Traveling Wavefronts for the Discrete Fisher′s Equation

Uniqueness of travelling waves for nonlocal monostable equations

“…One can then obtain the existence of a solution for c = c 0 using a limiting argument, as is done in, e.g., [4]. In [15], the authors construct travelling wave solutions for (1.2) with suppJ = {−1, 1} and c ≥ c 0 using a degree argument. For (1.2) with J such that suppJ contains 1 or two relatively prime integers, one can first use either of the above techniques to first obtain a solution such that u ≤ 0, and then add a comparison argument as in [2], p. 290, to show that actually u < 0.…”

Pushed and pulled fronts in a discrete reaction–diffusion equation

“…Pulled fronts have c * (µ) and λ * (µ) given by (4), (5) with λ * real, (5) being the repeated-root condition (cf. Zinner et al [17], equation (6), the supremum in which is related to the minimum wavespeed statement that led to (5) above); (4) has two real roots for c > c * > 0 (these also being its roots with the smallest positive real part) and none for c < c * . Pushed fronts are those for which f (u; a) is such that there is a c = c † > c * for which the exponential corresponding to the smaller positive real root of (4) is absent as z → +∞ in the solution to (26), (27) (i.e.…”

“…Here, however, our focus is on the propagation of heteroclinic connections between unstable and stable states (rather than stable-stable connections), for which travelling wave solutions to the continuous Fisher (or Kolmogorov-Petrovskii-Piskunov (KPP)) equation, in particular, have been extremely widely studied. In the discrete case, Zinner et al [17] considered the existence of travelling-wave solutions to the discrete Fisher-KPP equation on a one-dimensional lattice, obtaining a constraint on the coupling strength for which strictly increasing travelling wavefronts of speed c > 0 exist; more recent work on related systems includes that of Guo and Morita [18] and Hakberg [19].…”

“…In the discrete case we have been unable to identify any f (u; a) for which c † can be determined explicitly for a pushed front (a somewhat abortive attempt to generalise (17) in this direction, exploiting a corresponding quadratically nonlinear form, is given in Appendix C); explicit solutions can be constructed by specifying a suitable specific profile for U (z) and then determining the nonlinearity f from (26), but the resulting f will in general then depend on µ, so this procedure is not suitable for the current purposes. For example, proceeding in this way we find that for…”

Existence of traveling wave solutions in m-dimensional delayed lattice dynamical systems with competitive quasimonotone and global interaction

“…They studied the long time behavior of solutions to (1.1) for some nonlinear function f. When the nonlinear term f is a monostable/bistable type, there are extensive results about the traveling wave solutions for equation (1.1), some of which have revealed some essential differences between a discrete model and its corresponding continuous one. For details, see for example, [4,5,23,24]. Taking into account time delay in population dynamics, Wu and Zou [21] considered the delayed lattice differential equations and studied the existence of traveling wave solutions.…”