Traveling waves in the nonlocal KPP-Fisher equation: Different roles of the right and the left interactions
| Autor: | Sergei Trofimchuk, Petra Nábělková, Karel Hasik, Jana Kopfová |
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| Rok vydání: | 2016 |
| Předmět: |
Applied Mathematics
010102 general mathematics Mathematical analysis Fisher equation 01 natural sciences 010101 applied mathematics Monotone polygon Mathematics - Classical Analysis and ODEs 34K12 35K57 92D25 Classical Analysis and ODEs (math.CA) FOS: Mathematics Traveling wave Uniqueness 0101 mathematics Analysis Kernel (category theory) Lyapunov–Schmidt reduction Mathematical physics Mathematics |
| Zdroj: | Journal of Differential Equations. 260:6130-6175 |
| ISSN: | 0022-0396 |
| DOI: | 10.1016/j.jde.2015.12.035 |
| Popis: | We consider the nonlocal KPP-Fisher equation $u_t(t,x) = u_{xx}(t,x) + u(t,x)(1-(K *u)(t,x))$ which describes the evolution of population density $u(t,x)$ with respect to time $t$ and location $x$. The non-locality is expressed in terms of the convolution of $u(t, \cdot)$ with kernel $K(\cdot) \geq 0,$ $\int_{\mathbb{R}} K(s)ds =1$. The restrictions $K(s), s \geq 0,$ and $K(s), s \leq 0,$ are responsible for interactions of an individual with his left and right neighbors, respectively. We show that these two parts of $K$ play quite different roles as for the existence and uniqueness of traveling fronts to the KPP-Fisher equation. In particular, if the left interaction is dominant, the uniqueness of fronts can be proved, while the dominance of the right interaction can induce the co-existence of monotone and oscillating fronts. We also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on $K$. 38 pages, submitted |
| Databáze: | OpenAIRE |
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