A mathematical model of algae growth in a pelagic benthic coupled shallow aquatic ecosystem

Transcription

1 J. Math. Biol. 218) 76: Mathematical Biology A mathematical model of algae growth in a pelagic benthic coupled shallow aquatic ecosystem Jimin Zhang 1 Junping Shi 2 Xiaoyuan Chang 3 Received: 25 February 217 / Revised: 25 June 217 / Published online: 1 August 217 Springer-Verlag GmbH Germany 217 Abstract A coupled system of ordinary differential equations and partial differential equations is proposed to describe the interaction of pelagic algae, benthic algae and one essential nutrient in an oligotrophic shallow aquatic ecosystem with ample supply of light. The existence and uniqueness of non-negative steady states are completely determined for all possible parameter range, and these results characterize sharp threshold conditions for the regime shift from extinction to coexistence of pelagic and benthic algae. The influence of environmental parameters on algal biomass density is also considered, which is an important indicator of algal blooms. Our studies suggest that the nutrient recycling from loss of algal biomass may be an important factor in the algal blooms process; and the presence of benthic algae may limit the pelagic algal biomass density as they consume common resources even if the sediment nutrient level is high. Keywords Reaction diffusion model Pelagic algae Benthic algae Nutrients Environmental parameters Algal biomass density Mathematics Subject Classification 35J25 92D25 35A1 Partially supported by NSFC , NSFHLJ-A21414, NSFHLJ-QC2165, HLJUF-DYS-JCL2123, and NSF-DMS B Junping Shi jxshix@wm.edu 1 School of Mathematical Sciences, Heilongjiang University, Harbin, Heilongjiang 158, People s Republic of China 2 Department of Mathematics, College of William and Mary, Williamsburg, VA , USA 3 School of Applied Sciences, Harbin University of Science and Technology, Harbin, Heilongjiang 158, People s Republic of China

2 116 J. Zhang et al. 1 Introduction Algae are important primary producers of organic compounds through photosynthesis and chemosynthesis, and they form the base of the food chain in the lakes, oceans and other aquatic ecosystems. The growth of algae depends on light intensity, temperature and nutrients in the water, thus the vertical distribution of algae in the water column usually has a stratified structure. Especially, in shallow lakes or littoral zones, there are two distinct layers of algae: pelagic algae and benthic algae Jäger and Diehl 214; Scheffer et al. 23; Vasconcelos et al. 216). Pelagic algae, typically phytoplankton, drift in the water column of lakes and oceans and provide significant biomass to aquatic ecosystems. It has been recognized that the growth of pelagic algae needs two essential resources: light and nutrients. The interaction between pelagic algae and its resources has been studied extensively in three possible ways, in both the theory and the real practical applications. One extreme case is in oligotrophic aquatic ecosystems with ample supply of light such that pelagic algae tend to compete only for nutrients see Hsu et al. 213; Mei et al. 216; Nie et al. 216, 215; Wang et al. 215); and the other extreme case is in eutrophic ecosystems with ample nutrient supply such that pelagic algae tend to compete only for light see Du and Hsu 21; Du and Mei 211; Du et al. 215; Hsu and Lou 21; Huisman et al. 1999; Kolokolnikov et al. 29; Mei and Zhang 212a; Peng and Zhao 216). In some aquatic ecosystems, pelagic algae compete for light and nutrients simultaneously see Du and Hsu 28a, b; Huisman et al. 26; Jäger et al. 21; Kerimoglu et al. 212; Klausmeier and Litchman 21; Ryabov et al. 21; Yoshiyama et al. 29; Zagaris et al. 29; Zagaris and Doelman 211). Benthic algae typically microalgae) are located in the bottom of lakes or oceans, and they attach to the surface of other plants and rocks or root in the sediment. There is growing evidence to show that benthic algae in some shallow, oligotrophic, clear-water ecosystem not only is an important food source of aquatic animals, but also influence water quality, energy cycle, and various biogeochemical interactions Jäger and Diehl 214; Scheffer et al. 23; Vadeboncoeur et al. 28). But there have been very few studies of the important role of benthic algae. When both of pelagic and benthic algae are presented in an aquatic ecosystem, they may have an intense competition for light and nutrition. There have been existing dynamical models characterizing this relation between pelagic and benthic algae by using ordinary differential equations see Jäger and Diehl 214; Scheffer et al. 23; Vasconcelos et al. 216; Vadeboncoeur et al. 28), in which an important assumption is that the pelagic habitat is well mixed to produce homogeneous vertical distributions of pelagic algae and nutrients in shallow water columns. However there is increasing recognition that the distributions of pelagic algae show strong spatial heterogeneity, both vertically and horizontally see Du and Hsu 28a, b; Huisman et al. 22, 26; Jäger et al. 21; Klausmeier and Litchman 21; Yoshiyama et al. 29). Therefore it is important and of great interest to explore the effect of spatial heterogeneity on the pelagic and benthic algae growth, which has been neglected in previous studies, and establish some threshold conditions that cause a regime shift between coexistence and extinction of pelagic algae and benthic algae.

3 Algae growth in a pelagic benthic aquatic ecosystem 1161 Motivated by the existing studies and the above considerations, in this study, we establish a coupled system of ordinary differential equations and partial differential equations to describe the interactions of pelagic algae, benthic algae and one essential nutrient in an oligotrophic shallow aquatic ecosystem with ample supply of light. The new model reveals the effect of the spatial heterogeneity of pelagic algae and gives threshold conditions of coexistence or regime shifts between pelagic algae and benthic algae. In addition, from the perspective of preventing algae blooms, we explore the influence of environmental parameters on algal biomass density. The present paper only focuses on the case where pelagic algae and benthic algae compete for an essential nutrient, while other possible cases compete for light or for light and nutrients simultaneously) will be considered in forthcoming studies. The rest of the paper is organized as follows. In Sect. 2, we derive a mathematical model of pelagic algae, benthic algae and nutrients consisting of two ordinary differential equations and two partial differential equations. In Sect. 3, we investigate dynamical properties of this model including the existence, uniqueness and stability of steady states, which are complemented by numerical simulations under reasonable parameter values from literature. In Sect. 4, we consider the influence of environmental parameters on algal biomass density via a systematic sensitivity analysis. In the discussion section, we summary our findings and state some biologically motivated mathematical questions for future study. 2 Model construction In this section, we establish a mathematical model to describe the interactions between pelagic algae and benthic algae in an oligotrophic shallow aquatic ecosystem with ample supply of light, which means that pelagic algae and benthic algae tend to compete only for nutrients. We assume that the entire shallow aquatic area consists of two layers of habitat with uniform depth: pelagic habitat and benthic habitat. Let z denote the depth coordinate. We assume that z = is the surface of the water, z = L 3 is the sediment surface bottom of the lake/ocean), and z = L 1, L 3 ) is the interface between the pelagic and benthic habitats. Hence the positive z direction is from the surface of water to the bottom of lake/ocean. Here L 1 and L 2 = L 3 L 1 are the thickness of pelagic habitat and benthic habitat respectively see Fig. 1). In general, the benthic habitat closely contacts with the sediment and its thickness is far less than the depth of the pelagic habitat so L 2 L 1. Therefore, here we assume that dissolved nutrients in the benthic habitat are well mixed and homogeneous in space. A coupled system of two ordinary differential equations ODE) and two partial differential equations PDE) is established below to describe the dynamics of biomass density of pelagic algae U), biomass density of benthic algae V ), concentration of dissolved nutrients in the pelagic habitat R), concentration of dissolved nutrients in the benthic habitat W ). All the variables and parameters of the system and their biological significance are listed in Table 1.

