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doi: 10.3934/dcdsb.2022018
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Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect

a. 

College of Mathematics and Statistics, Fujian Normal University, Fuzhou 350007, China

b. 

Center for Applied Mathematics of Fujian Province (FJNU), Fuzhou 350117, China

c. 

Fujian Key Laboratory of Mathematical Analysis and Applications, Fuzhou 350117, China

d. 

College of Information and Statistics, Guangxi University of Finance and Economics, Nanning 530003, China

*Corresponding author: Jianhe Shen

Received  June 2021 Revised  November 2021 Early access February 2022

Fund Project: The research was supported by the Natural Science Foundation of China (NO. 11771082) and Guangxi College Enhancing Youths Capacity Project (2021KY0653)

This paper studies bifurcations of canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect. Based on geometric singular perturbation theory (GSPT) and canard theory, canard explosion is observed and the associated bifurcation curve is determined. Due to the canard point, a homoclinic orbit with slow and fast segments and homoclinic to a saddle can also exist, in which, the stable and unstable manifolds of the saddle are connected under certain parameter value. By analyzing the slow divergence integral, it is proved that the cyclicity of canard cycles in this model is at most four. Finally, by calculating the entry-exit function explicitly, a unique, orbitally stable canard relaxation oscillation passing through a transcritical bifurcation point is detected. All these theoretical predictions on the birth of canard explosion, canard limit cycles and homoclinic orbits are verified by numerical simulations.

Citation: Liang Zhao, Jianhe Shen. Canards and homoclinic orbits in a slow-fast modified May-Holling-Tanner predator-prey model with weak multiple Allee effect. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022018
References:
[1]

P. AguirreE. Gonzalezolivares and E. Saez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM J. Appl. Math., 69 (2009), 1244-1262.  doi: 10.1137/070705210.

[2]

S. Ai and S. Sadhu, The entry-exit theorem and relaxation oscillations in slow-fast planar systems, J. Differential Equations, 268 (2020), 7220-7249.  doi: 10.1016/j.jde.2019.11.067.

[3]

M. Andersson and S. Erlinge, Influence of predation on rodent populations, Oikos, 29 (1977), 591-597. 

[4]

C. Arancibia-Ibarra, The basins of attraction in a modified May-Holling-Tanner predator-prey model with allee affect, Nonlinear Anal., 185 (2019), 15-28.  doi: 10.1016/j.na.2019.03.004.

[5]

C. Arancibia-Ibarra and J. Flores, Stability analysis of a modified Leslie-Gower predation model with weak Allee effect on the prey, preprint, arXiv preprint, arXiv: 2009.02478.

[6]

C. Arancibia-IbarraJ. D. FloresM. Bode and G. Pettet, A May-Holling-Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 943-962.  doi: 10.3934/dcdsb.2020148.

[7]

C. Arancibia-Ibarra, J. Flores, G. Pettet and P. van Heijster, A Holling-Tanner predator-prey model with strong Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1930032, 16 pp. doi: 10.1142/S0218127419300325.

[8]

A. Arsie, C. Kottegoda and C. Shan, Multiple limit cycles and heteroclinic loops in a predator-prey system with Allee effects in prey, preprint, arXiv preprint, arXiv: 2007.06806.

[9]

A. Atabaigi amd A. Barati, Relaxation oscillations and canard explosion in a predator-prey system of Holling and Leslie types, Nonlinear Anal. Real World Appl., 36 (2017), 139-153.  doi: 10.1016/j.nonrwa.2017.01.006.

[10]

R. BastiaansenP. Carter and A. Doelman, Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems, Nonlinearity, 32 (2019), 2759-2814.  doi: 10.1088/1361-6544/ab1767.

[11]

L. BerecE. Angulo and F. Courchamp, Multiple allee effects and population management, Trends in Ecology and Evolution, 22 (2007), 185-191. 

[12]

P. N. DavisP. van HeijsterR. Marangell and M. R. Rodrigo, Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis, Journal of Dynamics and Differential Equations, 12 (2021), 1-23. 

