MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION INDUCED BY PHOTOINHIBITION

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1 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION INDUCED BY PHOTOINHIBITION YIHONG DU, SZE-BI HSU AND YUAN LOU Abstract. We study the effect of photoinhibition in a nonlocal reaction-diffusionadvection equation, which models the dynamics of a single phytoplankton species in a water column where the growth of the species depends solely on light. Our results show that, in contrast to the case of no photoinhibition, where at most one positive steady state can eist, the model with photoinhibition possesses at least two positive steady states in certain parameter ranges. Our approach involves bifurcation theory and perturbation-reduction arguments. 1. Introduction Phytoplankton are microscopic plants that float in oceans and lakes and form the base of the aquatic food chain. Since they transport significant amounts of atmospheric carbon dioide into the deep oceans, they may also play a crucial role in the climate dynamics. Phytoplankton species typically compete for nutrients and light [4, 5, 16, 19, 8, 9]. But in oligotrophic ecosystems with ample supply of light, they tend to compete only for nutrients [, ], and in eutrophic environments with ample nutrients supply, they compete only for light [8, 15]. In a water column, a phytoplankton population diffuses due to turbulent miing caused by wind and wave actions. In many cases, phytoplankton also sinks due to its own weight. In this paper, we consider a single sinking phytoplankton species in a eutrophic water column. Our analysis is based on a nonlocal reaction-diffusion-advection model given by Huisman and colleagues in [8, 14], but the growth function g(i) of phytoplankton species in the model is modified to include photoinhibition into consideration. Photoinhibition is characterized by a decreasing rate of photosynthesis with increasing light, which occurs in many phytoplankton species that are sensitive to strong light. This phenomenon is caused by damage to the photosynthetic machinery of cells and by protective mechanisms to avoid this damage ([3, 7]). Date: August 1, Mathematics Subject Classification. 35J55, 35J65, 9D5. Key words and phrases. phytoplankton, photoinhibition, reaction-diffusion equation, steady state. Department of Mathematics, School of Science and Technology, University of New England, Armidale, NSW 351, Australia. ydu@turing.une.edu.au. Department of Mathematics and National Center for Theoretical Sciences, National Tsing-Hua University, Hsinchu, Taiwan 3, Republic of China. sbhsu@math.nthu.edu.tw. Institute for Mathematics Sciences, Renmin University of China, PRC and Department of Mathematics, Ohio State University, Columbus, OH 431, USA. lou@math.ohio-state.edu. Y. Du and Y. Lou thank the National Center for Theoretical Sciences for financial support and hospitality. Y. Du s research was supported by the Australian Research Council. S-B Hsu s research was supported by National Council of Science, Taiwan, Republic of China. Y. Lou s research was partially supported by The Recruitment Program of Global Eperts and NSF grant DMS

2 Y. DU, S.-B. HSU AND Y. LOU Without photoinhibition, the growth function g(i) is generally assumed to be strictly increasing in I, representing the fact that increase of the light level I leads to better growth of the phytoplankton. In such a case this model was investigated recently through rigorous mathematical analysis in [6, 7, 13, 1] (see also earlier work in [17, 18, 6] and references therein), which show that the phytoplankton population either stabilizes at a unique positive steady-state or converges to as time goes to infinity, depending on whether the loss rate is below or above a critical level. With photoinhibition, observations in many laboratory studies ([1, 11, 4]) suggest that the function g(i) should be increasing before I reaches a certain critical level I > where g(i) has a maimum, after which g(i) decreases and converges to as I. In a completely mied water column, the reaction-diffusion-advection model reduces to an ODE model, and the effect of photoinhibition was studied in the recent papers [9, 1]. With a growth function g(i) as above, [9] demonstrates that the phytoplankton population may have two stable steady-states (one positive, the other ), plus another unstable positive steady state, causing a bistable dynamical behavior with the phytoplankton population stabilizing at one or the other stable steady state, depending on its initial value. (The multi-species case was also considered in [9] and [1].) In this paper, we eamine the effect of photoinhibition in a incompletely mied water column, through a single species reaction-diffusion-advection model, where photoinhibition is incorporated into the growth function g(i) as described above. We show that the phenomenon of multiple positive steady-states observed in completely mied water column persists, and their stability suggests a bistable dynamical behavior. Multiplicity results for similar reaction-diffusion equations are usually obtained by making use of the upper and lower solution technique, combined with tools from global analysis (such as the topological degree theory or global bifurcation theory). However, such upper and lower solution techniques are difficult to apply here due to the nonlocal nature of the problem. To overcome this difficulty, apart from employing local and global bifurcation analysis, we also use a perturbation and reduction approach, which is new to this kind of nonlocal reaction-diffusion equations. It is our hope that the mathematical techniques developed here may find more applications in phytoplankton models. We now describe the model in more detail. Consider a vertical water column with a cross section of one unit area and depth h. Let p(, t) be the population density of the phytoplankton at depth [, h] and time t. Then the change of density is governed by the following reaction-diffusion-advection problem [ p t = Dp σp + g (I e k k ) ] p(s,t)ds d p, < < h, t >, (1.1) Dp (, t) σp(, t) =, = or h, t >, p(, ) = p (), h, where d > is the loss rate of the species, the positive constants D, σ represent the diffusion rate and the sinking rate, respectively. The term (1.) I(, t) = I e k k p(s,t)ds is known as the light intensity, with k the background turbidity, k > the light attenuation coefficient of the phytoplankton species, and I > the incident light intensity.