4 1162 J. Zhang et al. Fig. 1 Interactions of pelagic algae, benthic algae and one essential nutrient in a shallow aquatic ecosystem 2.1 Pelagic algae and benthic algae Let Uz, t) denote the biomass density of pelagic algae at depth z [, L 1 ] and time t. The intrinsic growth rate of pelagic algae depends on the concentration of dissolved nutrients Rz, t) in the pelagic habitat, and it takes a Michaelis Menten type functional response form r u R/R + γ u ), where r u is the maximum production rate of pelagic algae, and γ u is the half-saturation constant. On the other hand, the pelagic algal biomass density is lost at a density-independent rate m u, caused by processes such as respiration, death and grazing. Pelagic algal transport is governed by passive movement due to turbulence with a depth independent turbulent diffusion coefficient D u and also active movement due to sinking or buoyant with speed s. Taking together these assumptions results in the following reaction diffusion advection equation of U with no-flux boundary condition: Uz, t) = turbulent diffusion sinkingbuoyant) + growth loss t 2 U = D u z 2 s U ) z + ru R m u U, z, L 1 ), R + γ u D u U z ) su) =, D u U z L 1 ) sul 1 ) =. 2.1) The algae in the benthic habitat attach to the surface of other plants, rocks or roots in the sediment. This implies that they move very slowly or are motionless. Also the thickness of the benthic habitat is far less than the one of pelagic habitat, hence we assume that the density function V is spatially uniformly distributed. That is V z, t) V t) for z L 1, L 3 ). The change in the benthic algal biomass density comes from two processes: growth and loss. The intrinsic growth rate of benthic algae is governed by the concentration of dissolved nutrients W z, t) W t) in the benthic habitat, again with a Michaelis Menten type functional response r v W/W +γ v ), where r v is the maximum production rate of the benthic algae, and γ v is the half-saturation

5 Algae growth in a pelagic benthic aquatic ecosystem 1163 Table 1 System variables and parameters with biological meanings and values Symbol Meaning Value Unit Source t Time Day z Depth m U Biomass density of pelagic algae mgc/m 3 V Biomass density of benthic algae mgc/m 3 R Concentration of dissolved nutrients in the pelagic habitat W Concentration of dissolved nutrients in the benthic habitat mgp/m 3 mgp/m 3 Du Vertical turbulent diffusivity of pelagic algae ) m 2 /day Huisman et al. 22, 26), Jäger et al. 21), Kerimoglu et al. 212), Klausmeier and Litchman 21), Ryabov et al. 21) Dr Vertical turbulent diffusivity of dissolved nutrients in the pelagic habitat ) m 2 /day Huisman et al. 22, 26), Jäger et al. 21), Kerimoglu et al. 212), Klausmeier and Litchman 21), Ryabov et al. 21) s Sinking or buoyant velocity of pelagic algae.5 5 5) m/day Huisman et al. 22, 26), Jäger et al. 21), Jäger and Diehl 214),Kerimoglu et al. 212),Klausmeier and Litchman 21), Ryabov et al. 21), Vasconcelos et al. 216) ru Maximum specific production rate of pelagic algae 1 Day 1 Huisman et al. 22, 26), Jäger and Diehl 214), Ryabov et al. 21) rv Maximum specific production rate of benthic algae 1 Day 1 Jäger and Diehl 214) mu, mv Loss rate of pelagic and benthic algae, respectively.1 Day 1 Jäger and Diehl 214), Vasconcelos et al. 216) γu Half saturation constant for nutrient-limited production of pelagic algae 3 mgp/m 3 Jäger and Diehl 214), Vasconcelos et al. 216)

6 1164 J. Zhang et al. Table 1 continued Symbol Meaning Value Unit Source γv Half saturation constant for nutrient-limited production of benthic algae 5 mgp/m 3 Jäger and Diehl 214), Vasconcelos et al. 216) cu Phosphorus to carbon quota of pelagic algae.8 mgp/mgc Vasconcelos et al. 216) cv Phosphorus to carbon quota of benthic algae.15 mgp/mgc Vasconcelos et al. 216) a Nutrient exchange rate between sediment and benthic habitat b Nutrient exchange rate between sediment and benthic habitat.5 m/day Jäger and Diehl 214), Vasconcelos et al. 216).5 m/day Jäger and Diehl 214), Vasconcelos et al. 216) L1 Depth of the pelagic habitat below water surface) 2 m Jäger and Diehl 214) L2 Vertical extent of the benthic habitat.1 m Jäger and Diehl 214), Vasconcelos et al. 216) Wsed Concentration of dissolved nutrients in the sediment 1.3 5) mgp/m 3 Jäger and Diehl 214), Vasconcelos et al. 216) βu Nutrient recycling proportion from loss of pelagic algal biomass βv Nutrient recycling proportion from loss of benthic algal biomass.5 1) Jäger and Diehl 214), Kerimoglu et al. 212), Klausmeier and Litchman 21), Ryabov et al. 21), Vasconcelos et al. 216).3.9) Vasconcelos et al. 216)

7 Algae growth in a pelagic benthic aquatic ecosystem 1165 constant. The loss rate of benthic algal biomass density is scaled by a parameter m v. Combining these assumptions give the following ODE for benthic algae: dvt) dt = growth loss = ) rv W m v V. 2.2) W + γ v 2.2 Nutrients in the pelagic and benthic habitats The function Rz, t) describes the concentration of dissolved nutrients in the pelagic habitat at depth z [, L 1 ] and time t, and W t) is the concentration of dissolved nutrients in the benthic habitat at time t. Here again due to the slow movement in the benthic habitat, the nutrient concentration in the benthic habitat is spatially uniform. The nutrients in the whole shallow aquatic ecosystems are supplied from the sediment with a fixed concentration W sed there. The change of dissolved nutrients in the pelagic habitat depends on turbulent diffusion with a diffusion coefficient D r, consumption by pelagic algae, recycling from the loss of pelagic algal biomass with carbon ratio c u and proportion β u [, 1], and nutrients exchange between the pelagic and benthic habitat at z = L 1 with nutrient exchange rate a. The dynamics of Rz, t) is given by Rz, t) = turbulent diffusion + recycling consumption t 2 R = D r z 2 + c uβ u m u U c ur u RU, z, L 1 ), R + γ u R z, t) =, D r R z L 1, t) = aw t) RL 1, t)) nutrients exchange). 2.3) The benthic nutrient W t) could change as a result of consumption by benthic algae, recycling from the loss of benthic algal biomass with carbon ratios c v and proportion β v [, 1], nutrients exchange between the pelagic and benthic habitat, and supplying from the sediment. Thus the dynamics of W t) is described as dwt) = supplying nutrients exchange + recycling consumption dt = b W sed W ) a W RL 1, t)) + c v β v m v V c vr v WV. L 2 L 2 W + γ v 2.4) In Eqs. 2.3) and 2.4), we include the recycle of nutrients c u β u m u U and c v β v m v V. This is because high temperature can cause the rapid decomposition of algae, and thus promote more nutritious to be recycled. In previous studies, there are very few dynamical results on recycling. Here our subsequent studies show that the nutrient recycling from loss of algal biomass may be an important factor in the existence and uniqueness of non-negative steady state solutions and the algal blooms process. Also in 2.1) and 2.2), we do not include the exchange of algaes between pelagic habit and benthic habit, as the experiments in Jäger et al. 21), Scheffer et al. 23), Vasconcelos et al. 216) show that there is no algal exchange between the pelagic and benthic habitat found in these situations.

