The Impact of Variable Stoichiometry on Predator‐Prey Interactions

vol. 162, no. 1 the american naturalist july 2003

The Impact of Variable Stoichiometry on Predator-Prey Interactions: A Multinutrient Approach

James P. Grover*

Biology Department, University of Texas, P.O. Box 19498, (N), and phosphorus (P). Many published stoichiometric Arlington, Texas 76019 models examine nutrient-limited autotrophic prey (e.g., algae or plants) and their herbivorous predators. Building Submitted June 18, 2002; Accepted January 30, 2003; Electronically published June 18, 2003 from a seminal analysis of herbivorous zooplankton by Sterner (1990), many recent models assign constant pro- portions of nutrient elements to predator biomass, em- bodying an assumption of homeostasis in composition (Andersen 1997; Hessen and Bjerking 1997; Elser and Ur- abstract: A model for prey and predators is formulated in which abe 1999; Loladze et al. 2000; Muller et al. 2001; but see three essential nutrients can limit growth of both populations. Prey take up dissolved nutrients, while predators ingest prey, assimilate a Kooijman 1995; Kooijman et al. 1999). Some models even fraction of ingested nutrients that depends on their current nutrient assume constant element composition of algal or plant status, and recycle the balance. Although individuals are modeled as prey (e.g., Daufresne and Loreau 2001; Grover 2002), ig- identical within populations, amounts of nutrients within individuals noring well-documented variations in autotrophic popu- vary over time in both populations, with reproductive rates increasing lations (Morel 1987; Grover 1991; Spijkerman and Coesel with these amounts. Equilibria and their stability depend on nutrient 1996, 1998; Ducobu et al. 1998). supply conditions. When nutrient supply increases, unusual results can occur, such as a decrease of prey density. This phenomenon Despite the potential limitations of models with con- occurs if, with increasing nutrient, predators sequester rather than stant composition of predators or prey, they have yielded recycle nutrients. Furthermore, despite use of a linear functional significant insight into coupling of population dynamics response for predators, high nutrient supply can destabilize equilibria. and nutrient recycling, especially for lakes where herbi- Responses to nutrient supply depend on the balance between assim- vores of the genus Daphnia prey on nutrient-limited algae ilation and recycling of nutrients by predators, which differs de- (Andersen 1997; Elser and Urabe 1999). Theoretically, pending on the identity of the limiting nutrient. Applied to microbial ecosystems, the model predicts that the efficiency of organic carbon when prey are severely nutrient limited, their quality as mineralization is reduced when supply of mineral nutrients is low food for predators may be so low that predator populations and when equilibria are unstable. The extent to which predators cannot persist (Andersen 1997; Hessen and Bjerking 1997; recycle or sequester limiting nutrients for their prey is of critical Loladze et al. 2000; Muller et al. 2001). Nutrient limitation importance for the stability of predator-prey systems and their re- severe enough to eliminate Daphnia occurs in a minor but sponse to enrichment. significant proportion of lakes (Hessen and Faafeng 2000). Keywords: ecological stoichiometry, microbial ecology, predation, Models predict that predator extinction due to low food nutrient recycling. quality depends on initial conditions, which is consistent with experimental studies of Daphnia (Sommer 1992). On-line enhancement: appendix. Stoichiometric theory also predicts that predator nutrient recycling often controls the identity of limiting nutrients Ecologists have a growing interest in relations between for prey (Daufresne and Loreau 2001), with possible con- population dynamics and nutrient recycling (e.g., Hessen sequences for the outcome of prey competition for those 1997; Elser et al. 2000). This interest has inspired “stoi- nutrients (Sterner 1990; Andersen 1997; Elser 1999; Grover chiometric” models, which couple population dynamics 2002). to fluxes of nutrient elements such as carbon (C), nitrogen In contrast to previous work, the model formulated here assumes that both predator and prey have variable contents * E-mail: [email protected]. of the elements C, N, and P. It is motivated by observations Am. Nat. 2003. Vol. 162, pp. 29–43. ᭧ 2003 by The University of Chicago. that among microorganisms, high population growth rates 0003-0147/2003/16201-020234$15.00. All rights reserved. require high cellular contents of limiting nutrients, and

This content downloaded from 128.101.152.236 on March 26, 2020 07:40:29 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 30 The American Naturalist low growth rates occur when one or more nutrients have dB p m B Ϫ DB Ϫ mBϪ aBZ,(1) reduced content (Droop 1974; Morel 1987; Vadstein 2000). dt BB Although individuals within each population at any time are assumed to have identical nutrient contents, these con- dZ p Ϫ Ϫ mZZZ DZ mZ.(2) tents vary over time depending on nutrient availability. dt This framework is used to build a model of bacterial prey attacked by a flagellate predator—a situation found in vir- The functions mB and mZ represent the per capita repro- tually every aquatic ecosystem known (Sherr and Sherr duction rates of prey and predator, D is the dilution rate 1984; Fenchel 1986). Such prey consume nutrient elements of the chemostat, and mB and mZ are the per capita rates from their environment, and their predators obtain nu- of prey and predator mortality to causes not explicitly trients by ingesting prey, recycling ingested nutrients that represented in the model; all these quantities have di- Ϫ1 are not assimilated. For predator and prey, nutrient ac- mensions time . The final term in equation (1) is the prey quisition is governed both by availability in food or dis- mortality rate due to predation, assuming a linear func- solved pools, respectively, and by current nutrient status. tional response based on a simple mass-action encounter process: a is the “attack” or “clearance” rate of the predator The resulting model can have limit cycle dynamics, despite Ϫ1 Ϫ1 an assumption of linear dependence of the predator’s in- (volume cell time ). gestion rate on prey density. The occurrence and amplitude of cycles are affected by both absolute and relative sup- Reproductive Rates plies of the nutrients C, N, and P. Influences of microbial Reproductive rates mB and mZ of the prey and predator predator-prey interactions on ecosystem processes such as Ϫ1 organic C mineralization and nutrient recycling are also depend on cellular nutrient contents (in mol cell )or suggested by the model. “quotas”Qj,i , where j indexes the nutrient elements C, N, and P, whilei p B or Z indexes prey and predator. The reproductive rate increases as the quota of the limiting nutrient increases from a minimal value at which repro- min duction ceases (Qj,i ), according to a rectangular hyper- bola (Droop 1974). The nutrient whose quota is lowest in Model Formulation relation to the minimum quota determines reproductive rate, in accordance with Liebig’s law of the minimum. For Overview multiple nutrients, Thingstad (1987) combined Droop’s equation and Liebig’s law of the minimum as