3 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 3 The function g(i) governs the growth rate according to the change of light level I. We always assume that g is C 1. Taking into account of photoinhibition we assume, as in [9], that g(i) has the following properties: (1.3) (i) g() =, (ii) there eists I > such that (I I)g (I) > for I I, (iii) lim I g(i) =. The boundary conditions at = and = h imply that there is no population flu at the surface or bottom of the water column. We are interested in the multiplicity of positive steady states of (1.1). To simplify notations we assume that D = h = I = 1. We stress that this is for simplicity of the notations only; our method can deal with the general case without etra difficulties. Thus we will study the positive solutions of the nonlocal elliptic boundary value problem { p σp + p [g(i) d] =, < < 1, (1.4) p () σp() =, p (1) σp(1) =, with (1.5) I = I() = e k k p(s)ds. Our first eistence and multiplicity result is obtained by a standard argument involving local and global bifurcation theory of Crandall and Rabinowitz [1,, 5]. The multiplicity result is local in nature. Theorem 1.1. Suppose that (1.3) holds and I < e k. Then there eist some positive constants d < d < g(i ) such that (1.4) has at least one positive solution for d (, d ], two positive solutions for d (d, d ), and no positive solution for d > g(i ). Here d is uniquely determined by an eigenvalue problem, and d > d is sufficiently close to d, whose eistence arises from a bifurcation analysis along the line of trivial solutions {(d, ) : d R 1 }, which shows that a branch of positive solutions {(d, p)} bifurcates from the line of trivial solutions at (d, ), and it goes rightward initially but has to become unbounded through d converging to. It can be shown that for d (d, d ), one of the positive steady state is unstable (see Lemma.3), but we have no information on the stability of the other positive steady state, though we believe it is stable. By making use of a perturbation-reduction approach, we can obtain a multiplicity result which is global in nature, together with information on the asymptotic profile and stability of the solutions, but only for large σ. Theorem 1.. Suppose that (1.3) holds and I < e k. (i) Define G(µ) = 1 µ µ g(e k s )ds. Then there eists a unique µ > ln(e k /I ) such that G (µ ) =, G (µ)(µ µ) > for µ (, + ) \ {µ }. Moreover, for each d (g(e k ), G(µ )), the equation d = G(µ) has eactly two positive solutions µ 1, µ, and < µ 1 < µ < µ.

4 4 Y. DU, S.-B. HSU AND Y. LOU (ii) For each d (g(e k ), G(µ )), there eists ɛ > such that for every σ > 1/ɛ, (1.4) has two positive solutions of the form p 1 () = µ 1 k σeσ( 1) + z 1,σ (), p () = µ k σeσ( 1) + z,σ (), with z i,σ satisfying lim σ 1 z i,σ() d =, i = 1,. Remark 1.3. If I e k, we can show that (1.4) has at most one positive solution for any σ. Indeed, we can modify g(i) for I > I to obtain a new C 1 function g(i) which is strictly increasing in I for all I >. When I e k, it is easily seen that if p is a positive solution of (1.4), then it is also a positive solution of (1.4) with g(i) replaced by g(i). Hence we are back in the no-photoinhibition case and can apply the result in [13, 7] to conclude that there is at most one positive solution, and the dynamics of (1.1) is simple. Biologically this fact is rather natural, as e k is the highest possible level of light intensity felt by the species in the water column, so only the values of g(i) for I (, e k ] contribute to the growth of the species. The results in Theorem 1. suggest that for large σ, the two solutions p 1 and p form a -shaped curve in the (d, p)-space as d is varied in the range (g(e k ), G(µ )), which resembles the solution curve of the equation d = G(µ) in the (d, µ)-plane. Figure 1 shows the graph of the curve G(µ) and the two solutions µ 1 and µ of d = G(µ) for the case I < e k. G(µ ) * d -k₀ g(e ) G(μ) μ1 μ μ μ Figure 1. Illustrative graph of G(µ) and the solutions of d = G(µ). It is interesting to note that, by Theorem 3. of [13], as σ +, the bifurcation value d in Theorem 1.1 converges to g(e k ), suggesting that the global bifurcation curve in Theorem 1.1 for large σ looks like the solution curve of the equation d = G(µ) in the (d, µ)-plane, and the two positive solutions in Theorem 1. are from the lower and upper branches of the global bifurcation curve.

5 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 5 Let us also note that for large σ, the solutions p 1 () and p () are well approimated by µ 1 k σeσ( 1) and µ k σeσ( 1), respectively, which have values close to away from = 1, while their values at = 1 go to as σ. The fact that the populations concentrate at the bottom of the water column is due to the assumption of large sinking rate σ. The information on the asymptotic profiles of p 1 () and p () enables us to investigate their stability, see Theorem 4.1 in section 4, which suggests that p 1 is unstable, p is stable, as epected for the bistable phenomenon. (Note that is a stable steady state for d in the range of Theorem 1..) The rest of the paper is organized as follows. In Section we use a bifurcation approach to prove Theorem 1.1, with d as the bifurcation parameter. To overcome some of the limitations encountered in Section in the bifurcation approach, in Section 3 we use a perturbation and reduction approach to study the steady-state solutions of (1.1) with large sinking rate and prove Theorem 1.. In Section 4 we consider the linearized stability of the two positive steady states found in Section 3. We thank the referee for helpful suggestions on the presentation of the paper.. Multiple steady-states through a bifurcation approach In this section we use a bifurcation approach to prove Theorem 1.1. We focus on the eistence of positive solutions by studying the bifurcation of a branch of positive solutions of (1.4) from the trivial solution p =, with coefficient d as the bifurcation parameter. We first consider the stability of p =, which is determined by the sign of the largest eigenvalue, denoted by λ = d, of the linear eigenvalue problem { ϕ σϕ + g(e k )ϕ = λϕ, < < 1, (.1) ϕ () σϕ() =, ϕ (1) σϕ(1) =. By the well-known Krein-Rutman theorem it is known that d eists and is positive. Its corresponding eigenfunction can be chosen to be positive in [, 1], denoted by ϕ, which is uniquely determined by the normalization ma [,1] ϕ = 1. To investigate the structure of the set of solutions of (1.4) near (d, p) = (d, ), we first introduce a few notations. Set X = {u C ([, 1]) : Du σu = at =, 1}, Y = C([, 1]). Define mapping F (d, p) : (, ) X Y by Clearly, F (d, ) = for d (, ). Since F (d, p) = p σp + p [g(i) d]. (.) F p (d, )ϕ = ϕ σϕ + [ g(e k ) d ] ϕ, we see that (i) the kernel of F p (d, ) is spanned by ϕ, and is thus one dimensional; (ii) the range of F p (d, ), denoted by R(F p (d, )), is given by { 1 } ζ Y : e σ ϕ ζd =, and is thus of co-dimension one. Furthermore, F pd (d, )ϕ = ϕ R(F p (d, )). By Theorem 1.7 of [1], we obtain the result: Lemma.1. Let Z be any complement of span of {ϕ } in X. Then there eists some δ > and continuously differentiable functions d : ( δ, δ) R and ψ : ( δ, δ) X such that d() = d, ψ() =, and F (d(s), p(s)) =, where p(s) = sϕ + sψ(s). Moreover,

