Appendix from K. Yoshiyama et al., Phytoplankton Competition for Nutrients and Light in a Stratified Water Column

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1 2009 by The University of Chicago. All rights reserved. DO: / Appendix from K. Yoshiyama et al. Phytoplankton Competition for Nutrients and Light in a Stratified Water Column (Am. Nat. vol. 174 no. 2 p. 190) Steady State Distributions and Their Regions of Stability We first show calculations of steady state and the region of stability from the steady state conditions (13) and the stability criteria (14). n the following analyses all parameters are positive. n the main text we occasionally consider a bg p 0 (figs. 2 5) as a simplified case which does not affect results in this appendix. From the stability criteria following properties can be easily derived for the stable steady state: (1) each species has a single layer with positive biomass at most (2) the surface-layer maximum [S] is colimited by nutrients and light or limited by light only (3) the deep-layer maximum [D] is colimited by nutrients and light and (4) and the benthic-layer maximum [] is limited by nutrients. From now on we refer to the biomass of species i as a vertically integrated cell density multiplied by the attenuation coefficient a i z i 0 p ab(z)dz. (A1) i i Stable Steady State of a onoculture n the following we consider the steady state of culture growth and the region of stability. We introduce useful functions. The first one describes the balance of biomass yield from light limitation (the first two terms of the right-hand side) and nutrient limitation (the third term of the right-hand side) at depth z: ( ) gd(r ) Q (z) p log abgz. (A2) t is easy to see that dq /dz! 0 for 0 z z when R 1 R. The second one describes net growth in the surface layer ( ḡ m): 1 g(r ) m H(R ) p d. (A3) log ( / ) Specifically denotes the zero net growth isocline (ZNG) in the surface layer; H(R ) p 0 (R ) is a point on the ZNG (i.e. H(R ) p 0). The partial derivatives of H(R ) are written as 1

2 H 1 gd p (A4) R log ( / ) R H 1 g(r ) m g(r ) m p d. (A5) 2 [log ( / )] log ( / ) When H(R ) p 0 the first term of the right-hand side of equation (A5) is 0 and the second term is negative when R 1 R because! in this case. According to the implicit function theorem we can define (R) for R 1 R that satisfies H(R (R)) p 0. From equations (A4) and (A5) the derivative is written as H(R )/ R ecause g/ R 0 and g(r ) m! 0 for R 1 R we have d gd p p. (A6) dr H(R )/ g(r ) m R d 0. dr (A7) No iomass [ ] When cells are absent in the water column the steady state is expressed by b(z) { 0 R(z) { R (A8) (z) p exp ( abgz) for 0 z z. ecause R(z) is constant and (z) decreases over z we have an inequality g g(r(z) (z)) for z! z z. Thus [ ] is stable if ḡ m 0. This condition is written as H[R exp ( a z )] 0. (A9) bg enthic-layer iomass [] f the benthic-layer maximum [] is limited by R then [] satisfies the stability criteria. The stable steady state is expressed by 0 for 0 z! z b(z) p { for z p z R(z) { R exp ( abgz) for 0 z! z (z) p { exp [ (a z )] for z p z bg (A10) where 2

3 gd(r ) p. (A11) 1/h For the existence of positive steady state 1 0 and g(r (z )) m p 0. The former inequality implies that R 1 R and the latter implies that (z ) ; that is exp [ (abgz )]. (A12) f we substitute with equation (A11) easy manipulation of inequality (A12) gives Q (z ) 0. Therefore the stability conditions can be expressed by {R 1 R } {Q (z ) 0}. (A13) Deep-Layer aximum [D] f the deep-layer maximum [D] is colimited by R and then the steady state satisfies the stability criteria. When biomass maximum is at z p z the stable steady state is expressed by 0 for 0 z! z z! z z b(z) p { for z p z { R for 0 z! z R R (z z ) for z z z z 1/h z R(z) p (A14) exp ( abgz) for 0 z! z (z) p { exp [ a (z z )] for z z z bg where From equation (A15) we have an equality ( ) p gd(r ) p log ( a z ) bg. (A15) gd(r ) bg log a z p 0. (A16) z 1/h z Thus z satisfies Q (z ) p 0. The steady state exists if 1 0 and z! z! z. ecause Q (z) is a decreasing function for 0 z z when R 1 R the above conditions can be expressed by {R 1 R } {Q (z ) 1 0} {Q (z )! 0}. (A17) Surface-Layer aximum [S] With nutrients in the surface layer R and light (z ) p steady state [S] is expressed by 3

