## Abstract

Contemporary theory of predator coexistence through relative non-linearity in their functional responses strongly relies on the Rosenzweig and MacArthur (1963) equations in which the (autotrophic) prey exhibits logistic growth in the absence of the predators. This implies that the prey is limited by a resource such as light or space, which availability is independent of the predators. This assumption does not hold under nutrient limitation where both prey and predators bind resources such as nitrogen and phosphorus in their biomass. Furthermore, the prey’s resource uptake rate is assumed to be linear and the predator-prey system is considered to be closed. All these assumptions are unrealistic for many natural systems. Here, we show that nutrient retention in predator biomass strongly hampers the coexistence of two predators on one prey because it stabilizes the dynamics. In contrast, a non-linear resource uptake rate of the prey slightly promotes predator coexistence. Our study highlights that predator coexistence does depend not only on differences in the curvature of their functional responses but also on the type of resource constraining the growth of their prey. This has far-reaching consequences for the relative importance of fluctuation-dependent and fluctuation-independent mechanisms of species coexistence in natural systems where autotrophs experience light or nutrient limitation.

This is a preview of subscription content, access via your institution.

## References

Abrams PA (2004) When does periodic variation in resource growth allow robust coexistence of competing consumer species. Ecology 85:372–382

Abrams PA (2006) The prerequisites for and likelihood of generalist-specialist coexistence. Am Nat 167:329–342

Abrams PA (2009) When does greater mortality increase population size? The long history and diverse mechanisms underlying the hydra effect. Ecol Lett 12:462–474

Abrams PA, Holt RD (2002) The impact of consumer-resource cycles on the coexistence of competing consumers. Theor Popul Biol 62:281–295

Armstrong RA, Mcgehee R (1980) Competitive-exclusion. Am Nat 115:151–170

Barabás G, D’Andrea R, Stump SM (2018) Chesson’s coexistence theory. Ecol Monogr 88:277–303

Bar-On YM, Phillips R, Milo R (2018) The biomass distribution on Earth. Proc Natl Acad Sci U S A 115:6506–6511

Boraas ME, Seale DB, Horton JB (1990) Resource competition between two rotifer species (Brachionus rubens and B. calyciflorus): an experimental test of mechanistic model. J Plankton Res 12:77–78

Brose U, Williams RJ, Martinez ND (2006) Allometric scaling enhances stability in complex food webs. Ecol Lett 9:1228–1236

Burkepile DE (2013) Comparing aquatic and terrestrial grazing ecosystems: is the grass really greener? Oikos 122:306–312

Butler GJ, Waltman P (1981) Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat. J Math Biol 12:295–310

Butler GJ, Hsu SB, Waltman P (1983) Coexistence of competing predators in a Chemostat. J Math Biol 17:133–151

Chase JM, Leibold MA (2003) Ecological niches: linking classical and contemporary approaches. The University of Chicago Press, Chicago

Chesson P (2000) Mechanisms of maintenance of species diversity. Annu Rev Ecol Syst 31:343–366

Chesson P (2003) Quantifying and testing coexistence mechanisms arising from recruitment fluctuations. Theor Popul Biol 64:345–357

Chesson P (2018) Updates on mechanisms of maintenance of species diversity. J Ecol 106:1773–1794

Chesson P, Warner RR (1981) Environmental variability promotes coexistence in lottery competitive systems. Am Nat 117:923–943

DeAngelis DL (1992) Dynamics of nutrient cycling and food webs. Chapman & Hall, New York

del Giorgio PA, Gasol JM (1995) Biomass distribution in freshwater plankton communities. Am Nat 146:135–152

Droop MR (2003) In defence of the cell quota model of micro-algal growth. J Plankton Res 25:103–107

Dugdale RC (1967) Nutrient limitation in the sea: dynamics, identification, and significance. Limnol Oceanogr 12:685–695

Ellner SP, Snyder RE, Adler PB (2016) How to quantify the temporal storage effect using simulations instead of math. Ecol Lett 19:1333–1342

Ellner SP, Snyder RE, Adler PB, Hooker G (2018) An expanded modern coexistence theory for empirical applications. Ecol Lett 22:3–18

Elser, J.J., I. Loladze, A. L. Peace and Y. Kuang (2012) Lotka re-loaded: Modeling trophic interactions under stoichiometric constraints. Ecological Modelling 245: 3–11

Eppley RW, Rogers JN, McCarthy JJ (1969) Half-saturation constants for uptake of nitrate and ammonium by marine phytoplankton. Limnol Oceanogr 14:912–920

Fredrickson AG, Stephanopoulos G (1981) Microbial competition. Science 213:972–979

Gaedke U, Hochstädter S, Straile D (2002) Interplay between energy limitation and nutritional deficiency: empirical data and food web models. Ecol Monogr 72:251–270

Gause GF (1932) Experimental studies on the struggle for existence. J Exp Biol 9:389–402

Grover JP (1988) Dynamics of competition in a variable environment: experiments with two diatom species. Ecology 69:408–417

Grover JP (1997) Resource competition. Chapman & Hall, London

Grover JP, Holt RD (1998) Disentangling resource and apparent competition: realistic models for plant-herbivore communities. J Theor Biol 191:353–376

