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. Limnol. Oceanogr., 39(3), 1994, 597-608 0 1994, by the American Society of Limnology and Oceanography, Inc. Grazing limitation and nutrient limitation in marine ecosystems: Steady state solutions of an ecosystem model with multiple food chains Robert A. Armstrong Program in Atmospheric Jersey 08544-07 10 and Oceanic Sciences, Sayre Hall, P.O. Box CN7 10, Princeton University, Princeton, New Abstract Here I develop a model with multiple phytoplankton-zooplankton food chains where each food chain is based on a different size class of algae. Model parameters were chosen to allow food chains based on smaller algal size classes to dominate under oligotrophic conditions, with larger size classes being added sequentially with increased nutrient loading. The model makes use of allometric relations among size classes to minimize the number of free parameters, facilitating numerical exploration of its steady state behavior. The model is simple, yet it is complex enough to allow simultaneous predator limitation of algal size classes (or species) and nutrient limitation of total phytoplankton biomass. This perspective, not obtainable from models with single phytoplankton and zooplankton size classes, is applied to the particular case of elevated nitrate levels in the equatorial Pacific. The model shows clearly that it is possible for each phytoplankton size class to be limited by its herbivores, while at the same time micronutrient (notably iron) dcliciency may limit the number of size classes that can exist in the community and hence the total phytoplankton biomass that can be supported. Empirical evidence and theoretical argument suggest that a common ecosystem structure is one wherein the density of each plant species is limited by its herbivores while total plant biomass is limited by the availability of space or nutrients (Hairston et al. 1960; Power 1992; Strong 1992). Marine ecosystem models containing a single “phytoplankton” variable and a single “zooplankton” variable cannot easily convey this distinction, however, since the phytoplankton variable in such models simultaneously represents a single phytoplankton species and the entire phytoplankton trophic level, confounding the limitation of species’ population sizes with the limitation of total algal abundance. The inability of models with single phytoplankton (P) and single zooplankton (Z) variables (1 P 1Z models) to convey this distinction limits their utility in exploring important empirical questions. Consider, for example, the discussion by Frost and Franzen (1992) of simultaneous nutrient limitation and grazing control of phyAcknowledgments I thank Steve Carson, Jorge Sarmiento, and Robbie Toggweiler for helpful suggestions on the manuscript. This research was funded in part by grant NA26RGO 10201 from the National Oceanic and Atmospheric Administration. Additional support was provided by NASA grant NAGW-3 137 and NSF grant OCE 90- 12333 to J. L. Sarmiento. 597 toplankton standing stock and growth rates in the equatorial Pacific. Frost and Franzen rely on a simple model of pelagic food chain to observe that nutrient limitation acting alone would result in algal standing stocks that are too high and algal growth rates that are too low. Then they rely on verbal arguments to further conclude that limitation by micronutrients may be necessary to account for growth rates that seem low by a factor of two (Cullen et al. 1992) as well as to account for the absence of large algal species, which are thought to be more sensitive to low micronutrient levels than species smaller are (Hudson and Morel 1990; Morel et al. 1991). This reliance on verbal arguments to account for the absence of large phytoplankton size classes underscores the incompleteness of their 1P 1Z model: it would be more satisfying, and more instructive, if the model were capable of portraying not only the simultaneous effects of herbivory and nutrient limitation on resident phytoplankton populations but could also account naturally for the absence of larger size classes. Here, I develop a more complete model that allows the interaction of nutrient control and grazing control to be examined more directly. This model is a simplified version of the model of Moloney and Field (199 1). The model consists of multiple food chains, each based on a different size class of algae. The parameters of 598 Armstrong each size class are related to those of other size classes by allometric relations, imparting a unified structure to the model and limiting the number of free parameters. Numerical steady state solutions of this model show that as nutrient loading is increased, larger algae and zooplankton size classes are added sequentially until virtually all limiting nutrient is exhausted or until no larger algal size class can grow fast enough to invade the existing community. In this context, the problem of high ambient nitrate levels in the equatorial Pacific is reduced to the question: What keeps larger algal size classes from invading? (Miller et al. 199 1). On a broader scale, the model also shows that while predators may limit the population sizes of their respective prey, the total biomass of the system and its productivity