8 1166 J. Zhang et al. 2.3 The full model Combining all Eqs. 2.1) 2.4), we have the following full system of pelagic algaebenthic algae-nutrients model: U 2 U = D u t z 2 s U z + r u RU m u U, < z < L 1, t >, R + γ u dv = r vwv m v V, t >, dt W + γ v R t dw dt U D u z, t) su, t) =, D u R z, t) =, D r 2 R = D r z 2 + c uβ u m u U c ur u RU, < z < L 1, t >, R + γ u = b W sed W ) a W RL 1, t)) + c v β v m v V c vr v WV L 2 L 2, t >, W + γ v U z L 1, t) sul 1, t) =, t >, R z L 1, t) = aw t) RL 1, t)), t >. 2.5) In consideration of the biological significance of 2.5), we assume that s R, β u,β v [, 1] and the remaining parameters are all positive constants. Furthermore, we consider the solutions of 2.5) with nonnegative initial values, i.e. Uz, ) = U z), Rz, ) = R z), z L 1, V ) = V, W ) = W. 2.6) In the following we study the dynamics of 2.5). In particular, we are interested in the existence, uniqueness, and stability of non-negative steady state solutions Uz), V, Rz), W ) which satisfy the following steady state system: ) D u U z) su ru Rz) z) + m u Uz) =, < z < L 1, ) Rz) + γ u rv W V m v =, W + γ v D r R z) + c u β u m u Uz) c ur u Rz)Uz) =, < z < L 1, Rz) + γ u bw sed W ) aw RL 1 )) + c v L 2 β v m v r ) vw V =, W + γ v D u U ) su) = D u U L 1 ) sul 1 ) =, R ) =, D r R L 1 ) = aw RL 1 )). 3 Existence and stability of steady states The main purpose of this section is to investigate the existence, uniqueness and local/global stability of non-negative steady state solutions of 2.5). The possible non-negative steady states of 2.5) are listed below: 1. Nutrient-only semi-trivial steady state E 1 :,, R 1 z), W 1 ), where R 1 z), W 1 ) solves R z) =, < z < L 1, bw sed W ) aw RL 1 )) =, 3.1) R ) =, D r R L 1 ) = aw RL 1 ));

9 Algae growth in a pelagic benthic aquatic ecosystem Benthic algae-nutrient semi-trivial steady state E 2 :, V 2, R 2 z), W 2 ), where V 2, R 2 z), W 2 ) solves r v W m v =, W + γ v R z) =, < z < L 1, bw sed W ) aw RL 1 )) + c v L 2 β v m v r ) vw V =, W + γ v R ) =, D r R L 1 ) = aw RL 1 )); 3.2) 3. Pelagic algae-nutrient semi-trivial steady state E 3 : U 3 z),, R 3 z), W 3 ), where U 3 z), R 3 z), W 3 ) solves ) D u U z) su ru Rz) z) + m u Uz) =, < z < L 1, Rz) + γ u D r R z) + c u β u m u Uz) c ur u Rz)Uz) =, < z < L 1, Rz) + γ u bw sed W ) aw RL 1 )) =, D u U ) su) = D u U L 1 ) sul 1 ) =, R ) =, D r R L 1 ) = aw RL 1 )); 3.3) 4. Coexistence steady state E 4 : U 4 z), V 4, R 4 z), W 4 ), where U 4 z), V 4, R 4 z), W 4 ) solves ) D u U z) su ru Rz) z) + m u Uz) =, < z < L 1, Rz) + γ u r v W m v =, W + γ v D r R z) + c u β u m u Uz) c ur u Rz)Uz) =, < z < L 1, Rz) + γ u 3.4) bw sed W ) aw RL 1 )) + c v L 2 β v m v r ) vw V =, W + γ v D u U ) su) = D u U L 1 ) sul 1 ) =, R ) =, D r R L 1 ) = aw RL 1 )). In the following subsections, we will discuss the existence, uniqueness and local stability of steady states in each form categorized above, and also discuss the implication of such steady states to the whole dynamics of 2.5). To establish the local stability of the above steady states, we linearize the system 2.5) about a steady state ūz), v, rz), w) and obtain an eigenvalue problem ) λϕz) = D u ϕ z) sϕ ru rz) z) + m u ϕz) + r uγ u ūz) rz) + γ u rz) + γ u ) 2 φz), < z < L 1, ) rv w λξ = m v ξ + r vγ v v w + γ v w + γ v ) 2 ζ, λφz) = c u β u m u c ) ur u rz) ϕz) + D r φ z) c ur u γ u ūz) rz) + γ u rz) + γ u ) 2 φz), < z < L 3.5) 1, λζ = c v β v m v c ) vr v w ξ c vr v γ v v w + γ v w + γ v ) 2 ζ + a ) a + b φl 1 ) ζ, L 2 L 2 D u ϕ sϕ z=,l1 =, φ ) =, D r φ L 1 ) = aζ φl 1 )).

10 1168 J. Zhang et al. A steady state solution of 2.5) is locally asymptotically stable if all eigenvalues of 3.5) have negative real part, and it is unstable if at least one eigenvalue has positive real part. Note that here for 2.5), the linear stability defined by 3.5) consisting of reaction diffusion equations and ordinary differential equations) implies the nonlinear local stability uniformly asymptotically stable) in some proper function spaces see Henry 1981, Chapter 5). Similar stability of steady state solutions of shadow systems of reaction diffusion systems have been considered in Ni et al. 21a, b). Before we discuss each possible steady state as introduced above, we notice the following observation for the dynamics of the system 2.5). Proposition 3.1 Let Uz, t), V t), Rz, t), W t)) be a solution of 2.5) with initial condition specified as in 2.6). 1. If m u > r u, then lim t Ux, t) = uniformly for x [, L 1 ]; 2. If m v > r v, then lim t V t) =. Proof From the equation of V t),wehavev t) r v m v )V t), thus it is clear that if m v > r v, then lim t V t) =. On the other hand, from U t D u 2 U z 2 s U z + r u m u )U, and m u > r u, we obtain that ux, t) converges to uniformly for x [, L 1 ] as t by the comparison theorem of parabolic equations. The result in Proposition 3.1 indicates that when m u > r u, the dynamics of 2.5) is effectively reduced to benthic algae-nutrients subsystem, while when m v > r v,the dynamics of 2.5) is effectively reduced to pelagic algae-nutrients subsystem. 3.1 Nutrient-only semi-trivial steady state For any parameter value, there is a unique nutrient-only steady state that is in balance with the sediment nutrient concentration W sed, and it is at least locally asymptotically stable when the algae s loss rates are high. Theorem 3.2 The system 2.5) has a unique nutrient-only steady state solution E 1,, W sed, W sed ). 3.6) Moreover if m u > r uw sed and m v > r vw sed, 3.7) W sed + γ u W sed + γ v then E 1 is locally asymptotically stable with respect to 2.5), and if then E 1 is unstable. < m u < r uw sed W sed + γ u or < m v < r vw sed W sed + γ v, 3.8)

