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June  2021, 26(6): 3381-3407. doi: 10.3934/dcdsb.2020236

Qualitative properties and bifurcations of a leaf-eating herbivores model

Jiyu Zhong ,

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

*Corresponding author: matzhjy@sina.com

Received  February 2020 Revised  May 2020 Published  June 2021 Early access  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.

Keywords: Center manifold, normal form, bifurcation, invariant circle, periodic-n cycle.
Mathematics Subject Classification: Primary: 37G10, 39A28; Secondary: 58K50.
Citation: Jiyu Zhong. Qualitative properties and bifurcations of a leaf-eating herbivores model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3381-3407. doi: 10.3934/dcdsb.2020236
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. 

[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. 

[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.

Figure 1.  Diagram of bifurcation for system (2)
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Figure 2.  Bifurcation diagram of system (2) near the point $ GF $
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Figure 3.  Bifurcation diagram of system (31) for small $ |\beta| $
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Figure 4.  Flip bifurcation route to chaos for $ k = 0.3 $
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Figure 5.  An invariant circle $ \Gamma $ produced from the Neimark-Sacker bifurcation
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Figure 6.  A saddle period-five cycle on the invariant circle $ \Gamma $
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Figure 7.  A saddle period-six cycle on the invariant circle $ \Gamma $
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Table 1.  Topological types of fixed point $ E $ in the hyperbolic case
Conditions $ E_1 $ Cases
$ 0<k<1 $ $ 0<b< 4k $ stable focus $ \mathfrak{D}_5 $
$ 4k\leq b<4/(2-k) $ stable node $ \mathfrak{D}_1 $
$ b>4/(2-k) $ saddle point $ \mathfrak{D}_2 $-I
$ k=1 $ $ b>4 $ saddle point $ \mathfrak{D}_2 $-II
$ 1<k<2 $ $ 0<b<4k $ unstable focus $ \mathfrak{D}_4 $-I
$ 4k\leq b<4/(2-k) $ unstable node $ \mathfrak{D}_3 $-I
$ b>4/(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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Conditions $ E_1 $ Cases
$ 0<k<1 $ $ 0<b< 4k $ stable focus $ \mathfrak{D}_5 $
$ 4k\leq b<4/(2-k) $ stable node $ \mathfrak{D}_1 $
$ b>4/(2-k) $ saddle point $ \mathfrak{D}_2 $-I
$ k=1 $ $ b>4 $ saddle point $ \mathfrak{D}_2 $-II
$ 1<k<2 $ $ 0<b<4k $ unstable focus $ \mathfrak{D}_4 $-I
$ 4k\leq b<4/(2-k) $ unstable node $ \mathfrak{D}_3 $-I
$ b>4/(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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