The Rosenzweig–MacArthur system via reduction of an individual based model
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.
KeywordsPredator–prey model Holling disk function Functional response Singular perturbation theory Tikhonov–Fenichel parameters
Mathematics Subject Classification92D40 34C20
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.
- Arnold VI (1973) Ordinary differential equations. MIT Press, CambridgeGoogle 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
- Gotelli NJ (1995) A primer of ecology. Sinauer Associates, SunderlandGoogle Scholar
- Gurney WSC, Nisbet RM (1998) Ecological dynamics. Oxford Univ. Press, OxfordGoogle Scholar
- Kruff N, Walcher S (2017) Coordinate-independent criteria for Hopf bifurcations. Preprint. arXiv:1708.06545
- 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
- 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