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The impact of toxins on competition dynamics of three species in a polluted aquatic environment
Qualitative properties and bifurcations of a leaf-eating herbivores model
School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China |
In this paper, we discuss the dynamics of a discrete-time leaf-eating herbivores model. First of all, to investigate the bifurcations of the model, we study the qualitative properties of a fixed point, including hyperbolic and non-hyperbolic. Secondly, applying the center manifold theorem, we give the conditions that the model produces a supercritical flip bifurcation and a subcritical flip bifurcation respectively, from which we find a generalized flip bifurcation point. And then, we prove rigorously that the model undergoes a generalized flip bifurcation and give three parameter regions that the model possesses two period-two cycles, one period-two cycles and none respectively. Next, computing the normal form, we prove that the model undergoes a subcritical Neimark-Sacker bifurcation and produces a unique unstable invariant circle near the fixed point. Finally, by numerical simulations, we not only verify our results but also show a saddle period-five cycle and a saddle period-six cycle on the invariant circle.
References:
[1] |
L. J. S. Allen, M. K. Hannigan and M. J. Strauss, Mathematical analysis of a model for a plant-herbivore system, Bull. Math. Biol., 55 (1993), 847-864. Google Scholar |
[2] |
J. Carr, Application of Center Manifold Theory, , Springer, New York, 1981. |
[3] |
V. Castellanos and F. Sánchez-Garduño,
The existence of a limit cycle in a pollinator-plant-herbivore mathematical model, Nonlinear Anal. Real World Appl., 48 (2019), 212-231.
doi: 10.1016/j.nonrwa.2019.01.011. |
[4] |
F. M. Dannan, S. N. Elaydi and V. Ponomarenko,
Stability of hyperbolic and nonhyperbolic fixed points of one-dimensional maps, J. Difference Equ. Appl., 9 (2003), 449-457.
doi: 10.1080/1023619031000078315. |
[5] |
L. Edelstein-Keshet, Mathematical Models in Biology, Society for industrial and Applied Mathematics, Philadelphia, 2005.
doi: 10.1137/1.9780898719147. |
[6] |
S. Elaydi, An Introduction to Difference Equations, 3$^rd$ edition, Springer, New York, 2005. |
[7] |
M. Erb and P. Reymond,
Molecular interactions between plants and insect herbivores, Annu. Rev. Plant Biol., 70 (2019), 527-557.
doi: 10.1146/annurev-arplant-050718-095910. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. Huang, S. Liu, S. Ruan and D. Xiao,
Bifurcations in a discrete predator-prey model with nonmonotonic functional response, J. Math. Anal. Appl., 464 (2018), 201-230.
doi: 10.1016/j.jmaa.2018.03.074. |
[10] |
R. R Kariyat and S. L. Portman,
Plant-herbivore interactions: Thinking beyond larval growth and mortality, Am. J. Bot., 103 (2016), 789-791.
doi: 10.3732/ajb.1600066. |
[11] |
A. Q. Khan and M. N. Qureshi, Stability analysis of a discrete biological model, Int. J. Biomath., 9 (2016), 1650021, 19 pp.
doi: 10.1142/S1793524516500212. |
[12] |
A. Q. Khan, J. Ma and D. Xiao,
Bifurcations of a two-dimensional discrete time plant-herbivore system, Commun. Nonlinear Sci. Numer. Simul., 39 (2016), 185-198.
doi: 10.1016/j.cnsns.2016.02.037. |
[13] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2$^nd$ edition, Springer, New York, 1998. |
[14] |
Y. Li, Z. Feng, R. Swihart, J. Bryant and N. Huntly,
Modeling the impact of plant toxicity on plant-herbivore dynamics, J. Dyn. Differ. Equ., 18 (2006), 1021-1042.
doi: 10.1007/s10884-006-9029-y. |
[15] |
L. Li, J. Zhen and L. Jing,
Periodic solutions in a herbivore-plant system with time delay and spatial diffusion, Appl. Math. Model., 40 (2016), 4765-4777.
