STABILITY AND ASYMPTOTIC PROFILE OF STEADY STATE SOLUTIONS TO A REACTION-DIFFUSION PELAGIC-BENTHIC ALGAE GROWTH MODEL

By using bifurcation theory, we investigate the local asymptotical stability of non-negative steady states for a coupled dynamic system of ordinary differential equations and partial differential equations. The system models the interaction of pelagic algae, benthic algae and one essential nutrient in an oligotrophic shallow aquatic ecosystem with ample supply of light. The asymptotic profile of positive steady states when the diffusion coefficients are sufficiently small or large are also obtained.

1. Introduction.In this paper, we consider the following coupled system of two ordinary differential equations and two parabolic partial differential equations: which was proposed and analyzed in [33].Here all the variables and parameters of the model (1) and their biological significance are listed in Table 1, and we assume that s ∈ R, β u , β v ∈ [0, 1] and the remaining parameters are all positive constants.Model (1) characterizes the interactions of pelagic algae, benthic algae and one essential nutrient in an oligotrophic shallow aquatic ecosystem with ample supply of light (see Fig. 1 of [33]).In view of practical biological facts in model (1), we have three basic assumptions: (i) L 2 L 1 ; (ii) the benthic habitat closely contacts with the sediment and dissolved nutrients in the benthic habitat are well mixed and homogeneous in space; (iii) benthic algae move very slowly or are motionless, so they are spatially uniformly distributed.There is accumulating evidence suggesting that the distributions of pelagic algae in aquatic ecosystems exhibit strong spatial heterogeneity [3,4,12,13,15,30].In [33], the model ( 1) is established to consider the effect of spatial heterogeneity on the interactions of pelagic algae, benthic algae and one essential nutrient.The existence, uniqueness and classification of non-negative steady states are obtained in [33] to characterize sharp threshold conditions for the regime shift from extinction to coexistence of pelagic and benthic algaes.
The present paper is a continuation of studies in [33], and here we provide the answer to the following two questions: 1. the local asymptotic stability of non-negative steady states in model ( 1) by applying bifurcation theory and associated linear stability theory; 2. the asymptotic profile of positive steady states when the diffusion coefficients D u , D r are sufficiently small or large in model (1).It has long been recognized that pelagic algae and benthic algae are both potentially important primary producers in the aquatic ecosystem.As a good indicator of water quality and climate change, pelagic algae generally drift in the water column of lakes and oceans ecosystem, and compete with each other for essential resources such as nutrition and light [3,4,12,30,31,32].It should be noted that the types of pelagic algae competing major resources are not the same in different aquatic environments.In an eutrophic aquatic environment, pelagic algae tend to compete only for light [5,6,7,9,11,16,18,21], while in a shallow or oligotrophic aquatic environments, pelagic algae tend to compete only for nutrients [10,19,20,26].In the streams, rivers or shallow lakes, benthic algae provide the main energy base in driving production for higher trophic levels.Accordingly, benthic algae are often more important than pelagic algae in these situations.Especially, in some shallow and clear-water aquatic environments, both planktonic algae and benthic algae exist simultaneously and compete fiercely for nutrition and light [8,14,22,24,25].This competitive relationship as one of the challenges associated with understanding benthic-pelagic coupling has been described by using ordinary differential equations [14,22,24,25].
The rest of the paper is organized as follows.In Section 2, we introduce some basic preliminary results on bifurcation analysis in order to establish the local asymptotic stability of non-negative steady states in model (1).Section 3 is devoted to establishing the locally asymptotically stable results of non-negative steady states in model ( 1) by applying the bifurcation theorems.In Section 4, we investigate the asymptotic profile of positive steady states when the diffusion coefficients are sufficiently small or large in model (1).

2.
Preliminaries.In this section, we give a short overview on some notations, definitions and well-known results for bifurcation theory that are important for the present study.
Let (X, • ), (Y, • ) be Banach spaces and X is continuously embedding in Y .For a linear operator L, we denote N (L) as the null space of L and R(L) as the range space of L. Also L[w] denotes the image of w under L, and if L is a multilinear operator, Consider a steady state equation where F : R × X → Y is a nonlinear mapping and sufficiently smooth.For a given (λ 0 , u 0 ) ∈ R × X, let U be a neighborhood of (λ 0 , u 0 ) in R × X.The following bifurcation theorems are well-known, and we recall them for the convenience of readers.The first result is the local bifurcation theory known as "bifurcation from simple eigenvalue", and the second result shows the stability of bifurcating solutions obtained in the first one.
Theorem 2.4 (Theorem 4.4 in [23]).Assume that all conditions in Theorem 2.3 hold.If and (λ, u) are both in U .