6 6 Y. DU, S.-B. HSU AND Y. LOU F 1 ({}) near (d, ) consists precisely of the curves p = and (d(s), p(s)), s ( δ, δ). Furthermore, 1 1 ( ) (.3) d () e σ ϕ d = k e σ e k g (e k )ϕ ϕ d. Proof. It suffices to check (.3). Dividing F (d(s), p(s)) = by s and differentiating the result with respect to s at s =, using p = sϕ + sψ(s) we have (ψ ()) σ(ψ ()) + ψ () [ [ ] ] g(e k ) d + ϕ kg (e k )e k ϕ d () =. Multiplying the above equation by e σ ϕ and integrating by parts we obtain (.3). In the terminology of [], is a F pd (d, )-simple eigenvalue of the operator F p (d, ). By Corollary 1.13 and Theorem 1.16 of [] we have Proposition.. There eist some positive constants δ 1 and δ and continuously differentiable functions γ : (d δ 1, d + δ 1 ) R, µ : ( δ, δ ) R, v : (d δ 1, d + δ 1 ) X, w : ( δ, δ ) X such that (.4) F p (d, )v(d) = γ(d)v(d), F p (d(s), p(s))w(s) = µ(s)w(s), sd (s)γ (d ) lim s,µ(s) µ(s) where γ(d ) = µ() =, v(d ) = w() = ϕ. The net result suggests that for s > small, the nontrivial (positive) solution p(s) = sϕ + sψ(s) is unstable under suitable conditions. Lemma.3. Suppose that (1.3) holds and I < e k. Then for any sufficiently small s >, µ(s) <. Proof. By (.) and the definition of d, we see that γ(d) = d d. If (1.3) holds, g (e k ) < for (, 1), which together with (.3) implies that d () >. By (.4), we see that µ(s) < for s > small. = 1, Lemma.4. Suppose that (1.3) holds. solution. If d (, g(i )), then (1.4) has no positive Proof. Let p denote a positive solution of (1.4). Integrating (1.4) in (, h) and applying the boundary condition in (1.4), we have 1 p[g(i) d] d =. Since g(i) (, g(i )) and I I, we see that d (, g(i )). Therefore, (1.4) has no positive solution when d (, g(i )). Lemma.5. Given any η >, there eists some positive constant C(η) such that every positive solution p of (1.4) with d η satisfies p L (,1) C(η).

7 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 7 The proof of Lemma.5 is identical to that of Lemma 4. of [13] and is omitted. Proof of Theorem 1.1. By Lemma.1 and the global bifurcation result of Rabinowitz [5], (1.4) has an unbounded connected branch of positive solutions, denoted by Γ = {(d, p)} R C 1 ([, 1]), which bifurcates from the trivial solution branch {(d, )} at (d, ). Since (1.4) has no positive solutions when d (, g(i )) (Lemma.4) and all positive solutions of (1.4) are uniformly bounded when d is positive and bounded away from zero (Lemma.5), we see that Γ can only become unbounded as d +. As Γ is connected, (1.4) has at least one positive solution for every d (, d ). Denote {(d, p) Γ : < d < d } by Γ 1. By Lemma.1, Γ contains a branch of positive solutions, denoted by Γ, which is given by (d(s), p(s)) for s > small. By (1.3) and I < e k, we have d () >. In particular, there eists some d > d such that the projection of Γ onto the d-ais is given by (d, d ). Hence Γ 1 and Γ must be disjoint. As Γ is connected and Γ 1, Γ Γ, we see that the projection of Γ \ Γ onto the d-ais must contain (, d ), i.e. (1.4) has at least two positive solutions for d (d, d ). By Lemma.4, (1.4) has no positive solution for d > g(i ). Thus we complete the proof of Theorem 1.1. The proof of Theorem 1.1 indicates that there eists d [d, g(i )) such that (1.4) has no positive solution lying on Γ for d > d, and it has at least one positive solution for d (, d ]. In such a case, it is natural to epect that there eist two positive solutions for d (d, d ), and the global bifurcation curve is -shaped, with a turning point at d = d. Moreover, one epects the following typical conclusions: For d (d, d ), there are two positive solutions, and the solution on the upper branch of the global bifurcation curve is stable, while that located on the lower branch is unstable. Note also that the zero solution p = is always stable for d > d. Such a global bifurcation picture can be partially proved by techniques of [3] for increasing operators, if the problem at hand has the usual order-preserving property. Unfortunately, due to the nonlocal nature of our problem, this nice property is lost and we cannot use such tools involving the order-preserving property as in [3] or the usual upper and lower solution technique. We remark that even the modified order-preserving property used in [6] is lost here due to the fact that g(i) is no longer increasing in I for all I >. To overcome these difficulties, in the net two sections, we employ a perturbation and reduction approach, which strongly suggests the validity of the global bifurcation picture described above, at least for large σ. 3. Multiple steady-states via a perturbation-reduction approach In this section, we use a perturbation and reduction approach to study the positive solutions of (1.4). Their stability will be considered in Section 4 later. We will eamine the problem with a large σ, and write it in the form σ = ɛ 1, with ɛ > small. Let p() be a positive solution of (1.4), and define A simple calculation shows that u() = p(ɛ + 1)e /.