11 Algae growth in a pelagic benthic aquatic ecosystem 1169 Proof It is easy to see that E 1 given in 3.6) is the unique solution of 3.1). For the stability of E 1,, W sed, W sed ),itfollowsfrom3.5) that the stability of E 1 is determined by the eigenvalue problem λϕz) = D u ϕ z) sϕ ru W sed z) + ) rv W sed λξ = m v W sed + γ v λφz) = c u β u m u c ) ur u W sed W sed + γ u m u ) ϕz), < z < L 1, 3.9a) ξ, 3.9b) W sed + γ u ϕz) + D r φ z), < z < L 1, a + b λζ = c v β v m v c ) vr v W sed ξ + a φl 1 ) W sed + γ v L 2 L 2 D u ϕ sϕ z=,l1 =, φ ) =, D r φ L 1 ) = aζ φl 1 )). To establish the local stability of E 1,weset h 1 = r uw sed W sed + γ u m u, h 2 = r vw sed W sed + γ v m v. 3.9c) ) ζ, 3.9d) 3.9e) Let λ 1 be the largest eigenvalue of 3.9), and let ϕ,ξ,φ,ζ) be the corresponding eigenfunction. We consider the following three cases: a 1 ) ϕ ; a 2 ) ϕ = and ξ = ; or a 3 ) ϕ = and ξ =. Case a 1 ): ϕ. In this case, the stability of E 1 is completely described by characteristic equations 3.9a), 3.9c), 3.9d). Let ϕ = e s/d u)z ϕ. Then 3.9a) translates into { λ ϕz) = D u ϕ z) + s ϕ z) + h 1 ϕz), < z < L 1, ϕ ) = ϕ 3.1) L 1 ) =. It is easy to see that the dominant eigenvalue of 3.1) ish 1 and the corresponding eigenfunction is ϕ 1 z) = 1. Then the dominant eigenvalue of 3.9a) isλ 1 = h 1 and the corresponding eigenfunction is ϕz) = e sz/d u. Substituting λ 1 and ϕz) into 3.9c) and 3.9d) we have ζ = a a + b + h 1 L 2 φl 1 ), φz) = c um u 1 β u ) + h 1 )e sz/d u h 1 φ ) =, D r φ L 1 ) = ab + h 1L 2 ) a + b + h 1 L 2 φl 1 ). Solving 3.11b) one has that 3.11a) + D r φ z) h 1, < z < L 1, 3.11b) 3.11c) φ 1 z) =ĉ 1 e z h 1 /D r +ĉ 2 e z h 1 /D r + c u D 2 u m u1 β u ) + h 1 )e sz/d u s 2 D r h 1 D 2 u, ĉ 1, ĉ 2 R,

12 117 J. Zhang et al. when h 1 /D r = s/d u,or φ 1 z) =ĉ 3 e z h 1 /D r +ĉ 4 e z h 1 /D r + c u D u m u 1 β u ) + h 1 )ze sz/d u 2sD r, ĉ 3, ĉ 4 R, when h 1 /D r = s/d u. It follows from the boundary conditions 3.11c) that there exist constants ĉ i i = 1, 2, 3, 4) such that φ 1 z) satisfies 3.11c). This shows that there exists a solution ζ, φz)) satisfying Eq. 3.11). Therefore, in case a 1 ), λ 1 = h 1 determines the stability of E 1. Case a 2 ): ϕ = and ξ. In this case it follows from 3.9b) that λ 1 = h 2 with eigenfunction ξ = 1. Combining 3.9c) with 3.9d), we have a ζ = φl 1 ) c v L 2 h 2 + m v 1 β v )), a + b + h 2 L 2 a + b + h 2 L 2 φz) = D r φ z), < z < L 1, h 2 φ ) =, D r φ L 1 ) = a [ b + h2 L 2 a + b + h 1 L 2 φl 1 ) + c v L 2 h 2 + m v 1 β v )) a + b + h 1 L 2 ]. 3.12) It is straightforward to show that there exists a solution ζ, φz)) satisfying Eq. 3.12). This implies that λ 1 = h 2 determines the stability of E 1 in case a 2 ). Case a 3 ): ϕ = and ξ =. Now 3.9) reduces to λφz) = D r φ z), < z < L 1, λζ = aφl 1) a + b)ζ, 3.13) L 2 L 2 φ ) =, D r φ L 1 ) = aζ φl 1 )). If ζ = in3.13), then φ satisfies λφz) = D r φ z), < z < L 1,φ ) =, φ L 1 ) =, φl 1 ) =, 3.14) which implies that φz) =. Hence ζ =, and consequently φz) and φ, ζ ) satisfies a ζ = φl 1 ), a + b + λl 2 λφz) = D r φ z), < z < L 1, 3.15) φ ) =, D r φ L 1 ) = ab + λl 2) φl 1 ). a + b + λl 2 If λ> is an eigenvalue of 3.15), then from φ ) = wehaveφz) = coshωz) for ω = λ/d r. But from the boundary condition at x = L 1, we get D r sinhωl 1 ) = ab + λl 2) a + b + λl 2 cosωl 1 ),

13 Algae growth in a pelagic benthic aquatic ecosystem 1171 that is a contradiction. It is also easy to see λ = cannot be an eigenvalue of 3.15). Thus the eigenvalues of 3.15) must be negative. Indeed for λ<, we have φz) = cosωz) for ω = λ/d r since φ ) =. From the boundary condition at x = L 1, we find that tanωl 1 ) = a/d r )b ω 2 D r L 2 ) ωa + b ω 2 D r L 2 ). 3.16) Then the dominant eigenvalue λ 1 of 3.15)is D r ω1 2, where ω 1 is the smallest positive root of 3.16). Summarizing above discussions, we conclude that in case a 3 ), λ 1 is negative. In view of case a 1 ) a 3 ), we conclude that λ 1 = max{h 1, h 2, D r ω1 2 }, and if 3.7) holds, then λ 1 < and E 1 is locally asymptotically stable. On the other hand, if 3.8) holds, then E 1 is unstable. The condition 3.7) implies that large algal loss rates in both the pelagic and benthic habitats lead to extinction of both algae population. Indeed we prove next that the extinction is global for all initial conditions if a stronger condition on the loss rates is satisfied. Theorem 3.3 Suppose that m u > r u, and m v > r v, 3.17) then the nutrient-only steady state solution E 1,, W sed, W sed ) is globally asymptotically stable for 2.5) with respect to any nonnegative initial value. Proof From Proposition 3.1, wehave lim ux, t) = uniformly for x [, L 1] and t lim V t) = provided 3.17) holds. From the theory of asymptotical autonomous t systems Mischaikow et al. 1995), 2.5) reduces to a limiting system R t = D r 2 R z 2, < z < L 1, t >, dw = b W sed W ) a W RL 1, t)), t >, dt L 2 L 2 R z, t) =, D R r z L 1, t) = aw t) RL 1, t)), t >. In order to obtain our results, we construct a Lyapunov function for system 3.18): V R, W ) = 1 2 L1 Rz) W sed ) 2 dz + L 2 2 W W sed) ) Let Rz, t), W t)) be an arbitrary solution of 3.18) with nonnegative initial values. Then dvr, t), W t)) dt = L1 Rz, t) W sed ) R t dz + L 2W t) W sed ) dw dt

14 1172 J. Zhang et al. = L1 Rz, t) W sed ) 2 R z 2 dz bw t) W sed) 2 aw t) W sed )W t) RL 1, t)) R = D r z R W sed) L L1 ) 1 R 2 dz z bw t) W sed ) 2 aw t) W sed )W t) RL 1, t)) L1 ) R 2 = aw t) RL 1, t))rl 1, t) W sed ) dz z bw t) W sed ) 2 aw t) W sed )W t) RL 1, t)) L1 ) R 2 = aw t) RL 1, t)) 2 bw sed W t)) 2 dz. z Note that dv )/dt = holds if and only if W t) = W sed, RL 1, t) = W t), and R/ z =, that is, W t) Rz, t) W sed. It follows from the LaSalle s Invariance Principle that Rz, t), W t)) converges to W sed, W sed ) uniformly for x [, L 1 ] as t for 3.18), and it also follows that any solution of 2.5) with nonnegative initial value converges to E 1 as t. We remark that the global stability of W sed, W sed ) in the nutrient-only subspace always holds without the condition 3.17), as shown in the proof of Theorem 3.3. The global stability in Theorem 3.3 also implies that under the condition 3.17), there exist no any other steady state solutions such as E 2, E 3, E 4 as mentioned in the beginning of this section. 3.2 Benthic algae-nutrient semi-trivial steady state In this subsection, we prove that when the benthic algal loss rate m v is not large, then benthic algae is able to grow to a positive equilibrium level. In fact, we show that m v = r vw sed /W sed + γ v ) is a sharp threshold for the persistence/extinction of benthic algae. Theorem 3.4 The system 2.5) has a positive benthic algae-nutrient semi-trivial steady state E 2 if and only if β v < 1, m u >, < m v < r vw sed W sed + γ v. 3.19) Whenever E 2 exists, it is unique and it is given by E 2, V 2, R 2 z), W 2 ) =, bw sed W 2 ) c v m v L 2 1 β v ), Moreover if in addition to 3.19), we also have γ v m v r v m v, γ v m v r v m v ). 3.2) m u > r u γ v m v γ v m v + γ u r v m v ), 3.21)