In the model, physiological realism is attempted for the coupling of nutrient content to growth and the acquisition Q min p max Ϫ j,i of nutrients for both predator and prey. Ingestion of prey miim 1 max (3) []j ()Qj,i by predators is handled more crudely with a linear functional response to prey density, assuming that a simple p p max fori B, Zj andC, N, P . The parameter mi is the encounter mechanism governs prey consumption. Nutri- “apparent” maximal rate of reproduction achieved as- ent supply and boundary fluxes are represented by a simple ymptotically for an infinite quota. It is higher than the type of open habitat—a chemostat (Monod 1950; Smith true maximal rate of reproduction achieved at a finite and Waltman 1995) in which nutrients are supplied by quota. The assumptions of equation (3) are experimentally inflow, with a balancing outflow that removes nutrients supported for algae (Droop 1974; Rhee 1974, 1978), and organisms. Mortality apart from predation on prey is though extension to bacteria and flagellates requires fur- density independent for both predator and prey. The goal ther exploration. here is to explore the implications of the physiological assumptions governing population growth and nutrient content, acquisition, and recycling. Thus, simple and the- Bacterial Quota Dynamics oretically well-understood assumptions are made about Bacteria obtain nutrients from dissolved pools, whose con- ingestion rate, boundary fluxes, and mortality rates. centrations are denoted C, N, and P, respectively. This The model represents cell densities (cells volumeϪ1), radically simplifies nutrient chemistry: aquatic ecosystems where B represents bacterial prey and Z represents flagel- typically contain many distinct forms of organic C and late predators (table A1), which follow the differential both inorganic and organic forms of N and P. In this equations model, the dissolved pools for each nutrient are perhaps

best interpreted as representing labile organic C, inorganic j:aBQj,B , with dimensions of mol nutrient (predator phosphate P, and inorganic ammonium N. Nutrient up- cells)Ϫ1 timeϪ1. These ingested ﬂuxes increase the ﬂagellate take follows Michaelis-Menten kinetics but also decreases quota of a nutrient, which decreases due to utilization for linearly with quota (as found in algae and cyanobacteria; reproduction, at ratesmZjQ ,Z , and due to respiration of C

Morel 1987; Ducobu et al. 1998). When quota is near or recycling of N and P, at ratesR j,Z (with dimensions of minimal, uptake is high, and as quota approaches a max- mol nutrient [predator cells]Ϫ1 timeϪ1). Thus, three dif- imum, uptake vanishes (Thingstad 1987): ferential equations describe ﬂagellate quota dynamics:

dQ max Ϫ j,Z p Ϫ Ϫ [j] Qj,BjQ ,B aBQj,BZjm Q ,ZjR ,Z (7) V p V max (4) dt j,Bj,B ϩ maxϪ min ()()Kj,Bj[j] Q ,BjQ ,B forj p C, N, P . forj p C, N, P , whereV max is the maximal rate of uptake j,B Available studies of flagellates do not support specific (mol cellϪ1 timeϪ1),K is a half-saturation constant (mol j,B quantitative models of nutrient assimilation, recycling, and volumeϪ1),Q max is the maximal quota (mol cellϪ1), and j,B respiration. However, it is reasonable to assume that when [j] denotes the concentration of element j (mol volumeϪ1). a nutrient becomes severely limiting for flagellates and the The bacterial quota of a nutrient increases due to uptake respective quota approaches its minimum, the proportion and decreases due to utilization for reproduction at a rate of that nutrient assimilated from the ingested nutrient flux Q . Quota could also decrease due to respiration or mBj,B will reach a maximum (Andersen et al. 1986; Nakano recycling at rateR (mol cellϪ1 timeϪ1). Thus, three dif- j,B 1994a), denoted e . When a nutrient is not limiting and ferential equations describe bacterial quota dynamics: j the quota approaches its maximum, it is assumed that the assimilated proportion of the ingested nutrient flux ap- dQ j,B p Ϫ Ϫ Vj,BBjm Q ,BjR ,B (5) proaches 0. Based on these assumptions, the assimilated dt flux for nutrientj p C , N, P is forj p C, N, P . Recycling of N and P by bacteria is as- p p max sumed to be negligible,RN,BPR ,B 0 , reflecting current Q Ϫ Q A p ae BQ j,Zj,Z .(8) opinion that aquatic bacteria typically sequester rather j,Zjj,B maxϪ min ()Qj,ZjQ ,Z than recycle nutrients (Azam et al. 1983; Caron et al. 1988; Vadstein et al. 1988; Vadstein 2000). Respiration by bac- The termsQ max are maximal cell quotas of flagellates for teria (R ) has a growth-related component proportional j,Z C,B nutrients. The maximal assimilation efficiency e is as- to the rate at which C is utilized for reproduction j signed a value of 1 for N and P, permitting perfect assim- (m Q ) and a maintenance component proportional to BC,B ilation in the limiting case of severe nutrient limitation. the “surplus” cellular C above the minimal quota A lower value is assigned for C, because the inevitable (QQϪ min): C,BC,B respiratory costs of obtaining organic substrates limit the R p rmgmminQ ϩ r (Q Ϫ Q ), (6) efficiency of their assimilation. C,BBC,BC,BC,B The difference between the ingested flux of nutrient j Ϫ where rg (dimensionless) and rm (time 1) are coefficients (aBQj,B ) and the assimilated flux given above is the rate for growth-related and maintenance respiration, respec- of recycling for N and P and the rate of respiration for tively. According to equation (6), bacteria reduce their C: respiration rate when their reproductive rate is low and especially when it is limited by C (Thingstad 1987). Q max Ϫ Q R p aBQ 1 Ϫ e j,Zj,Z .(9) j,Zj,BjmaxϪ min []()Qj,ZjQ ,Z Flagellate Quota Dynamics As a nutrient becomes limiting for flagellates, equation (9) Flagellates obtain nutrients from ingesting bacterial prey, shows that the rate of recycling (or respiration for C) according to the linear functional response assumed in decreases. Under these conditions, flagellates sequester the equation (1). The per capita ingestion rate of flagellates is nutrient rather than recycle or respire it. When a nutrient aB, with dimensions prey cells (predator cells)Ϫ1 timeϪ1. is in excess for flagellates, their recycling (or respiration) p The corresponding fluxes of C, N, and P are the products rate increases. For N and P, settingej 1 implies that the of ingestion rate and the bacterial cell quota for nutrient recycling rate vanishes when either nutrient is severely