[13]

F. Dumortier, Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., 10 (2011), 65-85.  doi: 10.1007/s12346-011-0038-9.

[14]

F. Dumortier and R. Roussarie, Multiple canard cycles in generalized Li$\acute{e}$nard equations, J. Differential Equations, 174 (2001), 1-29.  doi: 10.1006/jdeq.2000.3947.

[15]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[16]

E. Gonz$\acute{a}$lez-OlivaresL. M. Gallego-BerrioB. Gonz$\acute{a}$lez-Ya$\tilde{n}$ez and A. Rojas-Palma, Consequences of weak Allee effect on prey in the May-Holling-Tanner predator-prey model, Math. Methods Appl. Sci., 39 (2016), 4700-4712.  doi: 10.1002/mma.3404.

[17]

E. Gonz$\acute{a}$lez-OlivaresB. Gonz$\acute{a}$lez-Y$\acute{a}$$\tilde{n}$ezJ. Mena-LorcaA. Rojas-Palma and J. D. Flores, Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Comput. Math. Appl., 62 (2011), 3449-3463.  doi: 10.1016/j.camwa.2011.08.061.

[18]

E. Gonz$\acute{a}$lez-Olivares and J. Huincahue-Arcos, Double allee effects on prey in a modified Rosenzweig-MacArthur predator-prey model, Computational Problems in Engineering, 307 (2014), 105-119. 

[19]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.

[20]

S.-B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911.  doi: 10.3934/dcdsb.2009.11.893.

[21]

T.-H. Hsu, On bifurcation delay: An alternative approach using geometric singular perturbation theory, J. Differential Equations, 262 (2017), 1617-1630.  doi: 10.1016/j.jde.2016.10.022.

[22]

T.-H. Hsu, Number and stability of relaxation oscillations for predator-prey systems with small death rates, SIAM J. Appl. Dyn. Syst., 18 (2019), 33-67.  doi: 10.1137/18M1166705.

[23]

T.-H. Hsu and G. S. K. Wolkowicz, A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1257-1277.  doi: 10.3934/dcdsb.2019219.

[24]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.

[25]

M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, 14 (2001), 1473-1491.  doi: 10.1088/0951-7715/14/6/304.

[26]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.  doi: 10.1137/S0036141099360919.

[27]

C. LiJ. LiZ. Ma and H. Zhu, Canard phenomenon for an SIS epidemic model with nonlinear incidence, J. Math. Anal. Appl., 420 (2014), 987-1004.  doi: 10.1016/j.jmaa.2014.06.035.

[28]

C. Li and K. Lu, Slow divergence integral and its application to classical Li$\acute{e}$nard equations of degree5, J. Differential Equations, 257 (2014), 4437-4469.  doi: 10.1016/j.jde.2014.08.015.

[29]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.

[30]

M. Y. LiW. LiuC. Shan and Y. Yi, Turning points and relaxation oscillation cycles in simple epidemic models, SIAM J. Appl. Math., 76 (2016), 663-687.  doi: 10.1137/15M1038785.

[31]

W. LiuD. Xiao and Y. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differential Equations, 188 (2003), 306-331.  doi: 10.1016/S0022-0396(02)00076-1.

[32]

P. D. MaesschalckF. Dumortier and R. Roussarie, Canard-cycle transition at a fast-fast passage through a jump point, C. R. Math., 352 (2014), 27-30.  doi: 10.1016/j.crma.2013.09.002.

[33]

P. D. Maesschalck and S. Schecter, The entry-exit function and geometric singular perturbation theory, J. Differential Equations, 260 (2016), 6697-6715.  doi: 10.1016/j.jde.2016.01.008.

[34]

L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters, Qual. Theory Dyn. Syst., 11 (2012), 167-198.  doi: 10.1007/s12346-011-0061-x.

[35]

N. Martinez-Jeraldo and P. Aguirre, Allee effect acting on the prey species in a Leslie-Gower predation model, Nonlinear Anal. Real World Appl., 45 (2019), 895-917.  doi: 10.1016/j.nonrwa.2018.08.009.

[36]

R. Ostfeld and C. Canham, Density-dependent processes in meadow voles: An experimental approach, Ecology, 76 (1995), 521-532. 