Haigh J, Maynard Smith J (1972) Can there be more predators than prey? Theor Popul Biol 3:290–299

Hall SR (2009) Stoichiometrically explicit food webs: feedbacks between resource supply, elemental constraints, and species diversity. Annu Rev Ecol Evol Syst 40:503–528

Hardin G (1960) The competitive exclusion principle. Science 131:1292–1298

Hassel MP, Lawton JH, Beddington JR (1977) Sigmoid functional responses by invertebrate predators and parasitoids. J Anim Ecol 46:249–262

Hasset RP, Cardinale B, Stabler LB, Elser JJ (1997) Ecological stoichiometry of N and P in pelagic ecosystems: comparison of lakes and oceans with emphasis on the zooplankton-phytoplankton interaction. Limnol Oceanogr 42:648–662

Hastings A, Powell T (1991) Chaos in a three-species food chain. Ecology 72:896–903

Hatton IA, McCann KS, Fryxell JM, Davies TJ, Smerlak M, Sinclair ARE, Loreau M (2015) The predator-prey power law: biomass scaling across terrestrial and aquatic biomes. Science 349:1070–1083

Hautier Y, Niklaus PA, Hector A (2009) Competition for light causes plant biodiversity loss after eutrophication. Science 324:636–638

Holling CS (1961) Principles of insect predation. Annu Rev Entomol 6:163–182

Hsu SB, Hubbell SP, Waltman P (1978) A contribution to the theory of competing predators. Ecol Monogr 48:337–349

Hsu SB, Ruan S, Yang T-H (2013) On the dynamics of two-consumers-one-resource competing systems with Beddington-Deangelis functional response. Discrete Continuous Dyn Syst Ser B 18:2331–2353

Hutchinson GE (1961) The paradox of the plankton. Am Nat 95:137–145

Jeschke JM, Kopp M, Tollrian R (2004) Consumer-food systems: why type I functional responses are exclusive to filter feeders. Biol Rev 79:337–349

Jones LE, Ellner SP (2007) Effects of rapid prey evolution on predator–prey cycles. J Math Biol 55:541–573

Keener JP (1985) Oscillatory coexistence in a food chain model with competing predators. J Math Biol 22:123–135

Klauschies T, Vasseur DA, Gaedke U (2016) Trait adaptation promotes species coexistence in diverse predator and prey communities. Ecol Evol 6:4141–4159

Koch AL (1974) Competitive coexistence of two predators utilizing the same prey under constant environmental conditions. J Theor Biol 44:387–395

Kooi BW, Boer MP, Kooijman SALM (1998a) On the use of the logistic equation in models of food chains. Bull Math Biol 60:231–246

Kooi BW, Poggiale JC, Auger P (1998b) Aggregation methods in food chains. Math Comput Model 27:109–120

Kooi BW, Poggiale JC, Auger P, Kooijman SALM (2002) Aggregation methods in food chains with nutrient recycling. Ecol Model 157:69–86

Kooijman SALM (2000) Dynamic energy and mass budgets in biological systems. Cambridge University Press, New York

Kuijper LDJ, Kooi BW, Anderson TR, Kooijman SALM (2004) Stoichiometry and food-chain dynamics. Theor Popul Biol 66:323–339

Lampert W, Sommer U (2007) Limnoecology. The ecology of lakes and streams. Oxford University Press, New York

Lemoine NP, Giery ST, Burkepile DE (2014) Differing nutritional constraints of consumers across ecosystems. Oecologia 174:1367–1137

Leroux SJ, Schmitz OJ (2015) Predator-driven elemental cycling: the impact of predation and risk effects on ecosystem stoichiometry. Ecol Evol 5:4976–4988

Letten AD, Ke P-J, Fukami T (2017) Linking modern coexistence theory and contemporary niche theory. Ecol Monogr 87:161–177

Litchman E, Klausmeier CA (2001) Competition of phytoplankton under fluctuating light. Am Nat 157:170–187

Loladze I, Kuang Y, Elser JJ, Fagan WF (2004) Competition and stoichiometry: coexistence of two predators on one prey. Theor Popul Biol 65:1–15

Lotka AJ (1925) Elements of physical biology. Baltimore, Williams and Wilkins. (reprinted as Elements of mathematical biology (1956) Dover, NY)

MacArthur R, Levins R (1967) Limiting similarity convergence and divergence of coexisting species. Am Nat 101:377–385

McCann K, Yodiz P (1994) Biological conditions for chaos in a three-species food chain. Ecology 75:561–564

McCauley DJ, Gellner G, Martinez ND, Williams RJ, Sandin SA, Micheli F, Mumby PJ, McCann KS (2018) On the prevalence and dynamics of inverted trophic pyramids and otherwise top-heavy communities. Ecol Lett 21:439–454

Michalis L, Menten ML (1913) Die Kinetik der Invertinwerkung. Biochem Z 49:333–369

Mittelbach GG (2012) Community ecology. Oxford Univeristy Press, New York

Moe SJ, Stelzer RS, Forman MR, Harpole WS, Daufresne T, Yoshida T (2005) Recent advances in ecological stoichiometry: insights for population and community ecology. Oikos 109:29–39

Murdoch WW (1973) Functional response of predators. J Appl Ecol 10:335–342

Murray JD (2002) Mathematical biology. I: an introduction. Springer-Verlag, New York

