Ψ = −af (S 0 )u 0 φ > 0 in (x p , 1), and L 2 U 0 = 0 in (x p , 1).

Proof. (i) is a direct result of Lemma 3.1(ii).
(ii) It follows from Lemma 3.1 that the fixed points of A in Ω are two types, which are the trivial fixed point (0, 0) and the positive fixed points (χ, U ). It follows from Lemma 3.3 that any positive fixed points (χ 0 , U 0 ) of A is non-degenerative and index(A, (χ 0 , U 0 ), W ) = 1. Meanwhile, by the compactness argument on the operator A and the non-degeneracy of its fixed points (including (0, 0) and positive fixed points), one knows that there are at most finitely many positive fixed points in Ω. Let them be (χ i , U i )(i = 1, 2, · · · , l). Then index(A, (χ i , U i ), W ) = 1 for i = 1, 2, · · · , l. By the additivity property of the fixed point index and Lemma 3.2, we have That is, for any δ > 0, there exists a unique positive solution of (15) if δ ≤ d < f (1). Proof. The continuity of the map d → (S d (x), u d (x)) from [δ 0 , f (1)−δ 0 ] to C 1 [0, 1]× C 1 [0, 1] follows from a standard compactness and uniqueness consideration. Indeed, if d n → d 0 ∈ [δ 0 , f (1) − δ 0 ], then there exists a sequence of (S dn (x), u dn (x)) converges in C 1 [0, 1] × C 1 [0, 1] to a positive solution of (15) with d = d 0 . By the uniqueness, this positive solution must be (S d0 (x), u d0 (x)). Therefore the entire sequence converges to (S d0 (x), u d0 (x)). Moreover, from the equations of S dn (x) and u dn (x), we easily see that (S dn  Next, we study the dynamical behavior of the solution (S(x, t), u(x, t)) of (14). It follows from Lemma 2.4 that for every initial value function (S 0 , u 0 ) ∈ W , the system (14)    Proof. By the maximum principle of the parabolic equation, it is easy to see that the solution (S(x, t), u(x, t)) of (14) satisfies S(x, t) > 0, u(x, t) > 0. Moreover, it is easy to see that lim sup t→∞ S(x, t) ≤ 1, which implies for > 0 small there exists Noting that d > f (1), there is small enough such that d > f (1 + ). Hence the comparison principle leads to U (x, t) → 0 as t → ∞ uniformly on [0, 1]. Thus lim t→∞ u(x, t) = 0 uniformly on [0, 1] provided d > f (1), which leads to there exists S (1, t)), t > T 2 S (x, T 2 ) = S(x, T 2 ), x ∈ [0, 1].
Proof. We prove it by making use of the abstract persistence theory, see [18]. Let Ψ(t) be the solution semiflow generated by the system (14) on the state space W .
(iii) The positive solutions (S, u, v) with S, u, v > 0 on [0, 1], which is the focus to study the properties of nonnegative solutions of the steady state system (7)- (8).
Proof. Noting that the system (7)-(8) is equivalent to (22), we only need to show there exists a continuum of positive solutions to (22), denoted bỹ which bifurcates from the semi-trivial solution branch 0), and meets the other semi-trivial solution branch ). To this end, for any δ > 0 and d 1 ∈ [δ, f 1 (1)) fixed, we construct the global bifurcation which corresponds to positive solutions by treating d 2 as a bifurcation parameter.