Then each of the sets C + and C − satisfies one of the following: (i) it is not compact; (ii) it contains a point (λ * , u 0 ) with λ * = λ 0 ; or (iii) it contains a point (λ, u 0 + z), where z = 0 and z ∈ Z.
3. Bifurcation analysis for the algae growth model.In this section, we investigate the local asymptotical stability of the non-negative steady state solutions of model (1) by using bifurcation method.We first recall the following possible non-negative steady state solutions of model (1).Let E 1 = (0, 0, R 1 , W 1 ) be the nutrient-only semi-trivial steady state, where (R 1 , W 1 ) solves In fact, by (2), we have E 1 = (0, 0, W sed , W sed ).Let E 2 = (0, V 2 , R 2 , W 2 ) be the benthic algae-nutrient semi-trivial steady state, where (V 2 , R 2 , W 2 ) satisfies By solving (3), we find ) be the pelagic algae-nutrient semi-trivial steady state, where (U 3 , R 3 , W 3 ) solves From ( 4), we obtain Proposition 3.1 in [33] shows that a coexistence steady state can only exist when 0 < m u ≤ r u and 0 < m v ≤ r v .By solving (5), we have From Lemma 3.10 in [33], we have 0 The local asymptotically stability results of E 1 and E 2 have been established in [33] (see Theorems 3.2 and 3.4).The existence of E 3 and E 4 were proved in [33] by using a priori estimates and degree theory, and it is also known that each of E 3 and E 4 is unique and non-degenerate (see Theorems 3.8 and 3.11 in [33]).We now are concerned with the local asymptotical stability of E 3 and E 4 with the help of bifurcation analysis.In the following discussion, taking m u as the bifurcation parameter, we explore the following two cases: In this subsection, we consider the bifurcation of pelagic algae-nutrient semi-trivial steady state E 3 from nutrient-only semi-trivial steady state E 1 at m u = m * u .We first investigate the local bifurcation theorem and local asymptotical stability of E 3 .For the convenience of the following discussion, we denote then there is a smooth curve Γ E3 of positive solutions of (4) bifurcating from the line of trivial solutions ΓE1 = {(m u , 0, W sed , W sed ) : m u > 0} at m u = m * u .Moreover, 1. near {(m * u , 0, W sed , W sed )}, there exists a positive constant δ > 0 such that all the positive solutions of (4) lie on a smooth curve asymptotically stable with respect to the following reduced equation without benthic algae: 3. If in addition m v > r v W sed /(W sed + γ v ), then the bifurcating steady state solution E 3 (τ ) = (m u (τ ), U (τ, z), 0, R(τ, z), W (τ )) is locally asymptotically stable with respect to the full system (1) for τ ∈ (0, δ).
The result in part 2 here shows that the bifurcating pelagic-algae-only steady state solution E 3 is locally asymptotically stable in the absence of initial benthic algae (in such case, the system (1) is effectively reduced to (11).On the other hand, if initially there is benthic algae but the death rate of the benthic algae m v is large, then part 3 shows that the bifurcating pelagic-algae-only steady state solution E 3 is locally asymptotically stable with respect to the full system.We prove part 1 and 2 of Theorem 3.1 here, and postpone the proof of part 3 to subsection 3.2.
Proof of Theorem 3.1 part 1 and 2. Let Denote X := X 1 ×X 2 ×R, and define a nonlinear mapping ) It is clear that F (m u , 0, W sed , W sed ) = 0 which implies that the assumption (a 1 ) holds in Theorem 2.1.
We next consider the codimension of R(L).Suppose that (f Multiplying both sides of ( 15) and ( 19) by ϕ(z) and Φ(z), respectively, subtracting and integrating on [0, L 1 ], also combining the boundary conditions in (15) and (19), we have This shows that and codimR(L) = 1.From ( 13), we have . This implies that the assumption (a 3 ) holds in Theorem 2.1.
Remark 3.1.In Theorem 3.1, we assume that 0 ≤ β u < 1.This is because that if This means that pelagic algae and dissolved nutrients in the pelagic habitat constitute a closed system with internal continuous cycle in ecology.In this case, authors in [33] showed that lim t→∞ 1 |Ω| Ω U (z, t)dz = ∞ if β u = 1 by numerical method.Considering practical biological significance, here we assume that 0 ≤ β u < 1.
Next we prove the global bifurcation property of the branch ΓE3 .First we have the following a priori estimates for positive solutions (U 3 , R 3 , W 3 ) of (4).
