Citation: |
[1] |
A. Aguilera-Moya and E. González-Olivares, A Gause type model with a generalized class of nonmonotonic functional response, in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini ), E-Papers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 206-217. |
[2] |
P. Aguirre, E. González-Olivares and E. Sáez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69 (2009), 1244-1262.doi: 10.1137/070705210. |
[3] |
D. K. Arrowsmith and C. M. Place, "Dynamical Systems. Differential Equations, Maps and Chaotic Behaviour," Chapman and Hall, 1992. |
[4] |
A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations," World Scientific, 1998.doi: 10.1142/9789812798725. |
[5] |
L. Berec, E. Angulo and F. Courchamp, Multiple Allee effects and population management, Trends in Ecology and Evolution, 22 (2007), 185-191. |
[6] |
D. S. Boukal and L. Berec, Single-species models and the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218 (2002), 375-394.doi: 10.1006/jtbi.2002.3084. |
[7] |
C. Chicone, "Ordinary Differential Equations with Applications," (2nd edition), Texts in Applied Mathematics 34, Springer, 2006. |
[8] |
C. W. Clark, "Mathematical Bioeconomic: The Optimal Management of Renewable Resources," (2nd edition), John Wiley and Sons, 1990. |
[9] |
C. W. Clark, "The Worldwide Crisis in Fisheries: Economic Model and Human Behavior," Cambridge University Press, 2007. |
[10] |
C. S. Coleman, Hilbert's 16th. problem: How many cycles?, in "Differential Equations Model" (eds. M. Braun, C. S. Coleman and D. Drew ), Springer Verlag, (1983), 279-297. |
[11] |
J. B. Collings, The effect of the functional response on the bifurcation behavior of a mite predator-prey interaction model, Journal of Mathematical Biology, 36 (1997), 149-168.doi: 10.1007/s002850050095. |
[12] |
E. D. Conway and J. A. Smoller, Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46 (1986), 630-642.doi: 10.1137/0146043. |
[13] |
F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse dependence and the Allee effect, Trends in Ecology and Evolution, 14 (1999), 405-410. |
[14] |
F. Courchamp, L. Berec and J. Gascoigne, "Allee effects in Ecology and Conservation," Oxford University Press, 2008. |
[15] |
F. Dumortier, J. Llibre and J. C. Artés, "Qualitative Ttheory of Planar Differential Systems," Springer, 2006. |
[16] |
H. I. Freedman, "Deterministic Mathematical Model in Population Ecology," Marcel Dekker, 1980. |
[17] |
H. I. Freedman and G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisted, Bulletin of Mathematical Biology, 48 (1986), 493-508.doi: 10.1016/S0092-8240(86)90004-2. |
[18] |
V. A. Gaiko, "Global Bifurcation Theory and Hilbert's Sixteenth Problem," Mathematics an its applications, 559, Kluwer Academic Publishers, 2003. |
[19] |
J. C. Gascoigne and R. N. Lipcius, Allee effects driven by predation, Journal of Applied Ecology, 41 (2004), 801-810. |
[20] |
E. González-Olivares, B. González-Yañez, E. Sáez and I. Szantó, On the number of limit cycles in a predator prey model with non-monotonic functional response, Discrete and Continuous Dynamical Systems, 6 (2006), 525-534.doi: 10.3934/dcdsb.2006.6.525. |
[21] |
E. González-Olivares, B. González-Yañez, J. Mena-Lorca and R. Ramos-Jiliberto, Modelling the Allee effect: Are the different mathematical forms proposed equivalents?, in "Proceedings of the 2006 International Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-papers Serviços Editoriais Ltda. Rio de Janeiro, (2007), 53-71. |
[22] |
E. González-Olivares, H. Meneses-Alcay, B. González-Yañez, J. Mena-Lorca, A. Rojas-Palma and R. Ramos-Jiliberto, Multiple stability and uniqueness of limit cycle in a Gause-type predator-prey model considering Allee effect on prey, Nonlinear Analysis: Real World and Applications, 12 (2011), 2931-2942.doi: 10.1016/j.nonrwa.2011.04.003. |
[23] |