O’Dwyer JP (2018) Whence Lotka-Volterra? Conservation laws and integrable systems in ecology. Theor Ecol 11:441–452

Polis GA (1999) Why are parts of the world green? Multiple factors control productivity and the distribution of biomass. Oikos 86:3–15

Quévreux P, Barot S, Thébault É (2018) Impact of nutrient cycling on food web stability. bioRxiv 276592; https://doi.org/10.1101/276592

Rosenzweig ML (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171:385–387

Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions of predator-prey interactions. Am Nat 97:209

Rothhaupt KO (1988) Mechanistic resource competition theory applied to laboratory experiments with zooplankton. Nature 333:660–662

Santiago LS, Goldstein G (2016) Is photosynthesis nutrient limited in tropical trees? In: Goldstein G, Santiago L (eds) Tropical tree physiology. Springer, Cham, pp 299–315. Tree physiology, volume 6.

Sarma SSS, Iyer N, Dumont HJ (1996) Competitive interactions between herbivorous rotifers: importance of food concentration and initial population density. Hydrobiologia 331:1–7

Sarma SSS, Araiza MAF, Nandini S (1999) Competition between Brachionus calyciflorus Pallas and Brachionus patulus (Müller) (Rotifera) in relation to algal food concentration and initial population density. Aquat Ecol 33:339–345

Sarnelle O, Wilson AE (2008) Type III functional response in daphnia. Ecology 89:1723–1732

Schmitz OJ, Hawlena D, Trussell GC (2010) Predator control of ecosystem nutrient dynamics. Ecol Lett 13:1199–1209

Schoener TW (1974) Some methods of calculating competition coefficients from resource-utilization spectra. Am Nat 108:332–340

Scranton K, Vasseur DA (2016) Coexistence and emergent neutrality generate synchrony among competitors in fluctuating environments. Theor Ecol 9:353–363

Seifert LI, de Castro F, Marquart A, Gaedke U, Weithoff G, Vos M (2014) Heated relations: temperature-mediated shifts in consumption across trophic levels. PLoS One 9:e95046

Shurin JB, Gruner DS, Hillebrandt H (2006) All wet or dried up? Real differences between aquatic and terrestrial food webs. Proc R Soc B 273:1–9

Sibly RM, Hone J (2002) Population growth rate and its determinants: an overview. Philos Trans R Soc B 357:1153–1170

Sieber M, Hilker FM (2012) The hydra effect in predator–prey models. J Math Biol 64:341–360

Smith HL, Waltman P (1995) The theory of the chemostat. Dynamics of microbial competition. Cambridge University Press, New York

Sommer U (1985) Comparison between steady-state and nonsteady state competition-experiments with natural phytoplankton. Limnol Oceanogr 30:335–346

Sommer U (2005) Biologische Meereskunde. Springer-Verlag, Berlin Heidelberg, New York

Sommer, U. and H. Stibor (2002) Copepoda – Cladocera – Tunicata: The role of three major mesozooplankton groups in pelagic food webs. Ecological Research 17: 161–174

Stemberger RS, Gilbert JJ (1985) Body size, food concentration and population growth in planktonic rotifers. Ecology 66:1151–1159

Sterner RW, Elser JJ (2002) Ecological stoichiometry. The biology of elements from molecules to the biosphere. Princeton University Press, New Jersey

Vance RR (1984) Interference competition and the coexistence of two competitors on a single limiting resource. Ecology 65:1349–1357

Vanni MJ (2002) Nutrient cycling by animals in freshwater ecosystems. Annu Rev Ecol Evol Syst 33:341–370

Volterra V (1928) Variations and fluctuations of the number of individuals in animal species living together. ICES J Mar Sci 3:3–51

Williams RJ, Martinez ND (2008) Success and its limits among structural models of complex food webs. J Anim Ecol 77:512–519

Xiao X, Fussmann GF (2013) Armstrong-McGehee mechanism revisited: competitive exclusion and coexistence of non-linear consumers. J Theor Biol 339:26–35

Yamamichi M, Yoshida T, Sasaki A (2011) Comparing the effects of rapid evolution and phenotypic plasticity on predator-prey dynamics. Am Nat 178:287–304

## Acknowledgements

We thank Michael Sieber, Ruben Ceulemans, Alice Boit and two anonymous reviewers for helpful comments and suggestions.

## Funding

Toni Klauschies was funded by the German Research Foundation (DFG, GA 401/26-1).

## Author information

### Authors and Affiliations

### Contributions

Toni Klauschies conceived the idea of this study, developed the mathematics included, coded and simulated the model, analysed the data and wrote the manuscript. Ursula Gaedke commented on the manuscript and improved the writing.

### Corresponding author

## Appendices

### Appendix 1. Necessary conditions and parameter range of potential species coexistence

Following the general methodology of Hsu et al. (1978) and Xiao and Fussmann (2013), we here derive ecological conditions for species coexistence that determine a range of parameters for which species coexistence might be possible. Although these conditions are not sufficient, they are necessary, implying that outside this parameter range species coexistence is impossible. For simplicity, we will assume predator one (*C*_{1}) to exhibit a linear functional response and predator two (*C*_{2}) to have a (non-linear) functional type II response.

