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Uniqueness of limit cycles and multiple attractors in a Gausetype predatorprey model with nonmonotonic functional response and Allee effect on prey
1.  Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile, Chile, Chile 
2.  Department of Mathematics, The University of South Dakota, Vermillion, SD 570692390, United States 
We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions.
Trajectories starting at one side of this separatrix curve have the equilibrium point $(0,0)$ as their $\omega $limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point $(K,0)$ in which the predators disappear and prey attains their carrying capacity.
We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong influence on the system dynamics, resulting in multiple domains of attraction.
Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.
References:
[1] 
A. AguileraMoya and E. GonzálezOlivares, 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 ), EPapers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 206217. 
[2] 
P. Aguirre, E. GonzálezOlivares and E. Sáez, Three limit cycles in a LeslieGower predatorprey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69 (2009), 12441262. 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), 185191. 
[6] 
D. S. Boukal and L. Berec, Singlespecies models and the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218 (2002), 375394. 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), 279297. 
[11] 
J. B. Collings, The effect of the functional response on the bifurcation behavior of a mite predatorprey interaction model, Journal of Mathematical Biology, 36 (1997), 149168. doi: 10.1007/s002850050095. 
[12] 
E. D. Conway and J. A. Smoller, Global analysis of a system of predatorprey equations, SIAM Journal on Applied Mathematics, 46 (1986), 630642. doi: 10.1137/0146043. 
[13] 
F. Courchamp, T. CluttonBrock and B. Grenfell, Inverse dependence and the Allee effect, Trends in Ecology and Evolution, 14 (1999), 405410. 
[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, Predatorprey systems with group defence: The paradox of enrichment revisted, Bulletin of Mathematical Biology, 48 (1986), 493508. doi: 10.1016/S00928240(86)900042. 
[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), 801810. 
[20] 
E. GonzálezOlivares, B. GonzálezYañez, E. Sáez and I. Szantó, On the number of limit cycles in a predator prey model with nonmonotonic functional response, Discrete and Continuous Dynamical Systems, 6 (2006), 525534. doi: 10.3934/dcdsb.2006.6.525. 
[21] 
E. GonzálezOlivares, B. GonzálezYañez, J. MenaLorca and R. RamosJiliberto, 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), Epapers Serviços Editoriais Ltda. Rio de Janeiro, (2007), 5371. 
[22] 
E. GonzálezOlivares, H. MenesesAlcay, B. GonzálezYañez, J. MenaLorca, A. RojasPalma and R. RamosJiliberto, Multiple stability and uniqueness of limit cycle in a Gausetype predatorprey model considering Allee effect on prey, Nonlinear Analysis: Real World and Applications, 12 (2011), 29312942. doi: 10.1016/j.nonrwa.2011.04.003. 
[23] 
E. GonzálezOlivares and A. RojasPalma, Multiple limit cycles in a Gause type predatorprey model with Holling type III functional response and Allee effect on prey, Bulletin of Mathematical Biology, 73 (2011), 13781397. doi: 10.1007/s1153801095775. 
[24] 
E. GonzálezOlivares, J. MenaLorca, A. RojasPalma and J. D. Flores, Dynamical complexities in the LeslieGower predatorprey model considering a simple form to the Allee effect on prey, Applied Mathematical Modelling, 35 (2011), 366381. doi: 10.1016/j.apm.2010.07.001. 
[25] 
B. GonzálezYañez and E. GonzálezOlivares, Consequences of Allee effect on a Gause type predatorprey model with nonmonotonic functional response, in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), EPapers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 358373. 
[26] 
K. Hasík, On a predatorprey system of Gause type, Journal of Mathematical Biology, 60 (2010), 5974. doi: 10.1007/s0028500902578. 
[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 Gausetype models of predatorprey systems, Mathematical Biosciences, 88 (1988), 6784. doi: 10.1016/00255564(88)900491. 