may ultimately be set by the number of food chains that can be supported at a given nutrient loading (Hairston et al. 1960; Power 1992; Strong 1992). A model with multiple food chains In a review of the effects of size on algal physiology and phytoplankton ecology, Chisholm (1992, p. 222) noted that the total amount of chlorophyll in each of several successively larger size fractions seems to have an upper limit and that “beyond certain thresholds, chlorophyll can only be added by adding a larger size class of cells.” Raimbault et al. (1988), for example, noted that there appears to be a maximum of roughly 0.8 mg Chl a m-3 in the l-3-pm phytoplankton size class and 0.7 mg Chl a m-3 in the 3-lo-pm size class. These size classes are of approximately equal width on a logarithmic scale, suggesting roughly equal maximum chlorophyll per logarithmic increment in size. This pattern can be produced quite naturally by models with multiple food chains, where the steady state phytoplankton concentration in each size class is set by the herbivorous predator on that size class and where parameters are chosen to produce equal phytoplankton biomass in each logarithmic size interval. Further, to allow food chains based on smaller phytoplankton to dominate under oligotrophic conditions, with chains based on larger phytoplankton appearing only at increased nutrient loadings, chains based on small phytoplankton and zooplankton must be able to invade at lower nutrient concentrations than can chains based on larger size classes of algae. These simple requirements define the structure of the model developed below. Independent food chains -The simplest model containing multiple food chains is one in which the food chains are parallel and independent (Fig. la). In the most general case, each food chain could include its own triplet (phytoplankton, zooplankton, detritus), with appropriate growth, mortality, remineralization, and export rates. However, for clarity of presentation, I suppress the role of detritus and assume instant remineralization of dead organic matter. This model (Fig. la) is defined by the equations dPi = PiPi - XiPi - ZiHi, dt (la) dZ. 2 = YiZiHi dt (lb) - GiZi, and S=T-CPi-CZi (2) for food chains i = 0, 1, 2, . . . , n - 1. Here phytoplankton biomass Pi, zooplankton biomass Zi, free nutrient (substrate) concentration S, and total nutrient concentration T are all measured in terms of their nutrient (here nitrogen) equivalents (see notation list). Furthermore, in these equations pi is the growth rate of phytoplankton size class i as a function of substrate concentration S; Xi is a background loss rate of this phytoplankton size class due to predation by generalized herbivores or other causes; Hi is the harvest rate (per unit zooplankton) of phytoplankton size class i by the zooplankton size class that feeds on it; yi is the assimilation efficiency; and 6i is the death rate (including excretory losses) of this zooplankton size class. Nutrient is assumed to be instantaneously remineralized upon phytoplankton loss or zooplankton death so that total nutrient is conserved. For purposes of numerical investigation, the growth and harvest functions are specified to have the following forms: CL,= Pnx3x.iS/(Ks,i + AS); W Multiple The Monod function (Eq. 3a), with maximum growth rate pmax,iand half-saturation constant Ks,i, is a standard choice for describing algal growth (e.g. DeAngelis 1992). However, the piecewise-linear form (Eq. 3b) chosen for the harvest function is not standard; it was chosen. over the Monod form both because it simplifies the algebra (making the model considerably easier to solve numerically) and because it yields a stable predator-prey relationship in the single-chain case (see Armstrong 1976; Wolkowicz 1989) without the necessity of adding a feeding threshold with its attendant additional parameter (cf. Frost and Franzen 1992). Note particularly that the parameter Kp,i in Eq. 3b is the phytoplankton concentration at which zooplankton feeding is fully saturated, not half-saturated. For numerical investigation, we substitute Eq. 3 into Eq. 1, yielding (W Note that only the ascending limb of Eq. 3b has been used, since unique steady state values of the Pi can exist only for YiHmax,i 1 6i and Pi/ Kp,i < 1. The parameters of each food chain are next specified by means of allometric (power law) relations among food chains based on different size classes of algae (Moloney and Field 1989, 199 1). In particular, rate constant Ri for the ith food chain is related to the corresponding rate constant R,, for a reference food chain by Ri = Ro(LiILo)‘R (5) where, without loss of generality, we have taken the reference chain to be the smallest food chain, i = 0. In Eq. 5, PR is the allometric constant for rate constant R, and Li and Lo are the characteristic lengths of phytoplankton at the bases of food chain i and reference food chain 0, respectively. It is important to understand how parameter values influence steady state phytoplankton and zooplankton concentrations. Setting dPi/dt = 599 food chains zo Zl 25.. PO PI P2 . . . zo - Zl - 22 . . . PO Pi P2= mm (a) w Fig. 1. Food-web structures. [a.] A food web with multiplc indepcndcnt food chains. Each zooplankton size class (Z”, 5, -G * * .) can only feed on a single size class of phytoplankton (P,,, P,, Pz, . . .). [b.] A food web where larger zooplankton size classes can also prey on smaller zooplankton size classes. 