[37]

P. J. Pal and T. Saha, Qualitative analysis of a predator-prey system with double Allee effect in prey, Chaos Solitons and Fractals, 73 (2015), 36-63.  doi: 10.1016/j.chaos.2014.12.007.

[38]

S. H. PiltzF. VeermanP. K. Maini and M. A. Porter, A predator-2 prey fast-slow dynamical system for rapid predator evolution, SIAM J. Appl. Dyn. Syst., 16 (2017), 54-90.  doi: 10.1137/16M1068426.

[39]

C. H. Shan, Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases, Discrete and Continuous Dynamical Systems-B, preprint. doi: 10.3934/dcdsb.2021097.

[40]

J. H. Shen, Canard limit cycles and global dynamics in a singularly perturbed predator-prey system with non-monotonic functional response, Nonlinear Anal. Real World Appl., 31 (2016), 146-165.  doi: 10.1016/j.nonrwa.2016.01.013.

[41]

J. Shen and M. Han, Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Li$\acute{e}$nard systems, Discrete Contin. Dyn. Syst., 33 (2013), 3085-3108.  doi: 10.3934/dcds.2013.33.3085.

[42]

J. H. ShenC. H. Hsu and T. H. Yang, Fast-slow dynamics for intraguild predation models with evolutionary effects, J. Dynam. Differential Equations, 32 (2020), 895-920.  doi: 10.1007/s10884-019-09744-3.

[43]

M. SenM. Banerjee and A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecological Complexity, 11 (2012), 12-27. 

[44]

C. Wang and X. Zhang, Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type Ⅲ, J. Differential Equations, 267 (2019), 3397-3441.  doi: 10.1016/j.jde.2019.04.008.

[45]

C. Wang and X. Zhang, Stability loss delay and smoothness of the return map in slow-fast systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 788-822.  doi: 10.1137/17M1130010.

[46]

M. Wechselberger, Geometric Singular Perturbation Theory Beyond the Standard Form, Frontiers in Applied Dynamical Systems: Reviews and Tutorials, 6, Springer, Cham, 2020. doi: 10.1007/978-3-030-36399-4.

[47]

Z. Wei, Y. H. Xia and T. H. Zhang, Stability and bifurcation analysis of an amensalism model with weak Allee effect, Qual. Theory Dyn. Syst., 19 (2020), Art. 23, 15 pp. doi: 10.1007/s12346-020-00341-0.

[48]

J. Y. Xu, T. H. Zhang and M. A. Han, A regime switching model for species subject to environmental noises and additive Allee effect, Phys. A, 527 (2019), 121300, 16 pp. doi: 10.1016/j.physa.2019.121300.

[49]

Y. Zhang, Y. Zhou and B. Tang, Canard phenomenon in an SIRS epidemic model with nonlinear incidence rate, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050073, 19 pp. doi: 10.1142/S021812742050073X.

show all references

References:
[1]

P. AguirreE. Gonzalezolivares and E. Saez, Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect, SIAM J. Appl. Math., 69 (2009), 1244-1262.  doi: 10.1137/070705210.

[2]

S. Ai and S. Sadhu, The entry-exit theorem and relaxation oscillations in slow-fast planar systems, J. Differential Equations, 268 (2020), 7220-7249.  doi: 10.1016/j.jde.2019.11.067.

[3]

M. Andersson and S. Erlinge, Influence of predation on rodent populations, Oikos, 29 (1977), 591-597. 

[4]

C. Arancibia-Ibarra, The basins of attraction in a modified May-Holling-Tanner predator-prey model with allee affect, Nonlinear Anal., 185 (2019), 15-28.  doi: 10.1016/j.na.2019.03.004.

[5]

C. Arancibia-Ibarra and J. Flores, Stability analysis of a modified Leslie-Gower predation model with weak Allee effect on the prey, preprint, arXiv preprint, arXiv: 2009.02478.

[6]

C. Arancibia-IbarraJ. D. FloresM. Bode and G. Pettet, A May-Holling-Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 943-962.  doi: 10.3934/dcdsb.2020148.

[7]

C. Arancibia-Ibarra, J. Flores, G. Pettet and P. van Heijster, A Holling-Tanner predator-prey model with strong Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1930032, 16 pp. doi: 10.1142/S0218127419300325.

