Stability in Kolmogorov-type quadratic systems describing interactions among two species. A brief revision


  • Eduardo González-Olivares Pontificia Universidad Católica de Valparaíso, Chile.
  • Alejandro Rojas-Palma Departamento de Matemática, Física y Estadística, Facultad de Ciencias Básicas, Universidad Católica del Maule, Talca, Chile.



Predator-prey model, functional response, cycles, separatrix curve, stability, Lyapunov function


Population dynamics is a relevant topic in Biomathematics, being the study of the long-term behavior of interaction models between species, one of its central problems. A large part of these relationships are described by ordinary differential equations (ODE), having as main objectives the study of the stability of their solutions.

In this document we mainly describe the dynamic behavior of the Volterra predation model. In addition, we make a review of some derived predation models and a brief review of the dynamical properties of models describing other interactions between species such as: competition, mutualism, amensalism, and commensalism; also described by nonlinear ODE systems of the second order of Kolmogorov-type.

For each of these models, the non-existence of limit cycles can be demonstrated and in most of  them, there is a globally stable equilibrium point. In one of them, there are  conditions in the parameters for which the only positive equilibrium point is a center, as in the original Lotka-Volterra model. 

The methodology used is the usual one for the analysis of models with hyperbolic equilibrium points, but it can guide the analysis of other more complicated models.


Bazykin AD. Nonlinear Dynamics of interacting populations. Singapore: World Scientific Publishing Co. Pte. Ltd. 1998.

Berryman AA, Gutierrez AP, Arditi R. Credible, parsimonious and useful predator-prey models - A reply to Abrams, Gleeson and Sarnelle, Ecology. 1995 76:1980-1985.

Birkhoff G, Rota GS. Ordinary Differential Equations (4th ed.) New York: John Wiley & Sons; 1989.

Braun M. Ecuaciones diferenciales y sus aplicaciones. California: Grupo Editorial Iberoamérica; 1990.

Cheng KS. Uniqueness of a limit cycle for a predator-prey system, SIAM J. on Math. Anal. 1981; 12:541–548.

Chicone C. Ordinary differential equations with applications (2nd edition), New YorK: Springer; Texts in Applied Mathematics 34; 2006.

Clark CW. Bioeconomic modelling and fisheries managements. New York: John Wiley and Sons; 1985.

Clark CW. Mathematical Bioeconomic: The optimal management of renewable resources, (second edition). New York: John Wiley and Sons; 1990.

Coleman CS. Hilbert’s 16th. Problem: How Many Cycles? In: M. Braun, CS. Coleman and D. Drew (Ed). Differential Equations Model, Springer Verlag. 1983; 279-297.

Edelstein-Keshet L. Mathematical Models in Biology. SIAM; Classics in Applied Mathematics, 46; 2005.

Dumortier F, Llibre J, Art´es JC. Qualitative theory of planar differential systems. Berlin: Springer-verlag; 2006.

Freedman HI. Deterministic Mathematical Model in Population Ecology. New York: Marcel Dekker; 1980.

Gaiko VA. Global Bifurcation theory and Hilbert sixteenth problem, Mathematics and its Applications 559. New York: Kluwer

Academic Publishers; 2003.

Goh B-S. Management and Analysis of Biological Populations. New York: Elsevier Scientific Publ. Co.; 1980.

González-Olivares E y Mena-Lorca J. Análisis cualitativo de un modelo de pesquerías de acceso abierto. Invest. Mar., Valparaíso. 1994; 22:3-11.

González-Olivares E, Valenzuela-Figueroa S. and Rojas-Palma A. A simple Gause type predator-prey model considering social

predation, Mathematical Methods in the Applied Sciences, 2019; 42:5668-5686.

Hasík K. On a predator-prey system of Gause type. J. of Mathematical Biology. 2010(1):60:59-74.

Iannelli M, Pugliese A. An Introduction to Mathematical Population Dynamics. Italia: Springer; 2014.

Kot M. Elements of Mathematical Ecology. Cambridge: Cambridge University Press; 2003.

Kuang Y, Freedman HI. Uniqueness of limit cycles in Gause-type models of predator-prey systems. Mathematical Biosciences, 1988; 88:67-84.

Leslie PH. Some further notes on the use of matrices in population mathematics, Biometrica. 1948; 35:213-245.

May RM. Stability and complexity in model ecosystems (2nd edition), Princeton: Princeton University Press; 2001.

Maynard Smith J. Models in Ecology, New York: University Press, 1974.

Murray JD. Mathematical Biology I. An Introduction, (3rd ed.), New York: Springer; 2002.

Neal D. Introduction to Population Biology. Cambridge: Cambridge University Press; 2003.

Perko L. Differential equations and dynamical systems (3rd ed.) New York: Springer; 2001.

Schlomiuk D, Vulpe N. Global classification of the planar Lotka-Volterra differential systems according to their configurations of invariant straight lines, J. of Fixed Point Theory and Applications, 2010; 8:177-245.

Taylor RJ. Predation. London: Chapman and Hall; 1984.

Teixeira Alves M, Hilker FM. Hunting cooperation and Allee effects in predators, J. of Theoretical Biology, 2017; 419: 13-22.

Turchin P. Complex population dynamics. A theoretical/empirical synthesis, Monographs in Population Biology 35. Princeton:

Princeton University Press; 2003.



How to Cite

González-Olivares, E., & Rojas-Palma, A. (2021). Stability in Kolmogorov-type quadratic systems describing interactions among two species. A brief revision. Selecciones Matemáticas, 8(01), 131 - 146.

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