4 where The point (R ) for 0 z z b(z) p z {0 for z! z z { R (z z ) for z! z z z 1/h z R for 0 z! z R(z) p R (A18) exp [ log ( / )(z/z )] for 0 z z (z) p { exp [ a (z z )] for z! z z bg gd(r ) p z 1/h z ( ) p log a z. (A19) is a point on the ZNG and satisfies an equality obtained from equation (A19): ( ) bg H(R ) p 0 gd(r ) log a bg z p 0. (A20) f we solve equation (A20) simultaneously (R ) is obtained. The steady state exists if 1 0. This implies R! R. To satisfy the stability criteria the growth in the surface layer should be colimited by nutrients and light or limited by light only. This implies R R and. Thus [S] is a stable steady state if equation (A20) has a root such that R R! R and. These conditions are satisfied if the two inequalities hold which can be rewritten as ( ) gd(r ) bg log a z 0 (A21) z 1/h z H[R exp ( a z )] 1 0 (A22) bg {Q (z ) 0} {H[R exp ( abgz )] 1 0}. (A23) Two-Species Stable Steady State Here we consider stable steady state of two species. Species 1 is a good nutrient competitor and species 2 is a good light competitor ( R 1! R2 and 1 1 2). At steady state the vertical distribution of each species is either [ ] [] [D] or [S]. Out of all 16 combinations it is theoretically impossible for four cases to be stable steady state. f [-] is stable growth of both species should be limited by nutrients according to the stability criteria. This requires R(z ) p R1 p R2 which contradicts R 1! R2. When species 1 forms [D] R(z) p R1 for 0 z! z1. Above the layer of species 1 the growth rate of species 2 is negative because of nutrient limitation. Thus steady states where species 2 resides above species 1 ([-D] [-S] [D-S]) are impossible. Hence the above four cases are eliminated. Here instead of presenting solutions of all 12 steady states we select two coexistence cases [D-D] and [S-D] 4

5 and discuss the details of their existence and stability. The surface coexistence case [S-S] was analyzed by Huisman and Weissing (1995). Other cases can be solved similarly and much more easily than these three cases. A function H i(r ) is defined for each species ( i p 1 2) according to equation (A3). For R 1 Ri i(r) can be defined such that H i(r i(r)) p 0. When H 1(R 2 2) 1 0 1(R 2)! 2 p 2(R 2) ; this case corresponds to high conditions (fig. 1A). When and f 1()/m1 f 2()/m2 for 1 0 it is easy to see that 1( ) 1 2( ). Thus under high conditions two ZNGs have an intersection at R 1 R2 where the ZNG of species 2 is steeper than that of species 1 (i.e. Fd 1/dRF! Fd 2/dRF). This intersection corresponds to the surface coexistence state [S-S]. n addition we introduce two useful functions: ( ) 1 R2 R21 ( ) ( z z ) g1d(r 2 1) comp bg 1 1 z2 z1 Q (z z ) p log a z (A24) comp Q 2 (z 1 z 2) p log a bg(z 2 z 1) g2d. (A25) The above functions describe the balance of biomass yields of species 1 (eq. [A24]) and species 2 (eq. [A25]) from light limitation and from nutrient limitation when competing species are present. Coexistence [D-D] Steady state [D-D] satisfies the stability criteria if both species are colimited by nutrients and light at their biomass maxima. The stable steady state is expressed by 0 for 0 z! z i z i! z z b i(z) p { for z p zi R1 for 0 z! z1 R2 1 R(z) p R1 (z z 1) for z1 z! z 2. (A26) z2 z1 R 2 {R 2 (z z 2) for z2 z z 2 { 2 bg 2 2 exp ( a z) for 0 z! z bg 1 (z) p 1exp [ a bg(z z 1)] for z1 z! z 2 exp [ a (z z )] for z z z The biomass of each species is obtained by nutrient flux and light attenuation: g1d(r 2 1) 1 z 2 z1 ( ) bg 1 1 p p log a z z 1/h z2 z2 z1 The above equalities (A27) and (A28) are equivalent to 5 ( ) ( 1 ) bg 2 1 (A27) R R p g D (A28) p log a (z z ). 2