0 and Zi = 0 in Eq. 4a, we calculate the minimum substrate concentrations Smin,i at which phytoplankton size class i can exist in the community in the absence of its herbivorous prcdator as s.,= KSi m1n’2 (/dmax,i/Xi) - 1 * (6) The allometric constants associated with the three parameters pmax,Ks,i, and Xi are then chosen SO that Smini increases with algal size, allowing smaller size classes to invade at lower nutrient concentrations than are required for the invasion of larger size classes. Next, we set dZildt = 0 in zooplankton Eq. 4b to determine steady state phytoplankton concentrations Pi”, which represent both the minimum concentrations of phytoplankton in size class i that are necessary for zooplankton size class i to invade and the steady state phytoplankton concentrations once invasion has occurred. These relationships are defined by pi” - ‘iKPPi Yi (7) Hmax,i and are independent of zooplankton densities Zi”. Finally, the parameters of phytoplankton 600 Armstrong Eq. 4a determine steady state zooplankton concentrations (when zooplankton are present) as zi* = dKPi H max,i S - Xi m (8) pmaxdKs,i + s > Consider now the simplest case (the case explored by Frost and Franzen 1992), where only a single food chain (i = 0) is present. These equations generated a simple pattern that has three parts (e.g. see DeAngelis 1992). First, when T is less than the critical value Smin,O(Eq. 6) for phytoplankton invasion, neither phytoplankton nor zooplankton will be present, and S = T. In the range Smin,O< T < Smin,O+ PO* (Eq. 6 and 7), the steady state value of S is pinned at Smin,O,and phytoplankton concentration PO steadily rises as T - Smin,O.In this range, the phytoplankton concentration is less than that required for zooplankton to invade, and Z,* = 0. Above T = Smin,O+ PO*, however, zooplankton are present, their grazing fixes the phytoplankton density at PO*, and S* and Z,* rise together (Eq. 8) with increasing total nutrient concentration T. Finally, as the algal growth rate approaches pmax,O,Z. approaches its maximum value (Eq. 8), and all additional increase in T goes to free nutrient S. Next, consider a case with multiple food chains (Fig. 2). Assume that the Smin,i (Eq. 6) increases with i. Then as nutrient input is increased, phytoplankton density in the smallest size class will increase until PO* is reached, when the first predator (Zo) will invade and present further increase. At this point, Zo* and nutrient concentration S both start to rise until of the S min, 1 is reached, where phytoplankton next larger size class P, start to increase in density until they in turn arc limited by their predator. Phytoplankton and zooplankton size classes are then added sequentially in pairs in order of size, and the contribution of each size class to biomass is constrained in accordance with Chisholm’s (1992) model. A generalization: The food-web model of Moloney and Field-The model developed above can be generalized by allowing each zooplankton to graze on multiple zooplankton size classes as well as on multiple phytoplankton size classes. Moloney and Field (199 1) proposed such a general model but restricted their attention to the situation in which each zoo- plankton size class grazes only a single size class of phytoplankton (as in the model of the previous section) and also grazes only the next smaller zooplankton size class (Fig. 1b). In this generalized model, Eq. 1 for phytoplankton and zooplankton must be replaced by dP, pi L = PiPi - XiPi - Zi Hi, dt Pi + 4Zi-1 (94 and dZ,- - TiZiHi - SiZi - Zi+I dt 4zi H, Pi+1 + +Zi ‘+l’ Pb) In this equation, the parameter 4 characterizes the relative efficiencies of grazing on zooplankton and phytoplankton; in particular, for 6 = 0, the model reduces to the independent food chains model of the previous section. For numerical investigation, the harvest function Hi is defined in analogy with Eq. 3b as Hi = Hmax,i(Pi + @Zi- 1 YKp,i, i H maxJ7 Pi + +Zi- 1 5 Pi + $Zi-, Substituting Kp,i > Kp,i. (10) Eq. 3a and 10 into 9 yields dP. S L = Pi p-L,a,iv Ks,i f S dt (1 la> and dZ. (= dt H zi yiL ,““y i (Pi + @Zi-1) f,l - 6i - Zi+,$y H . (llb) P,r+l Parameter values-The pattern of invasion of phytoplankton and zooplankton size classes is affected both qualitatively and quantitatively by the choice of model parameters. To illustrate the range of possible behaviors, I will use representative parameter values from Moloney and Field (1989, 199 I), Chisholm (1992), Ducklow and Fasham (1992), and Frost and Franzen (1992). Parameter definitions and their units are summarized in the list of notation. Multiple 601 food chains 10 IO 8. 8 6 6 (w '4 4 2=22 22z2 ...-,& f .*’ 22 ...-.. 22122= *......-#Z . . . . y;,......... -mm $,.. ,?fff~~::;::::::::::::::::::::: 22. .;PPPPPPPPPPPPPP Zr::.ppPPpPPbBb~iiiie............... 2 2 .......................................... .............................................. tn 2 8 0 0 0 2 4 6 8 -IO . zg+~PPQ pP$P . . . . . . . . . ... .. ... ... .. ,I’ ... ... .. ... ... .. 0 2 .. .. .. .. . . . ..a...................... ..