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The Impact of Variable Stoichiometry on Predator‐Prey Interactions

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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 func- tional 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 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). Stoichiometry and Predation 31 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).
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