Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations
Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations
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DOI：
10.3934/dcdsb.2020323
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Zhenzhen Wang;Tianshou Zhou
中科院分区：
文献类型：
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作者：
Zhenzhen Wang;Tianshou Zhou
Fisher-KPP方程是一类重要的有实际背景的数学模型。以往的研究通过对增长函数施加递减约束，分析了波前前后的渐近行为，证明了随机行波的存在性。对于增长率随机波动的Fisher-KPP方程，我们发现，如果减少限制被删除，相同的结果仍然成立。此外，我们表明，随着噪声强度的增加，原来的方程与Fisher-KPP非线性演化到第一个与退化的Fisher-KPP非线性，然后与Nagumo非线性。对于受环境噪声影响的Fisher-KPP方程，即使不考虑增长函数的下降约束，波前的渐近性态仍然成立。如果这个约束被删除，然而，建立的波前背面的渐近行为将不再保持，这意味着增长函数的减少约束是一个充分和必要的条件，以确保波前背面的渐近行为。在这两种情况下的噪音，系统可以允许随机行波。
Fisher-KPP equations are an important class of mathematical models with practical background. Previous studies analyzed the asymptotic behaviors of the front and back of the wavefront and proved the existence of stochastic traveling waves, by imposing decrease constraints on the growth function. For the Fisher-KPP equation with a stochastically fluctuated growth rate, we find that if the decrease restrictions are removed, the same results still hold. Moreover, we show that with increasing the noise intensity, the original equation with Fisher-KPP nonlinearity evolves into first the one with degenerated Fisher-KPP nonlinearity and then the one with Nagumo nonlinearity. For the Fisher-KPP equation subjected to the environmental noise, the established asymptotic behavior of the front of the wavefront still holds even if the decrease constraint on the growth function is ruled out. If this constraint is removed, however, the established asymptotic behavior of the back of the wavefront will no longer hold, implying that the decrease constraint on the growth function is a sufficient and necessary condition to ensure the asymptotic behavior of the back of the wavefront. In both cases of noise, the systems can allow stochastic traveling waves.