A Case Study of Multiple Wave Solutions in a Reaction-Diffusion System Using Invariant Manifolds and Global Bifurcations
Abstract.
A thorough analysis is performed to find traveling waves in a qualitative reaction-diffusion system inspired by a predator-prey model. We provide rigorous results coming from a standard local stability analysis, numerical bifurcation analysis, and relevant computations of invariant manifolds to exhibit homoclinic and heteroclinic connections, and periodic orbits in the associated traveling wave system with four components. In so doing, we present and describe a wide range of different traveling wave solutions. In addition, homoclinic chaos is manifested via both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An actual computation of global invariant manifolds near a focus-focus homoclinic bifurcation is also presented to unravel a multiplicity of wave solutions in the model.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
V. S. Afraimovich, S. V. Gonchenko, L. M. Lerman, A. L. Shilnikov, and D. V. Turaev, Scientific heritage of L.P. Shilnikov, Regul. Chaotic Dyn., 19 (2014), pp. 435–460.
2.
P. Aguirre, Bifurcations of two-dimensional global invariant manifolds near a noncentral saddle-node homoclinic orbit, SIAM J. Appl. Dyn. Syst., 14 (2015), pp. 1600–1643.
3.
P. Aguirre, E. Doedel, B. Krauskopf, and H. M. Osinga, Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields, Discrete Contin. Dyn. Syst., 29 (2010), pp. 1309–1344.
4.
P. Aguirre, J. D. Flores, and E. González-Olivares, Bifurcations and global dynamics in a predator-prey model with a strong allee effect on the prey, and a ratio-dependent functional response, Nonlinear Anal. Real World Appl., 16 (2014), pp. 235–249.
5.
P. Aguirre, B. Krauskopf, and H. M. Osinga, Global invariant manifolds near homoclinic orbits to a real saddle: (Non)orientability and flip bifurcation, SIAM J. Appl. Dyn. Syst., 12 (2013), pp. 1803–1846.
6.
P. Aguirre, B. Krauskopf, and H. M. Osinga, Global invariant manifolds near a Shilnikov homoclinic bifurcation, J. Comput. Dyn., 1 (2014), pp. 1–38.
7.
L. A. Belyakov, Bifurcation of systems with homoclinic curve of a saddle-focus with saddle quantity zero, Mat. Zametki, 36 (1984), pp. 681–689.
8.
V. Breña-Medina, A. R. Champneys, C. Grierson, and M. J. Ward, Mathematical modeling of plant root hair initiation: Dynamics of localized patches, SIAM J. Appl. Dyn. Syst., 13 (2014), pp. 210–248.
9.
V. Breña-Medina and A. Champneys, Subcritical turing bifurcation and the morphogenesis of localized patterns, Phys. Rev. E, 90 (2014), 032923.
10.
A. R. Champneys, Y. A. Kuznetsov, and B. Sandstede, A numerical toolbox for homoclinic bifurcation analysis, Int. J. Bifurcat. Chaos, 6 (1996), pp. 867–887.
11.
D. Contreras-Julio, P. Aguirre, J. Mujica, and O. Vasilieva, Finding strategies to regulate propagation and containment of dengue via invariant manifold analysis, SIAM J. Appl. Dyn. Syst., 19 (2020), pp. 1392–1437.
12.
A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. Meijer, and B. Sautois, New features of the software MatCont for bifurcation analysis of dynamical systems, Math. Comput. Model. Dyn. Syst., 14 (2008), pp. 147–175, https://doi.org/10.1080/13873950701742754.
13.
E. J. Doedel, Lecture notes on numerical analysis of nonlinear equations, in Numerical Continuation Methods for Dynamical Systems, Springer, New York, 2007, pp. 1–49.
14.
E. J. Doedel, T. F. Fairgrieve, B. Sandstede, A. R. Champneys, Y. A. Kuznetsov, and X. Wang, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Concordia University, Montreal, 2007.
15.
S. R. Dunbar, Traveling waves in diffusive predator-prey equations: Periodic orbits and point-to-periodic heteroclinic orbits, SIAM J. Appl. Math., 46 (1986), pp. 1057–1078.
16.
G. B. Ermentrout and D. H. Terman, Mathematical Foundations of Neuroscience, Springer, New York, 2010.
17.