### Persistence of predator-prey systems in the absence of exploitative competition

Predator coexistence necessarily requires persistence of the two different predator-prey systems comprising either *C*_{1} or *C*_{2} next to the prey. Hence, each predator has to be able to invade the system that only includes the prey. In the absence of *C*_{1} and *C*_{2}, the prey is approaching its single-species equilibrium\( {R}_S^{\ast }=1-\frac{\delta }{\left(1-{h}_R\delta \right)} \). The two predators thus exhibit positive invasion growth rates when their grazing rates for \( {R}_S^{\ast } \) are larger than the sum of their different loss rates through mortality \( \left({d}_{C_i}\right) \) and dilution (*δ*), respectively, i.e. \( {a}_{C_1}{R}_S^{\ast }>{d}_{C_1}+\delta \) for *C*_{1} and \( \frac{a_{C_2}{R}_S^{\ast }}{1+{h}_{C_2}{a}_{C_2}{R}_S^{\ast }}>{d}_{C_2}+\delta \) for *C*_{2}. Consequently, the death rates that allow persistence of the two different predator-prey systems are limited to \( {d}_{C_1}<{a}_{C_1}\left(1-\frac{\delta }{\left(1-{h}_R\delta \right)}\right)-\delta \) for *C*_{1} and to \( {d}_{C_2}<\frac{a_{C_2}\left(1-\left({h}_R+1\right)\delta \right)}{\left(1-{h}_R\delta \right)+{h}_{C_2}{a}_{C_2}\left(1-\left({h}_R+1\right)\delta \right)}-\delta \) for *C*_{2}. These conditions put natural limits on the two-dimensional parameter space in the \( {d}_{C_1}-{d}_{C_2} \) plane for which predator coexistence might be possible. The first necessary condition for predator coexistence is thus given by the following:

with \( {d}_{C_1}^U={a}_{C_1}\left(1-\frac{\delta }{\left(1-{h}_R\delta \right)}\right)-\delta \) and \( {d}_{C_2}^U=\frac{a_{C_2}\left(1-\left({h}_R+1\right)\delta \right)}{\left(1-{h}_R\delta \right)+{h}_{C_2}{a}_{C_2}\left(1-\left({h}_R+1\right)\delta \right)}-\delta \) denoting the upper limits of the death rates of the two different predators *C*_{1} and *C*_{2}, respectively.

### Invasion of *C*_{2}

The linear functional response of *C*_{1} ensures that the *C*_{1} − *R* system will approach a stable equilibrium with prey density \( {R}_{C_1}^{\ast }=\frac{d_{C_1}+\delta }{a_{C_1}} \). A necessary condition for predator coexistence is that *C*_{2} is able to invade the *C*_{1} − *R* system. Hence, the invasion growth rate of *C*_{2} for \( {R}_{C_1}^{\ast } \) has to be larger than 0, i.e. \( \frac{1}{C_2}\frac{d{C}_2}{dt}=\frac{a_{C_2}{R}_{C_1}^{\ast }}{1+{h}_{C_2}{a}_{C_2}{R}_{C_1}^{\ast }}-{d}_{C_2}-\delta >0 \), which corresponds to \( \frac{a_{C_2}\left({d}_{C_1}+\delta \right)}{a_{C_1}+{h}_{C_2}{a}_{C_2}\left({d}_{C_1}+\delta \right)}-{d}_{C_2}-\delta >0 \). Therefore, the second necessary condition for predator coexistence is given by the following:

This implies that the equilibrium density of the prey in the *C*_{2} − *R* system has to be smaller than \( {R}_{C_1}^{\ast } \), i.e. \( {R}_{C_2}^{\ast }=\frac{d_{C_2}+\delta }{\left({a}_{C_2}-\left({d}_{C_2}+\delta \right){h}_{C_2}{a}_{C_2}\right)}<{R}_{C_1}^{\ast } \). In other words, the minimum prey requirement of *C*_{2} has to be lower than the minimum prey requirement of *C*_{1}. In contrast, when \( {R}_{C_2}^{\ast }>{R}_{C_1}^{\ast } \), *C*_{2} cannot invade the *C*_{1} − *R* system.

### Competitive exclusion of *C*_{1}

Since the curvature of the type II functional response is concave, the time-averaged per capita net growth rate of *C*_{2} will always be larger than or equal to the time-averaged per capita net growth rate of a corresponding predator that has a linear functional response with the same per capita net growth rates at the minimum (*R*_{min} = 0) and the maximum \( \left({R}_{\mathrm{max}}={R}_S^{\ast }=1-\frac{\delta }{\left(1-{h}_R\delta \right)}\right) \) feasible prey densities as *C*_{2}. Hence, the following inequality holds:

with \( {\overline{R}}_{C_2}=\int R(t) dt \)being the time-averaged prey density of the *C*_{2} − *R* system. Since \( \int \frac{1}{C_2}\frac{d{C}_2}{dt} dt=0 \) at equilibrium, the following inequality holds: \( \frac{a_{C_2}{\overline{R}}_{C_2}}{1+{h}_{C_2}{a}_{C_2}{R}_S^{\ast }}-{d}_{C_2}-\delta \le 0 \). Setting the term \( \frac{a_{C_2}}{1+{h}_{C_2}{a}_{C_2}{R}^{\ast }}{\overline{R}}_{C_2}-{d}_{C_2}-\delta \) to 0 and solving for \( {\overline{R}}_{C_2} \) thus gives us an upper limit for the temporal average of the prey density in the corresponding predator-prey system that is denoted by\( {\overline{R}}_{C_2}^U=\frac{\left({d}_{C_2}+\delta \right)}{a_{C_2}}\left(1+{h}_{C_2}{\mathrm{a}}_{C_2}{R}_S^{\ast}\right) \). Therefore, whenever *C*_{1} has a minimum prey requirement \( {R}_{C_1}^{\ast } \)higher than \( {\overline{R}}_{C_2}^U \), it will inevitably be outcompeted by *C*_{2}. Consequently, the following inequality has to be met for the two predators to be able to coexist:

This leads to the following third necessary condition of predator coexistence:

Nevertheless, \( {R}_{C_1}^{\ast }<{\overline{R}}_{C_2}^U \)does not ensure that *C*_{1} can invade the *C*_{2} − *R* system. For this to happen, the minimum prey requirement of *C*_{1} has to be lower than the actual temporal average of the prey density in the corresponding predator-prey system, i.e. \( {R}_{C_1}^{\ast }<{\overline{R}}_{C_2} \).

### Hopf bifurcation in predator-prey models with type II functional response

Predator coexistence through relative non-linearity in their functional responses requires oscillations in the abundances of the different species. It is well known from the theory of non-linear dynamics that limit cycles only emerge in systems of ordinary differential equations with more than 2 dimensions when there is some degree of non-linearity, i.e. non-linear density-dependence, in the per capita rates of the corresponding state variables. Hence, in the predator-prey system defined by Eqs. 9a, 9b and 9c, oscillations can only occur in the presence of *C*_{2}. Therefore, we performed a linear stability analysis of the two-dimensional *R* − *C*_{2} system, in order to evaluate the influence of nutrient retention by the predators, the effect of a non-linear resource uptake rate of the prey and the impact of a shared loss rate for all species on the stability of the predator-prey dynamics. For simplicity, we will neglect the index of predator two for its state variable and parameters and thus consider the following predator-prey model:

Throughout the analysis, we will focus on parameter values where the predator is able to persist, i.e. *d* < *d*^{U}. To start, we calculate the interior equilibrium of the prey (*R*^{∗}) and predator (*C*^{∗}) species for the different model parametrizations used in the main text by setting Eqs. 11a and 11b to 0, i.e. \( \frac{dR}{dt}=0 \) and \( \frac{dC}{dt}=0 \), and solving for the corresponding values of the state variables satisfying these two conditions (for the calculated equilibria see Appendix Table 1). Subsequently, we will calculate the entries of the Jacobian matrix of the linearized predator-prey system and evaluate them at the interior equilibrium in order to describe the local behaviour of the predator-prey system closely around its equilibrium state and to determine its local stability. The stability of the interior equilibrium depends on the properties of the Jacobian matrix:

The predator-prey model defined by Eqs. 9a, 9b and 9c (main text) does not account for interference or cooperation among individual predators or any other kind of direct density-dependence in *C*. Consequently, the lower right entry of *J* is 0, i.e. \( {\left.\partial \dot{C}/\partial C\right|}_{R={R}^{\ast },C={C}^{\ast }}=0 \). Furthermore, an increase in the density of the prey has always a positive effect on the predator density due to its type II functional response, which is a monotonically increasing function in *R*. Hence, the lower left entry of *J* is strictly positive, i.e. \( {\left.\partial \dot{C}/\partial R\right|}_{R={R}^{\ast },C={C}^{\ast }}>0 \). In contrast, an increase in the predator density will always have a negative effect on the prey density through grazing and nutrient retention. Therefore, the upper right entry of *J* is strictly negative, i.e. \( {\left.\partial \dot{R}/\partial C\right|}_{R={R}^{\ast },C={C}^{\ast }}<0 \). Finally, the sign of the upper left entry of *J*, i.e. \( {\left.\partial \dot{R}/\partial R\right|}_{R={R}^{\ast },C={C}^{\ast }} \), will depend on the actual parameter values, e.g. the death rate of the predator. Hence, the entries of *J* possess the following signs:

According to the Routh-Hurwitz criterion for a two-dimensional system of ordinary differential equations, the interior equilibrium is stable when the determinant of the Jacobian matrix evaluated at the interior equilibrium (*J*) is positive and the corresponding trace is negative (Murray 2002). The determinant of the Jacobian matrix is given by the following:

In line with our reasoning above, the sign of Det(*J*) is as follows:

Therefore, the determinant of the Jacobian will always be positive. Consequently, the stability of the interior equilibrium entirely depends on the trace of the Jacobian matrix, which is given by the following:

Since \( {\left.\partial \dot{C}/\partial C\right|}_{R={R}^{\ast },C={C}^{\ast }}=0 \), the trace of the Jacobian is given by the derivative of the rate of change of the prey with respect to its own density. Hence, we calculated \( {\left.\partial \dot{R}/\partial R\right|}_{R={R}^{\ast },C={C}^{\ast }} \) for the different model parametrizations used in the main text and generally solved for the death rate at which the trace of the Jacobian is 0. At this particular point (*d*^{H}), a Hopf bifurcation occurs, separating two regions for which the system exhibits either a stable equilibrium (*d* > *d*^{H}) or a limit cycle (*d* < *d*^{H}) (cf. Fig. 4c, d). In general, the Hopf bifurcation point *d*^{H} has to be calculated numerically. However, in some cases, explicit expressions for \( {d}_{C_2}^H \) can be obtained (for details, see Appendix Table 1).