The results have been proved in Lemma 3.6 of [33] except the statement that U mu 3 ∞ → ∞ as m u → 0. Suppose this is not true, then there exists a sequence of m u , denoted by m n := m n u , and corresponding positive solutions (U n 3 , R n 3 , W n 3 ) of ( 4) such that m n → 0 and U n 3 ∞ → C < ∞ as n → ∞.By using L p theory for elliptic operators and the Sobolev embedding theorem, after passing to a subsequence if necessary, we may assume that . Integrating the first equation of ( 4) on [0, L 1 ], we have From part (i) and ( 26), we obtain that On the other hand, integrating the second equation of ( 4) on [0, L 1 ], we get which contradicts with ( 26)-( 27) and R n 3 is strictly increasing on [0, L 1 ] showed in Lemma 3.6 of [33].Therefore U mu 3 ∞ → ∞ as m u → 0. Now we state the global bifurcation theorem of the steady state solution E 3 .Theorem 3.3.Let S + be the set of positive solutions to (4).Then S + is a smooth curve in R + × X in form satisfying lim Proof.From Theorem 3.3 and Remark 3.4 of [23], it is easy to check that for any fixed ( Ũ (z), R(z), W ) ∈ X , is a Fredholm operator with index zero.By applying Theorem 2.3, we obtain a connected component C of the set S of all solutions to (4) emanating from It follows from Theorem 2.4 that each of C + and C − satisfies one of the following three cases: (1) It is not compact in X ; (2) It contains a point ( mu , 0, Without loss of generality, we take where Φ(z), Ψ(z), Θ are given in Theorem 3.1.We only consider C + .From the strong maximum principle and connectedness of C + , all solutions (U (z), R(z), W ) of ( 4) on C + satisfies U (z) > 0, R(z) < W sed and W < W sed from Lemma 3.2.
If case (3) holds, then there exists mu ∈ (0, m * u ), such that ( Ũ (z), W sed + R(z), W sed + W ) is positive and it satisfies F ( mu , Ũ (z), W sed + R(z), W sed + W ) = 0. From Lemma 3.2, we have W sed + R(z) < W sed , W sed + W < W sed and Ũ (z) > 0. Thus Ũ (z) > 0, R(z) < 0 and W < 0 and it implies that since Φ(z) > 0, Ψ(z) < 0 and Θ < 0 from the proof of Theorem 3.1.But (30) contradicts with (29), which implies that case (3) cannot happen either.Hence case (1) must occur for C + .According to case (1), C + is not compact in X , which implies that it is unbounded in X by the elliptic regularity theory.By Lemma 3.2, if m u ∈ [ε, m * u ) for any ε > 0, then (U (z), R(z), W ) is bounded.And also when m u = 0, (4) has no positive solution.Thus, the projection of C + on the m u -axis must be the interval (0, m * u ), and as shown in Lemma 3.2, ||U || ∞ → ∞ as m u → 0 + .Now from Theorem 3.8 of [33], the positive solution of ( 4) is indeed unique and non-degenerate.Therefore C + must be a smooth curve in form of (28) from the implicit function theorem, and S + = C + .Theorem 3.3 shows the continuous increase of the pelagic algae from 0 at m u = m * u to ∞ as m u = 0, and the stability proved in Theorem 3.1 shows that this steady state E 3 is locally asymptotically stable near m u = m * u for the dynamics of ( 11) and the dynamics of (1) if m v is large.The stability of E 3 for m u not near the bifurcation point is still not known.
In this subsection, we still use m u as a bifurcation parameter, and consider the bifurcation of positive solution E 4 from the branch of semi-trivial solutions Theorem 3.4.Assume that Then there is a smooth curve Γ E4 of positive solutions of (5) bifurcating from the line of trivial solutions , there exists δ > 0 such that all the positive solutions of (5) lie on a smooth curve ) and h 4 (0) = 0; 2. for τ ∈ (0, δ), the bifurcating solution (m u (τ ), U (τ, z), V (τ ), R(τ, z), W (τ )) is locally asymptotically stable with respect to (1).
It follows from Lemma 3.10 in [33] and similar arguments in Lemma 3.2, we obtain the following a priori estimates for positive solutions (U 4 , V 4 , R 4 , W 4 ) of (5).
(ii) for any ε > 0, there exists a positive constant B(ε By applying Lemma 3.5 and similar arguments as in Theorem 3.3, we have the following conclusion.Theorem 3.6.Let S+ be the set of positive solutions to (5).
We now prove the part 3 of Theorem 3.1 by using the setting in the proof of Theorem 3.4.
Adding ( 50) and (51) together, we get Thus R 0 (z) = C (a constant) for any z ∈ [0, L 1 ].An elementary argument shows that R n (z) → R 0 (z) uniformly in any compact subset of [0, L 1 ).Moreover integrating the first equation in (61) on [0, L 1 ], and observing that the right hand side of the first equation in (61) tends to 0, we obtain that R n (z) is small for z where the last equality follows from the Lebesgue dominated convergence theorem.This completes the proof.

Table 1 .
Variables and parameters of model (1) with biological meanings.