[29] 
M. Liermann and R. Hilborn, Depensation: Evidence, models and implications, Fish and Fisheries, 2 (2001), 3358. 
[30] 
L. Perko, "Differential Equations and Dynamical Systems," (3rd ed), Texts in Applied Mathematics 7, SpringerVerlag, 2001. 
[31] 
A. RojasPalma, E. GonzálezOlivares and B. GonzálezYañez, Metastability in a Gause type predatorprey 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), Epapers Serviços Editoriais Ltda., (2007), 295321. 
[32] 
S. Ruan and D. Xiao, Global analysis in a predatorprey system with nonmonotonic functional response, SIAM Journal of Applied Mathematics, 61 (2001), 14451472. 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), 401405. 
[34] 
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185190. 
[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 predatorprey systems with a strong Allee effect, Mathematical Biosciences, 209 (2007), 451469. doi: 10.1016/j.mbs.2007.02.006. 
[38] 
S. VélizRetamales and E. GonzálezOlivares, Dynamics of a Gause type preypredator model with a rational nonmonotonic consumption function, in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), EPapers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 181192. 
[39] 
J. Wang, J. Shi and J. Wei, Predatorprey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291331. doi: 10.1007/s0028501003321. 
[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 predatorprey system involving group defense, SIAM Journal on Applied Mathematics, 48 (1988), 592606. doi: 10.1137/0148033. 
[42] 
D. Xiao and S. Ruan, Bifurcations in a predatorprey system with group defense, International Journal of Bifurcation and Chaos, 11 (2001), 21232131. doi: 10.1142/S021812740100336X. 
[43] 
D. Xiao and Z. Zhang, On the uniquenes and nonexsitence of limit cycles for predatorprey systems, Nonlinearity, 16 (2003), 11851201. doi: 10.1088/09517715/16/3/321. 
[44] 
H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predatorprey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 63 (2002), 636682. doi: 10.1137/S0036139901397285. 
[45] 
J. Zu and M. Mimura, The impact of Allee effect on a predatorprey system with Holling type II functional response, Applied Mathematics and Computation, 217 (2010), 35423556. doi: 10.1016/j.amc.2010.09.029. 
show all references
References:
[1] 
A. AguileraMoya and E. GonzálezOlivares, 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 ), EPapers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 206217. 
[2] 
P. Aguirre, E. GonzálezOlivares and E. Sáez, Three limit cycles in a LeslieGower predatorprey model with additive Allee effect, SIAM Journal on Applied Mathematics, 69 (2009), 12441262. 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), 185191. 
[6] 
D. S. Boukal and L. Berec, Singlespecies models and the Allee effect: Extinction boundaries, sex ratios and mate encounters, Journal of Theoretical Biology, 218 (2002), 375394. 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), 279297. 
[11] 
J. B. Collings, The effect of the functional response on the bifurcation behavior of a mite predatorprey interaction model, Journal of Mathematical Biology, 36 (1997), 149168. doi: 10.1007/s002850050095. 
[12] 
E. D. Conway and J. A. Smoller, Global analysis of a system of predatorprey equations, SIAM Journal on Applied Mathematics, 46 (1986), 630642. doi: 10.1137/0146043. 
[13] 
F. Courchamp, T. CluttonBrock and B. Grenfell, Inverse dependence and the Allee effect, Trends in Ecology and Evolution, 14 (1999), 405410. 
[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, Predatorprey systems with group defence: The paradox of enrichment revisted, Bulletin of Mathematical Biology, 48 (1986), 493508. doi: 10.1016/S00928240(86)900042. 
[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), 801810. 
[20] 
E. GonzálezOlivares, B. GonzálezYañez, E. Sáez and I. Szantó, On the number of limit cycles in a predator prey model with nonmonotonic functional response, Discrete and Continuous Dynamical Systems, 6 (2006), 525534. doi: 10.3934/dcdsb.2006.6.525. 
[21] 
E. GonzálezOlivares, B. GonzálezYañez, J. MenaLorca and R. RamosJiliberto, 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), Epapers Serviços Editoriais Ltda. Rio de Janeiro, (2007), 5371. 