doi: 10.1016/j.apm.2015.12.003. |
[16] |
S. Li and W. Zhang,
Bifurcations of a discrete prey-predator model with Holling type Ⅱ functional response, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 159-176.
doi: 10.3934/dcdsb.2010.14.159. |
[17] |
X. Liu and D. Xiao,
Bifurcations in a discrete time Lotka-Volterra predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 559-572.
doi: 10.3934/dcdsb.2006.6.559. |
[18] |
R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis,
Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 245 (2008), 442-467.
doi: 10.1016/j.jde.2007.10.034. |
[19] |
E. Lorenz,
Computational chaos - a prelude to computational instability, Physica D, 35 (1989), 299-317.
doi: 10.1016/0167-2789(89)90072-9. |
[20] |
J. L. Maron, A. A. Agrawal and D. W. Schemske, Plant-herbivore coevolution and plant speciation, Ecology, 100 (2019), e02704 (33pages).
doi: 10.1002/ecy.2704. |
[21] |
R. M. May,
Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos, Science, 186 (1974), 645-647.
doi: 10.1126/science.186.4164.645. |
[22] |
G.-Q. Sun, A. Chakraborty, Q.-X. Liu, Z. Jin, K. E. Anderson and B.-L. Li,
Influence of time delay and nonlinear diffusion on herbivore outbreak, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1507-1518.
doi: 10.1016/j.cnsns.2013.09.016. |
[23] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. |
[24] |
J. Zhang and J. Zhong,
Qualitative structures of a degenerate fixed point of a Ricker competition model, J. Difference Equ. Appl., 25 (2019), 430-454.
doi: 10.1080/10236198.2019.1581181. |
[25] |
Y. Zhao, Z. Feng, Y. Zheng and X. Cen,
Existence of limit cycles and homoclinic bifurcation in a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 258 (2015), 2847-2872.
doi: 10.1016/j.jde.2014.12.029. |
[26] |
J. Zhong and J. Zhang,
The stability of a degenerate fixed point for Guzowska-Luis-Elaydi Model, J. Differenc Equ. Appl., 24 (2018), 409-424.
doi: 10.1080/10236198.2017.1411909. |
show all references
References:
[1] |
L. J. S. Allen, M. K. Hannigan and M. J. Strauss, Mathematical analysis of a model for a plant-herbivore system, Bull. Math. Biol., 55 (1993), 847-864. Google Scholar |
[2] |
J. Carr, Application of Center Manifold Theory, , Springer, New York, 1981. |
[3] |
V. Castellanos and F. Sánchez-Garduño,
The existence of a limit cycle in a pollinator-plant-herbivore mathematical model, Nonlinear Anal. Real World Appl., 48 (2019), 212-231.
doi: 10.1016/j.nonrwa.2019.01.011. |
[4] |
F. M. Dannan, S. N. Elaydi and V. Ponomarenko,
Stability of hyperbolic and nonhyperbolic fixed points of one-dimensional maps, J. Difference Equ. Appl., 9 (2003), 449-457.
doi: 10.1080/1023619031000078315. |
[5] |
L. Edelstein-Keshet, Mathematical Models in Biology, Society for industrial and Applied Mathematics, Philadelphia, 2005.
doi: 10.1137/1.9780898719147. |
[6] |
S. Elaydi, An Introduction to Difference Equations, 3$^rd$ edition, Springer, New York, 2005. |
[7] |
M. Erb and P. Reymond,
Molecular interactions between plants and insect herbivores, Annu. Rev. Plant Biol., 70 (2019), 527-557.
doi: 10.1146/annurev-arplant-050718-095910. |
[8] |
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
doi: 10.1007/978-1-4612-1140-2. |
[9] |
J. Huang, S. Liu, S. Ruan and D. Xiao,
Bifurcations in a discrete predator-prey model with nonmonotonic functional response, J. Math. Anal. Appl., 464 (2018), 201-230.
