×

zbMATH — the first resource for mathematics

Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator-prey systems. (English) Zbl 1459.92072
MSC:
92D25 Population dynamics (general)
34D15 Singular perturbations of ordinary differential equations
Software:
MATCONT
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexandra, E., Lutscher, F. and Seo, G., Bistability and limit cycles in generalist predator-prey dynamics, Ecol. Complex.14 (2013) 48-55.
[2] Ambrosio, B., Aziz-Alaoui, M. A. and Yafia, R., Canard phenomenon in a slow-fast modified Leslie-Gower model, Math. Biosci.295 (2018) 48-54. · Zbl 1380.92050
[3] Arnold, V. I., Dynamical Systems V: Bifurcation Theory and Catastrophe Theory, , Vol. 5 (Springer, Berlin, 1994).
[4] Atabaigi, A. and Barati, A., Relaxation oscillations and canard explosion in a predator-prey system of Holling and Leslie types, Nonlinear Anal.: Real World Appl.36 (2017) 139-153. · Zbl 1362.34074
[5] Benot, E., Callot, J. L., Diener, F. and Diener, M., Chasse au canard, Collect. Math.31-32(1-3) (1981) 37-119.
[6] De Maesschalck, P., Dumortier, F. and Roussarie, R., Canard cycle transition at a slow-fast passage through a jump point, C. R. Math. Acad. Sci. Paris352(4) (2014) 317-320. · Zbl 1291.34097
[7] Dhooge, A., Govaerts, W. and Kuznetsov, Y. A., MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs, ACM Trans. Math. Softw.29(2) (2003) 141-164. · Zbl 1070.65574
[8] Dumortier, F., Panazzolo, D. and Roussarie, R., More limit cycles than expected in Liénard equations, Proc. Amer. Math. Soc.135(6) (2007) 1895-1904. · Zbl 1130.34018
[9] Dumortier, F. and Roussarie, R. H., Canard Cycles and Center Manifolds, Vol. 577. (American Mathematical Society, 1996). · Zbl 0851.34057
[10] Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations31(1) (1979) 53-98. · Zbl 0476.34034
[11] Hsu, S. and Shi, J., Relaxation oscillation profile of limit cycle in a predator-prey system, Discrete Contin. Dyn. Syst. Ser. B11 (2009) 893-911. · Zbl 1176.34049
[12] Jones, C. K. R. T., Geometric Singular Perturbation Theory, (Springer, Berlin, 1995), pp. 44-118. · Zbl 0840.58040
[13] Krupa, M. and Szmolyan, P., Relaxation oscillation and canard explosion, J. Differential Equations174(2) (2001) 312-368. · Zbl 0994.34032
[14] Krupa, M. and Szmolyan, P., Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal.33(2) (2001) 286-314. · Zbl 1002.34046
[15] Kuehn, C., Multiple Time Scale Dynamics, Vol. 191 (Springer, Berlin, 2015). · Zbl 1335.34001
[16] Kuznetsov, Y. A., Elements of Applied Bifurcation Theory, 3rd edn. , Vol. 112 (Springer, New York, 2004). · Zbl 1082.37002
[17] Lamontagne, Y., Coutu, C. and Rousseau, C., Bifurcation analysis of a predator-prey system with generalized Holling type III function response, J. Dynam. Differential Equations20 (2008) 535-571. · Zbl 1160.34047
[18] Li, C. and Zhu, H., Canard cycles for predator-prey systems with Holling types of functional response, J. Differential Equations254(2) (2013) 879-910. · Zbl 1257.34035
[19] Liu, W., Xiao, D. and Yi, Y., Relaxation oscillations in a class of predator-prey systems, J. Differential Equations188 (2003) 306-331. · Zbl 1094.34025
[20] May, R. M., Limit cycles in predator-prey communities, Science177(4052) (1972) 900-902.
[21] Roberts, A. and Gendinning, P., Canard-like phenomena in piecewise-smooth Van der Pol systems, Chaos: Interdiscipl. J. Nonlinear Sci.24(2) (2014) 023138. · Zbl 1345.34070
[22] Rosenzweig, M. L., Paradox of enrichment: destabilization of exploitation ecosystems in ecological time, Science171(3969) (1971) 385-387.
[23] Shen, J., 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. · Zbl 1375.92056
[24] Wang, C. and Zhang, X., Canards, heteroclinic and homoclinic orbits for a slow-fast predator-prey model of generalized Holling type III, J. Differential Equations267(6) (2019) 3397-3441. · Zbl 1418.34103
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.