DOI: http://dx.doi.org/10.18273/revint.v35n1-2017008

Original article

A model of competing species that exhibits zip bifurcation

Un modelo de especies en competencia que exhibe bifurcación zip

 

Luis F. Echeverri¹

Óscar I. Giraldo²,

Edwin Zarrazola¹

 

¹Universidad de Antioquia, Instituto de Matemáticas, Medellín, Colombia

²Universidad Nacional de Colombia, Escuela de Matemáticas, Medellín, Colombia

 

E-mail: lfecheve@mat.ucm.es

 

Abstract:

The purpose of this paper is to present a concrete model of competing population species that exhibits a phenomenon called zip bifurcation. The Zip Bifurcation was introduced by Farkas in 1984 for a three dimensional ODE prey-predator system describing a chemostat. We will study a three dimensional system of ordinary differential equations that model the competition of two predators species for one single prey species. The system is based on concrete trigonometric functions modeling the growth rate of the prey and the functional response of the predator. The model exhibits different kinds of behavior and shows examples of the so called “competitive exclusión principle,” and the competition of one “r-strategist” and one “K-strategist.” Additionally, in order to illustrate the zip bifurcation, we will present some numerical simulations for our model.

Keywords: Predator prey model, Zip bifurcation, r-strategist, K-strategist

 

Resumen:

El objetivo de este trabajo es presentar un modelo concreto de poblaciones de especies en competición que exhibe la bifurcación Zip. La bifurcación zip fue introducida por Farkas en 1984 para un sistema tridimensional de ecuaciones diferenciales ordinarias que describe un quimiostato. Estudiaremos un sistema tridimensional de ecuaciones diferenciales ordinarias que modela la competición de dos poblaciones distintas de predadores por una única población presa. El sistema usa funciones trigonométricas concretas para representar la tasa de crecimiento de la presa y la respuesta funcional del predador. El modelo exhibe diferentes clases de comportamientos y muestra ejemplos de los llamados principio de exclusión competitiva y la competición

Palabras clave: Modelo depredador-presa, Bifurcación Zip, r-estratega, K-estratega

 

Received: 20 June 2016,

Accepted: 22 June 2017.

 

Texto complete disponible en PDF

 

 

References

1. Angulo F., Olivar G., Osorio G.A., Escobar C.M., Ferreira J.D. and Redondo J.M., “Bifurcations of non-smooth systems”, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), No. 12, 4683–4689.

2. Bocsó A. and Farkas M.,“Political and economic rationality leads to velcro bifurcation”, Appl. Math. Comput. 140 (2003), No. 2–3, 381–389.

3. Butler G.J. and Waltman P., “Bifurcation from a limit cycle in a two predator one prey ecosystem modeled in a chemostat”, J. Math. Biol. 12 (1981), No. 3, 295–310.

4. Butler G.J., Hsu S.B. and Waltman P., “Coexistence of competing predators on a chemostat”, J. Math. Biol. 17 (1983), No. 2, 133–151.

5. Butler G.J., “Competitive predator-prey systems and coexistence”, in: Population Biology Proceedings, Edmonton 1982, Lecture Notes in Biomath. 52, Springer-Verlag (1983), 210– 217.

6. Escobar-Callejas C.M., Gonzalez-Granada J.R. and Posso-Agudelo A.E., “Atractividad local en la bifurcación zip”, Ingeniería y Ciencia 6 (2010), No. 12, 11–41.

7. Farkas M., “Zip Bifurcation in a competition model”, Nonlinear Anal. 8 (1984), No. 11, 1295–1309.

8. Farkas M., “Competitive exclusion by zip bifurcation”, in: Dynamical systems, IIASAWorkshop, Sopron, Lecture Notes in Econom. and Math. Systems 287, Springer-Verlag (1987), 165–178.

9. Farkas M., Periodic Motions, Springer, New York, 1994.

10. Farkas M., Dynamical Models in Biology, Academic Press, San Diego, 2001.

11. FarkasM., Sáez E. and Szántó I., “Velcro bifurcation in competition models with generalized Holling functional response”, Miskolc Math. Notes 6 (2005), No. 2, 185–195.

12. Farkas M. and Ferreira J.D., “Zip bifurcation in a reaction-diffusion system”, Differ. Equ. Dyn. Syst. 15 (2007), No. 3-4, 169–183.

13. Ferreira J.D., “Zip bifurcation in an ample class of competitive systems”, Miskolc Math. Notes 8 (2007), No. 2, 147–156.

14. Ferreira J.D. and De Oliveira L.A.F., “Zip bifurcation in a competitive system with diffusion”, Differ. Equ. Dyn. Syst. 17 (2009), No. 1–2, 37–53.

15. Ferreira J.D., and Sree Hari Rao V., “Unsustainable zip-bifurcation in a predator-prey model involving discrete delay”, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), No. 6, 1209–1236.

16. Hardin G., “The competitive exclusion principle”, Science 131 (1960), No. 3409, 1292–1297.

17. Kalmykov L. and Kalmykov V., “Verification and reformulation of the competitive exclusión principle”, Chaos Solitons Fractals 56 (2013), 124–131.

18. May R.M. and McLean A.R., Theoretical ecology. Principles and applications, Oxford University Press, Oxford, 2007.

19. Sáez E., Stange E. and Szántó I., “Simultaneous zip bifurcation and limit cycles in three dimensional competition models”, SIAM J. Appl. Dyn. Syst. 5 (2006), No. 1, 1–11.

20. Smith H.L. and Waltman P., The theory of the chemostat. Dynamics of microbial competition, Cambridge Studies in Mathematical Biology 13, Cambridge University Press, Cambridge, 1995.

21. Waltman P., Hsu S.H. and Hubbell S.P., “Theoretical and experimental investigations of microbial competition in continuous cultures. Modelling and differentiel equations in Biology”, in Lecture Notes in Pure and Applied Mathematics 58, Dekker (1980), 107–152.

 

Para citar este artículo: L.F. Echeverri, Ó.I. Giraldo, E. Zarrazola, A model of competing species that exhibits zip bifurcation, Rev. Integr. Temas Mat. 35 (2017), No. 1, 127–141.