! limiting, while settingej 1 implies a finite lower bound terial P nutrition (Vadstein 2000). Also, the apparent max- max Ϫ1 to the respiration rate. imal reproductive ratemB was set to 6 d , half that proposed by Thingstad (1987), to produce the low reproductive rates observed in field studies of aquatic bacterioplankton (Vadstein et al. 1988; Morris and Lewis 1992; Dissolved Nutrient Dynamics Chrzanowski and Grover 2001). With the other assigned The inflowing flux (mol volumeϪ1 timeϪ1) for nutrient j parameter values, the true maximal reproductive rate for Ϫ1 is D[j]in, where [j]in is the supply concentration (mol vol- bacteria is 2.667 d , and bacteria are C limited in this Ϫ ume 1) of the nutrient. The outflowing flux of dissolved state. The respiratory parameter rm was reduced to 25% Ϫ Ϫ nutrient j is D[j] (mol volume 1 time 1). The concentra- of the value proposed by Thingstad (1987) because slow- tion of nutrient j also decreases due to bacterial uptake growing bacteria should have slow C turnover and low Ϫ1 Ϫ1 according to termsBVj,B (mol volume time ). For sim- maintenance respiration. plicity, it is assumed that the entire nutrient content of With the assigned values, C and N quotas vary twofold cells lost due to mortality is instantaneously recycled, lead- at most. When all quotas are minimal, bacteria have Red- ing to termsmBQBj,BZj and mZQ,Z in the three differential field (1958) stoichiometry, with C : N : P of 106 : 16 : 1. equations for nutrient dynamics: When all quotas are maximal, bacteria have excess P due to modest capacity to store this element (Vadstein 2000), d[j] p Ϫ Ϫ ϩ ϩ ϩ D([j]in [j]) BVj,BjZY ,ZZjmZQ,ZBjmBQ,B (10) with C : N : P of 106 : 16 : 6. The model allows deviations dt from Redfield stoichiometry, depending on growth conditions: C : N ranges 4# from 3.3 to 13.3, while C : P and forj p C, N, P . The terms ZY represent nutrient recy- j, Z N : P range about 12# from 18 to 212 and from 2.7 to cling. For N and P,Y p R , the per capita recycling j, Zj, Z 32, respectively. These patterns are consistent with exper- rates of flagellates given above (eq. [9]). Respiratory C imental studies of aquatic bacteria (Vadstein and Olsen fluxes are assumed to leave the system by setting Y p C, Z 1989; Chrzanowski and Kyle 1996; Goldman and Dennett 0. 2000). Flagellate parameter were assigned to represent the well- studied genus Paraphysomonas (Caron et al. 1985, 1986; Mass Balances Goldman and Caron 1985; Goldman et al. 1985, 1987; Andersen et al. 1986; Nakano 1994a; Eccleston-Parry and The full model consists of 11 differential equations: those Leadbeater 1995). When all quotas are minimal, flagel- for nutrient dynamics (eq. [10]) together with equations (1), (2), (5), and (7). As with other chemostat models, it late composition has a stoichiometry with C : N : P of is instructive to examine system-level balances of nutrient 92 : 32 : 1. When all quotas are maximal, flagellates have mass (Smith and Waltman 1995). The total mass concen- C : N : P of 78 : 25 : 1. The assigned parameters allow trations of N and P approach the supply concentrations much variation in composition depending on growth con- Ϫ # N and P with a characteristic time of D 1, but the total ditions: C : N ranges 23 from 0.6 to 13.9, C : P ranges in in # # mass concentration of C is always below the supply con- 4.8 from 16 to 78, and N : P ranges 26 from 5.6 to 147. Based on Caron et al. (1985), the apparent maximal centration Cin (see app. B in the on-line edition of the max American Naturalist). Organic C is supplied externally, and reproductive ratemZ was adjusted to produce a true max- Ϫ1 the activities of the resident heterotrophs can only reduce imal reproductive rate of 2.506 d , given other parameter the total mass of organic carbon, leading to net miner- values. Flagellates are N limited when reproducing alization of organic C in the ecosystem. The “missing” C maximally. disappears from the model’s accounting into dissolved and The biological parameters discussed above were not var- atmospheric reservoirs of CO2. ied in simulations presented here. The dilution rate D was usually set to 0.2 or 1.5 dϪ1, while the supply concentra-

