# The Rosenzweig–MacArthur system via reduction of an individual based model

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## Abstract

The Rosenzweig–MacArthur system is a particular case of the Gause model, which is widely used to describe predator–prey systems. In the classical derivation, the interaction terms in the differential equation are essentially derived from considering handling time vs. search time, and moreover there exist derivations in the literature which are based on quasi-steady state assumptions. In the present paper we introduce a derivation of this model from first principles and singular perturbation reductions. We first establish a simple stochastic mass action model which leads to a three-dimensional ordinary differential equation, and systematically determine all possible singular perturbation reductions (in the sense of Tikhonov and Fenichel) to two-dimensional systems. Among the reductions obtained we find the Rosenzweig–MacArthur system for a certain choice of small parameters as well as an alternative to the Rosenzweig–MacArthur model, with density dependent death rates for predators. The arguments to obtain the reductions are intrinsically mathematical; no heuristics are employed.

## Keywords

Predator–prey model Holling disk function Functional response Singular perturbation theory Tikhonov–Fenichel parameters## Mathematics Subject Classification

92D40 34C20## Notes

### Acknowledgements

We thank two anonymous referees and the Editor-in-chief, Mats Gyllenberg, for helpful criticism and comments on earlier versions, and in particular for alerting us to a number of references. The competitiveness of system 3 was noted by the second referee, as well as the applicability of Poincaré-Bendixson theory.