6 comp Q 1 (z 1 z 2) p 0 comp Q 2 (z 1 z 2) p 0. (A29) y solving equation (A29) we obtain z1 and z2. The existence of [D-D] requires and z! z 1! z 2! z. From equation (A27) because R! R and z! z. From equation (A28) 1 0 when f we rearrange Q comp (z z ) p 0 z is expressed by y substituting z R R 1 0. (A30) z2 z1 g1d(r 2 1) 2 1 log ( / 1 ) abgz1 z p z. (A31) in inequality (A30) with equation (A31) we have Rearranging inequality (A32) we have an inequality R 2 log ( / 1 ) abgz (A32) g1d(r 2 R 1) g1 D 1 log ( / ) a z 1 bg 1 Q 1 (z 1)! 0 (A33) where Q 1 (z) is defined in equation (A2) which is a decreasing function of z. Therefore the inequality (A33) is equivalent to z1 1 z 1 (A34) where z1 is a root of Q 1 (z) p 0. This condition implies that the biomass maximum of species 1 is deeper when competing with species 2 than the culture growth. Now the conditions for existence of [D-D] are simplified to z! z 1! z 2(z 1)! z z 1! z 1 (A35) comp where z 2(z 1) is defined by the right-hand side of equation (A31) and satisfies Q 1 (z 1 z 2(z 1)) p 0. From equation (A29) we have an equality Tedious manipulations of the derivative of comp Q 2 (z 1 z 2(z 1)) p 0. (A36) comp Q 2 give Thus the conditions for existence of [D-D] are summarized by comp comp comp dq2 Q2 Q2 dz2 p! 0. (A37) dz1 z1 z2 dz1 comp comp 1 {Q 2 (z z 2(z )) 1 0} {Q 2 (z 2 (z ) z )! 0} comp {Q 2 (z 1 z 2(z 1 )) 1 0}. (A38) comp The first boundary of conditions (A38) Q (z z (z )) p 0 divides [D-D] and [S-D]. The second boundary 2 2 6

7 comp 1 comp Q 2 (z 2 (z ) z ) p 0 divides [D-D] and [D-]. The third boundary Q 2 (z 1 z 2(z 1 )) p 0 divides [D-D] and [D- ]. Therefore transition between [D-D] and [ -D] is not possible. comp comp y adding Q (z z ) p 0 and Q (z z ) p 0 we have ( ) ( ) [ ( ) ] g2 1 g2 g2d(r 2) bg 2 1 g1 1 2 g1 z 1/h z 2 log log a z 1 z p 0. (A39) Specifically when equality (A39) is modified to g1 p g2 Q 2 (z 2) p 0. This implies that the depth of the biomass maximum of species 2 is unchanged by the presence of species 1 when g1 p g2 while the depth of species 1 is deepened by the presence of species 2 (inequality [A34]). Next we examine the effect of changing g i on the biomass maxima z i. f we take parameters g 1 and g 2 as variables satisfies z 1 comp Q 2 (z 1 z 2(z 1; g 1); g 2) p 0. (A40) Derivatives of z and z with respect to g 2 and g 1 are expressed by 1 2 comp z 1 ( Q 2 / g 2) comp comp p (A41) g ( Q / z ) ( Q / z )( z / z ) z z z z g z g g p ( Q / z )( z / g ) z z p (A42) ( Q / z ) ( Q / z )( z / z ) z g comp comp comp [ ] comp z 2 ( Q 2 / z 2)( z 2/ z 1) comp comp p 1 g ( Q / z ) ( Q / z )( z / z ) Easy manipulations gives the inequalities z 1! 0 g 2 z 2! 0. g 1 (A43) The above inequalities describe how vertical positions of biomass maxima are affected by changing the g of competing species. Therefore the following properties are derived for vertical positions z1 and z2: (1) z1 is deepened by the presence of species 2 ( z1 1 z1 ) (2) z2 is not affected by species 1 when g1 p g2 ( z 2 p z 2 ) (3) species 1 moves down when g 2 decreases ( z 1/ g 2! 0) and (4) species 2 moves up when g 1 increases ( z / g! 0). 2 1 Coexistence [S-D] Steady state [S-D] is expressed by 7