For instance, in the absence of nutrient retention by the predators (*η* = 0) and in case of a linear resource uptake rate of the prey (*h*_{R} = 0) and batch culture conditions (*δ* = 0), the Hopf bifurcation point *d*^{H} of the classical predator-prey system (Rosenzweig and MacArthur 1963) with logistically growing prey in the vacancy of the predator is given by the following (see also Abrams and Holt 2002):

Since *d* > 0, a Hopf bifurcation only occurs when *ah* > 1, so that *d*^{H} is also positive.

Furthermore, nutrient retention by the predators generally reduces the Hopf bifurcation point of the corresponding *C* − *R* system and thus stabilizes its population dynamics. This can be seen when choosing a particular value for the nutrient to carbon ratio *η* of the predator, e.g. *η* = 1. In this case, the Hopf bifurcation point *d*^{H} can be written as follows:

The stabilizing effect of nutrient retention on the predator-prey dynamics is stronger, when the non-linearity of the type II functional response is weaker. In contrast to the stabilizing effect of nutrient retention by the predators, a non-linear resource uptake rate of the prey has a destabilizing effect on the predator-prey dynamics. When accounting for non-linearity in the preys’ resource uptake rate (*h*_{R} > 0), the Hopf bifurcation point of the corresponding predator-prey system is given as follows:

Hence, *d*^{H} increases with an increase in *h*_{R}. The impact of *h*_{R} on the overall stability of the predator-prey dynamics depends on the curvature of the type II functional response of the predator. The larger the product *ah* is, the larger the factor \( {\left(1+{h}_R\left(1+\frac{1}{ah}\right)\right)}^{-\frac{1}{2}} \) becomes, and thus, the smaller the term \( 1-{\left(1+{h}_R\left(1+\frac{1}{ah}\right)\right)}^{-\frac{1}{2}} \) will be. Hence, the destabilizing effect of *h*_{R} on the predator-prey dynamics is larger when the non-linearity of the type II functional response of the predator is weaker.

Finally, a common death rate of all species through the dilution rate *δ* is stabilizing the predator-prey dynamics. The Hopf bifurcation point of the corresponding *C* − *R* system, i.e. when *δ* > 0, is given by the following:

Hence, *d*^{H} decreases when *δ* is increasing. The stabilizing effect of *δ* on the predator-prey dynamics is larger when the non-linearity of the type II functional response of the predator is stronger and, thus, when the handling time *h* is higher. This happens, because the dilution rate does affect not only the stability of the predator-prey dynamics but also the persistence of the corresponding *C* − *R* system.

### Quantifying the size of the parameter range of potential species coexistence

The necessary conditions *N*_{2}, *N*_{3} and *N*_{4} of predator coexistence derived above determine a range of parameters that may permit the coexistence of the two different predators (Fig. 6a). In order to quantify the size of this parameter region, we first calculated three different threshold values of the death rate of *C*_{1}, \( {d}_{C_1}^{T_i} \), that allows piecewise integration of the relevant functions. The first threshold value \( {d}_{C_1}^{T_1} \) is given by the death rate \( {d}_{C_1} \) for which the function \( {N}_2\left({d}_{C_1}\right)=\frac{a_{C_2}\left({d}_{C_1}+\delta \right)}{a_{C_1}+{h}_{C_2}{a}_{C_2}\left({d}_{C_1}+\delta \right)}-\delta \) (reflecting the second necessary condition) intersects with the Hopf bifurcation point \( {d}_{C_2}^H \) (denoting the fourth necessary condition), i.e. \( {N}_2\left({d}_{C_1}^{T_1}\right)={d}_{C_2}^H \) (cf. Fig. 6b). Solving this equation with respect to \( {d}_{C_1}^{T_1} \) leads to the following expression of the first threshold value:

Accordingly, the second threshold value \( {d}_{C_1}^{T_2} \) is given by the intersection point of the function \( {N}_3\left({d}_{C_1}\right)=\frac{\left({d}_{C_1}+\delta \right){a}_{C_2}}{a_{C_1}\left(1+{h}_{C_2}{a}_{C_2}{R}_S^{\ast}\right)}-\delta \) (representing the third necessary condition) with the Hopf bifurcation point \( {d}_{C_2}^H \), i.e. \( {N}_3\left({d}_{C_1}^{T_2}\right)={d}_{C_2}^H \) (cf. Fig. 6c). Evaluating this equation with respect to \( {d}_{C_1}^{T_2} \) results into the following expression of the second threshold value:

Finally, the third threshold value \( {d}_{C_1}^{T_3} \) is given by the value of \( {d}_{C_1} \) at which the function \( {N}_3\left({d}_{C_1}\right) \) equals 0, i.e. \( {N}_2\left({d}_{C_1}^{T_3}\right)=0 \) (cf. Fig. 6d). Solving the latter for \( {d}_{C_1}^{T_3} \) results into the following expression of the third threshold value:

Using these three different threshold values, the absolute size of the parameter region of potential predator coexistence is given by the following sum of integrals (cf. Fig. 6):

After having solved the first integral with *Wolfram Alpha*, we obtain the following expression of *P*_{a}:

Note, although it is not explicitly stated, *P*_{a} also depends on \( {\eta}_{C_2} \) through the dependency of the Hopf bifurcation point \( {d}_{C_2}^H \) on \( {\eta}_{C_2} \). However, *P*_{a} does not depend on \( {\eta}_{C_1} \). This strongly contrasts with the fact that the parameter range of realized species coexistence will likely also depend on \( {\eta}_{C_1} \) and potential differences between \( {\eta}_{C_1} \)and \( {\eta}_{C_2} \).

### Appendix 2. Fluctuations in the prey abundance act as a stabilizing mechanism of predator coexistence

Contemporary theory suggests that species coexistence depends on the balance between stabilizing niche differences and destabilizing fitness differences among species (Chesson 2000; Barabás et al. 2018). Niche differences enable species coexistence by increasing intra- relative to interspecific competition, thereby facilitating the regrowth of species from low densities (Chesson 2018). In contrast, fitness differences promote competitive exclusion of inferior species in the absence of niche differentiation (Letten et al. 2017). Hence, two or more predators cannot coexist on one prey in the absence of niche differences, because the presence of inevitable fitness differences among the different predators will drive all but one of them extinct (Chesson 2000). Stabilizing niche differences between two different predators may arise through relative non-linearity in their functional responses and associated fluctuations in the prey abundance. We therefore here derive corresponding terms reflecting average fitness differences and stabilizing niche differences in our predator-prey model defined by Eqs. 9a, 9b and 9c. Following Chesson (2018), average fitness differences between the different predators can be inferred from their invasion growth rates in the absence of stabilizing niche differences. Hence, we calculated the invasion growth rates, i.e. temporal averages of the per capita net growth rates, of both predators for the corresponding resident system that comprises the other predator and the prey at its long-term equilibrium, which is assumed to be stable. Hence, the invasion growth rate of predator one for the equilibrium density of the prey in the corresponding resident predator-prey system comprising predator two is given by the following:

Similarly, the invasion growth rate of predator two for the equilibrium density of the prey in the corresponding resident predator-prey system comprising predator one is given by the following:

Comparing the invasion growth rates of both predators reveals that the competitive outcome entirely depends on the difference between \( {R}_{C_2}^{\ast } \) and \( {R}_{C_1}^{\ast } \). Hence, the average fitness differences (FD) between the two predators are given by the difference between the prey requirements of predator two \( \left({R}_{C_2}^{\ast}\right) \) and predator one \( \left({R}_{C_1}^{\ast}\right) \), i.e. \( {R}_{C_2}^{\ast }-{R}_{C_1}^{\ast } \).

According to the coexistence mechanism of relative non-linearity in the predator’s functional responses, stabilizing niche differences between the two different predators only arise through predator-prey oscillations. Given that the linear functional response of predator one ensures that the equilibrium of the corresponding predator-prey system is always stable, additional terms cannot arise in the invasion growth rate of predator two. However, the type II functional response of predator two may destabilize the equilibrium of the corresponding predator-prey system. Under such conditions, the invasion growth rate of predator one is given as follows:

Comparing the invasion growth rate of predator one for the resident system of predator two in the absence of predator-prey oscillations, which prevents coexistence, with the one in the presence of predator-prey oscillations, which potentially allows coexistence, shows that the stabilizing niche differences (ND) between the two predators are given by the difference between \( {\overline{R}}_{C_2} \) and \( {R}_{C_2}^{\ast } \), i.e. \( {\overline{R}}_{C_2}-{R}_{C_2}^{\ast } \). Hence, the net-stabilizing effect of relative non-linearity in the predators’ functional response on predator coexistence is thus given by the sum of stabilizing niche differences (+) and destabilizing fitness differences (−), i.e. \( ND+ FD={\overline{R}}_{C_2}-{R}_{C_2}^{\ast }+{R}_{C_2}^{\ast }-{R}_{C_1}^{\ast }={\overline{R}}_{C_2}-{R}_{C_1}^{\ast } \).

### Appendix 3. Fourier analysis of predator-prey oscillations

We found two different types of population dynamics in case of predator coexistence. For most parameter combinations, simple predator-prey oscillations emerged with the increase of the prey followed by an increase in the two predators (e.g. Fig. 5e). For some other parameter combinations, the competition between *C*_{1} and *C*_{2} for the shared prey generated much more complex predator-prey dynamics (e.g. Fig. 5f). To evaluate the differences in the population dynamics in more detail, we performed a discrete Fourier transformation by using the *fft* function implemented in MATLAB, version 7.13 (The MathWorks Inc., Natick, MA, 2011).