8 8 Y. DU, S.-B. HSU AND Y. LOU (3.1) u 1 [ ] 4 u + ɛ g(ĩ) d u =, (, ), u 1 u =, { ɛ 1, }, where ( ) Ĩ = ep k ɛk kɛ u(s)e s/ ds. We will look for a solution to (3.1) of the form with η > and z H := Since the function e / satisfies u() = ɛ 1 ηe / + z() { z : z()e / d =, } z() e / d <. u 1 4 u = in ( ɛ 1, ), u 1 u = at = ɛ 1,, we necessarily have z z = ɛ [ g(ĩ) d ] u, (, ), z 1 z =, { ɛ 1, }, with Ĩ as above, and u() = ɛ 1 ηe / + z(). We now define {g(ĩ)[ɛ 1 } f() = f(; ɛ, η, z) := ɛ ηe / + z()] + d[ɛ 1 ηe / + z()] (here u + = ma{u, }) and consider the auiliary problem z + 1 (3.) 4 z = f() λe/, (, ), z 1 z =, { ɛ 1, }, with λ determined by (3.3) λ e d = f()e / d. Then from (3.) we find that z H has the epression z() = f(y) sinh( y )dy λ e y/ sinh( y )dy + αe /, where sinh() = (e e )/ and α R 1 is chosen such that z()e/ d =. A simple calculation gives z() = f(y) sinh( y )dy + λe / ( 1 + e ) + αe /.

9 with MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 9 For z H, we now define the nonlinear operator F 1 ɛ,η(z)() := F ɛ,η (z)() := F 1 ɛ,η(z)() + α(ɛ, η, z)e /, f(y; ɛ, η, z) sinh( y )dy + λ(ɛ, η, z)e / ( 1 + e ), where λ(ɛ, η, z) is determined by (3.3) with f() = f(; ɛ, η, z), and α(ɛ, η, z) is determined by e / Fɛ,η(z)()d 1 + α e d =, that is, (3.4) α(ɛ, η, z) := (1 e ) 1 e / F 1 ɛ,η(z)()d. We will show that for all small ɛ > and all η > in a certain interval J, the operator F ɛ,η is a contraction mapping in H, and hence it has a unique fied point z = z ɛ,η. We will then choose η = η ɛ so that λ(ɛ, η, z) = for z = z ɛ,η and η = η ɛ. We will show that this gives a positive solution u = ɛ 1 η ɛ e / + z ɛ,ηɛ () to (3.1) for all small ɛ >. A key point in this approach is that with our assumptions on g(i), for each small ɛ >, for d > in a certain range, we always have two solutions for η ɛ, and hence this approach yields two positive steady-states for (3.1). We will show that for ɛ > small, the equation λ(ɛ, η, z ɛ,η ) = is a perturbation of the equation kη 1 d = kη g(ξ e s )ds, ξ = e k (, 1]. Denote µ µ = kη, G(µ) = 1 g(ξ e s )ds and G (µ) = µ then the above equation is equivalent to (3.5) d = G(µ). µ g(ξ e s )ds, In order to determine the range J for η so that F ɛ,η is a contraction mapping, and multiple solutions to (3.1) eist, we need to obtain several simple properties of the functions G(µ) and G (µ) first Properties of G (µ) and G(µ). Our assumptions on g imply that g(i) c I for some c > and all I >. It follows that Clearly and If I ξ, then G (+ ) = + g(ξ e s )ds (, ). G() = g(ξ ) >, G(µ) > for µ >, G(+ ) = G (µ) = µ [µg (µ) G (µ)] =: µ G(µ). G (µ) = g(ξ e µ ) >, G (µ) = g (ξ e µ )ξ e µ < µ.

10 1 Y. DU, S.-B. HSU AND Y. LOU Hence It follows that G (µ) = µg (µ) < for µ >. G(µ) < G() = for µ >. Thus G (µ) = µ G(µ) < for µ >. This implies that for each d (, d ) := (, g(ξ )), (3.5) has a unique solution µ >, and for each d d, (3.5) has no solution µ >. If I < ξ, then G (µ) > for µ, and Thus G (µ) > for µ (, ln(ξ /I )), G (µ) < for µ > ln(ξ /I ). G (µ) > for µ (, ln(ξ /I )), G (µ) < for µ > ln(ξ /I ). Using G (µ) < for all large µ > and G (+ ) >, it is easily seen that G(µ) < for all large µ. Thus there eists a unique µ > ln(ξ /I ) such that for µ (, µ ), G(µ) >, and for µ > µ, G(µ) <. It follows that G (µ) > for µ (, µ ), G (µ) < for µ > µ. Define ˆd := G(µ ); then for each d (g(ξ ), ˆd ), (3.5) has eactly two solutions µ 1 (, µ ) and µ (µ, + ), and (3.5) has a unique solution µ > for d { ˆd } (, g(ξ )). Moreover, (3.6) G (µ 1 ) > > G (µ ). From now on, we always assume that (3.7) I < ξ and d (g(ξ ), ˆd ). Thus (3.5) has eactly two solutions < µ 1 < µ, and apart from (3.6), we have (3.8) µ > G(µ) for µ (, µ 1 ) (µ, ), µ < G(µ) for µ (µ 1, µ ). Clearly part (i) of Theorem 1. follows from the above discussions. The rest of this section is devoted to the proof of part (ii) of Theorem 1.. We set [ µ1 J = k, µ ]. k 3.. F ɛ,η maps a subset of H into itself. It is easily seen that H = H ɛ endowed with the norm is a Banach space. z = z() e / d Lemma 3.1. There eists ɛ > and M > such that for each ɛ (, ɛ ] and η J, F ɛ,η maps Ω := {z H : z M} into itself. Proof. e / Fɛ,η(z)() 1 = e / f(y; ɛ, η, z) sinh( y )dy + λ(ɛ, η, z)[( 1)e + 1] = S 1 () + S ().