8 1 for 0 z z b (z) p z 1 {0 for z! z z 0 for 0 z! z 2 z 2! z z b 2(z) p { for z p z2 R for 0 z z R2 R(z) p R (z z ) for z! z! z 2 (A44) z2 z R 2 {R (z z ) for z z z { 2 bg 2 2 exp [ log ( / )(z/z )] for 0 z z (z) p exp [ a bg(z z )] for z! z! z 2. exp [ a (z z )] for z z z The biomass of each species is obtained by nutrient flux and light attenuation: p g1d(r 2) z z 1 2 ( ) bg R R2 2 2 p g2d( z z ) 2 2 ( ) bg 2 2 Solving the following equalities we obtain z and (R ): p log a z (A45) p log a (z z ). (A46) 2 H (R ) p 0 1 g1d(r 2 ) ( ) bg z2 z (A47) log a z p 0 (A48) ( ) ( ) R2 2 bg R log a (z z ) g D p 0. (A49) z z The steady state exists when and z! z 2! z and satisfies the stability criteria when R R1 and 1. From these conditions we obtain the region where [S-D] is the stable steady state. Unlike other steady states we could find simple expressions (e.g. conditions [A38]) for stability region of [S-D] only when two species have the same g. From now on we consider the case when g1 p g2 p g. For R 1 R1 is expressed as a function of R: p 1(R) such that H 1(R 1(R)) p 0. When we manipulate equation (A48) is expressed by z 2 gd(r 2 ) z 2 p z 2(R ) p z. (A50) log ( / ) abgz 8

9 Substituting and z in equation (A49) with (R) and z (R ) we have ( ) ( ) R2 1 2 bg 2 z z R log a (z z ) gd p 0. (A51) SD SD SD We define the left-hand side of equation (A51) as Q (R). Then R satisfies Q (R) p 0. A derivative of Q (R) is expressed by ( ) dq Q Q d Q z z d dr R dr R dr SD SD SD SD p 1 z 2 1 [ ] gd 1 d gd(r ) gd(r 2 ) 1 2 bg dr 2 2 p a (A52) z z ( ) (z z ) [ ] gd(r ) gd 2 d bg 1 1 bg # log ( / ) a z (log ( / ) a z ) dr When we rearrange equation (A52) the derivative is rewritten as [ ] SD dq gd(r 2) p a bg dr ( ) 2 2 { } gd(r ) gd 2 d bg 1 1 bg #. (A53) log ( / ) a z [log ( / ) a z ] dr SD SD SD ecause d 1/dR 0 we have dq /dr 1 0. Hence Q (R) p 0 has a root R [R 1 R 2) when Q (R 1) 0 and SD Q (R ) 1 0. The former inequality is identical to 2 When R p R and 1 0 z p z. Thus the latter inequality is rewritten as comp Q 2 (z z 2(z )) 0. (A54) ( ) ( ) (R ) R R2 SD Q (R 2) p log gd 1 0. (A55) 2 z 1/h z z2 z The second term of the left-hand side of inequality (A55) is equivalent to 2. Note that when 1(R 2)! 2 2 should be negative in order to satisfy inequality (A55). Using equation (A45) we modify inequality (A55) to [ ] [ ] 1(R 2) gd(r 2) bg 2 2 z 1/h z 1(R 2) log log a z p Q (z ) 1 0. (A56) As we have shown for [D-D] it is easy to see that z p z when g p g. Thus z! z when Q (z )! 0. When we have following equalities at steady state:

