Geodesic
Allee Effect in Gause Type Predator-Prey Models: Existence of Multiple Attractors, Limit cycles and Separatrix Curves. A Brief Review
E. González-Olivares1 ; A. Rojas-Palma1
1 Grupo de Ecología Matemática, Instituto de Matemáticas Pontificia Universidad Católica de Valparaíso, Chile.
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 6, pp. 143-164

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This work deals with the consequences on structural stability of Gause type predator-prey models, when are considered three standard functional responses and the prey growth rate is subject to an Allee effect. An important consequence of this ecological phenomenon is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane. The origin is an attractor for any set of parameters and the existence of heteroclinic curves can be also shown. Conditions on the parameter values are established to ensure the existence of a unique positive equilibrium, which can be either an attractor or a repellor surrounded by one or more limit cycles. The influence of the Allee effect on the number of limit cycles is analyzed and the results are compared with analogous models without this phenomenon, and which main features have been given in various above works. Ecological interpretations of these results are also given.
DOI : 10.1051/mmnp/20138610

E. González-Olivares 1 ; A. Rojas-Palma 1

1 Grupo de Ecología Matemática, Instituto de Matemáticas Pontificia Universidad Católica de Valparaíso, Chile. 
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E. González-Olivares; A. Rojas-Palma. Allee Effect in Gause Type Predator-Prey Models: Existence of Multiple Attractors, Limit cycles and Separatrix Curves. A Brief Review. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 6, pp. 143-164. doi: 10.1051/mmnp/20138610

[1] P. Aguirre, E. González-Olivares, E. Sáez SIAM J. Appl. Math. 2009 1244 1269

[2] E. Angulo, G. W. Roemer, L. Berec, J. Gascoigne, F. Courchamp Conser. Biol. 2007 1082 1091

[3] D. K. Arrowsmith, C. M. Place. Dynamical System. Differential equations, maps and chaotic behaviour, Chapman and Hall, London, 1992.

[4] A. D. Bazykin, F. S. Berezovskaya, A. S. Isaev, R. G. Khlebopros Dynamics of forest insect density: Bifurcation approach J. Theor. Biol. 1997 267 278

[5] A.D. Bazykin. Nonlinear dynamics of interacting populations. Nonlinear Sciences Series A Vol. 11, World Scientific, Singapore, 1998.

[6] L. Berec, E. Angulo, F. Courchamp Trends Ecol. Evol. 2007 185 191

[7] F. Berezovskaya, G. Karev, R. Arditi J. Math. Biol. 2001 221 246

[8] D. S. Boukal, L. Berec J. Theor. Biol. 2002 375 394

[9] D. S. Boukal, M. W. Sabelis, L. Berec Theor. Popul. Biol. 2007 136 147

[10] K. S. Cheng SIAM J. Math. Anal. 1981 541 548

[11] C. Chicone, Ordinary differential equations with applications (2nd edition). Texts in Applied Mathematics 34, Springer, New York, 2006.

[12] C. W. Clark SIAM Rev. 1979 81 99

[13] C. W. Clark, Mathematical Bioeconomics: The optimal management of renewable resources (2nd edition), John Wiley and Sons, New York, 1990.

[14] C. W. Clark. The Worldwide Crisis in Fisheries: Economic Model and Human Behavior. Cambridge University Press, Cambridge, 2007.

[15] C. S. Coleman. Hilbert’s 16th problem: How many cycles?. In: M. Braun, C. S. Coleman and D. Drew (Eds.) Differential Equations Models, Springer-Verlag, New York, (1983), 279-297.

[16] J. B. Collings, D. J. Wollkind SIAM J. Appl. Math. 1990 1348 1372

[17] E. D. Conway, J. A. Smoller SIAM J. Appl. Math. 1986 630 642

[18] F. Courchamp, T. Clutton-Brock, B. Grenfell Trends Ecol. Evol. 1999 405 410

[19] F. Courchamp, L. Berec, J. Gascoigne. Allee effects in Ecology and Conservation. Oxford University Press, Oxford, 2007.