11 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 11 We have, for ɛ > small, λ(ɛ, η, z) = (1 e ɛ 1 ) 1 ɛ { g( )[ɛ 1 ηe / + z()] + d[ɛ 1 ηe / + z()] } e / d ɛ 1 [c e ɛk ] (ɛ 1 ηe y/ z(y) )e y/dy + d (ɛ 1 ηe + z() e / )d ɛ ɛ 1 [ c e ɛk z + d ] (ɛηe + ɛ z() e / )d C(e ɛk z + 1)(ɛ + ɛ z ), for some C > independent of η J and small ɛ >. In the following, we will use C to denote a generic positive constant that is independent of η J and small ɛ >, whose value may change from line to line. From the above estimate we obtain For the term S 1 () we have S 1 () = e/ S () d λ(ɛ, η, z) ( 1)e + 1 d C(e ɛk z + 1)(1 + ɛ z ). f(y; ɛ, η, z) sinh( y )dy e / ɛ g( ) d ( ɛ 1 ηe y/ + z(y) ) sinh( y ) dy ɛ C(e ɛk z + 1) ( ɛ 1 e y + z(y) e y/) e y 1 dy C(e ɛk z + 1)(1 + ɛ z )ɛ. Thus We therefore have S 1 () d C(e ɛk z + 1)(1 + ɛ z ). F 1 ɛ,η(z) C(e ɛk z + 1)(1 + ɛ z ). By (3.4), we have α(ɛ, η, z)e / F 1 ɛ,η(z) C(e ɛk z + 1)(1 + ɛ z ). Therefore, taking M = 3C then from the above inequalities we can easily find ɛ > sufficiently small (depending on C) such that for all ɛ (, ɛ ] and η J, F ɛ,η (z) M if z M.

12 1 Y. DU, S.-B. HSU AND Y. LOU 3.3. F ɛ,η is a contraction mapping. In this subsection we show that there eists ɛ (, ɛ ] such that for each ɛ (, ɛ ] and each η J, F ɛ,η is a contraction mapping on Ω. Suppose that z 1, z Ω, and denote for i = 1,, Clearly I i () = ep ( k ɛk k ) ηe y dy g(i 1 ) g(i ) C I 1 I C e ɛk z 1 (y)e y/dy e ɛk We write [ F 1 ɛ,η (z 1 )() F 1 ɛ,η(z )() ] e / = ɛ e ɛk z i (y)e y/ dy. z (y)e y/ dy Ce ɛk( z 1 + z ) ɛk z 1 (y) z (y) e y/ dy Ce ɛkm ɛ z 1 z ɛc z 1 z. { g(i 1 ) ( ) ɛ 1 ηe y/ + + z 1 g(i ) ( ) ɛ 1 ηe y/ + } + z d(z1 z ) e y/ (e y 1)dy + [λ(ɛ, η, z 1 ) λ(ɛ, η, z )]e ( 1 + e ) = ɛ [g(i 1 ) g(i )] ( ) ɛ 1 ηe y/ + + z 1 e y/ (e y 1)dy + ɛ { [ (ɛ ) g(i ) 1 ηe y/ + ( ) + z 1 ɛ 1 ηe y/ + ] } + z d(z 1 z ) e y/ (e y 1)dy + [λ(ɛ, η, z 1 ) λ(ɛ, η, z )]e ( 1 + e ) = T 1 () + T () + T 3 (). Then Since T 1 () ɛ Cɛ z 1 z Cɛ z 1 z. ( ɛ 1 ηe y + z 1 e y/) dy g(i ) + d CI + d Ce ɛk z + d C, we have T () Cɛ { ( ɛ 1 ηe y/ + z 1 ) + ( ɛ 1 ηe y/ + z ) + + z1 z } e y/ e y 1 dy Cɛ z 1 (y) z (y) e y/ dy Cɛ z 1 z.

13 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 13 To estimate T 3 (), we notice that [ ] λ(ɛ, η, z 1 ) λ(ɛ, η, z ) e d ɛ 1 { = ɛ g(i 1 ) ( ) ɛ 1 ηe y/ + + z 1 g(i ) ( ) ɛ 1 ηe y/ + } + z d(z1 z ) e y/ dy = ɛ [g(i 1 ) g(i )] ( ) ɛ 1 ηe y/ + + z 1 e y/ dy ɛ 1 { [ (ɛ ) + ɛ g(i ) 1 ηe y/ + ( ) + z 1 ɛ 1 ηe y/ + ] + z Therefore similar to the estimates for T 1 () and T () above, we obtain λ(ɛ, η, z 1 ) λ(ɛ, η, z ) Cɛ z 1 z. Since e 1 + e C on [, ], we obtain It follows that By (3.4), we have T 3 () Cɛ z 1 z. Fɛ,η(z 1 1 ) Fɛ,η(z 1 ) ( T 1 () + T () + T 3 () )d Cɛ z 1 z. } d(z 1 z ) e y/ dy. α(ɛ, η, z 1 )e / α(ɛ, η, z )e / = F 1 ɛ,η(z 1 ) F 1 ɛ,η(z ) Cɛ z 1 z. Thus F ɛ,η is a contraction mapping on Ω provided that ɛ (, ɛ ] and η J with ɛ = min{ɛ, (3C) 1 }. Summarizing, we have proved the following result. Lemma 3.. There eists ɛ (, ɛ ] such that F ɛ,η is a contraction mapping on Ω for every ɛ (, ɛ ] and η J. Applying Banach s fied point theorem, we obtain Proposition 3.3. For each ɛ (, ɛ ] and η J, F ɛ,η has a unique fied point z ɛ,η Ω. Remark 3.4. Since F ɛ,η depends continuously on (ɛ, η), the uniqueness of the fied point implies that z ɛ,η also depends continuously on (ɛ, η) The reduced equation. Proposition 3.5. Suppose ɛ (, ɛ ]. If η J satisfies (3.9) λ(ɛ, η, z ɛ,η ) =, then is a positive solution to (3.1). u ɛ,η = ɛ 1 ηe / + z ɛ,η ()

