Topological horseshoes of traveling waves for a fast-slow predator-prey system. (English) Zbl 1143.34032
The authors apply a general theory developed by some of them [T. Gedeon, H. Kokubu, K. Mischaikow and H. Oka, J. Differ. Equations, to appear] to study traveling waves of the reaction diffusion system
\[ \varepsilon u_t=\varepsilon^2u_{xx}+u(1-u)(u-v), \quad v_t=v_{xx}+v(1.65u-0.25-v), \]
where \(\varepsilon>0\) is a small parameter. This particular problem examplifies the index techniques (derived from Conley index theory) used by the authors in the general theory. Basically, two existence results are obtained for \(\varepsilon>0\) small enough. First, the authors obtain the existence of two pairs of periodic wave solutions. The first pair consists of solutions with wave speed \(\vartheta =+ 0.25\), the second pair with wave speed \(\vartheta =- 0.25\). Second, they obtain two families of solutions, one of them with wave speed \(\vartheta = + 0.25\), the second one with wave speed \(\vartheta = - 0.25\). Each of these solutions are approximated by one of all the possible concatenations of the two periodic solutions restricted to their smallest period. These results are obtained using geometric objects, periodic corridors containing boxes, which are builded explicitely from numerical computations. This illustrates how to apply the somewhat technical conditions used in the general theory [Gedeon et al., op. cit.] .
\[ \varepsilon u_t=\varepsilon^2u_{xx}+u(1-u)(u-v), \quad v_t=v_{xx}+v(1.65u-0.25-v), \]
where \(\varepsilon>0\) is a small parameter. This particular problem examplifies the index techniques (derived from Conley index theory) used by the authors in the general theory. Basically, two existence results are obtained for \(\varepsilon>0\) small enough. First, the authors obtain the existence of two pairs of periodic wave solutions. The first pair consists of solutions with wave speed \(\vartheta =+ 0.25\), the second pair with wave speed \(\vartheta =- 0.25\). Second, they obtain two families of solutions, one of them with wave speed \(\vartheta = + 0.25\), the second one with wave speed \(\vartheta = - 0.25\). Each of these solutions are approximated by one of all the possible concatenations of the two periodic solutions restricted to their smallest period. These results are obtained using geometric objects, periodic corridors containing boxes, which are builded explicitely from numerical computations. This illustrates how to apply the somewhat technical conditions used in the general theory [Gedeon et al., op. cit.] .
Reviewer: Patrick Habets (Louvain-La-Neuve)
MSC:
| 34E15 | Singular perturbations for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 35B25 | Singular perturbations in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 92D25 | Population dynamics (general) |
Keywords:
singular perturbation; Conley index theory; traveling wave; periodic solution; symbolic dynamics; prey-predator systemSoftware:
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