# American Institute of Mathematical Sciences

2013, 10(2): 345-367. doi: 10.3934/mbe.2013.10.345

## Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey 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 57069-2390, United States

Received  October 2011 Revised  August 2012 Published  January 2013

The main purpose of this work is to analyze a Gause type predator-prey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation.
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.
Citation: Eduardo González-Olivares, Betsabé González-Yañez, Jaime Mena-Lorca, José D. Flores. Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey. Mathematical Biosciences & Engineering, 2013, 10 (2) : 345-367. doi: 10.3934/mbe.2013.10.345
##### References:
 [1] A. Aguilera-Moya and E. González-Olivares, A Gause type model with a generalized class of nonmonotonic functional response,, in, 2 (2004), 206. [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. 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. [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. doi: 10.1006/jtbi.2002.3084. [7] C. Chicone, "Ordinary Differential Equations with Applications,", (2nd edition), (2006). [8] C. W. Clark, "Mathematical Bioeconomic: The Optimal Management of Renewable Resources,", (2nd edition), (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, (1983), 279. [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. 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. 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. [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. 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 (2003). [19] J. C. Gascoigne and R. N. Lipcius, Allee effects driven by predation,, Journal of Applied Ecology, 41 (2004), 801. [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. 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, (2007), 53. [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. 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. 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. 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, 2 (2004), 358. [26] K. Hasík, On a predator-prey system of Gause type,, Journal of Mathematical Biology, 60 (2010), 59. 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. 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. [30] L. Perko, "Differential Equations and Dynamical Systems,", (3rd ed), (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, (2007), 295. [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. 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. [34] P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?,, Oikos, 87 (1999), 185. [35] R. J. Taylor, "Predation,", Chapman and Hall, (1984). [36] P. Turchin, "Complex Population Dynamics. A Theoretical/Empirical Synthesis,", Monographs in Population Biology 35, (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. 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, 2 (2004), 181. [39] J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey,, Journal of Mathematical Biology, 62 (2011), 291. doi: 10.1007/s00285-010-0332-1. [40] S. Wolfram, "Mathematica: A System for Doing Mathematics by Computer,", (2nd edition), (1991). [41] G. S. W. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defense,, SIAM Journal on Applied Mathematics, 48 (1988), 592. 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. 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. 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. 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. doi: 10.1016/j.amc.2010.09.029.

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##### References:
 [1] A. Aguilera-Moya and E. González-Olivares, A Gause type model with a generalized class of nonmonotonic functional response,, in, 2 (2004), 206. [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. 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. [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. doi: 10.1006/jtbi.2002.3084. [7] C. Chicone, "Ordinary Differential Equations with Applications,", (2nd edition), (2006). [8] C. W. Clark, "Mathematical Bioeconomic: The Optimal Management of Renewable Resources,", (2nd edition), (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, (1983), 279. [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. 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. 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. [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. 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 (2003). [19] J. C. Gascoigne and R. N. Lipcius, Allee effects driven by predation,, Journal of Applied Ecology, 41 (2004), 801. [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. 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, (2007), 53. [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. 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. 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. 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, 2 (2004), 358. [26] K. Hasík, On a predator-prey system of Gause type,, Journal of Mathematical Biology, 60 (2010), 59. 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. 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. [30] L. Perko, "Differential Equations and Dynamical Systems,", (3rd ed), (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, (2007), 295. [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. 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. [34] P. A. Stephens, W. J. Sutherland and R. P. Freckleton, What is the Allee effect?,, Oikos, 87 (1999), 185. [35] R. J. Taylor, "Predation,", Chapman and Hall, (1984). [36] P. Turchin, "Complex Population Dynamics. A Theoretical/Empirical Synthesis,", Monographs in Population Biology 35, (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. 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, 2 (2004), 181. [39] J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey,, Journal of Mathematical Biology, 62 (2011), 291. doi: 10.1007/s00285-010-0332-1. [40] S. Wolfram, "Mathematica: A System for Doing Mathematics by Computer,", (2nd edition), (1991). [41] G. S. W. Wolkowicz, Bifurcation analysis of a predator-prey system involving group defense,, SIAM Journal on Applied Mathematics, 48 (1988), 592. 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. 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. 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. 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. doi: 10.1016/j.amc.2010.09.029.
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