Geodesic
Stochastic Path Perturbation Approach Applied to Non–Local Non–Linear Equations in Population Dynamics
M. A. Fuentes1, 2, 3 ; M. O. Cáceres4
1 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
2 Instituto de Investigaciones Filosóficas, Bulnes 642, 1428 Buenos Aires, Argentina
3 Universidad San Sebastián, Lota 2465, Santiago 7500000, Chile
4 Centro Atómico Bariloche, Instituto Balseiro and CONICET, 8400 Bariloche, Argentina
Mathematical modelling of natural phenomena, Tome 10 (2015) no. 6, pp. 48-60

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We described the first passage time distribution associated to the stochastic evolution from an unstable uniform state to a patterned one (attractor of the system), when the time evolution is given by an integro-differential equation describing a population model. In order to obtain analytical results we used the Stochastic Path Perturbation Approach introducing a minimum coupling approximation into the nonlinear dynamics, and a stochastic multiscale perturbation expansion. We show that the stochastic multiscale perturbation is a necessary tool to handle other problems like: nonlinear instabilities and multiplicative stochastic partial differential equations. A small noise parameter was introduced to define the random escape of the stochastic field. We carried out Monte Carlo simulations in a non-local Fisher like equation, to show the agreement with our theoretical predictions.
DOI : 10.1051/mmnp/201510605

M. A. Fuentes 1, 2, 3 ; M. O. Cáceres 4

1 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
2 Instituto de Investigaciones Filosóficas, Bulnes 642, 1428 Buenos Aires, Argentina
3 Universidad San Sebastián, Lota 2465, Santiago 7500000, Chile
4 Centro Atómico Bariloche, Instituto Balseiro and CONICET, 8400 Bariloche, Argentina 
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M. A. Fuentes; M. O. Cáceres. Stochastic Path Perturbation Approach Applied to Non–Local Non–Linear Equations in Population Dynamics. Mathematical modelling of natural phenomena, Tome 10 (2015) no. 6, pp. 48-60. doi: 10.1051/mmnp/201510605

[1] A. Mogilner, L. Edelstein-Keshet J. Math. Biol. 1999 534 570

[2] A. Mogilner, L. Edelstein-Keshet, L. Bent, A. Spiros J. Math. Biol. 2003 353 389

[3] J.D. Murray. Mathematical Biology, Vol. 1 and Vol. 2. Third Edition, Barcelona, Springer, Berlin, 2007.

[4] E. Hernandez-Garcia, C. Lopez Physical Review E 2004 016216

[5] M.A. Fuentes, M.N. Kuperman, V.M. Kenkre Physical Review Letters 2003 1581041

[6] M.A. Fuentes, M.N. Kuperman, V.M. Kenkre The Journal of Physical Chemistry B 2004 10505 10508

[7] E. Ben-Jacob, I. Cohen, H. Levine Advances in Physics 2000 395 554

[8] M.G. Clerc, D. Escaff, V.M. Kenkre Physical Review E 2005 056217

[9] I. Demin, V. Volpert Mathematical Modelling of Natural Phenomena 2010 80 101

[10] B.L. Segal, V.A. Volpert, A. Bayliss Physica D: Nonlinear Phenomena 2013 12 22

[11] J.K. Hale, J.K. Kocak. Dynamics and Bifurcations. Texts in Applied Mathematics. Springer-Verlag. Corrected Edition, 1996.

[12] J. Guckenheimer, P. Holmes. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Applied Mathematical Sciences. Springer. 1st ed. 1983. Corr. 6th Edition, 2002.

[13] P.C. Bressloff, M.A. Webber SIAM Journal on Applied Dynamical Systems 2012 708 740

[14] C. Kuehn, M.G. Riedler J. Math. Neuroscience 2014 1 33

[15] V. Volterra J. Cons. Int. Explor. Mer 1928 3 51

[16] R.A. Fisher Annals of Eugenics 1937 355 369

[17] M.O. Cáceres Journal of Statistical Physics 2014 94 118

[18] N.G. van Kampen. Stochastic Processes in Physics and Chemistry, 2nd ed. North Holland, Amsterdam, 1992.

[19] M.O. Cáceres. Elementos de estadistica de no equilibrio y sus aplicaciones al transporte en medios desordenados. In Spanish, Reverté S.A., Barcelona, 2003.

[20] J. Garcia-Ojalvo, J. M. Sancho. Noise in Spatially Extended, Systems. Springer, Berlin, 2010.

[21] P. Colet, F. De Pasquale, M.O. Cáceres, M. San Miguel Physical Review A 1990 1901

[22] M.O. Cáceres Journal of Statistical Physics 2008 487 500

[23] M.O. Caceres, Ch. D. R. Rojas Physica A 2014 61 70

[24] M.A. Fuentes, M.O. Cáceres Central European Journal of Physics 2013 1623 1628

[25] M.O. Cáceres Physical Review E 2014 022137

[26] S. Gonçalves, G. Abramson, M.F.C. Gomes The European Physical Journal B 2011 363 371

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