G. B. Ermentrout and N. Kopell, Parabolic bursting in an excitable system coupled with a slow oscillation, SIAM J. Appl. Math., 46 (1986), pp. 233–253.
18.
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differential Equations, 31 (1979), pp. 53–98.
19.
A. Fowler and C. Sparrow, Bifocal homoclinic orbits in four dimensions, Nonlinearity, 4 (1991), pp. 1159–1182.
20.
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci. 42, Springer Science & Business Media, New York, 2013.
21.
J. Guckenheimer, B. Krauskopf, H. M. Osinga, and B. Sandstede, Invariant manifolds and global bifurcations, Chaos, 25 (2015), 097604.
22.
M. W. Hirsch, Differential Topology, Grad. Texts in Math. 33, Springer, New York, 1976.
23.
A. J. Homburg and B. Sandstede, Homoclinic and heteroclinic bifurcations in vector fields, Handbook Dyn. Syst., 3 (2010), pp. 379–524.
24.
C.-H. Hsu, C.-R. Yang, T.-H. Yang, and T.-S. Yang, Existence of traveling wave solutions for diffusive predator-prey type systems, J. Differential Equations, 252 (2012), pp. 3040–3075.
25.
J. Huang, G. Lu, and S. Ruan, Existence of traveling wave solutions in a diffusive predator-prey model, J. Math. Biol., 46 (2003), pp. 132–152.
26.
E. Izhikevich, Simple model of spiking neurons, IEEE Trans. Neural Networks, 14 (2003), pp. 1569–1572, https://doi.org/10.1109/TNN.2003.820440.
27.
B. Krauskopf and H. M. Osinga, Computing invariant manifolds via the continuation of orbit segments, in Numerical Continuation Methods for Dynamical Systems, Springer, New York, 2007, pp. 117–154.
28.
B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer, A. Vladimirsky, M. Dellnitz, and O. Junge, A survey of methods for computing (un) stable manifolds of vector fields, in Modeling and Computations in Dynamical Systems: In Commemoration of the 100th Anniversary of the Birth of John von Neumann, World Scientific, River Edge, NJ, 2006, pp. 67–95.
29.
B. Krauskopf and T. Rieß, A Lin’s method approach to finding and continuing heteroclinic connections involving periodic orbits, Nonlinearity, 21 (2008), 1655.
30.
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Appl. Math. Sci. 112, Springer Science & Business Media, New York, 2013.
31.
M. Lewis and B. Li, Spreading speed, traveling waves, and minimal domain size in impulsive reaction-Diffusion models, Bull. Math. Biol., 74 (2012), pp. 2383–2402.
32.
H. Li and H. Xiao, Traveling wave solutions for diffusive predator-prey type systems with nonlinear density dependence, Comput. Math. Appl., 74 (2017), pp. 2221–2230.
33.
X. Lin, P. Weng, and C. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dynam. Differential Equations, 23 (2011), pp. 903–921.
34.
M. Dolnik, A. M. Zhabotinsky, A. B. Rovinsky, and I. R. Epstein, Spatio-temporal patterns in a reaction-diffusion system with wave instability, Chem. Eng. Sci., 55 (2000), pp. 223–231.
35.
K. Manna, S. Pal, and M. Banerjee, Analytical and numerical detection of traveling wave and wave-train solutions in a prey-predator model with weak allee effect, Nonlinear Dyn., 100 (2020), pp. 2989–3006.
36.
J. Mujica, B. Krauskopf, and H. M. Osinga, A Lin’s method approach for detecting all canard orbits arising from a folded node, J. Comput. Dyn., 4 (2017), pp. 143–165.
37.
J. Mujica, B. Krauskopf, and H. M. Osinga, Tangencies between global invariant manifolds and slow manifolds near a singular Hopf bifurcation, SIAM J. Appl. Dyn. Syst., 17 (2018), pp. 395–1431.
38.
J. D. Murray, Mathematical Biology: I. An Introduction, Interdiscip. Appl. Math. 17, Springer, New York, 2000.
39.
J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Interdiscip. Appl. Math. 18, Springer, New York, 2001.
40.
H. M. Osinga, Two-dimensional invariant manifolds in four-dimensional dynamical systems, Comput. Graph., 29 (2005), pp. 289–297.
41.