14 14 Y. DU, S.-B. HSU AND Y. LOU Proof. Since F ɛ,η z ɛ,η = z ɛ,η and λ(ɛ, η, z ɛ,η ) =, from the definition of F ɛ,η we find that u = u ɛ,η satisfies, for (, ), u u = (z ɛ,η) z ɛ,η = f(; ɛ, η, z ɛ,η ). Using the definition of f we have ( ) 1 (3.1) u ɛ d u = ɛ g(ĩ)u+ in (, ), with ( ) Ĩ = ep k ɛk ɛk u(y)e y/ dy. Clearly u also satisfies the boundary condition of (3.1). Hence, due to ɛ d >, we can apply the maimum principle to (3.1) to conclude that u. Thus u + = u and u solves (3.1). To show u is a positive solution, by the strong maimum principle, it suffices to show that u. But this follows trivially from u()e / d = ɛ 1 ηe d >. We net eamine closely the reduced equation λ(ɛ, η, z ɛ,η ) =, that is { [ ] + [ (3.11) g(i ɛ,η ) ɛ 1 ηe / + z ɛ,η () d ɛ 1 ηe / + z ɛ,η ()] } d =, ɛ 1 e / with ( ) I ɛ,η () = ep k ɛk ɛk [ɛ 1 ηe y/ + z ɛ,η (y)]e y/ dy. Multiplying (3.11) by ɛη 1 we obtain { [ ] + [ = g(i ɛ,η ) e / + ɛη 1 z ɛ,η () d e / + ɛη 1 z ɛ,η ()] } d with with = We may write e / ɛ 1 ɛ 1 e [g(i ɛ,η ) d]d + δ 1, δ 1 = { [ ] + e / g(i ɛ,η ) e / + ɛη 1 z ɛ,η() e /} d ɛ 1 Cɛ Cɛ. z ɛ,η () e / d I ɛ,η () = ξ ep ( ɛk kηe ) m ɛ,η (), ( ) ( m ɛ,η () = ξ 1 e k ep kɛ z ɛ,η (y)e y/ dy ep kηe ɛ 1) 1

15 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 15 uniformly for [, ] and η J as ɛ. Therefore with It follows that where We thus have and with Finally we have = g(i ɛ,η ()) = g ( ξ e ɛk kηe ) + δ () δ () uniformly as ɛ. [ g ( ξ e ɛk kηe) d ] e d + δ 1 + δ, δ = d[1 ep( )] = d = = δ ()e d = o(1) as ɛ. g ( ξ e ɛk kηe) e d + δ 1 + δ, g ( ξ e ɛk kηe) e d + δ 1 + δ + δ 3 δ 3 = d ep( ) = o(1). g ( ξ e ɛk kηe) e d = 1 kη = 1 kη = 1 kη g(ξ e ɛk kη kη 1 kη ɛk kηe g(ξ e k +kη ep( ) kη 1 kη )(ɛk + kηe ) 1 kη d ɛk kηe g(ξ e µ )dµ )d g(ξ e µ kη ep( ɛk ) )dµ k k g(ξ e µ )dµ + o(1) kη k g(ξ e µ )dµ + o(1) g(ξ e µ )dµ + o(1). Therefore the reduced equation can be written as d = 1 kη kη g(ξ e s )ds + o(1), µ

16 16 Y. DU, S.-B. HSU AND Y. LOU where o(1) uniformly in η J as ɛ. Recalling the definition of G(µ) we find that the reduced equation can be written as (3.1) d = G(kη) + o(1). Fi η 1, η and η 3 in J such that ( µ1 η 1 k, µ ) ( 1 µ1, η k k, µ ) ( µ, η 3 k k, µ ). k Then from (3.8) we find that d > G(kη 1 ), d < G(kη ), d > G(kη 3 ). Since λ(ɛ, η, z ɛ,η ) is a continuous function in (ɛ, η), the term o(1) in (3.1) can be written as o(ɛ, η) which is continuous and satisfies o(ɛ, η) uniformly in η J as ɛ. Therefore, for all small ɛ >, say ɛ (, ɛ ], with some ɛ (, ɛ ), the equation d = G(kη) + o(ɛ, η) has at least two solutions in J: η ɛ 1 (η 1, η ) and η ɛ (η, η 3 ). Moreover, lim η1 ɛ = µ 1 ɛ k, lim ɛ ηɛ = µ k and G (kη1) ɛ > > G (kη). ɛ We have thus proved the following result. Theorem 3.6. Suppose that (3.7) holds. Then there eists ɛ > such that for each ɛ (, ɛ ], (3.1) has two positive solutions of the form with z ɛ 1, z ɛ H satisfying z ɛ i Cɛ, and Since, for i = 1,, u 1 = ɛ 1 (η ɛ 1e / + z ɛ 1), u = ɛ 1 (η ɛ e / + z ɛ ) lim η1 ɛ = µ 1 ɛ k, lim ɛ ηɛ = µ k, G (µ 1 ) > > G (µ ). u i = ɛ 1 ( η ɛ ie / + z ɛ i ) ( = ɛ 1 µi ) k e/ + z i ɛ with z ɛ i := z ɛ i +o(1)e /, we find that z ɛ i = o(1), and part (ii) of Theorem 1. now follows directly from Theorem Stability analysis In this section, we consider the linearized stability of the two solutions u 1 and u in Theorem 3.6. Let u = u ɛ denote either u 1 or u. The linearized eigenvalue problem of (3.1) at u = u is given by φ 1 φ + 4 ɛ [g(i ɛ ) d]φ (4.1) ɛ u g (I ɛ )I ɛ ɛ k φ(s)e s/ ds + λφ =, (, ), where φ 1 φ =, {, }, I ɛ () = e k ɛk kɛ u (s)e s/ ds.

