Properties of traveling waves for integrodifference equations with nonmonotone growth functions |
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Authors: | Zhi-Xian Yu and Rong Yuan |
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Institution: | (1) School of Mathematical Sciences, Nankai University, Tianjin, 300071, People’s Republic of China;(2) Department of Mathematics and Statistics, York University, Toronto, ON, Canada, M3J 1P3 |
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Abstract: | In this paper, we will establish some new properties of traveling waves for integrodifference equations with the nonmonotone
growth functions. More precisely, for c ≥ c
*, we show that either limx?+¥ f(x)=u*{\lim\limits_{\xi\rightarrow+\infty} \phi(\xi)=u*} or 0 < liminfx? + ¥ f(x) < u* < limsupx?+¥f(x) £ b,{0 < \liminf\limits_{\xi \rightarrow + \infty} \phi(\xi) < u* < \limsup \limits_{\xi\rightarrow+\infty}\phi(\xi)\leq b,} that is, the wave converges to the positive equilibrium or oscillates about it at +∞. Sufficient conditions can assure that
both results will arise. We can also obtain that any traveling wave with wave speed c > c* possesses exponential decay at −∞. These results can be well applied to three types of growth functions arising from population
biology. By choosing suitable parameter numbers, we can obtain the existence of oscillating waves. Our analytic results are
consistent with some numerical simulations in Kot (J Math Biol 30:413–436, 1992), Li et al. (J Math Biol 58:323–338, 2009) and complement some known ones. |
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