Bifurcations in a Leslie-Gower type predator-prey model with a rational non-monotonic functional response
Abstract
A Leslie-Gower type predator-prey model including group defense formation is analyzed. This phenomenon, described by a non-monotonic function originates interesting dynamics; positiveness, boundedness, permanence of solutions, and existence of up to three positive equilibria are established. The solutions are highly sensitive to initial conditions since there exists a separatrix curve dividing their behavior. Two near trajectories can have far omega-limit sets. The weakness of a singularity is established showing two limit cycles can exist. Numerical simulations endorse the analytical outcomes.
Keyword : limit cycles, stability, separatrix, predator-prey model, functional response
How to Cite
González-Olivares, E., Mosquera-Aguilar, A., Tintinago-Ruiz, P. ., & Rojas-Palma, A. (2022). Bifurcations in a Leslie-Gower type predator-prey model with a rational non-monotonic functional response. Mathematical Modelling and Analysis, 27(3), 510–532. https://doi.org/10.3846/mma.2022.15528
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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C. Chicone. Ordinary Differential Equations with Applications. Texts in Applied Mathematics. Springer, New York, 2008.
F. Dumortier, J. Llibre and J.C. Artés. Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin, Heidelberg, 2006.
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H.I. Freedman and G.S.K. Wolkowicz. Predator-prey systems with group defence: The paradox of enrichment revisited. Bulletin of Mathematical Biology, 48(5):493–508, 1986. https://doi.org/10.1016/S0092-8240(86)90004-2
V.A. Gaiko and C. Vuik. Global dynamics in the LeslieGower model with the allee effect. International Journal of Bifurcation and Chaos, 28(12):1850151, 2018. https://doi.org/10.1142/S0218127418501511
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B. González-Yañez, E. González-Olivares and J. Mena-Lorca. Multistability on a Leslie-Gower type predator prey model with nomonotonic functional response. In R. Mondaini and R. Dilao(Eds.), International Symposium on Mathematical and Computational, Biology, Biomat 2006, pp. 359–384. World Scientific Co., 2007. https://doi.org/10.1142/9789812708779_0023
E. Gonzlez-Olivares, P. Tintinago-Ruiz and A. Rojas-Palma. A LeslieGowertype predatorprey model with sigmoid functional response. International Journal of Computer Mathematics, 92(9):1895–1909, 2015. https://doi.org/10.1080/00207160.2014.889818
C.S. Holling. The components of predation as revealed by a study of smallmammal predation of the european pine sawfly. The Canadian Entomologist, 91(5):293–320, 1959. https://doi.org/10.4039/Ent91293-5
J. Huang, S. Ruan and J. Song. Bifurcations in a predatorprey system of Leslie type with generalized Holling type III functional response. Journal of Differential Equations, 257(6):1721–1752, 2014. https://doi.org/10.1016/j.jde.2014.04.024
M.C. Köhnke, I. Siekmann, H. Seno and H. Malchow. A type IV functional response with different shapes in a predatorprey model. Journal of Theoretical Biology, 505:110419, 2020. https://doi.org/10.1016/j.jtbi.2020.110419
Y. Kuznetsov. Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. Springer, New York, 2013. https://doi.org/10.1007/978-1-4757-3978-7
Y. Lamontagne, C. Coutu and C. Rousseau. Bifurcation analysis of a predatorprey system with generalised Holling type III functional response. Journal of Dynamics and Differential Equations, 20(3):535–571, 2008. https://doi.org/10.1007/s10884-008-9102-9
P.H. Leslie. Some further notes on the use of matrices in population mathematics. Biometrika, 35(3-4):213–245, 12 1948. https://doi.org/10.1093/biomet/35.3-4.213
P.H. Leslie and J.C. Gower. The properties of a stochastic model for the predatorprey type of interaction between two species. Biometrika, 47(3-4):219–234, 12 1960. https://doi.org/10.1093/biomet/47.3-4.219
Y. Li and D. Xiao. Bifurcations of a predator-prey system of Holling and Leslie types. Chaos, Solitons & Fractals, 34(2):606–620, 2007. https://doi.org/10.1016/j.chaos.2006.03.068
R.M. May. Stability and Complexity in Model Ecosystems. Princeton Landmarks in Biology. Princeton University Press, 2019. https://doi.org/10.2307/j.ctvs32rq4
P. Monzón. Almost global attraction in planar systems. Systems & Control Letters, 54(8):753–758, 2005. ISSN 0167-6911. https://doi.org/10.1016/j.sysconle.2004.11.014
L. Perko. Differential Equations and Dynamical Systems. Texts in Applied Mathematics. Springer, New York, 2012. https://doi.org/10.1007/978-1-4613-0003-8
L. Puchuri, E. González-Olivares and A. Rojas-Palma. Multistability in a LeslieGowertype predation model with a rational nonmonotonic functional response and generalist predators. Computational and Mathematical Methods, 2(2):e1070, 2020. https://doi.org/10.1002/cmm4.1070
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 R. Mondaini(Ed.), Proceedings of the 2006 International Brazilian Symposium on Mathematical and Computational Biology, pp. 295–321. E-Papers Servi¸cos Editoriais, Ltda., 2007.