tions (Cin, Nin, Pin) were varied through wide ranges. Mor- tality rates m and m were both usually set to 0.1 dϪ1, Parameterization and Numerical Analysis B Z though wide ranges were also explored. Realistic parameter values for aquatic bacteria and their The model was simulated using a well-tested Runge- ﬂagellate predators were assigned (table A1). The model Kutta algorithm with adaptive step size (Press et al. 1986). of bacterial dynamics used here is adapted from Thingstad The expected mass balances for P and N were always ver- (1987), and the same parameter values were used, with iﬁed in simulations, and correct dynamics were obtained some exceptions to accommodate recent studies on bac- in the few special cases where analysis is possible (such as

strong effect of food quality (bacterial N content) on ﬂagel- late growth and restricting recycling by the predator of the

nutrient limiting to prey. When Nin increased further to 3.5 mmol LϪ1, bacteria became C limited, and their decline

with enrichment continued (ﬁg 3A, inset). At Nin above about 4 mmol LϪ1, both prey and predator abundances became invariant. Reproduction of predators remained N limited even at very high N supplies, but C limitation of the prey constrained predators from responding numer- ically to increased N supply and stabilized equilibria, preventing limit cycles. When P supply alone was enriched, patterns resembled those for enrichment of N supply alone (results not shown). However, there was no range of P supply in which prey density decreased with enrichment. Predators were always N limited and, thus, recycled P when it was limiting to prey.

Figure 1: Two-dimensional bifurcation plot for a “Redﬁeld habitat” with resource supply at a molar C : N : P of 106 : 16 : 1. As P supply increases, C and N supplies increase according to this ratio. The solid line shows the boundary between predator extinction and persistence with stable equilibria, and the dashed line shows the boundary between stable equilibria and limit cycles. combined mortality in excess of the true maximal reproductive rate).

Results Population Dynamics In common with many other predator-prey models, this model predicts limit cycle (periodic) dynamics in resource- rich habitats. Three basal resources permit several types of enrichment gradients. In the simplest of these, C, N, and P were supplied at molar ratios of 106 : 16 : 1, the classical Redfield (1958) ratio thought to characterize typ- ical seawater. For such “Redfield habitats,” the presence of limit cycles depended on dilution rate (fig. 1), and at a fixed dilution rate, there was a bifurcation from extinction to persistence of flagellate predators at a relatively low nutrient supply and another bifurcation from a stable equilibrium to limit cycles at a sufficiently high nutrient supply. Amplitudes of limit cycles increased with enrichment (fig. 2). Limit cycles always had a single dominant frequency; complex aperiodic, quasiperiodic, or chaotic dynamics were never observed. Figure 2: One-dimensional bifurcation plot for a “Redfield habitat” with When N supply alone increased, the bifurcation to resource supply at a molar C : N : P of 106 : 16 : 1 and a dilution rate of p Ϫ1 1.5 dϪ1. Solid lines, equilibrium densities when stable, average densities predator persistence occurred at Nin 2.8 mmol L (C:N:P p 106 : 5.6 : 1), and the equilibrium predator in the limit cycle regime; dashed lines, minimum and maximum densities in the limit cycle regime. A logarithmic scale is used for the full range density increased with N supply (fig. 3B), while the equi- of resource supplies investigated, while insets show the region from 0–1 librium prey density decreased (fig. 3A). Both flagellate mmol P LϪ1 with an arithmetic scale. A, Density of bacterial prey. B, predators and bacterial prey were N limited, producing a Density of flagellate predators.

sembled enrichment with N alone, while enrichment with C and N or C and P resembled enrichment with C alone.

To explore a wide range of N : P supply ratios, log Nin

was defined as a decreasing function of log Pin, with Nin Ϫ1 increasing from 0.16 to 1,600 mmol L and Pin decreasing from 100 to 0.01 mmol LϪ1 (fig. 5). For N : P supply ratios below about 1, bacteria were N limited; they were P limited for N : P supply ratios above about 220, and they were C limited for intermediate N : P supply ratios. Predators per- sisted for a wide range of N : P supply ratios from 0.44 to 980 (fig. 5B). For a narrower range of N : P supply ratios from 1.1 to 256, there were limit cycles whose amplitude was restricted by C limitation of prey. For all supply conditions, flagellate predators were N limited, and their extinction at high N : P supply ratios was due to the failure of severely P-limited bacteria to take up and “package” N at a sufficient rate to support predators.

Figure 3: One-dimensional bifurcation plot for a gradient of N supply Ϫ1 and a dilution rate of 1.5 d ; Pin and Cin were ﬁxed at 0.5 and 53 mmol LϪ1, respectively. Solid lines, equilibrium densities. A logarithmic scale is used for the full range of resource supplies investigated, while insets show the region from 1.8–4.2 mmol N LϪ1 with an arithmetic scale. A, Density of bacterial prey. B, Density of ﬂagellate predators.