[8]

A. Arsie, C. Kottegoda and C. Shan, Multiple limit cycles and heteroclinic loops in a predator-prey system with Allee effects in prey, preprint, arXiv preprint, arXiv: 2007.06806.

[9]

A. Atabaigi amd A. Barati, Relaxation oscillations and canard explosion in a predator-prey system of Holling and Leslie types, Nonlinear Anal. Real World Appl., 36 (2017), 139-153.  doi: 10.1016/j.nonrwa.2017.01.006.

[10]

R. BastiaansenP. Carter and A. Doelman, Stable planar vegetation stripe patterns on sloped terrain in dryland ecosystems, Nonlinearity, 32 (2019), 2759-2814.  doi: 10.1088/1361-6544/ab1767.

[11]

L. BerecE. Angulo and F. Courchamp, Multiple allee effects and population management, Trends in Ecology and Evolution, 22 (2007), 185-191. 

[12]

P. N. DavisP. van HeijsterR. Marangell and M. R. Rodrigo, Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis, Journal of Dynamics and Differential Equations, 12 (2021), 1-23. 

[13]

F. Dumortier, Slow divergence integral and balanced canard solutions, Qual. Theory Dyn. Syst., 10 (2011), 65-85.  doi: 10.1007/s12346-011-0038-9.

[14]

F. Dumortier and R. Roussarie, Multiple canard cycles in generalized Li$\acute{e}$nard equations, J. Differential Equations, 174 (2001), 1-29.  doi: 10.1006/jdeq.2000.3947.

[15]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), 53-98.  doi: 10.1016/0022-0396(79)90152-9.

[16]

E. Gonz$\acute{a}$lez-OlivaresL. M. Gallego-BerrioB. Gonz$\acute{a}$lez-Ya$\tilde{n}$ez and A. Rojas-Palma, Consequences of weak Allee effect on prey in the May-Holling-Tanner predator-prey model, Math. Methods Appl. Sci., 39 (2016), 4700-4712.  doi: 10.1002/mma.3404.

[17]

E. Gonz$\acute{a}$lez-OlivaresB. Gonz$\acute{a}$lez-Y$\acute{a}$$\tilde{n}$ezJ. Mena-LorcaA. Rojas-Palma and J. D. Flores, Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Comput. Math. Appl., 62 (2011), 3449-3463.  doi: 10.1016/j.camwa.2011.08.061.

[18]

E. Gonz$\acute{a}$lez-Olivares and J. Huincahue-Arcos, Double allee effects on prey in a modified Rosenzweig-MacArthur predator-prey model, Computational Problems in Engineering, 307 (2014), 105-119. 

[19]

G. Hek, Geometric singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386.  doi: 10.1007/s00285-009-0266-7.

[20]

S.-B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 893-911.  doi: 10.3934/dcdsb.2009.11.893.

[21]

T.-H. Hsu, On bifurcation delay: An alternative approach using geometric singular perturbation theory, J. Differential Equations, 262 (2017), 1617-1630.  doi: 10.1016/j.jde.2016.10.022.

[22]

T.-H. Hsu, Number and stability of relaxation oscillations for predator-prey systems with small death rates, SIAM J. Appl. Dyn. Syst., 18 (2019), 33-67.  doi: 10.1137/18M1166705.

[23]

T.-H. Hsu and G. S. K. Wolkowicz, A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 25 (2020), 1257-1277.  doi: 10.3934/dcdsb.2019219.

[24]

M. Krupa and P. Szmolyan, Relaxation oscillation and canard explosion, J. Differential Equations, 174 (2001), 312-368.  doi: 10.1006/jdeq.2000.3929.

[25]

M. Krupa and P. Szmolyan, Extending slow manifolds near transcritical and pitchfork singularities, Nonlinearity, 14 (2001), 1473-1491.  doi: 10.1088/0951-7715/14/6/304.

[26]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points-fold and canard points in two dimensions, SIAM J. Math. Anal., 33 (2001), 286-314.  doi: 10.1137/S0036141099360919.

[27]

C. LiJ. LiZ. Ma and H. Zhu, Canard phenomenon for an SIS epidemic model with nonlinear incidence, J. Math. Anal. Appl., 420 (2014), 987-1004.  doi: 10.1016/j.jmaa.2014.06.035.