10 From equations (A57) and (A58) we have (z 2 z ) 2 1 R p R (A57) gd 1(R) p exp ( abgz )exp( 1). (A58) [ ] (z 2 z ) bg 1 R p exp ( a z )exp( ). (A59) gd When 1 is sufficiently small a Taylor expansion of equation (A59) gives d 1 (z 2 z ) bg 1 (R ) p exp ( a z )(1 ). (A60) dr gd f we rearrange equation (A60) 1 is expressed by exp ( abgz ) 1(R 2) 1 bg 1 2 p. (A61) exp ( a z ) (d /dr)(z z )/gd t is easy to see that the denominator of the right-hand side of equation (A61) is positive. Therefore in order for 1 0 the following inequality is satisfied: 1 exp ( abgz ) 1(R 2) 1 0. (A62) When the following inequality holds: R 2 R2 2 2 Substituting z in inequality (A63) with z (R ) we have 2 2 ( ) 1. (A63) z z g1d(r ) log a bg z! 0. (A64) Let R1 be R of the culture steady state [S] of species 1. Then the left-hand side equals 0 when R p R1. This implies R! R1 because the left-hand side increases with R. Thus when the following inequality is satisfied: f equation (A45) is applied the inequality (A64) can be also rearranged to SD Q (R 1 ) 1 0. (A65) R2 R!. (A66) z 2 z f the reciprocals of both sides of inequality (A66) are taken some rearrangement gives a bg(z 1/h z )(R2 ) a bg(z 2 z ) 1. (A67) R 10

11 From inequality (A67) and equation (A46) we have The above inequality is rewritten as a bg(z 1/h z )(R2 ) ( ) 2 2 R log 1. (A68) a bg(z 1/h z )(R2 ) ( ) 2 R log (A69) nequality (A69) describes a condition where species 2 has a positive biomass in the deep layer. To summarize conditions for stability of [S-D] when g1 p g2 can be expressed by comp {Q 2 (z z 2(z )) 0} {Q 2 (z ) 1 0} {Q 2 (z )! 0} SD { exp ( abgz ) 1(R 2) 1 0} {Q (R 1 ) 1 0}. (A70) 2 comp The first boundary Q 2 (z z 2(z )) p 0 divides [S-D] and [D-D]; the second Q 2 (z ) p 0 divides [S-D] and [ -S]; the third Q 2 (z ) p 0 divides [S-D] and [S-]; the fourth exp ( abgz ) 1(R 2) p 0 SD divides [S-D] and [ -D]; and the fifth Q (R 1 ) p 0 divides [S-D] and [S- ]. The second equality however is not one of the boundaries of [S-D] under high conditions (i.e. 1(R 2)! 2 ) because it requires 2! 0 in the neighborhood of the boundary (see inequality [A55] and the following explanation). Thus a transition between [S-D] and [ -S] is possible only under low conditions. Numerical Simulations Following the method of Huisman and Sommeijer (2002) we solved the partial differential equations (1) (5) by using the package DVODE (the double-precision version of VODE; software.html). The vertical profile of turbulent mixing is expressed by a sigmoid function D DD D(z) p DD (A71) 1 exp [a(z z )] where D D and D denote the vertical mixing coefficient in the deep and surface layers respectively; a p 20 is the gain of a sigmoid function. Parameter values used in numerical simulations can be found in table 1. For derivation of analytical results we assumed D /(vmaxz ) k 1 (well-mixed surface layer) and D D/[v max(z z )] K 1 (poorly mixed deep layer). n our numerical simulations these values are 1.67 and respectively. Therefore biomass is not homogeneously distributed in the surface layer and does not form an infinitely thin layer in the deep layer. Time to Equilibrium eckmann and Hense (2007) characterized timescales of processes involved in a similar system by using dimensional analysis and concluded that the timescale of vertical mixing is the longest in oceanic environments. 2 n single-species simulations time to equilibrium agrees with this timescale ( z /DD p 40 days). n contrast time to equilibrium can be much longer when two species compete ranging from less than 100 to more than 2000 days depending on initial conditions and parameter values. n general it takes a long time when parameter values are near a boundary of two basins of attraction. Figure A1A shows an example where it takes about 150 days until two species stably coexist in the deep layer. n figure A1 we took parameter values near the boundary of the [S- ] and [ -S] regions. As a result competitive exclusion took more than 2000 days (fig. A1). 11