E. Sáez and E. González-Olivares. Dynamics of a predator-prey model. SIAM Journal on Applied Mathematics, 59(5):1867–1878, 1999. https://doi.org/10.1137/S0036139997318457
G. Seo and D.L. DeAngelis. A predator-prey model with a Holling type I functional response including a predator mutual interference. Journal of Nonlinear Science, 21(6):811–833, 2011. https://doi.org/10.1007/s00332-011-9101-6
R.J. Taylor. Predation. Population and Community Biology. Springer, Netherlands, 2013.
P. Turchin. Complex Population Dynamics: A Theoretical/Empirical Synthesis. Monographs in Population Biology. Princeton University Press, 2013. https://doi.org/10.1515/9781400847280
E. Venturino and S. Petrovskii. Spatiotemporal behavior of a preypredator system with a group defense for prey. Ecological Complexity, 14:37–47, 2013. https://doi.org/10.1016/j.ecocom.2013.01.004. Special Issue on the occasion of Horst Malchow’s 60th birthday.
K. Vilches, E. González-Olivares and A. Rojas-Palma. Prey herd behavior modeled by a generic non-differentiable functional response. Mathematical Modelling of Natural Phenomena, 13(3):26, 2018. https://doi.org/10.1051/mmnp/2018038
Inc Wolfram Research. Mathematica: A System for Doing Mathematics by Computer. Wolfram Research, Incorporated, 1992.
G.S.K. Wolkowicz. Bifurcation analysis of a predator-prey system involving group defence. SIAM Journal on Applied Mathematics, 48(3):592–606, 1988. https://doi.org/10.1137/0148033
D. Xiao and S. Ruan. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM Journal on Applied Mathematics, 61(4):1445– 1472, 2001. https://doi.org/10.1137/S0036139999361896
D. Xiao and H. Zhu. Multiple focus and hopf bifurcations in a predator-prey system with nonmonotonic functional response. SIAM Journal on Applied Mathematics, 66(3):802–819, 2006. https://doi.org/10.1137/050623449
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(2):636–682, 2003. https://doi.org/10.1137/S0036139901397285
C. Arancibia-Ibarra and E. González-Olivares. The Holling-Tanner model considering an alternative food for predator. In J. Vigo-Aguiar(Ed.), Proceedings of the 2015 International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE 2015, pp. 130–141. CMMSE, 2015.
D. Arrowsmith and C.M. Place. Dynamical Systems: Differential Equations, Maps, and Chaotic Behaviour. Chapman Hall/CRC Mathematics Series. Taylor & Francis, 1992. ISBN 9780412390807.
K.-S. Cheng. Uniqueness of a limit cycle for a predator-prey system. SIAM Journal on Mathematical Analysis, 12(4):541–548, 1981. https://doi.org/10.1137/0512047
C. Chicone. Ordinary Differential Equations with Applications. Texts in Applied Mathematics. Springer, New York, 2008.
F. Dumortier, J. Llibre and J.C. Artés. Qualitative Theory of Planar Differential Systems. Universitext. Springer, Berlin, Heidelberg, 2006.
H.I. Freedman. Deterministic Mathematical Models in Population Ecology. Monographs and textbooks in pure and applied mathematics. Dekker, New York, 1980.