Our results show that for the simple predator-prey oscillations, most of the variation in the species’ abundances can be attributed to a single dominate time scale (Fig. 7a, c, e). Here, the two predators are strongly synchronized, exhibiting a negligible phase difference of 0.03 *π* (cf. Fig. 7b, d). In contrast, for the more complex population dynamics, our analysis reveals two dominant time scales (Fig. 8a, c, e), which can be most clearly seen from the amplitude spectrum of *C*_{1} (Fig. 8c). At the low frequency, the population dynamics of the two predators exhibit a phase difference of 0.74 *π* (cf. Fig. 8b, d). Hence, their abundances cycle almost anti-synchronously, meaning that an increase in the abundance of *C*_{1} coincides with a decrease in the abundance of *C*_{2} and vice versa (Fig. 8d). In contrast to the prey (*R*) and *C*_{1}, *C*_{2} showed hardly any variation in its abundance at the high frequency (Fig. 8f), although this shorter time scale in the population dynamics is generated by the predator-prey interaction between *R* and *C*_{2}. This discrepancy can be explained by the fact that the destabilization of the predator-prey oscillations by *C*_{2} selects immediately for the recovery of *C*_{1} (Fig. 8b).

### Appendix 4. The opposing effects of nutrient retention and nutrient recycling on the stability of predator-prey dynamics

Our results presented in the main text and Appendix 1 demonstrate that the retention of nutrients in the predator biomass stabilizes predator-prey dynamics because it effectively reduces the carrying capacity of the prey. In contrast, previous studies have shown that the recycling of nutrients from dead organisms destabilizes predator-prey dynamics (e.g. DeAngelis 1992; Kooi et al. 2002). Hence, non-consumptive effects on the prey may enhance or reduce the stability of food webs, depending on whether the stabilizing effects of nutrient retention or the destabilizing effects of nutrient recycling prevail.

To clarify the relative importance of nutrient retention and nutrient recycling for system stability, we extended the predator-prey model used in the main text by two detrital components, thereby incorporating a time lag in the recycling of nutrients from dead predators and prey (cf. DeAngelis 1992). We track the recycling from dead predators and prey separately because they generally contain different amounts of nutrients per unit of biomass. For convenience, we consider here one predator species only. Hence, the amount of dissolved inorganic nutrients (*N*) and the biomasses of the dead predators (*D*_{C}) and prey (*D*_{R}) and the living predators (*C*) and prey (*R*) are changing over time according to the following set of equations:

In contrast to Eqs. 1a, 1b, 1c and 1d of the main text, nutrient recycling does not occur immediately but rather passes through a detrital component. The time needed for the recycling of nutrients contained in the dead organisms into the inorganic nutrient pool scales with \( \frac{1}{m_R} \) and \( \frac{1}{m_C} \) for the prey and predators, respectively. A complete description of the other terms and parameters can be found in “Methods”.

The predator-prey system described by Eqs. 12a, 12b, 12c, 12d and 12e is mass balanced. Hence, the total amount of nutrients is given by *N*_{T} = *N* + *η*_{R}(*R* + *D*_{R}) + *η*_{C}(*C* + *D*_{C}). As a result, we can follow the general approach described in the methods and reduce the number of state variables of Eqs. 12a, 12b, 12c, 12d and 12e. The dimensionally reduced predator-prey system is then given as follows:

To further simply the analysis of our predator-prey system described by Eqs. 13a, 13b, 13c and 13d, we now consider a separation of time scales between the predator-prey dynamics (slow dynamics) and the recycling of nutrients from dead organisms (fast dynamics) (cf. Kooi et al. 1998b). Hence, we set \( \frac{d{D}_R}{dt} \) and \( \frac{d{D}_C}{dt} \) always equal to 0 and then substitute the resulting algebraic expression into Eq. 13c (cf. O’Dwyer 2018). This results in the following approximation of our predator-prey system:

In the main text and Appendix 1, we showed that an increase in *η*_{R} and *η*_{C} has a stabilizing effect on the predator-prey dynamics. Consequently, an increase in *m*_{R} and *m*_{C} will have a destabilizing effect on the population dynamics of our predator-prey system described by Eqs. 14a and 14b because *m*_{R} and *m*_{C} scale inversely with *η*_{R} and *η*_{C}, respectively. Hence, our results confirm previous findings by e.g. DeAngelis (1992) and Kooi et al. (2002) that nutrient recycling has a destabilizing effect on the predator-prey dynamics (Fig. 9). This happens because nutrient recycling effectively enhances the carrying capacity of the prey (cf. Eqs. 14a). In fact, in the absence of nutrient recycling (endogenously driven nutrient replenishment), i.e. *m*_{R} = 0 and *m*_{C} = 0, and dilution (exogenously driven nutrient replenishment), i.e. *δ* = 0, the amount of nutrients in the living predator and prey biomasses will decrease over time because the nutrients accumulate in the detrital compartments.

## Rights and permissions

## About this article

### Cite this article

Klauschies, T., Gaedke, U. Nutrient retention by predators undermines predator coexistence on one prey.
*Theor Ecol* **13, **183–208 (2020). https://doi.org/10.1007/s12080-019-00440-y

Received:

Accepted:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s12080-019-00440-y

### Keywords

- Coexistence
- Consumer-resource oscillations
- Predator-prey cycles
- Relative non-linearity between functional responses
- Nutrient retention
- Nutrient recycling