15 Algae growth in a pelagic benthic aquatic ecosystem 1173 then E 2 is locally asymptotically stable with respect to 2.5), while E 2 is unstable if < m u < r u γ v m v γ v m v + γ u r v m v ). 3.22) Proof The steady state equation 3.2) can be explicitly solved. The equation of R and its boundary conditions imply that W 2 = R 2.ThevalueofW 2 can be solved from the first equation of 3.2), and finally V 2 can be solved from the third equation of 3.2). Thus E 2 must be given by 3.2). And it is easy to verify that V 2, R 2, W 2 ) is positive if and only if 3.19) holds. Next we investigate the stability of E 2.From3.5), the stability of E 2 is determined by the eigenvalue problem where λϕz) = D u ϕ z) sϕ z) + h 3 ϕz), < z < L 1, 3.23a) λξ = r vγ v V 2 ζ, W 2 + γ v ) b) λφz) = [c u m u 1 β u ) + h 3 )]ϕz) + D r φ z), < z < L 1, 3.23c) λζ = c v m v β v 1)ξ c vr v γ v V 2 W 2 + γ v ) 2 ζ + a ) a + b φl 1 ) ζ, 3.23d) L 2 L 2 D u ϕ sϕ z=,l1 =, φ ) =, D r φ L 1 ) = aζ φl 1 )), 3.23e) h 3 = r uw 2 W 2 + γ u m u. Again let λ 1 be the largest eigenvalue of 3.23), and let ϕ,ξ,φ,ζ)be the corresponding eigenfunction. We consider two cases: b 1 ): ϕ, or b 2 ): ϕ. Case b 1 ): ϕ. Carrying out similar arguments as those of case a 1 ) in Theorem 3.2, we conclude that the dominant eigenvalue of 3.23a)isλ 1 = h 3 and the corresponding eigenfunction is ϕz) = e sz/d u. With this λ 1 and ϕz), ξ, φz), ζ ) in 3.23) can be uniquely solved. Case b 2 ): ϕ =. If ξ =, then it is clear that ζ = as well. Then 3.23) reduces into 3.14) again, thus the dominant eigenvalue is λ 1 = D r π 2 )/L 2 1 and the corresponding eigenfunction is φz) = cosπz/l 1 ).Ifξ =, then 3.23) isreformulated as λξ = h 4 ζ, λφz) = D r φ z), < z < L 1, λζ = c vm v h 4 β v 1)ζ c v h 4 ζ + a ) a + b φl 1 ) ζ, λ L 2 L 2 φ ) =, D r φ L 1 ) = al 2λ 2 + b + c v h 4 L 2 )λ + c v m v h 4 L 2 1 β v ) al 2 λ 2 + a + b + c v h 4 L 2 )λ + c v m v h 4 L 2 1 β v ) φl 1), 3.24)

16 1174 J. Zhang et al. where h 4 = r v γ v V 2 /W 2 + γ v ) 2. Then similar to case a 3 ) in Theorem 3.2,3.24) has only negative eigenvalues, and the dominant eigenvalue λ 1 = D r η 2 1, where η 1 is the smallest root of tan ηl 1 = a[d r L 2 η 4 b + c v h 4 L 2 )D r η 2 + c v m v h 4 L 2 1 β v )] D r η[d r L 2 η 4 a + b + c v h 4 L 2 )D r η 2 + c v m v h 4 L 2 1 β v )]. Combining the cases b 1 ) and b 2 ), we conclude that E 2 is locally asymptotically stable if h 3 < which is equivalent to 3.21), and E 2 is unstable if h 3 > which is equivalent to 3.22). Remark It follows from 3.2) that V 2 increases with respect to the recycling proportion β v and lim βv 1 V 2 =. This implies that the benthic algae-nutrient semi-trivial steady state E 2 does not exist when β v = 1. From the perspective of ecological point of view, if the recycling proportion β v from the loss of benthic algal biomass is high, then there is a benthic algal bloom in this oligotrophic shallow aquatic ecosystem with ample supply of light. 2. One can see that lim mv m E v 2 = E 1, where m v = r vw sed /W sed + γ v ). Hence m v = m v is a critical value for the existence/nonexistence of benthic algae-nutrient steady state E If m u > r u, then from Proposition 3.1, one has that lim t Ux, t) =. Thus the pelagic algae become extinct in this case, and the system 2.5) reduces to the subsystem of benthic algae and pelagic and benthic) nutrients. It is an interesting question whether E 2 is globally asymptotically stable in this situation, which is indicated by our numerical simulations. 3.3 Pelagic algae-nutrient semi-trivial steady state In the oligotrophic shallow aquatic ecosystem, it is also possible that pelagic algae can grow while benthic algae become extinct. In this subsection, we show the existence and uniqueness of pelagic algae-nutrients semi-trivial steady state, in which benthic algae is absent in the system. Such a steady state is in a form of E 3 = U 3,, R 3, W 3 ) with U 3, R 3, W 3 ) being a positive solution of 3.3). We first establish some aprioriestimates for positive solutions of 3.3). Lemma 3.6 Assume that U 3, R 3, W 3 ) C[, L 1 ]) C[, L 1 ]) R + is a positive solution of 3.3) and β u [, 1). Then i) < m u < r uw sed W sed + γ u ; ii) R 3 is a strictly increasing function on [, L 1 ] and β uγ u m u r u β u m u R 3 z) <W sed for all z [, L 1 ]; iii) < W 3 < W sed ; iv) U 3 z)e sz/d u is a strictly increasing function of z on, L 1 ), and for any ε >, there exists a positive constant Aε) such that U 3 Aε) if m u [ε, r u W sed /W sed + γ u )).

17 Algae growth in a pelagic benthic aquatic ecosystem 1175 Proof i): It follows from the first equation of 3.3) and its boundary conditions that ) D u U 3 z) + su 3 z) ru R 3 z) U 3 z) = m u U 3 z), < z < L 1, R 3 z) + γ u D u U 3 ) su 3) = D u U 3 L 1) su 3 L 1 ) =. 3.25) Hence the principal eigenvalue of 3.25) is λ 1 r ) u R 3 ) = m u, R 3 ) + γ u with principal eigenfunction U 3. From the monotonicity of the principal eigenvalue on the weight functions, we have r uw sed = λ 1 r uw sed W sed + γ u W sed + γ u ) <λ 1 r ) u R 3 ) <λ 1 ) =. R 3 ) + γ u This means that < m u < r u W sed /W sed + γ u ). ii) and iii): It follows from the first two equations of 3.3) and its boundary conditions that L1 ) ru R 3 z) m u U 3 z)dz =, R 3 z) + γ u L1 aw 3 R 3 L 1 )) + c u β u m u r u R 3 z) R 3 z) + γ u ) U 3 z)dz =. 3.26) Since β u [, 1), wehaver 3 L 1 )<W 3. From the third equation of 3.3), we have W 3 = ar 3L 1 ) + bw sed a + b < W sed. 3.27) This proves iii) as W 3 >. It follows from 3.27) that R 3 z) satisfies D r R 3 z) + c u β u m u r u R 3 z) R 3 z) + γ u R 3 ) =, D r R 3 L 1) = abw sed R 3 L 1 )) a + b ) U 3 z) =, < z < L 1, >. 3.28) To prove the upper bound of R 3 z),weset 1 := {z [, L 1 ]:β u m u r u R 3 z)/r 3 z) + γ u ), z 1 }, 2 := {z [, L 1 ]:β u m u > r u R 3 z)/r 3 z) + γ u ), z 2 }.