[20] F. Dumortier, J. Llibre, J. C. Artés.Qualitative theory of planar differential systems. Springer, Berlin, 2006.

[21] J. D. Flores, J. Mena-Lorca, B. González-Yañez, E. González-Olivares. Consequences of depensation in a Smithś bioeconomic model for open-access fishery. In (R. Mondaini and R. Dilao Eds.) Proceedings of International Symposium on Mathematical and Computational Biology. E-papers Serviços Editoriais Ltda., Río de Janeiro, (2007), 219-232.

[22] H. I. Freedman, Deterministic mathematical model in Population Ecology, Marcel Dekker, New York, 1980.

[23] G. F. Fussmann, B. Blasius Biol. Lett. 2004 9 12

[24] V. Gaiko. Global bifurcation Theory and Hilbertś sexteenth problem. Kluwer Academic Press, Boston, 2003.

[25] J. C. Gascoigne, R. Lipcius J. Appl. Ecol. 2004 801 810

[26] G. F. Gause. The Struggle for Existence. Dover Publications Inc, London, 1971.

[27] W. M. Getz Ecology 1996 2014 2026

[28] L. Ginzburg, M. Colyvan. Ecological Orbits: How Planets Move and Populations Grow. Oxford University Press, Oxford, 2004.

[29] B-S. Goh, Management and Analysis of Biological Populations, Elsevier Scientific Publishing Company, Amsterdam, 1980.

[30] E. González-Olivares, R. Ramos-Jiliberto Ecol. Model. 2003 135 146

[31] E. González-Olivares, B. González-Yañez, E. Sáez, I. Szantó N Discrete Cont. Dyn-B. 2006 525 534

[32] E. González-Olivares, B. González-Yañez, J. Mena-Lorca, R. Ramos-Jiliberto. Modelling the Allee effect: are the different mathematical forms proposed equivalents? In R. Mondaini (Ed.) Proceedings of International Symposium on Mathematical and Computational Biology. E-papers Serviços Editoriais Ltda., Río de Janeiro, (2007), 53-71.

[33] E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores Appl. Math. Model. 2011 366 381

[34] E. González-Olivares, H. Meneses-Alcay, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma, R. Ramos-Jiliberto Nonlinear Anal-Real. 2011 2931 2942

[35] E. González-Olivares, A. Rojas-Palma Bull. Math. Biol. 2011 1378 1397

[36] E. González-Olivares, A. Rojas-Palma Math. Method App. Sci. 2012 963 975

[37] E. González-Olivares, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma, J. D. Flores Consequences of double Allee effect on the number of limit cycles in a predator-prey model Comp. Math. App. 2011 3449 3463

[38] B. González-Yañez, E. González-Olivares, Consequences of Allee effect on a Gause type predator-prey model with nonmonotonic functional response, In R. Mondaini (Ed.) Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology, E-Papers Serviços Editoriais Ltda, Río de Janeiro, Vol. 2 (2004), 358-373.

[39] K. Hasík Math. Biosci. 2000 203 215

[40] K. Hasík J. Math. Biol. 2010 59 74

[41] M. Hesaaraki, S. M. Moghadas Ecol. Model. 2001 1 9

[42] F. M. Hilker, M. Langlais, H. Malchow Am. Nat. 2009 72 88

[43] X-C. Huang, Y. Wang, L. Zhu Math. Method App. Sci. 2007 501 511

[44] T-W. Hwang J. Math. Anal. Appl. 2004 113 122

[45] S-B. Hsu, T-W. Hwang, Y. Kuang Discrete Cont. Dyn. S. 2008 857 871

[46] C. Hui, Z. Li Popul. Ecol. 2004 55 63

[47] M. Kot, Elements of Mathematical Ecology, Cambridge University Press 2001.

[48] Y. Kuang, H. I. Freedman Uniqueness of limit cycles in Gause type models of predator-prey systems Math. Biosci. 1988 67 84

[49] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory (2nd Edition), Springer-Verlag, New York, 1998.