12 Comparison with Analytical Results n figure A2 results of numerical simulations are superimposed onto figure 4. Steady state distributions were obtained from numerical simulations and the integrated biomasses and depths of biomass maxima of two species were compared with the corresponding analytical results. Numerical results qualitatively agreed with analytical results at steady state (fig. A2). ecause the thickness of deep-layer maxima is not infinitely thin in numerical simulations there are differences in integrated biomasses and depths of biomass maxima when species form deep-layer maxima. n contrast they are relatively consistent with analytical results when species form surfacelayer maxima though the surface layer is not perfectly mixed in numerical simulations. Descending Thermocline Now we examine the effect of deepening z for competition of two species. We used the following parameter values: p 1000 a bg p 0.2 and R p 120. We first ran a numerical simulation by setting z p 0. After we obtained steady state distributions we let z increase by 0.1 m day 1 from t p 0 to 200 d. Although the dynamics never reached steady state trends in biomass increases/decreases were consistent with the corresponding stable steady state (fig. A3). Species 1 moved up to the surface layer as the stable steady state shifted from [D-D] to [S-D] with deepening z. The biomass of species 2 decreased and increased afterward while the biomass of species 1 showed the opposite trends following the shifts from [S-D] to [S- ] and to [ - S]. This numerical simulation indicates that our theoretical predictions for steady state can partly explain seasonal dynamics. Figure A1: Time course of vertically integrated biomasses multiplied by the attenuation coefficient (top) and depths of biomass maxima scaled by z (bottom) of two species. Solid lines indicate species 1 (a good nutrient competitor) and dashed lines indicate species 2 (a good light competitor). Depth of the mixed layer z /z is indicated by the thick dotted line. n A R mg P m 3 andin mg P m 3 p 130 R p 188 ; z p 6 m p 1000 mmol m 2 s 1 anda m 1 bg p 0. Other values are found in table 1. nitial conditions are R(z) { R and 6 b cells m 3 (p1 cell ml 1 i(z) { 10 ). These conditions allow species 2 to grow first with sufficient nutrients suppressing the growth of species 1 by light limitation. Time to equilibrium varies depending on initial conditions and parameter ranging from!100 to days. 12

13 Figure A2: Numerical simulation results superimposed onto figure 4. n A mmol m 2 s 1 p 1000 andin mmol m 2 s 1 ; m 1 p 200 a bg p 0 and z p 6 m. We ran a numerical simulation for each R p mgpm 3 and plotted vertically integrated biomasses (top) and depths of biomass maxima (middle). Filled circles indicate species 1 and open circles indicate species 2. ottom vertical distributions of species 1 (solid lines) and species 2 (dashed lines) for selected R values. 13

14 Figure A3: Time course of vertically integrated biomasses multiplied by the attenuation coefficient (top) depths of biomass maxima scaled by z (middle) and vertical distributions at t p and 200 days of two species. Solid lines indicate species 1 and dashed lines indicate species 2. The thick dotted line indicates the depth of the mixed layer ( z ). We first ran a simulation with parameter values m mmol m 2 s 1 /z z p 0 p 1000 a m 1 and mgpm 3 bg p 0.2 R p 120 until we obtained stable distributions. We then increased z by 0.1 m day 1 from t p 0 to 200 days. The corresponding stable steady states are shown at the top. Open inverted triangles indicate points where the stable steady state shifts. 14