H.I. Freedman and G.S.K. Wolkowicz. Predator-prey systems with group defence: The paradox of enrichment revisited. Bulletin of Mathematical Biology, 48(5):493–508, 1986. https://doi.org/10.1016/S0092-8240(86)90004-2
V.A. Gaiko and C. Vuik. Global dynamics in the LeslieGower model with the allee effect. International Journal of Bifurcation and Chaos, 28(12):1850151, 2018. https://doi.org/10.1142/S0218127418501511
G.F. Gause. The Struggle for Existence. The Struggle for Existence. The Williams & Wilkins company, Baltimore, 1934. ISBN 9780598976703. https://doi.org/10.5962/bhl.title.4489
B. González-Yañez, E. González-Olivares and J. Mena-Lorca. Multistability on a Leslie-Gower type predator prey model with nomonotonic functional response. In R. Mondaini and R. Dilao(Eds.), International Symposium on Mathematical and Computational, Biology, Biomat 2006, pp. 359–384. World Scientific Co., 2007. https://doi.org/10.1142/9789812708779_0023
E. Gonzlez-Olivares, P. Tintinago-Ruiz and A. Rojas-Palma. A LeslieGowertype predatorprey model with sigmoid functional response. International Journal of Computer Mathematics, 92(9):1895–1909, 2015. https://doi.org/10.1080/00207160.2014.889818
C.S. Holling. The components of predation as revealed by a study of smallmammal predation of the european pine sawfly. The Canadian Entomologist, 91(5):293–320, 1959. https://doi.org/10.4039/Ent91293-5
J. Huang, S. Ruan and J. Song. Bifurcations in a predatorprey system of Leslie type with generalized Holling type III functional response. Journal of Differential Equations, 257(6):1721–1752, 2014. https://doi.org/10.1016/j.jde.2014.04.024
M.C. Köhnke, I. Siekmann, H. Seno and H. Malchow. A type IV functional response with different shapes in a predatorprey model. Journal of Theoretical Biology, 505:110419, 2020. https://doi.org/10.1016/j.jtbi.2020.110419
Y. Kuznetsov. Elements of Applied Bifurcation Theory. Applied Mathematical Sciences. Springer, New York, 2013. https://doi.org/10.1007/978-1-4757-3978-7
Y. Lamontagne, C. Coutu and C. Rousseau. Bifurcation analysis of a predatorprey system with generalised Holling type III functional response. Journal of Dynamics and Differential Equations, 20(3):535–571, 2008. https://doi.org/10.1007/s10884-008-9102-9
P.H. Leslie. Some further notes on the use of matrices in population mathematics. Biometrika, 35(3-4):213–245, 12 1948. https://doi.org/10.1093/biomet/35.3-4.213
P.H. Leslie and J.C. Gower. The properties of a stochastic model for the predatorprey type of interaction between two species. Biometrika, 47(3-4):219–234, 12 1960. https://doi.org/10.1093/biomet/47.3-4.219
Y. Li and D. Xiao. Bifurcations of a predator-prey system of Holling and Leslie types. Chaos, Solitons & Fractals, 34(2):606–620, 2007. https://doi.org/10.1016/j.chaos.2006.03.068
R.M. May. Stability and Complexity in Model Ecosystems. Princeton Landmarks in Biology. Princeton University Press, 2019. https://doi.org/10.2307/j.ctvs32rq4
P. Monzón. Almost global attraction in planar systems. Systems & Control Letters, 54(8):753–758, 2005. ISSN 0167-6911. https://doi.org/10.1016/j.sysconle.2004.11.014
L. Perko. Differential Equations and Dynamical Systems. Texts in Applied Mathematics. Springer, New York, 2012. https://doi.org/10.1007/978-1-4613-0003-8
L. Puchuri, E. González-Olivares and A. Rojas-Palma. Multistability in a LeslieGowertype predation model with a rational nonmonotonic functional response and generalist predators. Computational and Mathematical Methods, 2(2):e1070, 2020. https://doi.org/10.1002/cmm4.1070
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 R. Mondaini(Ed.), Proceedings of the 2006 International Brazilian Symposium on Mathematical and Computational Biology, pp. 295–321. E-Papers Servi¸cos Editoriais, Ltda., 2007.
E. Sáez and E. González-Olivares. Dynamics of a predator-prey model. SIAM Journal on Applied Mathematics, 59(5):1867–1878, 1999. https://doi.org/10.1137/S0036139997318457
G. Seo and D.L. DeAngelis. A predator-prey model with a Holling type I functional response including a predator mutual interference. Journal of Nonlinear Science, 21(6):811–833, 2011. https://doi.org/10.1007/s00332-011-9101-6
R.J. Taylor. Predation. Population and Community Biology. Springer, Netherlands, 2013.
P. Turchin. Complex Population Dynamics: A Theoretical/Empirical Synthesis. Monographs in Population Biology. Princeton University Press, 2013. https://doi.org/10.1515/9781400847280
E. Venturino and S. Petrovskii. Spatiotemporal behavior of a preypredator system with a group defense for prey. Ecological Complexity, 14:37–47, 2013. https://doi.org/10.1016/j.ecocom.2013.01.004. Special Issue on the occasion of Horst Malchow’s 60th birthday.
K. Vilches, E. González-Olivares and A. Rojas-Palma. Prey herd behavior modeled by a generic non-differentiable functional response. Mathematical Modelling of Natural Phenomena, 13(3):26, 2018. https://doi.org/10.1051/mmnp/2018038
Inc Wolfram Research. Mathematica: A System for Doing Mathematics by Computer. Wolfram Research, Incorporated, 1992.
G.S.K. Wolkowicz. Bifurcation analysis of a predator-prey system involving group defence. SIAM Journal on Applied Mathematics, 48(3):592–606, 1988. https://doi.org/10.1137/0148033
D. Xiao and S. Ruan. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM Journal on Applied Mathematics, 61(4):1445– 1472, 2001. https://doi.org/10.1137/S0036139999361896
D. Xiao and H. Zhu. Multiple focus and hopf bifurcations in a predator-prey system with nonmonotonic functional response. SIAM Journal on Applied Mathematics, 66(3):802–819, 2006. https://doi.org/10.1137/050623449
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(2):636–682, 2003. https://doi.org/10.1137/S0036139901397285