    ## Qualitative properties and bifurcations of a leaf-eating herbivores model

 School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guangdong 524048, China

*Corresponding author: matzhjy@sina.com

Received  February 2020 Revised  May 2020 Published  August 2020

Fund Project: The paper was partially supported by the Characteristic innovation projects of colleges and universities in Guangdong Province (2019KTSCX088), the National Natural Science Foundation of China (11771197) and the Key Subject Program of Lingnan Normal University (1171518004)

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.

Citation: Jiyu Zhong. Qualitative properties and bifurcations of a leaf-eating herbivores model. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020236
##### References:
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##### References:
  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  J. Carr, Application of Center Manifold Theory, , Springer, New York, 1981. Google Scholar  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.  Google Scholar  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.  Google Scholar  L. Edelstein-Keshet, Mathematical Models in Biology, Society for industrial and Applied Mathematics, Philadelphia, 2005. doi: 10.1137/1.9780898719147.  Google Scholar  S. Elaydi, An Introduction to Difference Equations, 3$^rd$ edition, Springer, New York, 2005. Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  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.  Google Scholar  Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2$^nd$ edition, Springer, New York, 1998. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  E. Lorenz, Computational chaos - a prelude to computational instability, Physica D, 35 (1989), 299-317.  doi: 10.1016/0167-2789(89)90072-9.  Google Scholar  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. Google Scholar  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. Google Scholar  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.  Google Scholar  S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, 2nd edition, Springer, New York, 2003. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar An invariant circle $\Gamma$ produced from the Neimark-Sacker bifurcation
Topological types of fixed point $E$ in the hyperbolic case
 Conditions $E_1$ Cases $04/(2-k)$ saddle point $\mathfrak{D}_2$-I $k=1$ $b>4$ saddle point $\mathfrak{D}_2$-II $14/(2-k)$ saddle point $\mathfrak{D}_2$-III $k\geq2$ $0< b<4k$ unstable focus $\mathfrak{D}_4$-II $b\geq 2k$ unstable node $\mathfrak{D}_3$-II
 Conditions $E_1$ Cases $04/(2-k)$ saddle point $\mathfrak{D}_2$-I $k=1$ $b>4$ saddle point $\mathfrak{D}_2$-II $14/(2-k)$ saddle point $\mathfrak{D}_2$-III $k\geq2$ $0< b<4k$ unstable focus $\mathfrak{D}_4$-II $b\geq 2k$ unstable node $\mathfrak{D}_3$-II
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