18 1176 J. Zhang et al. It is clear that 1 2 =, 1 2 =[, L 1 ], inf R 3 z) 1 sup R 3 z) 2. z 1 z 2 We show that 2 =. We consider the following two cases: c 1 ) 1 ;c 2 ) 2. Case c 1 ): 1. For any z 1, R 3 z) and R 3 z) is increasing for z on 1. This shows that R 3 z) for all z 1 and R 3 z) is an increasing function of z on 1 since R 3 ) = holds. Note that inf z 1 R 3 z) 1 sup z 2 R 3 z) 2 and R 3 z) is continuous, then x 1 for any x, L 1 ] and 2 = ; Case c 2 ): 2. For any z 2, R 3 z) < and R 3 z) is a strictly decreasing function of z on 2.Itfollowsfrom 2 that R 3 z) is strictly decreasing for z on 2, then x 2 for any x, L 1 ]. But on the other hand L 1 1 since R 3 L 1)>. This is a contradiction. Therefore, / 2. Combining cases c 1 ) and c 2 ), we conclude that R 3 z) is strictly increasing for z [, L 1 ] as R 3 L 1)>. By the boundary conditions of 3.28) and the definition of 1,wehaveR 3 ) β u γ u m u /r u β u m u ) and R 3 L 1 )<W sed, which implies that ii) holds. iv): It follows from the first equality of 3.26) and U 3 > that r u R 3 /R 3 + γ u ) m u must change sign in, L 1 ). From the monotonicity of R 3, we conclude that r u R 3 z)/r 3 z) + γ u ) m u is an increasing function of z on, L 1 ). This means that there is a z, L 1 ) such that r u R 3 z)/r 3 z) + γ u ) m u < forz, z ) and r u R 3 z)/r 3 z)+γ u ) m u > forz z, L 1 ). Combining the first equation of 3.3) with its boundary conditions, we have D u U 3 z) su 3z) > for any z, L 1 ). Then U 3 z)e sz/d u is a strictly increasing function of z on, L 1 ). To establish the boundedness of U 3 z), for any ε>, we assume that there are a sequence m i u [ε, r uw sed /W sed + γ u )] and corresponding positive solutions U3 iz), Ri 3 z), W 3 i) of 3.3) such that U 3 i as i. Without loss of generality, we assume that m i u m u as i.letu i = U3 i/ U 3 i. Then u i satisfies D u u i z) su iz)) + m i u u ru R3 i iz) = z) ) R 3 i z) + γ u i z), < z < L 1, u D u u i ) su i) = D u u i L 1) su i L 1 ) =, L1 ru R3 i z) ) R3 i z) + γ m i u u i z)dz =. u It follows from ii) that < r u R3 i z)/ri 3 z) + γ u)<r u W sed /W sed + γ u ) for all z [, L 1 ], which means that we may assume that there is a function d 1 C[, L 1 ]) such that r u R3 i z)/ri 3 z) + γ u) d 1 z) in C[, L 1 ]) as i. Noting that {u i }, {m i u } are both bounded in L [, L 1 ],byusingl p theory for elliptic operators and the Sobolev embedding theorem, we may assume passing to a subsequence if necessary) that u i u in C 1 [, L 1 ]) as i, and u satisfies in the weak sense)

19 Algae growth in a pelagic benthic aquatic ecosystem 1177 D u u z) suz)) + m u uz) = d 1 z)uz), < z < L 1, D u u ) su) = D u u L 1 ) sul 1 ) =, L1 d 1 z) m u )uz)dz =. 3.29) Since u and u = 1, it follows from the strong maximum principle that u > on [, L 1 ]. On the other hand, R3 i satisfies D r R3 i ru R z)) 3 i = c z) ) u R3 i z) + γ β u m i u u i z) U3 i, < z < L 1, u R3 i ) ) =, D r R3 i ) L 1 ) = abw sed R3 i L 3.3) 1)). a + b Choosing d 2 C [, L 1 ]) with d 2 z=,l 1 = and d 2 > on[, L 1 ], and multiplying both sides of 3.3) byd 2 and integrating in, L 1 ),wehave ) ab L1 d 2 L 1 )W sed R3 i a + b L 1)) + R3 i z)d 2z)) dz L1 = c u U3 i ru R3 i z) ) R3 i z) + γ β u m i u u i z)dz. u Dividing by U i 3 on both sides of the above equality and letting i give = L1 d 1 z) β u m u )uz)dz = L1 1 β u ) m u uz)dz, since the third equation of 3.29) holds. This is a contradiction to β u [, 1), m u [ε, r u W sed /W sed + γ u )] and u > on[, L 1 ]. Hence the boundedness of U 3 holds for m u [ε, r u W sed /W sed + γ u )). It is noteworthy that if U 3, R 3, W 3 ) C[, L 1 ]) C[, L 1 ]) R + is a nonnegative solution of 3.3) with U 3, then U 3, R 3, W 3 ) is a positive solution of 3.3). In fact, we first claim that U 3 ) > and R 3 ) >. Suppose that U 3 ) =, then U 3 ) = from the boundary condition of U 3,soU 3 z) forz [, L 1 ] from 3.25). Hence we have U 3 ) >. Next we assume that R 3 ) =. From 3.28), we have D r R 3 ) = c uβ u m u U 3 ) <. Since R 3 ) = R 3 ) =, then R 3 z) <forz,δ)for some δ>, that is a contradiction to R 3 z) for all z [, L 1 ]. Therefore we must have R 3 ) >. By using the maximum principle, we get R 3 z) > for all z [, L 1 ]. On the other hand, similarly, we have U 3 ) >, U 3 1) > and D u U 3 z) + su 3 z) + m uu 3 z) = r u R 3 z)u 3 z), < z < L 1, R 3 z) + γ u D u U 3 ) su 3) = D u U 3 L 1) su 3 L 1 ) =, which imply that U 3 z) > for all z [, L 1 ] from the strong maximum principle.