[50] Y. Lamontagne, C. Coutu, C. Rousseau J. Dyn. Differ. Equ. 2008 535 571

[51] M. Liermann, R. Hilborn Fish Fish 2001 33 58

[52] D. Ludwig, D. D. Jones, C. S. Holling J. Anim. Ecol. 1978 204 221

[53] R. M. May, Stability and complexity in model ecosystems(2nd Edition), Princeton University Press, Princeton, NJ, 2001.

[54] H. Meneses-Alcay, E. González-Olivares. Consequences of the Allee effect on Rosenzweig-MacArthur predator-prey model. In R. Mondaini (ed.) Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology BIOMAT 2003. E-papers Serviços Editoriais Ltda., Río de Janeiro, Volumen 2 (2004), 264-277.

[55] S. J. Middlemas, T. R. Barton, J. D. Armstrong, P. M. Thompson P. Roy. Soc. B. 2006 193 198

[56] S. M. Moghadas, B. D. Corbett Chaos Solitons Fract. 2008 1343 1355

[57] A. Morozov, E. Arashkevich Math. Mod. Nat. Phenom. 2008 131 148

[58] W. W. Murdoch, C. J. Briggs, R. M. Nisbet. Consumer-Resources Dynamics. Monographs in Population Biology 36, Princeton University Press, Princeton, 2003.

[59] M. R. Myerscough, M. J. Darwen, W. L. Hogarth Ecol. Model. 1996 31 42

[60] L. Perko. Differential Equations and Dynamical Systems (3rd. Edition). Springer-Verlag, New York, 2001.

[61] A. Rojas-Palma, E. González-Olivares, B. González-Yañez, Metastability in a Gause type predator-prey models with sigmoid functional response and multiplicative Allee effect on prey, In R. Mondaini (Ed.) Proceedings of International Symposium on Mathematical and Computational Biology, E-papers Serviços Editoriais Ltda., Río de Janeiro, (2007), 295-321.

[62] G. Seo, M. Kot Math. Biosci. 2008 161 179

[63] M. Sen, M. Banerjee, A. Morozov Ecol. Complex. 2013 12 27

[64] P. D. Spencer, J. S. Collie ICES J. Mar. Sci. 1995 615 628

[65] P. A. Stephens, W. J. Sutherland Trends Ecol. Evol. 1999 401 405

[66] P. A. Stephens, W. J. Sutherland, R. P. Freckleton Oikos 1999 185 190

[67] J. Sugie, K. Miyamoto, K. Morino Appl. Math. Lett. 1996 85 90

[68] J. Sugie, R. Kohno, R. Miyazaki P. Am. Math. Soc. 1997 2041 2050

[69] R. J. Taylor. Predation. Chapman and Hall, New York,1984.

[70] P. Turchin. Complex population dynamics. A theoretical/empirical synthesis. Mongraphs in Population Biology 35 Princeton University Press, Princeton, NJ, 2003.

[71] G. A. K. Van Voorn, L. Hemerik, M. P. Boer, B. W. Kooi Math. Biosci. 2007 451 469

[72] K. Vilches-Ponce, E. González-Olivares. A Gause-type predator-prey model with a non-differentiable functional response. (2013) submitted.

[73] M-H. Wang, M. Kot Speeds of invasion in a model with strong or weak Allee effects Math. Bioci. 2001 83 97

[74] J. Wang, J. Shi, J. Wei J. Math. Biol. 2011 291 331

[75] S. Wolfram Research, Mathematica: A System for Doing Mathematics by Computer (2nd edition), Wolfram Research, Addison Wesley 1991.

[76] D. J. Wollkind, J. B. Collings, J. A. Logan Bull. Math. Biol. 1988 379 109

[77] D. Xiao, Z. Zhang Nonlinearity 2003 1185 1201

[78] J. Zu, M. Mimura Appl. Math. Comp. 2010 3542 3556

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