*......................... ..... ... ......... ... ... .... . ..... ... ......... ... ... .... . 4 6 8 IO a, 10 TTT TTTT TT 8 TTTT No TT TTTT 6 TT TT’ TT TT TT 2222222222222222222222222222222 *.,..............m........... z#fff~. *,............*......,............ * .8 0 6 8 TTTT I4 I2 I 10 4 TT TT F. .-* $ 3 pPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP . .. . .. . . . . .. . .. . .. . . . .. . . . . . . . .. . .. . . . . . #PPP . . . . . . . ..mm....m.............................. 2 TT 10 I I TT TT TT ~222222222222222222222222222222222222222 ,,?Jf # pPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPPP .. . . . . . . . .. . . . .. . .. . . .. . .. . . . . . .. . .. .. . .. .. . 7 0 Total Nitrogen [mmol 2 4 6 8 IO m”) Fig. 2. Plots showing nutrient concentrations in each phytoplankton and zooplankton size class for the case of multiple independent food chains (Fig. la). The row of P’s in each panel denotes the nutrient in total phytoplankton (on the ordinate) vs. total nutrient concentration (on the abscissa).! The dotted lines beneath the P rows then denote the cumulative concentrations of nutrient in successive phytoplankto ’ size classes. For example, the distance between zero 7 (on the ordinate) and the first dotted line (at 0.6 mmol N m-3) represents the nutrient in the smallest algal size class; the distance between this line and the second dotted line (at 1.2 mmol N m-3) represents the nutrient (also 0.6 mmol N m-3) in the second size class; and so forth. Next, the distance between the P row on each plot and the row of Z’s on the same plot represents the total nutrient in zooplankton; the distances between successive dotted lines again represent the amount of nutrient in individual zooplankton size classes. Finally, the distance between a Z row and a diagonal row of T’s (total nutrient) represents ambient nutrient concentration. [a.] The first standard case (p,,,,, = 1.4 standard case (p,,,,, = 1.4 d-l; BP,,. = -0.4). [c.] The first (“ligand d- ‘; Plh,, = -0.75). [b.] The second (“diatom”) limited”) micronutrient case (pm,, = 0.7 d I; ppmoX = - 1.0). [d.] The second (“diffusion limited”) micronutrient case b4n,x = 0.7 d-‘; PPmor= -2.0). We begin by dividing the phytoplankton size spectrum into discrete size classes. Following Moloney and Field (1991), we define phytoplankton size classes that are of equal width on a logarithmic scale. I found it most convenient to define size classes so that the nominal size of algae in adjacent size classes differs by a factor of 4. If we then choose the upper limit of the smallest size class to be 2 pm, upper limits on size classes will be 2 pm, 8 pm, 32 ,um, and so forth, and the corresponding nom- inal algal sizes in these classes will be 1 pm, 4 ,um, 16 I.cm, etc. Note next that the data of Raimbault et al. (1988) show that the picophytoplankton (< lpm diam) have a maximum of 0.5 mg Chl a m-3 while the size fraction < 10 pm in diameter has a maximum of 2.0 mg Chl a m-3, implying a concentration of 1.5 mg Chl a m-3 in this lo-fold range of diameters. Size classes whose boundaries differ by a factor of 4 should therefore contain roughly 1.5 x log,,4 z 0.9 602 Armstrong Notation PO z, S, T :;y-’ A, Smin.r H max,, KP,, YI iI7 Phytoplankton and zooplankton concentrations, mmol N m-3 Ambient and total nutrient concentrations, mmolNm 3 Maximum phytoplankton growth rates, d I Half-saturation nutrient concentrations for phytoplankton growth, mmol N m-’ Intrinsic phytoplankton loss rates, d-l Minimum ambient nutrient concentrations for invasion of phytoplankton size classes, mmol N m-3 Maximum zooplankton harvest rates, mmol P (mmol a-’ d-l Zooplankton full-saturation constants for harvest and growth, mmol P m-’ Zooplankton growth efficiencies, dimensionless Intrinsic zooplankton loss rates, d -I Allometric constant for rate process or saturation constant R, dimensionless mg Chl a m-3. Assuming that there is a maximum of roughly 1.6 mg Chl a to 1 mmol N (Frost and Franzen 1992), each size class should contain 0.6 mmol N m-3 at steady state. Zooplankton parameter values for the reference chain were set to make Pi* = 0.6 mmol N me3 for all i in the case of independent food chains (4 = 0). Values for the maximum harvest rate of the smallest food chain HmaX,O= 1.4 d-l) and for growth efficiency (yO = 0.4) were taken from Frost and Franzen (1992). However, the half-saturation constant for zooplankton feeding (0.2 mmol N m-3) used by Frost and Franzen (1992) was much lower than values obtained from other sources; indeed, it was too small to allow Pi” = 0.6 mmol N m-3 with the functional response used here (Eq. 3b and 9b), since the piecewise-linear form requires Pi < Kp,i at steady state. I therefore adopted the higher half-saturation value, 1.0 mmol N mB3 (used by Ducklow and Fasham 1992 for both their size classes of zooplankton), doubling it to Kp,i = 2.0 mmol N m-3 for use as the full-saturation value in the piecewise-linear functional response. These values imply a combined death-excretion rate of 6, = 0.168 d-l to achieve the desired value of Pi*. Allometric coefficients for growth and respiration were taken from Moloney and Field (1989, 199 l), multiplied by 3 to convert from mass to diameter (under the assumption that mass is proportional to the cube of the