20 1178 J. Zhang et al. It follows from 3.27), 3.28) and 3.25) that 3.3) is equivalent to W = arl 1) + bw sed a + b W sed and Uz), Rz)) satisfies D u U z) su ru Rz) z) + m u Rz) + γ u D r R z) + c u β u m u r u Rz) Rz) + γ u D u U ) su) = D u U L 1 ) sul 1 ) =, R ) =, D r R L 1 ) = abw sed RL 1 )). a + b ) Uz) =, < z < L 1, ) Uz) =, < z < L 1, 3.31) Hence the existence and uniqueness of positive solutions of 3.3) is reduced to the existence and uniqueness of positive solutions of 3.31). Note that 3.31) isan elliptic system with predator-prey type nonlinearity. We recall the following assertion for the non-degeneracy of positive solutions of such system. Lemma 3.7 If U 3, R 3 ) is a positive solution of 3.31), then the linearization of 3.31) with respect to U 3, R 3 ), which is given by D u ϕ z) sϕ ru R 3 z) z) + c u β u m u c ur u R 3 z) R 3 z) + γ u ) ϕz) + r uγ u U 3 z) R 3 z) + γ u ) 2 φz) =, < z < L 1, m u R 3 z) + γ u ) ϕz) + D r φ z) c ur u γ u U 3 z) R 3 z) + γ u ) 2 φz) =, < z < L 1, D u ϕ sϕ z=,l1 =, φ ) =, D r φ L 1 ) + ab a + b φl 1) =, only has the trivial solution. This means that U 3, R 3 ) is non-degenerate. The proof of Lemma 3.7 is similar to that of Nie et al. 215, Lemma 3.1) and López-Gómez and Pardo 1993, Lemma 3.1), so we omit it here. We now embed our problem into the framework of topological degree theory. We first assume that m u [ε, r u W sed /W sed + γ u )), β u [, 1) hold for some ε> and let ωz) = W sed Rz) for any z [, L 1 ] and < Rz) <W sed. Then there is a positive constant K 1 such that Let W sed ωz) W sed ωz) + γ u W sed W sed + γ u K 1 ωz). X := {μ, ν) C[, L 1 ]) C[, L 1 ]) : μ, ν }, := {μ, ν) X : μ<a + 1, ν<w sed + 1}.

21 Algae growth in a pelagic benthic aquatic ecosystem 1179 Then X is the positive cone in C[, L 1 ]) C[, L 1 ]) and is a bounded open subset of X. For any U,ω), we consider ) D u ϕ z) sϕ κru W sed ωz)) z)) + K 2 ϕz) = m u Uz) + K 2 Uz), < z < L 1, W sed ωz)) + γ u ) D r φ cu r u W sed ωz)) z) + K 2 φz) = c u β u m u Uz) + K 2 ωz), < z < L 1, W sed ωz)) + γ u D u ϕ sϕ z=,l1 =, φ ) =, D r φ L 1 ) + ab a + b φl 1) =, 3.32) where κ [, 1] and K 2 is large enough such that { K 2 > max m u κr } uw sed ωz)), c u r u K 1 ωu. W sed ωz)) + γ u Denote the solution operator ϕ, φ) = T κ U,ω)for any U,ω). Itfollowsfrom the strong maximum principle and standard elliptic regularity theory that T κ : X, compact and continuously differentiable for any κ [, 1]. Moreover, 3.31) has a nonnegative solution if and only if the operator T 1 has a fixed point in. Carrying out similar arguments as Theorem 3.4 in Nie et al. 215), we have the following conclusions: e 1 ) indext 1,,X) = 1 and indext 1,, ), X) = ; e 2 )ifû, ˆR) is a positive solution of 3.31) and non-degenerate, then indext 1,Û, ˆω), X) = 1, where ˆω = 1 ˆR. It follows from the compactness of T 1 and the non-degeneracy of its fixed points see Lemma 3.7) that the operator T 1 has at most finitely many positive fixed points in, denoted as Û i, ˆω i ), i = 1, 2,...,N. Frome 1 ) and e 2 ), we have N = indext 1,, ), X) + N indext 1,Û i, ˆω i ), X) = indext 1,,X) = 1, i=1 which implies that 3.31) has a unique positive solution. Therefore, we obtain the following conclusion. Theorem 3.8 The system 2.5) has a pelagic algae-nutrient semi-trivial steady state E 3 if and only if β u < 1, < m u < r uw sed W sed + γ u, m v >, 3.33) and whenever E 3 exists, it is unique and non-degenerate. Remark It follows from a standard bifurcation argument [see Du and Hsu 28a), Mei and Zhang 212b] that when m u, U m u 3.This indicates the occurrence of a pelagic algal bloom, which is a serious environmental problem.

22 118 J. Zhang et al. 2. If m v > r v, then from Proposition 3.1, one has that lim t V t) =. Thus the benthic algae become extinct in this case, and the system 2.5) reduces to the subsystem of pelagic algae and pelagic and benthic) nutrients. 3. Although the pelagic algae-nutrients semi-trivial steady state E 3 is unique and non-degenerate, its stability is not known just as other diffusive predator-prey systems). But for realistic environmental parameters, our numerical simulation shows that under 3.33), solutions of 2.5) converge to E In case of ample supply of light, recycling of nutrients does not alter the monotony of nutrition with water depth see ii) in Lemma 3.6), but it has an important impact on the existence and uniqueness of pelagic algae-nutrient semi-trivial steady state E 3 for β u [, 1) and for β u = 1) and pelagic algal biomass see Sect. 4). 3.4 Coexistence steady state A coexistence steady state solution of 2.5) is the one whose each component is positive. The result in Proposition 3.1 shows that a coexistence steady state can only exist when < m u r u and < m v r v. A direct calculation gives W 4 = γ vm v, V 4 = bw sed W 4 ) + aw 4 R 4 L 1 )) r v m v c v m v L 2 1 β v ) 3.34) since the second and fourth equations of 3.4) hold. Similar to 3.26), we conclude that R 4 L 1 ) W 4, and the equal sign holds if and only if β u = 1. It is clear that if V 4 is positive, then β v < 1, < m v < r vw sed γ v + W sed. It follows from 3.34) that the existence and uniqueness of positive solutions of 3.4) is equivalent to the existence and uniqueness of positive solutions of the following system D u U z) su z) + ru Rz) D r R z) + c u β u m u r u Rz) ) Uz) =, < z < L 1, ) Uz) =, < z < L 1, m u Rz) + γ u Rz) + γ u D u U ) su) = D u U L 1 ) sul 1 ) =, ) R ) =, D r R γv m v L 1 ) = a RL 1 ). r v m v We now state our main results in this subsection. 3.35) Lemma 3.1 Assume that U 4, V 4, R 4, W 4 ) C[, L 1 ]) R + C[, L 1 ]) R + is a positive solution of 3.4) and β u [, 1). Then v) β v < 1, < m u < r u γ v m v γ v m v + γ u r v m v ) and < m v < r vw sed W sed + γ v ;

23 Algae growth in a pelagic benthic aquatic ecosystem 1181 vi) R 4 z) is a strictly increasing function on [, L 1 ] and β uγ u m u R 4 z) < r u β u m u γ v m v for all z [, L 1 ]; r v m v vii) U 4 z)e sz/d u is a strictly increasing function of z on, L 1 ), and for any ε>, there exists a positive constant Bε) such that U 4 Bε) if m u [ε, r u γ v m v /γ v m v + γ u r v m v )). Theorem 3.11 The system 2.5) has a positive coexistence steady state E 4 if and only if r u γ v m v γ v m v + γ u r v m v ), < m v < r vw sed W sed + γ v, β u,β v < 1, < m u < 3.36) and whenever E 4 exists, it is unique and non-degenerate. The proofs of Lemma 3.1 and Theorem 3.11 aresimilartothoseoflemma3.6 and Theorem 3.8 respectively, and here we omit them. Remark This existence and uniqueness of coexistence steady state under 3.36) shows that pelagic algae and benthic algae competing for one essential nutrient can coexist in the oligotrophic shallow aquatic ecosystem with ample supply of light. 2. A bifurcation approach can be used to show that the coexistence steady state E 4 bifurcates from the benthic algae-nutrient semi-trivial steady state E 2 at m u = r u γ v m v γ v m v + γ u r v m v ) when < m v < r vw sed is satisfied. Similarly the W sed + γ v pelagic algae-nutrient semi-trivial steady state E 3 bifurcates from the nutrientonly semi-trivial steady state E 1 at m u = r uw sed for any m v >. The W sed + γ u bifurcation structure of E i i = 1, 2, 3, 4) and associated exchange of stability will be considered in a forthcoming paper. 3. If the loss of pelagic and benthic algal biomass is completely recycled back to pelagic and benthic nutrients β u = 1 and β v = 1), then coexistence steady state E 4 dose not exist. 3.5 Simulations of steady states In this subsection, we show some numerical simulations to illustrate our analysis of steady states for model 2.5). In order to facilitate our simulations below, we partition the parameter ranges in Table 2 and summarize our main results on the existence, uniqueness and stability of steady state solutions shown in previous subsections. Note that the parameter space of m u, m v ) is partitioned into the following regions according to Table 2 see Fig. 2): { 1 := m u, m v ) : m u > r uw sed r } vw sed, 2 := { m u, m v ) : m u >, m v > W sed + γ u W sed + γ v r u γ v m v γ v m v + γ u r v m v ), < m v < r vw sed W sed + γ v },