doi: 10.1016/j.jmaa.2018.03.074. |
[10] |
R. R Kariyat and S. L. Portman,
Plant-herbivore interactions: Thinking beyond larval growth and mortality, Am. J. Bot., 103 (2016), 789-791.
doi: 10.3732/ajb.1600066. |
[11] |
A. Q. Khan and M. N. Qureshi, Stability analysis of a discrete biological model, Int. J. Biomath., 9 (2016), 1650021, 19 pp.
doi: 10.1142/S1793524516500212. |
[12] |
A. Q. Khan, J. Ma and D. Xiao,
Bifurcations of a two-dimensional discrete time plant-herbivore system, Commun. Nonlinear Sci. Numer. Simul., 39 (2016), 185-198.
doi: 10.1016/j.cnsns.2016.02.037. |
[13] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2$^nd$ edition, Springer, New York, 1998. |
[14] |
Y. Li, Z. Feng, R. Swihart, J. Bryant and N. Huntly,
Modeling the impact of plant toxicity on plant-herbivore dynamics, J. Dyn. Differ. Equ., 18 (2006), 1021-1042.
doi: 10.1007/s10884-006-9029-y. |
[15] |
L. Li, J. Zhen and L. Jing,
Periodic solutions in a herbivore-plant system with time delay and spatial diffusion, Appl. Math. Model., 40 (2016), 4765-4777.
doi: 10.1016/j.apm.2015.12.003. |
[16] |
S. Li and W. Zhang,
Bifurcations of a discrete prey-predator model with Holling type Ⅱ functional response, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 159-176.
doi: 10.3934/dcdsb.2010.14.159. |
[17] |
X. Liu and D. Xiao,
Bifurcations in a discrete time Lotka-Volterra predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 559-572.
doi: 10.3934/dcdsb.2006.6.559. |
[18] |
R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis,
Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 245 (2008), 442-467.
doi: 10.1016/j.jde.2007.10.034. |
[19] |
E. Lorenz,
Computational chaos - a prelude to computational instability, Physica D, 35 (1989), 299-317.
doi: 10.1016/0167-2789(89)90072-9. |
[20] |
J. L. Maron, A. A. Agrawal and D. W. Schemske, Plant-herbivore coevolution and plant speciation, Ecology, 100 (2019), e02704 (33pages).
doi: 10.1002/ecy.2704. |
[21] |
R. M. May,
Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos, Science, 186 (1974), 645-647.
doi: 10.1126/science.186.4164.645. |
[22] |
G.-Q. Sun, A. Chakraborty, Q.-X. Liu, Z. Jin, K. E. Anderson and B.-L. Li,
Influence of time delay and nonlinear diffusion on herbivore outbreak, Commun. Nonlinear Sci. Numer. Simul., 19 (2014), 1507-1518.
doi: 10.1016/j.cnsns.2013.09.016. |
[23] |
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. |
[24] |
J. Zhang and J. Zhong,
Qualitative structures of a degenerate fixed point of a Ricker competition model, J. Difference Equ. Appl., 25 (2019), 430-454.
doi: 10.1080/10236198.2019.1581181. |
[25] |
Y. Zhao, Z. Feng, Y. Zheng and X. Cen,
Existence of limit cycles and homoclinic bifurcation in a plant-herbivore model with toxin-determined functional response, J. Differential Equations, 258 (2015), 2847-2872.
doi: 10.1016/j.jde.2014.12.029. |
[26] |
J. Zhong and J. Zhang,
The stability of a degenerate fixed point for Guzowska-Luis-Elaydi Model, J. Differenc Equ. Appl., 24 (2018), 409-424.
doi: 10.1080/10236198.2017.1411909. |







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Conditions | Cases | ||
stable focus | |||
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