## Supplementary material

## References

- Arnold VI (1973) Ordinary differential equations. MIT Press, CambridgeGoogle Scholar
- Arnold VI (1988) Geometrical methods in the theory of ordinary differential equations, 2nd edn. Springer, New YorkCrossRefGoogle Scholar
- Cosner C, DeAngelis DL, Ault JS, Olson DB (1999) Effects of spatial grouping on the functional response of predators. Theor Popul Biol 56:65–75CrossRefzbMATHGoogle Scholar
- Decker W, Greuel GM, Pfister G, Schönemann H (2016) Singular 4-1-0—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de Accessed 12 Dec 2017
- Dawes JHP, Souza MO (2013) A derivation of Holling’s type I, II and III functional responses in predator–prey systems. J Theor Biol 327:11–22MathSciNetCrossRefzbMATHGoogle Scholar
- Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equations. J Differ Equ 31(1):53–98MathSciNetCrossRefzbMATHGoogle Scholar
- Gause GF (1934) The struggle for existence. Williams and Wilkins, BaltimoreCrossRefzbMATHGoogle Scholar
- Geritz SAH, Gyllenberg M (2012) A mechanistic derivation of the DeAngelis–Beddington functional response. J Theor Biol 314:106–108MathSciNetCrossRefzbMATHGoogle Scholar
- Geritz SAH, Gyllenberg M (2013) Group defence and the predator’s functional response. J Math Biol 66:705–717MathSciNetCrossRefzbMATHGoogle Scholar
- Geritz SAH, Gyllenberg M (2014) The DeAngelis–Beddington functional response and the evolution of timidity of prey. J Theor Biol 359:37–44MathSciNetCrossRefGoogle Scholar
- Goeke A, Walcher S (2014) A constructive approach to quasi-steady state reduction. J Math Chem 52:2596–2626MathSciNetCrossRefzbMATHGoogle Scholar
- Goeke A, Walcher S, Zerz E (2015) Determining “small parameters” for quasi-steady state. J Differ Equ 259:1149–1180MathSciNetCrossRefzbMATHGoogle Scholar
- Goeke A, Walcher S, Zerz E (2017) Classical quasi-steady state reduction—a mathematical characterization. Physica D 345:11–26MathSciNetCrossRefzbMATHGoogle Scholar
- Gotelli NJ (1995) A primer of ecology. Sinauer Associates, SunderlandGoogle Scholar
- Gurney WSC, Nisbet RM (1998) Ecological dynamics. Oxford Univ. Press, OxfordGoogle Scholar
- Heijmans HJAM (1984) Holling’s “hungry mantid” model for the invertebrate functional response considered as a Markov process. Part III: stable satiation distribution. J Math Biol 21:115–143MathSciNetCrossRefzbMATHGoogle Scholar
- Heineken FG, Tsuchiya HM, Aris R (1967) On the mathematical status of the pseudo-steady state hypothesis of biochemical kinetics. Math Biosci 1:95–113CrossRefGoogle Scholar
- Hek G (2010) Geometric singular perturbation theory in biological practice. J Math Biol 60:347–386. https://doi.org/10.1007/s00285-009-0266-7 MathSciNetCrossRefzbMATHGoogle Scholar
- Jeschke JM, Kopp M, Tollrian R (2002) Predator functional responses: discriminating between handling and digesting prey. Ecol Monogr 72(1):95–112CrossRefGoogle Scholar
- Kruff N, Walcher S (2017) Coordinate-independent criteria for Hopf bifurcations. Preprint. arXiv:1708.06545
- Korobeinikov A, Shchepakina E, Sobolev V (2016) Paradox of enrichment and system order reduction: bacteriophages dynamics as case study. Math Med Biol 33(3):359–369MathSciNetCrossRefzbMATHGoogle Scholar
- Kurtz TG (1980) Relationships between stochastic and deterministic population models. In: Jäger W, Rost H, Tautu P (eds) Biological growth and spread: mathematical theories and applications, proceedings of a conference held at Heidelberg, July 16–21, 1979. Springer lecture notes in biomathematics (38). Springer, New York pp 449–467Google Scholar
- Logan JD, Ledder G, Wolesensky W (2009) Type II functional response for continuous, physiologically structured models. J Theor Biol 259:373–381MathSciNetCrossRefzbMATHGoogle Scholar
- Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, BaltimorezbMATHGoogle Scholar
- Marsden J, McCracken M (1976) The Hopf bifurcation and its applications. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Metz JAJ (2005) Eight personal rules for doing science. J Evol Biol 18(5):1178–1181CrossRefGoogle Scholar
- Metz JAJ, van Batenburg FHD (1985a) Holling’s “hungry mantid” model for the invertebrate functional response considered as a Markov process. Part I: the full model and some of its limits. J Math Biol 22:209–231MathSciNetCrossRefzbMATHGoogle Scholar
- Metz JAJ, van Batenburg FHD (1985b) Holling’s “hungry mantid” model for the invertebrate functional response considered as a Markov process. Part II: negligible handling time. J Math Biol 22:239–257MathSciNetCrossRefzbMATHGoogle Scholar
- Metz JAJ, Diekmann O (1986) A gentle introduction to structured population models: three worked out examples. In: Metz JAJ, Diekmann O (eds) The dynamics of physiologically structured populations, vol 68. Lecture notes in biomathematics. Springer, Berlin, pp 3–28CrossRefGoogle Scholar
- Murray JD (2002) Mathematical biology. I. An introduction, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
- O‘Malley RE (1991) Singular perturbation methods for ordinary differential equations. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Perko L (2001) Differential equations and dynamical systems, 3rd edn. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Poggiale JC (1998) Predator–prey models in heterogeneous environment: emergence of functional response. Math Comput Model 27(4):63–71MathSciNetCrossRefzbMATHGoogle Scholar
- Rosenzweig ML, MacArthur RH (1963) Graphical representation and stability conditions of predator–prey interactions. Am Nat 47:209–223CrossRefGoogle Scholar
- Real LA (1979) Ecological determinants of functional response. Ecology 60(3):481–485CrossRefGoogle Scholar
- Rosenzweig ML (1971) The paradox of enrichment. Science 171:385–387CrossRefGoogle Scholar
- Segel LA, Slemrod M (1989) The quasi-steady-state assumption: a case study in perturbation. SIAM Rev 31:446–477MathSciNetCrossRefzbMATHGoogle Scholar
- Smith HL, Waltman P (1995) The theory of the chemostat. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- Smith HL (1995) Monotone dynamical systems. American Mathematical Society, ProvidenceGoogle Scholar
- Tikhonov AN (1952) Systems of differential equations containing a small parameter multiplying the derivative. Math Sb 31:575–586 (in Russian)Google Scholar
- Turchin P (2003) Complex population dynamics. Princeton University Press, PrincetonzbMATHGoogle Scholar
- Verhulst F (2005) Methods and applications of singular perturbations. Boundary layers and multiple timescale dynamics. Springer, New YorkCrossRefzbMATHGoogle Scholar
- Volterra V (1931) Leçons sur la théorie mathématique de la lutte pour la vie. Gauthier-Villars, PariszbMATHGoogle Scholar
- Walcher S (1993) On transformations into normal form. J Math Anal Appl 180:617–632MathSciNetCrossRefzbMATHGoogle Scholar
- Walcher S (2000) On the Poincaré problem. J Differ Equ 166:51–78CrossRefzbMATHGoogle Scholar