diameter); the values chosen were PNmax= -0.75 and & = - 0.75. Finally, I assumed that assimilation efficiencies yi and feeding saturation constants Kp,i are constant with size, implying in the case of independent food chains (4 = 0) that the Pi” will be equal for all i. Phytoplankton parameters were chosen next. Maximum phytoplankton growth rate and its allometric coefficient were allowed to vary across cases. Following Frost and Franzen (19% 1 set pmax,o= 1.4 d-l in the standard case; and following Moloney and Field (199 1) and Chisholm (1992), I set pW,,, = -0.75 in this case also. However, Chisholm (1992, p. 222) also notes that when additional algal size classes are added, “Diatoms play a particularly important role in this ‘additional’ chlorophyll.” Since maximum algal growth rate declines less fast within taxa than across taxa, I also considered a second “diatom” standard case where pmaX,O= 1.4 d-l, but also where -0.4, the value given by Chisholm PGmax (1992; for diatoms; this case represents a situation in which larger algal size classes contain mostly diatoms. Two cases representing micronutrient (e.g. iron) limitation were also considered. In both of these cases, pmax was reduced to 0.7 d-’ (where the reduction represents a nutrient stress on even the smallest size classes: see Cullen et al. 1992; Frost and Franzen 1992) while Phoaxwas taken alternately to be - 1 (representing a case where micronutrient uptake is limited by ligand density on the cell surface: Morel et al. 199 1) or -2 (for uptake limited ,by diffusion to the ccl1 surface: Morel et al. 199 1). In all cases, the half-saturation constant for nutrient uptake Ks,i was taken to be 0.1 mmol N m-3, an appropriate value for small (“5 pm) oceanic species from oligotrophic waters (table 2 of Epplcy et al. 1969). Finally, 1 assumed a constant background loss rate Xi of 0.0 16 d-l and applied it equally to all size classes. Although Moloney and Field (199 1) suggested a higher sinking rate, and hence a higher loss rate, for larger size classes (see also Michaels and Silver 1988), such an assumption would be inappropriate in the present context, where nutrients are assumed to be instantly regenerated. The constancy of Xi and K,i across phytoplankton size classes implies that minimum nutrient concentrations Smin,l required for invasion of algal size class i in the absence of Multiple predation depend only on maximum phytoplankton growth rates ,umaxi (Eq. 6). In particular, ifk,ax,i declines with size, then Smin,imust increase with size, so that smaller size classes can invade at lower nutrient levels as required. Although one could include size dependence of Xi (Moloney and Field 199 1) and K,i (Eppley et al. 1969; Aksnes and Egge 1991; Maloney and Field 199 1) to produce more complicated patterns, the requirement that Smin,i values increase with size dictates that these patterns will not be qualitatively new. It may be desirable to allow these parameters to vary for quantitative comparison with data, but constancy will suffice for present purposes. 603 food chains Table 1. The four cases explored in the presentstudy are each characterized by a maximum algal growth rate for the standard chain P,,,,,,~ (d-l) and by the allometric coefficient pp,., for this growth rate (dimensionless). For each case, the minimum free nutrient concentration Sm1n.i (mmol N m-3) needed for invasion of size class i at an intrinsic loss rate of A, = 0.016 d-l is listed next to the nominal size of individuals in that size class. Size classes whose entries are dashes have values of Sn1in.rthat exceed 10 mmol N m-3; these classes cannot invade in the current study, since free nutrient S is always less than total nutrient 7’ I 10 mmol N m- J. Case a &nax,o PLhox 1.4 -0.75 Nominal size (pm) I 0.0012 4 0.0033 16 0.0101 64 0.0349 256 0.2723 1,024 4,096 - Case b 1.4 -0.4 0.0012 Case c 0.7 -1.0 0.0023 Case d 0.7 -2.0 0.0023 0.0577 0.0020 0.0101 Results 0.0036 0.0577 For each of the two standard cases and two 0.0064 micronutrient-limited cases and for values 6 0.0117 0.0224 = 0 and 4 = 1, the model equations were solved 0.0467 numerically over a range of total nutrient concentrations T I 10 mmol N m-3-the value used by Frost and Franzen (1992) for their subsurface nutrient concentration. (A C-lanclasses can invade (though only the first seven guage program for solving these equations is are included in Table 1). In this case, ambient available on request.) nutrient concentration at the highest nutrient Minimum substrate concentrations Smin,i loadings is low enough (0.03 mmol N m-3) at necessary for each phytoplankton size class to T = 10 mmol N m-3 that the seventh algal invade at a loss rate of 0.0 16 d- l in the absence size class (Smin,T= 0.0467 mmol Nm-3) cannot of herbivory were computed with Eq. 6 for invade, and only six of the nine possible food each of the four cases; these values are listed chains are present. in Table 1 beside the nominal size of each size The two micronutrient-limited cases (Fig. class. Differences in the numbers of phyto2c,d) contrast sharply with the standard cases. plankton size classes that can invade in each The number of chains that can invade is limcase (Table 1) lead to dramatic differences in ited to three and two