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Abstract
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.
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References
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- Briggs, G. E., and Haldane, J. B. A Note on the Kinetics of Enzyme Action, Biochem J 19, (1925) pp. 338-339.
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- Doedel, E. J., et al., 2002, AUTO: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), California Institute of Technology, Pasadena, California.
- Elettreby, M.F., Two-prey one-predator model Chaos, Solitons & Fractals 39 5 (2009), Pp. 2018-2027
- 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.
- 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.
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- tionary Computation (GECCO'11) (2011): 109-110.
- 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.
- Klebano, A., Hastings, A., Chaos in one-predator, two-prey models: General results from
- bifurcation theory Mathematical Biosciences, 1994, pp. 221-233
- 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.
- Kuznetsov, Y. A., 1998, Elements of Applied Bifurcation Theory, 2nd ed., Springer, New York.
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- 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.
- 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.
- 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
References
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.
A.A. Berryman, The origins and evolution of predator-prey theory, Ecology, 73(5),(1992), 15301535.
Briggs, G. E., and Haldane, J. B. A Note on the Kinetics of Enzyme Action, Biochem J 19, (1925) pp. 338-339.
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.
of the series Lecture Notes in Electrical Engineering (2014) pp 751-756.
Dineen, S. Multivariate Calculus and Geometry, third edition, Springer Undergraduate Mathematics Series, Springer (2014), London etc.
Doedel, E. J., et al., 2002, AUTO: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), California Institute of Technology, Pasadena, California.
Elettreby, M.F., Two-prey one-predator model Chaos, Solitons & Fractals 39 5 (2009), Pp. 2018-2027
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.
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.
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
tionary Computation (GECCO'11) (2011): 109-110.
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.
Klebano, A., Hastings, A., Chaos in one-predator, two-prey models: General results from
bifurcation theory Mathematical Biosciences, 1994, pp. 221-233
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.
Kuznetsov, Y. A., 1998, Elements of Applied Bifurcation Theory, 2nd ed., Springer, New York.
Lessard, J., Rigorous verication of saddlenode bifurcations in ODEs, Indagationes Mathematicae, 27, 2016, 1013-1026.
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.
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.
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