$m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection

Changqing Ji,Dandan Zhu,Jingli Ren

doi:10.5206/mase/13996

Abstract

In this paper, we investigate a $m$th-order Fisher-KPP equation with free boundaries and time-aperiodic advection. Considering the influence of advection term and initial conditions on the long time behavior of solutions, we obtain spreading-vanishing dichotomy, spreading-transition-vanishing trichotomy, and vanishing happens with the coefficient of advection term in small amplitude, medium-sized amplitude and large amplitude, respectively. Then, the appropriate parameters are selected in the simulation to intuitively show the corresponding theoretical results. Moreover, the wave-spreading and wave-vanishing cases of the solutions are observed in our study.

Highlights

Reaction-diffusion equation is widely used to investigate various phenomena in physics, chemistry and biology
The expansion of biological populations are often affected by advection; in the mean time, a biological species typically lives in a bounded domain with the boundary moving/expanding according some rules
When β(t) = β is constant and f (t, x, u) = f (u) is homogeneous in time and space and f (u) is nonlinearity of Fisher type, Gu et al [7] studied this equation with free boundaries and obtained the long time behaviour of the solutions

Summary

Introduction

Reaction-diffusion equation is widely used to investigate various phenomena in physics, chemistry and biology. Gu et al [5, 6] proved the spreading-vanishing dichotomy by considering this equation with small advection environment and Fisher-KPP type of nonlinearity. When β(t) = β is constant and f (t, x, u) = f (u) is homogeneous in time and space and f (u) is nonlinearity of Fisher type, Gu et al [7] studied this equation with free boundaries and obtained the long time behaviour of the solutions. Sun et al [18] proposed a T-periodic advection-reaction-diffusion problem with free boundaries They obtained the following results: (i) there is a spreading-vanishing dichotomy in the case of β ∈ [0, c); (ii) there is a spreading-transition-vanishing trichotomy in the case of β ∈ [c, B(β)); (iii) vanishing happens in the case of β ≥ B(β).

Global existence and uniqueness

Long time behavior of the solutions

Numerical simulation and analysis