Main Article Content


We propose by means of an example of applications of the classical Lagrange Multiplier Method for computing fold bifurcation point of an equilibrium ina one-parameter family of dynamical systems. We have used the fact that an equilibrium of a system, geometrically can be seen as an intersection between nullcline manifolds of the system. Thus, we can view the problem of two collapsing equilibria as a constrained optimization problem, where one of the nullclines acts as the cost function while the other nullclines act as the constraints.


Fold Bifurcation Constrained Extremum Dynamical Systems

Article Details

How to Cite
Marwan, M., Tuwankotta, J. M., & Harjanto, E. (2018). Application of Lagrange Multiplier Method for Computing Fold Bifurcation Point in A Two-Prey One Predator Dynamical System. Journal of the Indonesian Mathematical Society, 24(2), 7–19.


  1. F. K. Balagadde, H. Song, J. Ozaki, C.H. Collins, M. Barnet, F.H. Arnold, S.R. Quake, and L. You, A synthetic Escherichia coli predatorprey ecosystem, Mol. Syst. Biol. (2008), 4:187.
  2. A.A. Berryman, The origins and evolution of predator-prey theory, Ecology, 73(5),(1992), 15301535.
  3. Briggs, G. E., and Haldane, J. B. A Note on the Kinetics of Enzyme Action, Biochem J 19, (1925) pp. 338-339.
  4. Z. Cai, Q. Wang, and G. Lie, Modeling the Natural Capital Investment on Tourism Industry Using a Predator-Prey Model, in Advances in Computer Science and its Applications, Vol.
  5. of the series Lecture Notes in Electrical Engineering (2014) pp 751-756.
  6. Dineen, S. Multivariate Calculus and Geometry, third edition, Springer Undergraduate Mathematics Series, Springer (2014), London etc.
  7. Doedel, E. J., et al., 2002, AUTO: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), California Institute of Technology, Pasadena, California.
  8. Elettreby, M.F., Two-prey one-predator model Chaos, Solitons & Fractals 39 5 (2009), Pp. 2018-2027
  9. A. Fenton, and S.E. Perkins, Applying predator-prey theory to modeling immune-mediated, within-host interspecic parasite interactions, Parasitology 137(6) (2010 May): 1027-38.
  10. R.M. Goodwin, A Growth Cycle, in Feinstein, C. H. (ed), in Socialism, Capitalism, and Economic Growth, Cambridge, Cambridge University Press, (1967), pp. 54-58.
  11. C. Grimme, and J. Lepping, Integrating niching into the predator-prey model using epsilon-constraints, Proceedings of the 13th Annual Conference Companion on Genetic and Evolu
  12. tionary Computation (GECCO'11) (2011): 109-110.
  13. Harjanto, E., and Tuwankotta, J.M., Bifurcation of Periodic Solution in a Predator-Prey Type of Systems with Non-monotonic Response Function and Periodic Perturbation, International Journal of Non-Linear Mechanics, 85, 2016, 188-196.
  14. Klebano, A., Hastings, A., Chaos in one-predator, two-prey models: General results from
  15. bifurcation theory Mathematical Biosciences, 1994, pp. 221-233
  16. Koren, I., Feingold, G., Aerosolcloudprecipitation system as a predator-prey problem., Proceedings of the National Academy of Sciences 108, nr. 30 (2011): pp. 12227-12232.
  17. Kuznetsov, Y. A., 1998, Elements of Applied Bifurcation Theory, 2nd ed., Springer, New York.
  18. Lessard, J., Rigorous verication of saddlenode bifurcations in ODEs, Indagationes Mathematicae, 27, 2016, 1013-1026.
  19. Owen, L., and Tuwankotta, J.M., Bogdanov-Takens Bifurcations in Three Coupled Oscillators System with Energy Preserving Nonlinearity, J. Indones. Math. Soc., 18(2), 2012, 73-83.
  20. A. Sharma, and, N. Singh, Object detection in image using predator-prey optimization, Signal & Image Processing: An International Journal (SIPIJ), vol. 2 (1)(2011), 205-221.
  21. Tripathi, J. P., Syed Abbas, S., Manoj Thakur, M., Local and global stability analysis of a two prey one predator model with help Communications in Nonlinear Science and Numerical Simulation, 2014, pp. 3284-3297