[28]

C. Li and K. Lu, Slow divergence integral and its application to classical Li$\acute{e}$nard equations of degree5, J. Differential Equations, 257 (2014), 4437-4469.  doi: 10.1016/j.jde.2014.08.015.

[29]

C. Li and H. Zhu, Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations, 254 (2013), 879-910.  doi: 10.1016/j.jde.2012.10.003.

[30]

M. Y. LiW. LiuC. Shan and Y. Yi, Turning points and relaxation oscillation cycles in simple epidemic models, SIAM J. Appl. Math., 76 (2016), 663-687.  doi: 10.1137/15M1038785.

[31]

W. LiuD. Xiao and Y. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differential Equations, 188 (2003), 306-331.  doi: 10.1016/S0022-0396(02)00076-1.

[32]

P. D. MaesschalckF. Dumortier and R. Roussarie, Canard-cycle transition at a fast-fast passage through a jump point, C. R. Math., 352 (2014), 27-30.  doi: 10.1016/j.crma.2013.09.002.

[33]

P. D. Maesschalck and S. Schecter, The entry-exit function and geometric singular perturbation theory, J. Differential Equations, 260 (2016), 6697-6715.  doi: 10.1016/j.jde.2016.01.008.

[34]

L. Mamouhdi and R. Roussarie, Canard cycles of finite codimension with two breaking parameters, Qual. Theory Dyn. Syst., 11 (2012), 167-198.  doi: 10.1007/s12346-011-0061-x.

[35]

N. Martinez-Jeraldo and P. Aguirre, Allee effect acting on the prey species in a Leslie-Gower predation model, Nonlinear Anal. Real World Appl., 45 (2019), 895-917.  doi: 10.1016/j.nonrwa.2018.08.009.

[36]

R. Ostfeld and C. Canham, Density-dependent processes in meadow voles: An experimental approach, Ecology, 76 (1995), 521-532. 

[37]

P. J. Pal and T. Saha, Qualitative analysis of a predator-prey system with double Allee effect in prey, Chaos Solitons and Fractals, 73 (2015), 36-63.  doi: 10.1016/j.chaos.2014.12.007.

[38]

S. H. PiltzF. VeermanP. K. Maini and M. A. Porter, A predator-2 prey fast-slow dynamical system for rapid predator evolution, SIAM J. Appl. Dyn. Syst., 16 (2017), 54-90.  doi: 10.1137/16M1068426.

[39]

C. H. Shan, Slow-fast dynamics and nonlinear oscillations in transmission of mosquito-borne diseases, Discrete and Continuous Dynamical Systems-B, preprint. doi: 10.3934/dcdsb.2021097.

[40]

J. H. Shen, Canard limit cycles and global dynamics in a singularly perturbed predator-prey system with non-monotonic functional response, Nonlinear Anal. Real World Appl., 31 (2016), 146-165.  doi: 10.1016/j.nonrwa.2016.01.013.

[41]

J. Shen and M. Han, Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Li$\acute{e}$nard systems, Discrete Contin. Dyn. Syst., 33 (2013), 3085-3108.  doi: 10.3934/dcds.2013.33.3085.

[42]

J. H. ShenC. H. Hsu and T. H. Yang, Fast-slow dynamics for intraguild predation models with evolutionary effects, J. Dynam. Differential Equations, 32 (2020), 895-920.  doi: 10.1007/s10884-019-09744-3.

[43]

M. SenM. Banerjee and A. Morozov, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecological Complexity, 11 (2012), 12-27. 

[44]

C. Wang and X. Zhang, Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type Ⅲ, J. Differential Equations, 267 (2019), 3397-3441.  doi: 10.1016/j.jde.2019.04.008.

[45]

C. Wang and X. Zhang, Stability loss delay and smoothness of the return map in slow-fast systems, SIAM J. Appl. Dyn. Syst., 17 (2018), 788-822.  doi: 10.1137/17M1130010.

[46]

M. Wechselberger, Geometric Singular Perturbation Theory Beyond the Standard Form, Frontiers in Applied Dynamical Systems: Reviews and Tutorials, 6, Springer, Cham, 2020. doi: 10.1007/978-3-030-36399-4.