When C supply alone increased (fig. 4), the bifurcation p Ϫ1 to predator persistence occurred at C in 29 mmol L (C : N :P p 58 : 16 : 1), and the equilibrium predator density increased with C supply (fig. 4B), while the equilibrium prey density was constant (fig. 4A). Further enrichment with C destabilized the predator-prey equilibrium and increased the amplitude of the limit cycle. ≈ Ϫ1 However, at C in 150 mmol L , bacteria shifted from being C limited over most of a cycle to being predomi- nantly N limited. This restricted the cycle amplitude and ensured that minimal densities remained well above those Figure 4: One-dimensional bifurcation plot for a gradient of C supply Ϫ1 risking extinction (e.g., cf. amplitudes in figs. 2, 4). Flagel- and a dilution rate of 1.5 d ; Pin and Nin were fixed at 0.5 and 8 mmol late predators were always N limited for the supply con- LϪ1, respectively. Solid lines, equilibrium densities when stable, average ditions shown. densities in the limit cycle regime; dashed lines, minimum and maximum densities in the limit cycle regime. A logarithmic scale is used for the Enrichment gradients were also explored in which sup- full range of resource supplies investigated, while insets show the region plies of two nutrients increased, while one remained fixed from 0–48 mmol C LϪ1 with an arithmetic scale. A, Density of bacterial (results not shown). Enrichment with both N and P re- prey. B, Density of flagellate predators.

behavior in all but one case involving a glacially slow increase of predators, in which the ultimate state was not observed.

Carbon Mineralization In this model, some of the supplied organic C is mineralized by microbial respiration. When C supply alone was increased, the proportion of supplied C mineralized increased as long as the predator-prey equilibrium remained stable (ﬁg. 6A). With sufﬁcient enrichment of C supply, limit cycles ensued, and the average proportion of C mineralized then decreased. More dissolved C accumulated

Figure 5: One-dimensional bifurcation plot for a gradient of N : P supply Ϫ1 Ϫ1 ratio and a dilution rate of 1.5 d ; Cin was ﬁxed at 106 mmol L . Solid lines, equilibrium densities when stable, average densities in the limit cycle regime; dashed lines, minimum and maximum densities in the limit cycle regime. A, Density of bacterial prey. B, Density of ﬂagellate predators.

Very similar results to those just summarized were found at the much lower dilution rate of 0.2 dϪ1, though limit cycles occurred for wider ranges of supply conditions and had larger amplitudes. Previous stoichiometric models of predator-prey interactions reveal bistability when severe nutrient limitation of prey makes their food quality to predators so low that predator persistence is endangered. Sufficiently large initial predator populations can ingest enough prey and recycle enough nutrients to relieve limitation of prey and allow predator persistence. But small initial predator populations cannot. For the microbial model examined here, such bi- Figure 6: Proportion of C supply mineralized as a function of nutrient Ϫ1 stability was not found. Supply conditions under which supply at a dilution rate of 1.5 d . A, Gradient of C supply, with Pin and N fixed at 0.5 and 8 mol LϪ1. Solid line shows mineralization at previous theory suggests bistability were examined by in m stable equilibria; dashed line shows average mineralization in the limit varying initial predator density over five orders of mag- cycle regime. B, Gradients of N and P supply, with Cin fixed at 53 mmol nitude. Predators always increased from low density, and LϪ1. Thick solid line shows a gradient of N : P supply ratio similar to different initial densities converged to the same asymptotic that of figure 5.

This content downloaded from 128.101.152.236 on March 26, 2020 07:40:29 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 36 The American Naturalist during periods of low microbial abundance than could be Discussion mineralized during periods of high microbial abundance. This pattern was found at low dilution rate and for other Three notable results of this study are instability and limit enrichment gradients that produced limit cycles (results cycles in a model with a linear functional response, lack not shown). of bistability due to predator nutrient limitation, and de- With C supply fixed at a low level preventing limit cycles, creases of prey density with increasing nutrient levels. A the proportion of supplied C mineralized increased with simplified model of the food quality of prey in predator- both N and P supply, reaching a plateau when both sup- prey systems provides unified insight into these results plies were sufficient (fig. 6B). For some gradients of (app. C), which arise in other stoichiometric models and N : P supply ratio, C mineralization would thus reach a in models of food quality (Edelstein-Keshet and Rausher plateau at intermediate N : P supply ratios (e.g., heavy solid 1989; Van de Koppel et al. 1996; Huxel 1999). The sim- line in fig. 6B), with nutrient limitation of C mineralization plified model suggests that the processes of predator se- at extreme supply ratios. questration and recycling of nutrients are critical to the findings presented here. In much predator-prey theory, instability arises through a nonlinear functional response of the predator and is Nutrient Recycling sensitive to enrichment (e.g., Rosenzweig 1971). Here, the For this model, the N : P ratio of nutrients recycled by functional response was linear, and prey growth was den- flagellates is sity dependent, due to nutrient limitation. Absent other complications, stability is expected, and instability is sur- prising. The simplified model (app. C) shows that even with a linear functional response, stability depends criti- QQϪ QQmin maxϪ Q min N : P recycled p N,BN,ZN,ZP,ZP,Z . cally on whether predators sequester nutrients, making Ϫ min maxϪ min QQP,BP()(),ZPQQ,ZN,ZNQ ,Z them unavailable to prey, or recycle nutrients, making them more available to prey. In the simplified model, both (11) predator and prey growth increase with prey quality, consistent with stoichiometric theory in which prey nutrient In the simulations presented above, the ratio of N : P re- content governs quality. In turn, prey quality (nutrient cycled was generally lower than the Redfield ratio of content) depends on predator density positively or neg- 16 : 1, except for the extremely P-limited conditions (table atively, depending on whether nutrients are recycled or 1), and high dilution rate increased the ratio of N : P re- sequestered by predators. cycled (fig. 7A), as did high predator mortality (fig. 7B). In the simplified model with these features, stability is As either dilution or mortality rate increases, predators conditional even though there is a linear functional re- must reproduce faster to persist, and such rapid growth sponse, and it can be lost in two different ways (app. C). caused them to sequester more P relative to N. First, if predators strongly recycle nutrients, an unstable