E. González-Olivares and A. Rojas-Palma, Multiple limit cycles in a Gause type predator-prey model with Holling type III functional response and Allee effect on prey, Bulletin of Mathematical Biology, 73 (2011), 1378-1397.doi: 10.1007/s11538-010-9577-5. |
[24] |
E. González-Olivares, J. Mena-Lorca, A. Rojas-Palma and J. D. Flores, Dynamical complexities in the Leslie-Gower predator-prey model considering a simple form to the Allee effect on prey, Applied Mathematical Modelling, 35 (2011), 366-381.doi: 10.1016/j.apm.2010.07.001. |
[25] |
B. González-Yañez and E. González-Olivares, Consequences of Allee effect on a Gause type predator-prey model with nonmonotonic functional response, in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-Papers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 358-373. |
[26] |
K. Hasík, On a predator-prey system of Gause type, Journal of Mathematical Biology, 60 (2010), 59-74.doi: 10.1007/s00285-009-0257-8. |
[27] |
M. Kot, "Elementary Mathematical Ecology," Cambridge University Press, 2001.doi: 10.1017/CBO9780511608520. |
[28] |
Y. Kuang and H. I. Freedman, Uniqueness of limit cycles in Gause-type models of predator-prey systems, Mathematical Biosciences, 88 (1988), 67-84.doi: 10.1016/0025-5564(88)90049-1. |
[29] |
M. Liermann and R. Hilborn, Depensation: Evidence, models and implications, Fish and Fisheries, 2 (2001), 33-58. |
[30] |
L. Perko, "Differential Equations and Dynamical Systems," (3rd ed), Texts in Applied Mathematics 7, Springer-Verlag, 2001. |
[31] |
A. Rojas-Palma, E. González-Olivares and B. González-Yañez, Metastability in a Gause type predator-prey models with sigmoid functional response and multiplicative Allee effect on prey, in "Proceedings of the 2006 International Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-papers Serviços Editoriais Ltda., (2007), 295-321. |
[32] |
S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM Journal of Applied Mathematics, 61 (2001), 1445-1472.doi: 10.1137/S0036139999361896. |
[33] |
P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology and Evolution, 14 (1999), 401-405. |
[34] |
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190. |
[35] | |
[36] |
P. Turchin, "Complex Population Dynamics. A Theoretical/Empirical Synthesis," Monographs in Population Biology 35, Princeton University Press, 2003. |
[37] |
G. A. K. van Voorn, L. Hemerik, M. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Mathematical Biosciences, 209 (2007), 451-469.doi: 10.1016/j.mbs.2007.02.006. |
[38] |
S. Véliz-Retamales and E. González-Olivares, Dynamics of a Gause type prey-predator model with a rational nonmonotonic consumption function, in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), E-Papers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 181-192. |
[39] |
J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291-331.doi: 10.1007/s00285-010-0332-1. |
[40] |
S. Wolfram, "Mathematica: A System for Doing Mathematics by Computer," (2nd edition), Wolfram Research, Addison Wesley, 1991. |
[41] |
G. S. W. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defense, SIAM Journal on Applied Mathematics, 48 (1988), 592-606.doi: 10.1137/0148033. |
[42] |
D. Xiao and S. Ruan, Bifurcations in a predator-prey system with group defense, International Journal of Bifurcation and Chaos, 11 (2001), 2123-2131.doi: 10.1142/S021812740100336X. |
[43] |
D. Xiao and Z. Zhang, On the uniquenes and nonexsitence of limit cycles for predator-prey systems, Nonlinearity, 16 (2003), 1185-1201.doi: 10.1088/0951-7715/16/3/321. |
[44] |
H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 63 (2002), 636-682.doi: 10.1137/S0036139901397285. |
[45] |
J. Zu and M. Mimura, The impact of Allee effect on a predator-prey system with Holling type II functional response, Applied Mathematics and Computation, 217 (2010), 3542-3556.doi: 10.1016/j.amc.2010.09.029. |