17 MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 17 We will show that, for every small ɛ >, this problem has an eigenpair (λ, φ) of the form λ = ɛ µ ɛ, φ() = e / + ɛ w ɛ (), w ɛ H, with µ ɛ µ as ɛ and w ɛ C for all ɛ > small. Here H is the Banach space defined in the previous section. Moreover, we will show that µ < when u = u 1, and µ > when u = u. This implies that u 1 is linearly unstable. Although this does not prove that u is linearly stable, but as eplained below, it strongly suggests the validity of such a conclusion. The difficulty in proving the linearized stability of u is due to the fact that (4.1) is a nonlocal eigenvalue problem, and the corresponding linear operator to this problem is not self-adjoint, nor order-preserving. Therefore it is difficult to use variational characterization or the well-known Krein-Rutman theorem to know the relationship of the eigenvalue obtained above to the rest of the spectrum. However, since φ = e / + ɛw ɛ is a small perturbation of a positive function, it is reasonable to believe that λ = ɛµ ɛ behaves like a principle eigenvalue, with all other eigenvalues having real parts strictly greater than λ, which would imply the linearized stability of u. We now look for an eigenpair of (4.1) of the form λ = ɛ µ, φ = e / + ɛw, w H. Substituting these into (4.1) we obtain w 1 w + ɛ[g(i 4 ɛ) d](e / + ɛw) (4.) ɛ u g (I ɛ )I ɛ ɛ k [e s + ɛwe s/ ]ds + ɛµ(e / + ɛw) =, (, ), w 1 w =, {, }. Multiplying the first equation in (4.) by ɛ 1 e /, and integrating over (, ), we obtain (4.3) = µ [g(i ɛ ) d](e + ɛwe / )d e d = µ(1 e ). This defines µ as a functional of w and we may write µ = µ(w, ɛ). Much as in the previous section, from (4.) we obtain w() = ɛ ɛ +ɛ µ e / u g (I ɛ )I ɛ ɛ k [g(i ɛ ) d](e y/ + ɛw) sinh( y )dy ( y ) u g (I ɛ )I ɛ ɛ k (e y/ + ɛw) sinh( y [e s + ɛwe s/ ]ds with γ R 1 chosen such that e/ w()d =. For w H, we now define the operator )dy + γe /, [e s + ɛwe s/ ]dsd sinh( y )dy L ɛ (w)() := L 1 ɛ(w)() + L ɛ(w)() + L 3 ɛ(w)() + γ(w, ɛ)e /,

18 18 Y. DU, S.-B. HSU AND Y. LOU with L 1 ɛ(w)() : = ɛ L ɛ(w)() : = ɛ [g(i ɛ ) d](e y/ + ɛw) sinh( y )dy, ( y ) u g (I ɛ )I ɛ ɛ k [e s + ɛwe s/ ]ds L 3 ɛ(w)() : = ɛ µ(w, ɛ) (e y/ + ɛw) sinh( y )dy, sinh( y )dy, where µ(w, ɛ) is determined by (4.3), and [ ] 1 γ(w, ɛ) = e d e [ / L 1 ɛ(w)() + L ɛ(w)() + L 3 ɛ(w)() ] d. We are going to show that there eists C > such that for every small ɛ >, L ɛ maps B := {w H : w C} into itself, and is a contraction mapping. Therefore L ɛ has a unique fied point w ɛ in B: L ɛ (w ɛ ) = w ɛ. Clearly this gives an eigenpair to (4.1): From (4.3), we easily obtain Clearly e / It follows that Using (λ, φ) = (ɛ µ(w ɛ, ɛ), e / + ɛw ɛ ). µ(w, ɛ) C(1 + ɛ w ) for all small ɛ > and some C >. (e y/ + ɛw) sinh( y L 3 ɛ(w) = )dy [e y + ɛ w e y/ ] e y 1 dy C(1 + ɛ w ). e / L 3 ɛ(w)() d C(1 + ɛ w ). e / L 1 ɛ(w)() = ɛ e / [g(i ɛ ) d](e y/ + ɛw)(e y e y )dy ɛ ɛ C [g(i ɛ ) + d](e y + ɛ w e y/ ) e y 1 dy (e y + ɛ w e y/ )dy ɛ C(1 + ɛ w ), we deduce L 1 ɛ(w) C(1 + ɛ w ). To estimate L ɛ(w) we notice that y [e s + ɛwe s/ ]ds C(1 + ɛ w ) for all y (, ), and hence e / L ɛ(w)() ɛ C(1 + ɛ w ) u g (I ɛ )I ɛ ɛ k C(e y/ + ɛ z ɛ i (y) ), ( e y + ɛ z ɛ i (y) e y/) e y 1 dy ɛ C(1 + ɛ w ).

