Skip to main content
Log in

Complex dynamics in a ratio-dependent two-predator one-prey model

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

A three-species ratio-dependent predator prey model system is considered. The global dynamic behavior of the model is investigated through (local) stability results for its equilibriums and large time computer simulations. Irregular patterns and existence of complex dynamics are noticed for different sets of parameter values. The model is shown to have chaotic attractors for suitable choice of values of parameters.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  • Abrams PA (1994) The fallacies of ‘ratio-dependent’ predation. Ecology 75:1842–1850

    Article  Google Scholar 

  • Akcakaya HR, Arditi R, Ginzburg LR (1995) Ratio-dependent predation: an abstraction that works. Ecology 76:995–1004

    Article  Google Scholar 

  • Arditi R, Ginzburg LR (1989) Coupling in predator-prey dynamics: ratio dependence. J Theor Biol 139:311–326

    Article  Google Scholar 

  • Arditi R, Ginzburg LR, Akcakaya HR (1991) Variation in plankton densities among lakes: a case for ratio-dependent predation models. Am Nat 138:1287–1296

    Article  Google Scholar 

  • Arditi R, Ginzburg LR (2012) How species interact: altering the standard view of trophic ecology. Oxford University Press, Oxford

    Google Scholar 

  • Arino O, El Abdllaoul A, Micram J, Chattopadhyay J (2004) Infection in prey population may act as a biological control in ratio-dependent predator-prey models. Nonlinearity 17:1101–1116

    Article  MATH  MathSciNet  Google Scholar 

  • Aziz-Alaoui MA, Daher-Okiye M (2003) Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-typeII schemes. App Math Lets 16(7):1069–1075

    Google Scholar 

  • Baek S, Ko W, Ahn I (2012) Coexistence of a one-prey two-predators model with ratio-dependent functional responses. Appl Comput Math 219:1897–1908

    Article  MATH  MathSciNet  Google Scholar 

  • Berryman AA (1992) The origins and evolution of predator-prey theory. Ecology 73:1530–1535

    Article  Google Scholar 

  • Brauer F, Castillo-Chavez C (2001) Mathematical models in population biology and epidemiology, texts in Applied Mathematics, 40. Springer, New York

    Google Scholar 

  • Cosner C, DeAngelis DL, Ault JS, Olson DB (1999) Effects of spatial grouping on the functional response of predators. Theor Pop Biol 56(1):65–75

    Article  MATH  Google Scholar 

  • Freedman HI, Mathsen RM (1993) Persistence in predator-prey systems with ratio-dependent predator—prey influence. Bull Math Biol 55:817–827

    Article  MATH  Google Scholar 

  • Freedman HI (1980) Deterministic mathematical models in population ecology. Monographs and textbooks in pure and applied mathematics, 57. Marcel Dekker Inc., New York, p 254.

  • Gakkhar S, Naji RK (2003) Order and chaos in predator to prey ratio-dependent food chain. Chaos Solitons Fractals 18:229–239

    Article  MATH  MathSciNet  Google Scholar 

  • Gutierrez AP (1992) Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm. Ecology 73:1552–63

    Article  Google Scholar 

  • Hanski I (1991) The functional response of predators: worries about scale. Trends Ecol Evol 6(5):141–142

    Article  Google Scholar 

  • Haque M (2009) Ratio-dependent predator-prey models of interacting populations. Bull Math Biol 71:430–452

    Article  MATH  MathSciNet  Google Scholar 

  • Hassell MP (1978) The dynamics of arthropod predator-prey systems. Princeton University Press, Princeton

    MATH  Google Scholar 

  • Hsu SB, Hwang TW (1995) Global stability for a class of predator-prey systems, SIAM. J Appl Math 55:763–783

    MATH  MathSciNet  Google Scholar 

  • Hsu SB, Hwang TW, Kuang Y (2001a) Rich dynamics of a ratio-dependent one prey two predator model. J Math Biol 43:377–396

    Article  MATH  MathSciNet  Google Scholar 

  • Hsu SB, Hwang TW, Kuang Y (2001b) Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system. J Math Biol 42:489–506

    Article  MATH  MathSciNet  Google Scholar 

  • Jost C, Arino O, Arditi R (1999) About deterministic extinction in ratio-dependent predator-prey models. Bull Math Biol 61:19–32

    Article  Google Scholar 

  • Jost C, Arditi R (2000) Identifying predator-prey processes from time-series. Theor Popul Biol 57:325–337

    Article  MATH  Google Scholar 

  • Korobeinikov A (2001) A Lyapunov function for Leslie-Gower predator prey models. Appl Math Lett 14(6):697–699

    Article  MATH  MathSciNet  Google Scholar 

  • Kuang Y (1999) Rich dynamics of Gause-type ratio-dependent predator-prey system. Fields Inst Commun 21:325–337

    Google Scholar 

  • Leslie PH (1948) Some furthers notes on the use of matrices in population mathematics. Biometrica 35:213–245

    Article  MATH  MathSciNet  Google Scholar 

  • Lev R, Ginzburg H, Akcakaya R (1992) Consequences of ratio-dependent predation for steady-state properties of Ecosystems. Ecol. 73:1536–1543

    Article  Google Scholar 

  • Liang Z, Pan H (2007) Qualitative analysis of a ratio-dependent Holling-Tanner model. J Math Anal Appl 334(2):954–964

    Google Scholar 

  • Liu Z, Zhong S, Yin C, Chen W (2011) On the dynamics of an impulsive reaction-diffusion predator prey system with ratio-dependent functional response. Acta Appl Math 115:329–349

    Article  MATH  MathSciNet  Google Scholar 

  • Lundberg P, Fryxell JM (1995) Expected population density versus productivity in ratio-dependent and prey-dependent models. Am Nat 146:153–161

    Article  Google Scholar 

  • May RM (1974) Stability and complexity in model ecosystems. Princeton University Press, Princeton

    Google Scholar 

  • McCarthy MA, Ginzburg LR, Akçakaya HR (1995) Predator interference across trophic chains. Ecology 76(4):1310–1319

    Article  Google Scholar 

  • Murray JD (1989) Math Biol. Springer, New York

    Google Scholar 

  • Rosenzweig ML (1969) Paradox of enrichment: destabilization of exploitation systems in ecological time. Science 171:385–387

    Article  Google Scholar 

  • Saez E, Gonzalez-Olivares E (1999) Dynamics of a predator-prey model. SIAM J Appl Math 59:1867–1878

    Article  MATH  MathSciNet  Google Scholar 

  • Thieme H (1997), Mathematical Biology. An introduction via selected topics, lecture note at Arizona State University.

  • Wolkind DJ, Collings JB, Logan J (1988) Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees. Bull Math Biol 50:379–409

    Article  MathSciNet  Google Scholar 

  • Xiao D, Ruan S (2001) Global dynamics of ratio-dependent predator-prey system. J Math Biol 43:268–290

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Tanuja Agrawal.

Additional information

Communicated by Antonio José Silva Neto.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Agrawal, T., Saleem, M. Complex dynamics in a ratio-dependent two-predator one-prey model. Comp. Appl. Math. 34, 265–274 (2015). https://doi.org/10.1007/s40314-014-0115-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-014-0115-1

Keywords

Mathematics Subject Classification (2000)

Navigation