Table 1: N : P ratios of nutrient recycling due to incomplete assimilation by ﬂagellate predators Dilution rate and Range of N : P supply ratio Range of N : P recycling ratio supply gradient (molar)a (molar) 1.5 dϪ1: C only 16 2.9–50 N only 5.6–200 2.4–3.4 P only .8–62 2.5–18.5 N versus P .49–950 1.5–18.5 C, N, and P 16 2.4–5.4 .2 dϪ1: C only 16 .86–1.7 N only .44–200 .13–.64 P only .8–1000 .57–25 N versus P .0029–950 .12–1.8 C, N, and P 16 .4–3.2 a Only supply conditions allowing predator persistence are shown.

unstable predator-prey equilibrium arises when predators strongly sequester nutrients rather than recycling them. In many stoichiometric predator-prey models, this unstable equilibrium is a saddle point, associated with bistability and predator extinction for some initial conditions. Much previous work focuses on the herbivore Daphnia as predator and P-limited algae as prey, which are poor quality food for Daphnia (Andersen 1997; Elser and Urabe 1999; Loladze et al. 2000; Muller et al. 2001). In these models, some initial conditions allow the Daphnia population to recycle enough P to alleviate algal limitation, improving food quality sufficiently to persist. But for other initial conditions, Daphnia populations cannot accomplish this, eat food of insufficient quality to support reproduction, and perish. This phenomenon was never found in the parameterized model of microbial predators presented here, though it is not thereby ruled out. Unlike the crustacean predators represented in earlier stoichiometric theory, the microbial predators represented here have variable nutrient content rather than homeostatically regulated (fixed) content (Goldman et al. 1985; Andersen et al. 1986; Nakano 1994a). Predators with homeostatic nutrient content are very sensitive to reduced nutrient content of their prey, but predators that can reduce their own nutrient content are less sensitive. The Droop relationship used here (eq. [3]) implies that as predator nutrient content initially de- clines from its maximum, reproduction is maintained at rates approaching maximal, with sharper reductions in predator reproduction occurring only as their nutrient content approaches its minimum. This phenomenon is Figure 7: N : P ratio of nutrients recycled by flagellate predators due to incomplete assimilation in relation to dilution and predator mortality well established among the photosynthetic relatives of fla- p Ϫ1 rates under fixed nutrient supply conditions (Pin 0.5 mmol L ; gellate predators and is a major adaptation to nutrient p Ϫ1 p Ϫ1 Nin8 mmol L ; C in 53 mmol L ). A, Gradient of dilution rate (with limitation (Turpin 1988). It is plausible that flexible nu- Ϫ1 predator mortality rate fixed at 0.1 d ). B, Gradient of predator mortality trient content affords flagellates and other microbial pred- rate (with dilution rate fixed at 0.2 dϪ1). ators a greater capability to persist under nutrient limitation than occurs among homeostatic predators, such as equilibrium with locally oscillating dynamics can occur. the crustacean Daphnia. In this case, it is likely that enrichment is destabilizing and In classical theory, abundances of predator and prey that limit cycles occur. Unstable equilibria of this type respond to enrichment in a “stair-step” fashion (Oksanen commonly appear in the much more complex model de- et al. 1981; Thingstad and Sakshaug 1990). When the en- veloped in this article. In contemporary theory, functional vironment is so poor that predators cannot persist, en- responses receive more attention than other destabilizing richment increases prey abundance. Once predators can aspects of predator-prey interactions. However, it has long persist, further enrichment increases their abundance but been known that either Allee effects within prey (Rosen- not that of prey. The stoichiometric predator-prey model zweig 1969) or time lags (Wangersky and Cunningham presented here usually displayed such responses to en- 1957) can be destabilizing, even when there is a linear richment. However, enrichment sometimes caused declin- functional response. The model developed here calls at- ing prey abundance and increasing predator abundance tention to nutrient recycling, suggesting that it be explored (e.g., fig. 3). The simplified model constructed in appendix along with demographics and the functional response to C suggests that this behavior arises when predators se- distinguish among alternative destabilizing processes. quester nutrients rather than recycle them or recycle nu- The simplified model suggests that a second type of trients only weakly. In the full stoichiometric model, such