[47]

Z. Wei, Y. H. Xia and T. H. Zhang, Stability and bifurcation analysis of an amensalism model with weak Allee effect, Qual. Theory Dyn. Syst., 19 (2020), Art. 23, 15 pp. doi: 10.1007/s12346-020-00341-0.

[48]

J. Y. Xu, T. H. Zhang and M. A. Han, A regime switching model for species subject to environmental noises and additive Allee effect, Phys. A, 527 (2019), 121300, 16 pp. doi: 10.1016/j.physa.2019.121300.

[49]

Y. Zhang, Y. Zhou and B. Tang, Canard phenomenon in an SIRS epidemic model with nonlinear incidence rate, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050073, 19 pp. doi: 10.1142/S021812742050073X.

Figure 1.  The intersections between the curve $ v = V(u) $ and the line $ v = u+w $. In the figure, the intersection points are the equilibriums of model (1.2)
Figure 2.  The slow-fast limiting dynamics associated with (1.2). In the figure, the orbits with double arrow indicates the fast fibers while the ones marked with single arrow are the slow orbits
Figure 3.  (a)-(b): Canard cycles of system (1.2); (c): Singular homoclinic orbit of system (1.2). In the figure, the orbits with double arrow indicates the fast fibers while the ones marked with single arrow are the slow orbits
Figure 4.  (a) The canard cycle of system (1.2) when $ h\in(0,v_0+a/p\beta) $; (b) The canard cycle of system (1.2) when $ h\in(0,v_0-v^*_1) $
Figure 5.  The integral on the function $ \overline{\Phi}(v) $, i.e., the slow divergence integral $ I(h,\mu_0) $
Figure 6.  (a) A homoclinic orbit of system (1.2); (b) The birth of the homoclinic orbit in system (1.2)
Figure 7.  An orbit of system (4.1) entering at $ (\bar{u}_0,\bar{v}_0) $ and exiting at $ (\bar{u}_0, p_{\varepsilon}(\bar{v}_0)) $, where $ 0<\varepsilon\ll1 $
Figure 8.  A singular slow-fast cycle $ \Gamma_0: = PLRMP $ of system (1.2)
Figure 9.  Singular Hopf bifurcation curve and canard curve of (1.2) when $ a = -0.4,\,\,p = 0.4,\,\,\beta = 2,\,\,\varepsilon = 0.001 $
Figure 10.  The birth and bifurcation of canard limit cycle when $ a = -0.4,\,\,p = 0.4,\,\,\beta = 2,\,\,\varepsilon = 0.001 $. (a): The small amplitude Hopf cycle when $ w = 0.362 $, where the initial value is $ (0.2, 0.54159) $; (b): Local zooming of the small amplitude Hopf cycle in Fig. 10 (a); (c) The time history of the small amplitude Hopf cycle when $ w = 0.362 $; (d): The canard limit cycle when $ w = 0.375 $, where the initial value is $ (0.2, 0.5048) $; (e): Local zooming of the canard limit cycle displayed in Fig. (d); (f) The time history of the canard cycle when $ w = 0.375 $; (g): The canard relaxation oscillation passing through the transcritical bifurcation point when $ w = 0.381 $, where the initial value is $ (0.381, 0.51) $; (h): Local zooming of the canard relaxation oscillation displayed in Fig. (i); The time history of the canard relaxation oscillation when $ w = 0.381 $
Figure 11.  The whole bifurcation process of canard explosion
Figure 12.  Singular Hopf bifurcation curve and canard curve of (1.2) when $ a = -0.3,\,\,p = 0.15,\,\,\beta = 2.13,\,\,\varepsilon = 0.0055685 $
Figure 13.  (a) and (b): The homoclinic cycle of system (1.2) when $ a = -0.3,\,\,p = 0.15,\,\,\beta = 2.13,\,\,\varepsilon = 0.0055685 $ and $ w = 0.9364 $; (c): The local zooming of the homoclinic cycle near the saddle (0.001387109324, 0.9413871091); (d): The time history of the homoclinic orbit under the present parameter value
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