chains, respectively. Betotal phytoplankton density and ambient nu- cause the number of phytoplankton size trient concentration among the four cases. classes than can invade is limited, total bioIndependent food chains- Consider first the mass cannot increase enough to reduce nutricase with independent food chains (4 = 0) (Fig. ent levels to near-zero levels, and they remain la). In the first standard case (P,,.,~~,~ = 1.4 d-l; high in both simulations (6.3 and 7.7 mmol N m-3, respectively) when T = 10 mmol N rnm3. Pknnx= - 0.75; Fig. 2a), five chains can potentially invade (Table l), and all five that can Connected food web -The parameter values invade do invade. At T = 10 mmol N mW3, used in the previous section were chosen in each extant phytoplankton size class contains part to produce the Chisholm (1992) pattern the same amount of nutrient (0.6 mmol N m-3) in the independent chains model (Fig. la). per model design, accounting together for 3.0 However, a food web comprised of multiple mmol N m-3, and the zooplankton taken toindependent food chains is empirically ungether account for 6.4 mmol N me3, leaving realistic, so it is of considerable interest to see 0.6 mmol N me3 as free nutrient. In the second how much or how little the model and its pa(“diatom”) standard case (pmaX,o= 1.4 d-l; rameters must be changed to produce this patP+Mx = -0.4; Fig. 2b), algal growth rate de- tern in a connected food web (Fig. lb). I thereclines more slowly with size than in the first fore also investigated a food-web model with standard case; here the first nine algal size $ = 1, where phytoplankton and smaller zoo- 604 Armstrong plankton are equally available to their zooplankton predators. When the food chains are connected by the grazing of larger zooplankton on smaller zooplankton, this grazing becomes an additional source of mortality for the smaller zooplankton which in turn increases the steady state phytoplankton concentrations of the smallest algal size classes. To offset this tendency and retain the values Pi* E 0.6 mmol N rnm3, the values 6i of the intrinsic death-respiration terms must be decreased. Since zooplankton will now have two food sources of roughly equal abundance, the first change I made was to halve the intrinsic death-respiration rate to 6, = 0.084 d-l. This change allows zooplankton to exist at lower total nutrient loadings (T - 0.3 mmol N m-3 rather than 0.6 mmol N mV3), which in turn allows coexistence of phytoplankton and zooplankton at lower total nutrient loadings. The resulting system exhibits undesirable behavior. Since little nutrient is lost at each zooplankton size class, the number of zooplankton size classes can become very large; use of the lowered mortality (6, = 0.084 d-l) in combination with parameters from the first standard case, for example, produced a system with two phytoplankton size classes and 11 zooplankton size classes at T = 10 mmol N me3. To eliminate this behavior we must make the zooplankton death-respiration rate fall off less rapidly with size than does the harvest rate. To accomplish this change, we separate the zooplankton death-respiration term into two parts, one part representing respiration (retaining its allometric coefficient of -0.75) and a second part representing background mortality (with a separate allometric coefficient). The smallest zooplankton size classes are so small that they are effectively immobile, so that if size-independent predation by gelatinous predators or larger zooplankton is the cause of the background mortality on the phytoplankton, it would seem reasonable to make zooplankton subject to this same background mortality. We therefore assign the same background mortality to the smallest zooplankton classes as to the phytoplankton (0.0 16 d-l) and assign the residual part of the death-respiration term (0.068 d- l) to respiration. We could leave the allometric coefficient for the death term at zero, as for the phytoplankton, or we could let this coefficient decline with size, reflecting an increased ability of larger zooplankton to escape this generalized predation. An allometric coefficient for the zooplankton death term of -0.4 was used to produce the results shown in Fig. 3. The resulting model produces patterns that are superior in many ways to those from the independent chains model. First note that in the standard cases, the model no longer automatically produces equal phytoplankton densities in each phytoplankton size class (Fig. 3a), although with careful selection of parameter values, rough equality can be obtained (Fig. 3b). Next note that for 4 > 0 the number of zooplankton size classes may exceed the number of phytoplankton size classes, since the ability of zooplankton to use smaller zooplankton when Cp> 0 allows new trophic levels to be added without adding supporting phytoplankton size classes; this behavior is clearly impossible at 4 = 0 and adds realistic new possibilities to the model. This behavior is most evident in the micronutrient cases (Fig. 3c,d), where both cases converge to a food chain with a single phytoplankton size class and five zooplankton size classes. Third, the ability of zooplankton size classes to use smaller zooplankton as food when 4 > 0 causes phytoplankton size classes to “wink” in and out at different nutrient densities. (Note in Fig. 3b the appearance of phytoplankton size class 5 at T = 2.9 mmol N m-3, its disappearance at T = 4.4 mmol N m-3, and its reappearance at T = 6.4 mmol N m-3.) As T is increased, algal productivity increases, leading to a higher concentration of small zooplankton. This increased zooplankton concentration in turn allows larger zooplankton to survive at steady state on lower concentrations of phytoplankton. When small zooplankton become abundant enough, larger zooplankton are able to survive without their corresponding phytoplankton size classes, and the latter disappear from the steady state solution. Finally, total phytoplankton biomass rises more smoothly (less stepwise) when 4 = 1. The connected food-web model produces the Chisholm (1992) pattern with low (0.03 mmol N md3) residual nitrate at T = 10 mmol N rnH3 in a reasonable standard case (Fig. 3b) while producing high (5.3 mmol N m-3) re- Multiple 605 food chains 10 8 -8 (b) T E 0 E 6 6 4 4 2 ,$f $T’ gz . . ,. *z= .* . zzz= ..* ..-- . ..’ . . . z~?+/.....~ ..a.9 ,...*- ...*. ..--. . . ZZ . . . . . .. -. ZI ..-*.---. ...’ zz,...‘m’ *........z. z* ,...‘:.*#,..’/p~~~PPPPPPPPPP ,,s;~~~~~~-~~~~~~~~~~~~~~~~~~::~::::::::::; 2 _E _ . . ..*.*......................... .. .. . . . . . . . . .. . . . .. . . . .. .. . . . . . . ,&y,::....... &......,....“..’ 0 0 2 4 6 8 10 10 8. -TTTTT TT (d) 8. TT TT 6. 0 2 4 6 8 10 -TT 0 2 4 TT 6 T” TT TT TT 8 10 Total Nitrogen (mmol mm3) Fig. 3. As Fig. 2, but for a case (4 = 1) where larger zooplankton classes (Fig. lb). sidual nitrate at the same nutrient loading in both micronutrient cases (Fig. 3c,d); this is especially reassuring since the changes that were needed to allow this to happen (splitting the zooplankton respiration-mortality term into components and identifying the background mortality term on the smallest zooplankton size class with the loss rate of phytoplankton) increases model realism rather than decreasing it. Discussion Ecosystems are highly structured. Two critical dimensions of pelagic ecosystem structure are size (Moloney and Field 199 1; Chisholm 1992) and nutritional requirements (e.g. silicon for diatoms). Yet often it is supposed that the essence of ecosystem function can be captured by extremely simple models in which the size classes can also eat smaller zooplankton size entire grazing system is represented by a single predator-prey pair (e.g. Evans and Parslow 1985; Fasham et al. 1990; Frost and Franzen 1992), by a single predator on multiple prey (Armstrong 1979; Evans 1988), by a linear food chain (Thingstad and Sakshaug 1990), or by two size classes of predators and prey interacting in a way not easily generalized to multiple size classes (Taylor and Joint 1990; Ducklow and Fasham 1992). When more complete ecosystem structures involving multiple food chains are proposed, they are often so complicated that their behaviors can only be explored in specialized symmetric cases (Armstrong 1982, 1983) or by simulation (Moloney and Field 1991). Here, I proposed a model that includes asymmetric interactions among multiple food chains, yet is structured in a way that mini- 606 Armstrong mizes the number of free parameters, facilitating numerical exploration of its steady state behaviors. The model does not include explicit consideration of detritus or other parts of the decomposition system. This omission limits its use as a quantitative model, since typically a large part of the nutrient in the system will be either sequestered in the decomposition system or exported as detritus and other components, where export itself may have a strong size dependence (Michaels and Silver 1988). The present model would, however, be particularly useful in driving a decomposition model with multiple detrital size classes derived from the multiple size classes of phytoplankton and zooplankton. Models with multiple size classes provide a natural way to capture the essential difference in phytoplankton size between regions dominated by small cells (domains 1 and 2 of Banse 1992) and regions having at least seasonal blooms of much larger cells (domain 3 of Banse 1992). In domain 1 (subtropical gyres), nutrients are scarce, limiting the number of food chains that can be supported. In domain 2 (the North Pacific, equatorial Pacific, and Southern Ocean), macronutrient (e.g. nitrate) concentrations are high, but phytoplankton abundances are low. In this region, the model suggests either that background mortality (Xi) on larger size classes must be large (e.g. because of generalized grazing by unselective predators, Banse 1992) or that low maximum growth rates (p.,,.J of larger cells in this domain limit the extent to which they can invade; low pm,,,i may in turn be due to micronutrient-imposed limits on fundamental growth rate P,~~,~,to an increased rate of decrease of growth rate p@,,, with size, or both. Finally, domain 3 is characterized by abundant nutrient delivery that is virtually all used during the growing season; here, multiple size classes can survive at steady state or under bloom conditions. The fundamental characteristics of these regions cannot be captured easily in a model without size structure, since in one-plankton-one-zooplankton (1PlZ) models, steady state phytoplankton density is completely determined by the grazing characteristics of the (single) zooplankton size class and is independent of nutrient loading. In contrast, the present model is well suited to capturing the essential features of all three domains. A particularly important difference between 1P 1Z models and models based on multiple food chains is that 1P 1Z models with constant mortality predict that ambient nutrient concentrations must reach high levels at high nutrient loadings, while multiple-phytoplankton multiple-zooplankton models predict that ambient nutrient concentrations will rise appreciably only when the next available size class of phytoplankton cannot grow fast enough to invade the existing community. Ultimately, therefore, high ambient nutrient concentrations during the growing season must be due to low growth rates, high background death rates, or high half-saturation constants of the next higher (missing) phytoplankton size class. This perspective offers insight into the hypothesis (Martin et al. 199 1) that micronutrient (notably iron) limitation is responsible for elevated nitrate levels in the equatorial Pacific. The resident phytoplankton in these regions are small (Chavez 1989; Banse 1992; Frost and Franzen 1992 for review), suggesting that the larger size classes that would normally use the nitrate are missing for some reason. It is possible, of course, that the (missing) “next larger” phytoplankton size classes in these systems are missing due to abnormally high background death rates (Miller et al. 199 1) or because the half-saturation constants for nitrate uptake in the equatorial Pacific are abnormally high or increase more rapidly with size than do those elsewhere in the world ocean. However, the appearance of larger size classes of algae upon addition of iron in “grow-out” experiments (Price et al. 199 1; Frost and Franzen 1992 for review) in which cultures are allowed to incubate for several days argues strongly for micronutrient limitation of maximum growth rates of larger size classes. I have modeled the action of micronutrient (iron) limitation as a generalized debilitating effect on algal growth (Harrison and Morel 1986; Greene et al. 199 I), and I have modeled its greater effect on larger algae as being caused by transport limitation of micronutrient uptake (Morel et al. 1991). Additional complications arise when the distinct roles of different nitrogenous substrates are recognized. Many algae appear to prefer ammonium as a growth substrate (Dortch 1990), and iron is needed more comfor the enzyme nitrate reductase. plete description of the effect of size on growth A Multiple rate might therefore need to include a general debilitating effect of iron limitation, competition for scarce supplies of ammonium (where smaller algae are expected to be superior competitors: Hudson and Morel 1990), and a combination of effects on uptake and nitrate reduction that tends to make nitrate relatively unusable, again disproportionately so for larger cells. Because each of these mechanisms might have its own characteristic allometric constant, the total response of algal growth rate to size might be considerably more complicated that the situation modeled above. From the perspective of ecosystem modeling, the tight constraint imposed by 1P 1Z models with constant zooplankton mortality on phytoplankton densities can lead to undesirable behaviors. For example, the 1PlZ structure of the Fasham et al. (1990) ecosystem model, used by Sarmiento et al. (1993) in their study of nutrient cycling in the North Atlantic, fixed steady state summer chlorophyll concentrations at high latitudes at constant values and likely caused at least part of the mismatch betwecn model predictions and satellite chlorophyll measurements (Armstrong et al. in press). This model also produced ammonium concentrations one to two orders of magnitude higher than reasonable when Armstrong et al. (in press) attempted to assimilate satellite data into it; multiple-predator/muItiplc-prey models alleviated these problems by allowing chlorophyll concentrations to increase to required levels (through the addition of larger phytoplankton size classes) with increased nutrient loading. From a broader perspective, models with multiple phytoplankton and zooplankton size classes provide a natural framework for disentangling limitation of particular size classes from limitation of total plant and animal biomass. Although alternative ways exist within the 1P 1Z structure to allow total phytoplankton and zooplankton biomass to rise with increases in nutrient loading (e.g. Ginzburg and Argakaya 1992; Steele and Henderson 1992), none allows limitation of species abundances to be considered separately from limitation of trophic levels. This ability to disentangle limitation of whole trophic levels from limitation of their constituent parts (while simultaneously producing observed patterns in algal size distributions and other community propcrtics) food chains 607 provides a strong conceptual reason to prefer models based on multiple phytoplankton and zooplankton size classes. References AKSNES, D. L., AND J. K. EGGE. 1991. 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