This content downloaded from 128.101.152.236 on March 26, 2020 07:40:29 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 38 The American Naturalist behavior occurred when N limited both predator and prey intraspecific variation has been documented for P. imper- and N recycling by predators was weak. forata (Goldman et al. 1985; Andersen et al. 1986). Further Many relationships between ecosystem functions and characterization of the N : P recycling ratio within and population dynamics could be explored for microbial food among flagellate species would be desirable. If a low webs with the model presented here, and two were pursued N : P ratio of recycling proves to be a general characteristic here: mineralization of dissolved organic C (DOC) and of bacterivorous flagellates, then their ecosystem role con- the ratio of N : P recycled by consumers (heterotrophic trasts with that of daphnids, which recycle at a high flagellates). The predicted increase of DOC mineralization N : P ratio (Sterner 1990; Elser and Urabe 1999). Inter- with the supply of mineral nutrients has been found in estingly, the model predicts that the ratio of N : P recycled empirical studies (Zweifel et al. 1995; Cotner et al. 1997; by bacterivorous flagellates increases with their growth rate Thingstad et al. 1999; Grover 2000). Thus, the model sup- (i.e., with dilution or mortality rates; fig. 7). If this result ports the notion that decomposition by microbial food accurately reflects the physiology of bacterivorous flagel- webs can be limited by availability of mineral nutrients. lates, then it supports the suggestion that demand for P Novel predictions are that the efficiency of DOC miner- rises more with growth rate than demand for N, raising alization is maximal at intermediate supply ratios of min- the N : P recycling ratio of rapidly growing organisms (El- eral nutrients and that predator-prey cycles reduce the ser et al. 1996). efficiency of DOC mineralization. Extreme N : P supply In conclusion, the extent to which predators recycle or ratios predicted to reduce the efficiency of DOC miner- sequester limiting nutrients for their prey is critical to alization appear to be rare in natural waters (Guildford stability and responses to enrichment. Sometimes these and Hecky 2000). However, flagellate predators and bac- effects will agree with classical expectations of predator- terial prey may be prone to cyclic dynamics (Fenchel prey theory, and sometimes they will show contrary pat- 1986), so that reduced DOC mineralization for this reason terns such as unexpected instability or decreases of prey could thus be common. abundance with enrichment. These results arise from in- As modeled, flagellates have a high demand for N under direct effects of predators on the nutrient status and food most growth conditions, and the bacterial N : P ratio is quality of their prey, and such indirect effects are likely generally lower than the Redfield ratio, so that bacterial widespread in nature. prey supply flagellates with less N and more P than they need. Thus, under most conditions, the model predicts Acknowledgments that bacterivorous flagellates strongly sequester N while strongly recycling P, so that the N : P ratio of recycling is This work was conducted as part of the Competition The- lower than the Redfield ratio (16 : 1). The relative amounts ory Working Group supported by the National Center for of N and P recycled by flagellates vary widely among spe- Ecological Analysis and Synthesis, a center funded by the cies. Eccleston-Parry and Leadbeater (1995) found that National Science Foundation (NSF; grant DEB-0072909), Paraphysomonas imperforata had the lowest N : P ratio of the University of California, and the Santa Barbara cam- recycling (about seven) among four species examined. pus. I thank the members of the working group for dis- Bodo designis had a similar low N : P ratio of recycling cussions and advice. Partial support was also provided by (about 11), while Jakoba libera and Stephanoeca diplocos- NSF grant DMS-0107439. Comments by T. H. Chrzanow- tata had ratios exceeding 200. Nakano (1994b) found that ski, H. V. Kojouharov, and two anonymous reviewers im- Spumella sp. recycles at N : P ratios of about 18–30. Wider proved the manuscript.

APPENDIX A

Table A1: Notation Symbol Deﬁnition Units Assigned value Indices: i Index for population (B, Z) j Index for nutrient (C, N, P) State variables: B Bacterial prey density cells mϪ3 Z Flagellate predator density cells mϪ3 Ϫ1 Qj, i Quota (content per cell) of nutrient j for population i mol nutrient cell

Table A1 (Continued) Symbol Definition Units Assigned value [j] Concentration of dissolved nutrient j, with specific nutrients mol mϪ3 denoted C, N, P Functions: Ϫ1 mB Per capita reproductive rate of bacteria d Ϫ1 mZ Per capita reproductive rate of flagellates d Ϫ1 Ϫ1 Vj, B Net uptake rate of nutrient j by bacteria mol nutrient cell d Ϫ1 Ϫ1 Rj, i Respiration or recycling rate of population i for nutrient j mol cell d Ϫ1 Ϫ1 Aj, Z Assimilation rate of nutrient j by flagellates mol cell d Parameters: D Dilution rate dϪ1 Varies, usually 1.5 or .2 Ϫ3 [j]in Supply concentration for nutrient j, with specific nutrients mol m Varies

denoted Cin, Nin, Pin Ϫ1 mB Per capita mortality rate of bacteria d .1 Ϫ1 mZ Per capita mortality rate of flagellates d .1 a Attack rate of flagellates on bacteria m3 cellϪ1 dϪ1 .81 # 10Ϫ10 max Ϫ1 mi Apparent maximal reproductive rate of population i d 6.0 for bacteria; 3.2 for flagellates min Ϫ1 # Ϫ15 Qj, B Minimal quota of nutrient j for bacteria mol nutrient cell 1.06 10 for C; .16 # 10Ϫ15 for N; .01 # 10Ϫ15 for P min Ϫ1 # Ϫ15 Qj, Z Minimal quota of nutrient j for flagellates mol nutrient cell 148 10 for C; 51 # 10Ϫ15 for N; 1.6 # 10Ϫ15 for P max Ϫ1 # Ϫ15 Qj, B Maximal quota of nutrient j for bacteria mol nutrient cell 2.12 10 for C; .32 # 10Ϫ15 for N; .06 # 10Ϫ15 for P max Ϫ1 # Ϫ15 Qj, Z Maximal quota of nutrient j for flagellates mol nutrient cell 708 10 for C; 235 # 10Ϫ15 for N; 9.1 # 10Ϫ15 for P max Ϫ1 Ϫ1 # Ϫ15 Vj, B Maximal uptake rate for nutrient j by bacteria mol nutrient cell d 52 10 for C; 7.7 # 10Ϫ15 for N; 1.25 # 10Ϫ15 for P Ϫ3 # Ϫ3 K j, B Half-saturation constant for uptake of nutrient j by bacteria mol nutrient m 1.0 10 for C; .1 # 10Ϫ3 for N and P rg Coefficient for growth-related bacterial respiration Dimensionless 1.0 rm Coefficient for maintenance-related bacterial respiration dϪ1 .3

ej Maximal assimilation efﬁciency for nutrient j by ﬂagellates Dimensionless .65 for C; 1 for N and P