24 1182 J. Zhang et al. Table 2 Existence, uniqueness and stability of steady states for model 2.5) Steady states Existence and uniqueness Local stability E 1 =,, R 1, W 1 ) Always m u > r u W sed W sed + γ u, m v > E 2 =, V 2, R 2, W 2 ) β v < 1, < m u, < m v < r vw sed W sed + γ v E 3 = U 3,, R 3, W 3 ) β u < 1, < m v, < m u < r u W sed W sed + γ u E 4 = U 4, V 4, R 4, W 4 ) β u,β v < 1, < m u < r u γ v m v γ v m v + γ u r v m v ), < m v < r vw sed W sed + γ v m u > Unknown Unknown r u γ v m v γ v m v + γ u r v m v ) r vw sed W sed + γ v Δ 3 Δ 1.8 m v.6.4 Δ 4.2 Δ m u Fig. 2 The parameter ranges in the m u, m v ) plane with different extinction/existence scenarios, as defined in Table 2. Here the parameter values see Table 1 { 3 := m u, m v ) : < m u < r uw sed, m v > r vw sed W sed + γ u W sed + γ v { 4 := m u, m v ) : < m u < }, r u γ v m v γ v m v + γ u r v m v ), < m v < r vw sed W sed + γ v }. Figures 3, 4, 5, 6 show the simulations of solutions of 2.5) for different algal loss rates m u, m v ) while other parameters are realistic ones from Table 1, and in each

25 Algae growth in a pelagic benthic aquatic ecosystem V Time t W Time t Fig. 3 Nutrient-only semi-trivial steady state E 1.Herem u = 1, m v = 1 and other parameters are from Table 1 case the solution converges to a steady state. The simulations appear to have same convergence results regardless of initial conditions. For the case of m u, m v ) = 1, 1), one can see that the extinction of both pelagic algae and benthic algae may arise from this model with the concentration of dissolved nutrients in the pelagic habitat and benthic habitat reaching the concentration of dissolved nutrients in the sediment see Theorem 3.2, 1 in Figs. 2, and 3). This means that in the absence of algae, dissolved nutrients is distributed evenly over the whole shallow aquatic area. For m u, m v ) = 1,.4), the pelagic algae becomes extinct and the benthic algae persists see Theorem 3.4, 2 in Figs. 2, and 4). In this case, the equilibrium nutrient levels in two habitats are still the same but are considerably lower than the sediment concentration. If m u, m v ) =.2, 1), the benthic algae dies out and the pelagic algae reaches a high level see Theorem 3.8, 3 in Figs. 2, and 5). Also here the dissolved nutrient in pelagic habitat is much lower than the one in benthic habitat. The transition from Fig. 4 to Fig. 5 also indicates that there is a regime shift between pelagic algae and benthic algae, where the dominance of benthic algae transforms into the dominance of pelagic algae in a shallow aquatic ecosystem. Finally for m u, m v ) =.1,.4), both of pelagic and benthic algae in the habitat maintain a positive level see Theorem 3.11, 4 in Figs. 2, and 6). From Figs. 7 and 8, we can see that U 3 z), R 3 z)) and U 4 z), R 4 z)) are both nonconstant steady states. Indeed in both cases, the pelagic algae Uz) and pelagic nutrient Rz) appear to be

26 1184 J. Zhang et al V Time t W Time t Fig. 4 Benthic algae-nutrient semi-trivial steady state E 2.Herem u = 1, m v =.4 and other parameters are from Table 1 increasing from the water surface z = to the pelagic-benthic interface z = L 1, which verify the monotonicity results shown in Lemmas 3.6 and Influence of environmental parameters on algal biomass The algal biomass density in an aquatic ecosystems is an important index for evaluating water quality and protecting biological diversity. Especially, algal blooms, exhibited by excessive proliferation of algae on account of the excessive amounts of nitrogen and phosphorus in the water, is a secondary pollution and may produce great harm to environment and human health. Therefore, in this section, we will explore the influence of model parameters in 2.5) on the pelagic algal biomass density and benthic algal biomass density. In order to facilitate the discussion below, we use the spatial average of Ux, t) and Rx, t) defined as Ut) = 1 L1 Ut, z)dz, Rt) = 1 L1 Rt, z)dz. L 1 L 1 In figures below, we compare the spatial averaged) coexistence steady states U 4, V 4, R 4, W 4 ) for different parameter values.

27 Algae growth in a pelagic benthic aquatic ecosystem V Time t W Time t Fig. 5 Pelagic algae-nutrient semi-trivial steady state E 3.Herem u =.2, m v = 1 and other parameters are from Table 1 First we observe the effect of nutrient recycling proportion β u and β v from loss of algal biomass. The parameters β u,β v are closely related and proportional to the ambient temperature or light intensity. We assume that parameters m u, m v ) are chosen so that the coexistence steady state E 4 can be achieved, then we vary β u and keep β v = to compare the biomass of E 4.FromFig.9 left panel, one can observe that pelagic algal biomass mean density increases and benthic algal biomass density almost keeps unchanged with the increase of β u. Note that in Theorem 3.11, the existence of E 4 is only shown when β u < 1. When β u = 1 the loss of pelagic algal biomass is completely recycled back to pelagic nutrients), the pelagic algae biomass appears to be increasing indefinitely see Fig. 9 right panel). This shows that algal blooms may still occur even in nutrient-poor aquatic ecosystems here W sed = 1), and it happens because rapid decomposition of dead algae in high temperature leads to adequate nutrient supply, which in turn causes algal blooms. The similar phenomenon also occurs for benthic algae when β v increases to 1 see Fig. 1). Indeed the expression of V 4 in 3.34) explicitly shows that V 4 as β v 1. These indicate that the nutrient recycling from loss of algal biomass may be an important factor in the algal blooms process. Our theoretical results in Section 3 have shown that the change of mortality rates m u, m v can cause the regime shift from one steady state to another. Here we observe the effect of m u and m v on the coexistence state algal biomass density. Comparing the changes in pelagic algal biomass mean density reveals that benthic algae could control

28 1186 J. Zhang et al V Time t W Time t Fig. 6 Coexistence steady state E 4.Herem u =.1, m v =.4 and other parameters are from Table 1 Steady state U 3 x) Space x Steady state R 3 x) Space x Fig. 7 Profile of pelagic algae-nutrient steady state E 3.Herem u =.2, m v = 1 and other parameters are from Table 1. Left U 3 x), Right R 3 x) pelagic algae, and vice versa see Fig. 11). This confirms that pelagic algae and benthic algae are able to control each other through the consumption of common resources even if they are located in different spatial positions. Therefore, in the presence of both algae in an aquatic ecosystem, it is possible to prevent the occurance of algal blooms. The pelagic habitat depth L 1 has no significant effect on pelagic algal biomass and benthic algal biomass see Fig. 12a), which is partly because that here we do not consider the role of light intensity, which has an important effect on pelagic and benthic