[22] 
E. GonzálezOlivares, H. MenesesAlcay, B. GonzálezYañez, J. MenaLorca, A. RojasPalma and R. RamosJiliberto, Multiple stability and uniqueness of limit cycle in a Gausetype predatorprey model considering Allee effect on prey, Nonlinear Analysis: Real World and Applications, 12 (2011), 29312942. doi: 10.1016/j.nonrwa.2011.04.003. 
[23] 
E. GonzálezOlivares and A. RojasPalma, Multiple limit cycles in a Gause type predatorprey model with Holling type III functional response and Allee effect on prey, Bulletin of Mathematical Biology, 73 (2011), 13781397. doi: 10.1007/s1153801095775. 
[24] 
E. GonzálezOlivares, J. MenaLorca, A. RojasPalma and J. D. Flores, Dynamical complexities in the LeslieGower predatorprey model considering a simple form to the Allee effect on prey, Applied Mathematical Modelling, 35 (2011), 366381. doi: 10.1016/j.apm.2010.07.001. 
[25] 
B. GonzálezYañez and E. GonzálezOlivares, Consequences of Allee effect on a Gause type predatorprey model with nonmonotonic functional response, in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), EPapers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 358373. 
[26] 
K. Hasík, On a predatorprey system of Gause type, Journal of Mathematical Biology, 60 (2010), 5974. doi: 10.1007/s0028500902578. 
[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 Gausetype models of predatorprey systems, Mathematical Biosciences, 88 (1988), 6784. doi: 10.1016/00255564(88)900491. 
[29] 
M. Liermann and R. Hilborn, Depensation: Evidence, models and implications, Fish and Fisheries, 2 (2001), 3358. 
[30] 
L. Perko, "Differential Equations and Dynamical Systems," (3rd ed), Texts in Applied Mathematics 7, SpringerVerlag, 2001. 
[31] 
A. RojasPalma, E. GonzálezOlivares and B. GonzálezYañez, Metastability in a Gause type predatorprey 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), Epapers Serviços Editoriais Ltda., (2007), 295321. 
[32] 
S. Ruan and D. Xiao, Global analysis in a predatorprey system with nonmonotonic functional response, SIAM Journal of Applied Mathematics, 61 (2001), 14451472. 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), 401405. 
[34] 
P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185190. 
[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 predatorprey systems with a strong Allee effect, Mathematical Biosciences, 209 (2007), 451469. doi: 10.1016/j.mbs.2007.02.006. 
[38] 
S. VélizRetamales and E. GonzálezOlivares, Dynamics of a Gause type preypredator model with a rational nonmonotonic consumption function, in "Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology" (ed. R. Mondaini), EPapers Serviços Editoriais Ltda, Río de Janeiro, 2 (2004), 181192. 
[39] 
J. Wang, J. Shi and J. Wei, Predatorprey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291331. doi: 10.1007/s0028501003321. 
[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 predatorprey system involving group defense, SIAM Journal on Applied Mathematics, 48 (1988), 592606. doi: 10.1137/0148033. 
[42] 
D. Xiao and S. Ruan, Bifurcations in a predatorprey system with group defense, International Journal of Bifurcation and Chaos, 11 (2001), 21232131. doi: 10.1142/S021812740100336X. 
[43] 
D. Xiao and Z. Zhang, On the uniquenes and nonexsitence of limit cycles for predatorprey systems, Nonlinearity, 16 (2003), 11851201. doi: 10.1088/09517715/16/3/321. 
[44] 
H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predatorprey system with nonmonotonic functional response, SIAM Journal on Applied Mathematics, 63 (2002), 636682. doi: 10.1137/S0036139901397285. 
[45] 
J. Zu and M. Mimura, The impact of Allee effect on a predatorprey system with Holling type II functional response, Applied Mathematics and Computation, 217 (2010), 35423556. doi: 10.1016/j.amc.2010.09.029. 
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