APPENDIX C nutrient content through expressions such as equation (3), motivating this simple predator-prey model: A Simpliﬁed Predator-Prey Model with Food Quality Effects dB p Bg(B, q) Ϫ aBZ, (C1a) dt The full stoichiometric model presented in this article is dZ too complex for conventional analysis of equilibria and p aZBf(q) Ϫ dZ.(C1b) stability. In this and other recent stoichiometric models, dt predator growth is inﬂuenced by both the quantity of prey and their quality as food. The latter property depends on Here, B and Z are prey and predator densities, and the their content (quota) of the growth-limiting nutrient for functionq(B, Z) determines prey quality, which depends predators. Prey growth is also positively related to prey on both prey and predator densities. The function

This content downloaded from 128.101.152.236 on March 26, 2020 07:40:29 AM All use subject to University of Chicago Press Terms and Conditions (http://www.journals.uchicago.edu/t-and-c). 40 The American Naturalist g(B, q) describes prey population growth in the absence of directions (becausedZ ∗∗/dB ! 0 ). Thus, the responses of predators. It decreases with prey density (Ѩg/ѨB ! 0 ) and equilibria in this food quality model can differ sharply increases with prey quality (Ѩg/Ѩq 1 0 ). For any fixed q, from the many predator-prey models for which the prey the prey growth function is positive forB p 0 and van- equilibrium is invariant to enrichment while the predator ishes for a positive value of B. For example,g(B, q) could equilibrium increases. be a logistic function of prey density whose elevation de- Standard phase plane analysis reveals the dynamics near pends on prey quality. Predators ingest prey according to equilibrium. Local stability requires that the associated Ja- a linear, type I functional response with attack rate a, cobian matrix has a negative trace and a positive deter- leading to the loss term ϪaBZ in equation (C1a). Predator minant. The determinant is positive if reproduction is the rate of ingestion multiplied by an efficiency function f(q) that depends on prey quality, and Ѩf Ѩq Ѩg Ѩq Ѩg Ѩq the per capita predator mortality rate is d. The function Baϩϩaf(q) Ϫ a 1 0. (C5) f(q) lies between 0 and 1 and is strictly increasing Ѩq ()()ѨB ѨB ѨZ Ѩq ѨZ ().df/dq 1 0 If prey quality q is constant, then a classical predator- Given the assumptions on the functions involved, either prey model is recovered, whose equilibrium is globally term in parentheses can be negative if predators strongly stable when it is feasible, due to the density-dependent sequester nutrients instead of recycling them, that is, if growth of prey. Making prey quality a function of B and Ѩq/ѨZ is negative and of sufficiently large magnitude. When Z encapsulates ecosystem feedbacks by which both pop- an equilibrium is unstable for this reason, it is a saddle ulations affect the nutrients available to prey and hence point. This type of unstable equilibrium has been found their food quality for predators. It is assumed that prey in many stoichiometric predator-prey models before, and Ѩ Ѩ quality decreases with prey density (q/ B ! 0 ) due to in these cases, it is associated with bistability such that competition among prey for limited nutrients. Prey quality predators go extinct for some initial conditions (Andersen Ѩ Ѩ ! decreases with predator density (q/ Z 0 ) if predators 1997; Elser and Urabe 1999; Loladze et al. 2000; Muller sequester nutrients more strongly than they recycle them, et al. 2001). decreasing availability to prey and thus reducing prey qual- Another type of unstable equilibrium arises when pred- Ѩ Ѩ ity. Prey quality increases with predator density ( q/ Z 1 ators recycle nutrients sufficiently strongly so that the de- 0) if predators recycle nutrients more strongly than they terminant of the Jacobian matrix is positive. Then, local sequester them, increasing availability to prey and thus stability holds if the trace of the Jacobian matrix is neg- increasing prey quality. ative, which requires The following equations define a nontrivial equilibrium point(B∗∗, Z ) for system (C1): Ѩg Ѩg Ѩq Ѩf Ѩq ϩϩaZ ! 0. (C6) ∗∗∗ ∗ Ѩ Ѩ Ѩ Ѩ Ѩ g(B , q(B , Z )) Ϫ aZ p 0, (C2a) B q B q Z aB∗∗∗ f(q(B , Z )) Ϫ d p 0. (C2b) The first two terms of this expression are always negative, but the third is positive when predators recycle nutrients Implicit differentiation of equation (C2a) reveals in part because thenѨq/ѨZ 1 0 . If this third term is large enough, how equilibrium densities of predator and prey respond then it outweighs the negative terms, and the equilibrium to parameter changes such as enrichment. The sign of ∗∗ is destabilized. This type of instability can depend on en- dZ /dB is positive if richment. IfѨq/ѨZ is large enough to satisfy condition

Ϫ1 (C3), then predator and prey density at equilibrium re- Ѩq Ѩg spond in the same direction to enrichment (dZ ∗∗/dB 1 0 ), 1 a .(C3) ѨZ ()Ѩq and we may define a richer habitat as one with higher densities. As a habitat becomes richer, Z∗ increases, making This condition is satisfied if predators are “strong recy- the third destabilizing term of condition (C6) larger. Thus, clers” of nutrients, having a sufficiently large positive effect recycling of nutrients by predators should be associated on prey quality. In such cases, predator and prey densities with behavior that follows the classical “paradox of en- at equilibrium respond to changes such as enrichment in richment” (Rosenzweig 1971), wherein a predator-prey the same direction: if one population increases, then both equilibrium is destabilized in richer habitats. When this populations increase. If predators sequester nutrients type of unstable equilibrium occurs, it is an unstable spiral, (Ѩq/ѨZ ! 0 ) or recycle only weakly, then predator and prey with oscillatory trajectories in its neighborhood. It is plau- densities at equilibrium respond to enrichment in opposite sible that there is then a limit cycle that is at least locally

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