19 It follows that MULTIPLE STEADY-STATES IN PHYTOPLANKTON POPULATION 19 From the definition of γ(w, ɛ) we obtain Therefore If w C, we obtain L ɛ(w) C(1 + ɛ w ). γ(w, ɛ)e / L 1 ɛ(w) + L ɛ(w) + L 3 ɛ(w) C(1 + ɛ w ). L ɛ (w) C(1 + ɛ w ). L ɛ (w) C(1 + ɛ C) C provided that ɛ (, ɛ ], with ɛ determined by (1 + ɛ C) =. Net we show that L ɛ is a contraction mapping on B H for all small ɛ >. Let w 1, w B. Then It follows easily that Similarly L 1 ɛ(w 1 ) L 1 ɛ(w ) = ɛ L ɛ(w 1 ) L ɛ(w ) = ɛ which gives Finally [g(i ɛ ) d]ɛ(w 1 w ) sinh( y L 1 ɛ(w 1 ) L 1 ɛ(w ) ɛ C w 1 w. )dy. ( y ) u g (I ɛ )I ɛ ɛ k ɛ(w 1 w )e s/ ds sinh( y )dy, L ɛ(w 1 ) L ɛ(w ) ɛ C w 1 w. L 3 ɛ(w 1 ) L 3 ɛ(w ) =ɛ µ(w 1, ɛ) ɛ(w 1 w ) sinh( y )dy + ɛ [ µ(w 1, ɛ) µ(w, ɛ) ] (e y/ + ɛw ) sinh( y )dy. By (4.3) we easily see that It then follows easily that We now obtain Thus we have γ(w 1, ɛ)e / γ(w, ɛ)e / µ(w, ɛ) µ(w 1, ɛ) ɛ C w 1 w. L 3 ɛ(w 1 ) L 3 ɛ(w ) ɛ C w 1 w. L 1 ɛ(w 1 ) L 1 ɛ(w ) + L 1 ɛ(w 1 ) L 1 ɛ(w ) + L 3 ɛ(w 1 ) L 3 ɛ(w ) ɛ C w 1 w. L ɛ (w 1 ) L ɛ (w ) ɛ C w 1 w 1 w 1 w for all w 1, w B provided that ɛ > is small enough, say < ɛ ɛ 1 ɛ.

20 Y. DU, S.-B. HSU AND Y. LOU We may now use the contraction mapping theorem to conclude that for every small ɛ >, say ɛ (, ɛ 1 ], L ɛ has a unique fied point w ɛ in B: L ɛ (w ɛ ) = w ɛ. It follows that, for such ɛ, (4.1) has an eigenpair of the form (λ, φ) = (ɛ µ(w ɛ, ɛ), e / + ɛw ɛ ). Let us now determine the sign of µ(w ɛ, ɛ) for u = u 1 and u = u, respectively. We will denote µ(w ɛ, ɛ) by µ i ɛ for u = u i, i = 1,. We define Iɛ() i from I ɛ () analogously. From its definition, we easily see that lim I i ɛ ɛ() = e k Thus we can use (4.3) to obtain Therefore lim ɛ µ i ɛ = = 1 kη ie s ds = ξ e µ ie uniformly in compact subsets of (, ]. [g(ξ e µ ie ) d]e d g(ξ e µ is )ds d 1 = G(µ i ) d + µ i G (µ i ) = µ i G (µ i ). η i ke g (ξ e µ ie )ξ e µ ie e d ξ µ i se µ is g (ξ e µ is )ds µ 1 ɛ = µ 1 G (µ 1 ) + o ɛ (1) <, µ ɛ = µ G (µ ) + o ɛ (1) >. Summarizing, we have proved the following result: Theorem 4.1. Let u 1 and u be given by Theorem 3.6. Then for each small ɛ >, the linearized eigenvalue problem (4.1) with u = u i has an eigenpair of the form (λ, φ) = (ɛ µ i ɛ, e / + ɛw i ɛ), i = 1,, with w i ɛ H having a uniform bound independent of ɛ, and µ 1 ɛ = µ 1 G (µ 1 ) + o ɛ (1) <, µ ɛ = µ G (µ ) + o ɛ (1) >. References [1] M.G. Crandall, P.H. Rabinowitz, Bifurcation from simple eigenvalues, J. Func. Anal., 8 (1971), [] M.G. Crandall, P.H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability, Arch. Rat. Mech. Anal., 5 (1973), [3] Y. Du, The structure of the solution set of a class of nonlinear eigenvalue problems, J. Math. Anal. Appl., 17 (199), [4] Y. Du, S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton I: Eistence, SIAM J. Math. Anal., 4 (8), [5] Y. Du, S.-B. Hsu, Concentration phenomena in a nonlocal quasi-linear problem modeling phytoplankton II: Limiting profile, SIAM J. Math. Anal., 4 (8), [6] Y. Du, S.-B. Hsu, On a nonlocal reaction-diffusion problem arising from the modeling of phytoplankton growth, SIAM J. Math. Anal., 4 (1), [7] Y. Du and L. Mei, On a nonlocal reaction-diffusion-advection equation modelling phytoplankton dynamics, Nonlinearity, 4 (11), [8] U. Ebert, M. Arrayas, N. Temme, B. Sommeijer, J. Huisman, Critical condition for phytoplankton blooms, Bull. Math. Biol., 63 (1), [9] D.J. Gerla, W.M. Wolf, J. Huisman, Photoinhibition and the assembly of light-limited phytoplankton communities, Oikos, 1 (11), [1] E.W. Helbling et al., UVR-induced photosynthetic inhibition dominates over DNA damage in marine dinoflagellates eposed to fluctuating solar radiation regimes, J. Ep. Mar. Biol. Ecol., 365 (8), 96-1.

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