I. Ovsyannikov and L. Shil’nikov, Systems with a homoclinic curve of multidimensional saddle-focus type, and spiral chaos, Math. USSR-Sb., 73 (1992), pp. 415–443.
42.
C. M. Postlethwaite and A. M. Rucklidge, A trio of heteroclinic bifurcations arising from a model of spatially-extended rock-paper-scissors, Nonlinearity, 32 (2019), pp. 1375–1407.
43.
B. Sandstede, Stability of n-fronts bifurcating from a twisted heteroclinic loop and an application to the Fitzhugh–Nagumo equation, SIAM J. Math. Anal., 29 (1998), pp. 183–207.
44.
B. Sandstede, Stability of travelling waves, in Handbook of Dynamical Systems, Vol. 2, Elsevier, Amsterdam, 2002, pp. 983–1055.
45.
B. Sandstede and A. Scheel, On the stability of periodic travelling waves with large spatial period, J. Differential Equations, 172 (2001), pp. 134–188.
46.
J. A. Sherratt and M. J. Smith, Periodic travelling waves in cyclic populations: Field studies and reaction-diffusion models, J. Roy. Soc. Interface, 5 (2008), pp. 483–505.
47.
A. Shilnikov and M. Kolomiets, Methods of the qualitative theory for the Hindmarsh-Rose model: A case study—a tutorial, Int. J. Bifurcat. Chaos, 18 (2008), pp. 2141–2168.
48.
L. P. Shilnikov, A case of the existence of a countable number of periodic orbits, Sov. Math. Dokl., 6 (1965), pp. 163–166.
49.
L. P. Shilnikov, A contribution to the problem of the structure of an extended neighborhood of a rough state to a saddle-focus type, Math. USSR-Sb., 10 (1970), pp. 91–102.
50.
L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. Chua, Methods of Qualitative Theory in Nonlinear Dynamics, Part II, World Sci. Ser. Nonlinear Sci. Ser. A 5, World Scientific, River Edge, NJ, 2001.
51.
L. T. Takahashi, N. A. Maidana, W. C. Ferreira, P. Pulino, and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind, Bull. Math. Biol., 67 (2005), pp. 509–528.
52.
S. Wiggins, Global Bifurcations and Chaos: Analytical Methods, Appl. Math. Sci. 73, Springer Science & Business Media, New York, 2013.
53.
X. Wu, Y. Luo, and Y. Hu Traveling waves in a diffusive predator-prey model incorporating a prey refuge, in Abstr. Appl. Anal. 2014, Hindawi, New York, 2014.
54.
W. M. S. Yamashita, L. T. Takahashi, and G. Chapiro, Traveling wave solutions for the dispersive models describing population dynamics of Aedes aegypti, Math. Comput. Simulation, 146 (2018), pp. 90–99.
Information & Authors
Information
Published In
SIAM Journal on Applied Dynamical Systems
Pages: 918 - 950
DOI: 10.1137/22M1474709
ISSN (online): 1536-0040
Copyright
© 2023 Society for Industrial and Applied Mathematics.
History
Submitted: 31 January 2022
Accepted: 8 November 2022
Published online: 8 June 2023
Keywords
MSC codes
Authors
Funding Information
Programa de Incentivo a la Iniciacion Cientifica (PIIC): DGIIP-UTFSM
Asociacion Mexicana de Cultura A.C.
Universidad Técnica Federico Santa María (UTFSM): PI-LI-19-06
Funding: The work of the first author was partially supported by Programa de Incentivo a la Iniciación Científica PIIC DGIIP-UTFSM and his work was carried out at the Departamento de Matemática, Universidad Técnica Federico Santa María. The work of the first and second authors was partially supported by Proyecto Interno UTFSM PI-LI-19-06. The work of the second author was supported by Proyecto Basal CMM Universidad de Chile. The work of the third author was supported by Asociación Mexicana de Cultura A.C.
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
- Local and global analysis of a discrete model describing the second-order digital filter with nonlinear signal processorsInternational Journal of Modern Physics C, Vol. 36, No. 01 | 21 June 2024
- Bifurcation and onset of chaos in an eco-epidemiological system with the influence of time delayChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 34, No. 6 | 7 June 2024
- The effect of nonlocal interaction on chaotic dynamics, Turing patterns, and population invasion in a prey–predator modelChaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 33, No. 10 | 20 October 2023
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].