<p>This paper is devoted to the study of a generalized nonlocal dispersal Fisher-KPP equation with spatio-temporal interaction (delay). It is well-known that the spatio-temporal interaction makes the propagation dynamics more complicated. The monotonicity and stability of regular traveling waves of this equation are challenging problems in the field of propagation dynamics. In this paper, we prove that any regular traveling wave is strictly monotone increasing. Moreover, we obtain the exponential convergence to the regular traveling wave for solutions of the Cauchy problem with wave-like initial functions. As far as we know, this is the first theoretical study on the stability and monotonicity of traveling waves of nonlocal dispersal equation with spatio-temporal interaction. Our results can be applied to many well-known models under the nonlocal dispersal setting.</p>

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Wave propagation for a generalized nonlocal dispersal Fisher-KPP equation

  • Zhaoquan Xu

摘要

This paper is devoted to the study of a generalized nonlocal dispersal Fisher-KPP equation with spatio-temporal interaction (delay). It is well-known that the spatio-temporal interaction makes the propagation dynamics more complicated. The monotonicity and stability of regular traveling waves of this equation are challenging problems in the field of propagation dynamics. In this paper, we prove that any regular traveling wave is strictly monotone increasing. Moreover, we obtain the exponential convergence to the regular traveling wave for solutions of the Cauchy problem with wave-like initial functions. As far as we know, this is the first theoretical study on the stability and monotonicity of traveling waves of nonlocal dispersal equation with spatio-temporal interaction. Our results can be applied to many well-known models under the nonlocal dispersal setting.