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.

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随机Fisher-KPP方程的渐近行为和随机行波

Fisher-KPP方程是一类具有实际背景的重要数学模型。先前的研究分析了波前和波前的渐近行为，并通过对增长函数施加减小约束来证明随机行波的存在。对于增长率具有随机波动的Fisher-KPP方程，我们发现，如果减少减少限制，则仍然可以得到相同的结果。此外，我们表明，随着噪声强度的增加，具有Fisher-KPP非线性的原始方程首先演化为具有退化的Fisher-KPP非线性的方程，然后发展为具有Nagumo非线性的方程。对于遭受环境噪声的Fisher-KPP方程，即使排除了对增长函数的减小约束，波前的已建立渐近行为仍然成立。但是，如果取消此约束，则已建立的波前背面渐近行为将不再成立，这意味着对增长函数的减小约束是确保波前背面渐近行为的充分必要条件。在两种情况下，系统都可以允许随机行波。