arXiv subjects

Ningkui Sun

Publications and source records attributed to Ningkui Sun.

4 recordsLinked to original sources

arXiv2019

Long-time behavior of a reaction-diffusion model with strong Allee effect and free boundary: effect of a protection zone

This paper concerns the effect of the (separated/connected) protection zone for the evolution of an endangered species on the reaction-diffusion equation with strong Allee effect and free boundary. We give a description of the long-time dynamical behavior of the problem of two types protection zones with the same length. Furthermore, the asymptotic profiles of solutions and the asymptotic spreading speed are estimated when spreading happens. Our results, together with those in previous papers [8, 12] on two other closely related models, show that the protection zone and the free boundary play an important role in the evolution of the endangered species.

math.AP↗

arXiv2018

Propagation dynamics of Fisher-KPP equation with time delay and free boundaries

Incorporating free boundary into time-delayed reaction-diffusion equations yields a compatible condition that guarantees the well-posedness of the initial value problem. With the KPP type nonlinearity we then establish a vanishing-spreading dichotomy result. Further, when the spreading happens, we show that the spreading speed and spreading profile are nonlinearly determined by a delay-induced nonlocal semi-wave problem. It turns out that time delay slows down the spreading speed.

math.AP↗

arXiv2018

The role of protection zone on species spreading governed by a reaction-diffusion model with strong Allee effect

It is known that a species dies out in the long run for small initial data if its evolution obeys a reaction of bistable nonlinearity. Such a phenomenon, which is termed as the strong Allee effect, is well supported by numerous evidence from ecosystems, mainly due to the environmental pollution as well as unregulated harvesting and hunting. To save an endangered species, in this paper we introduce a protection zone that is governed by a Fisher-KPP nonlinearity, and examine the dynamics of a reaction-diffusion model with strong Allee effect and protection zone. We show the existence of two critical values $0<L_*\leq L^*$, and prove that a vanishing-transition-spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition-spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. This suggests that the protection zone works when its length is larger than the critical value $L_*$. Furthermore, we compare two types of protection zone with the same length: a connected one and a separate one, and our results reveal that the former is better for species spreading than the latter.

math.AP↗

arXiv2016

A diffusive Fisher-KPP equation with free boundaries and time-periodic advections

We consider a reaction-diffusion-advection equation of the form: $u_t=u_{xx}-β(t)u_x+f(t,u)$ for $x\in (g(t),h(t))$, where $β(t)$ is a $T$-periodic function representing the intensity of the advection, $f(t,u)$ is a Fisher-KPP type of nonlinearity, $T$-periodic in $t$, $g(t)$ and $h(t)$ are two free boundaries satisfying Stefan conditions. This equation can be used to describe the population dynamics in time-periodic environment with advection. Its homogeneous version (that is, both $β$ and $f$ are independent of $t$) was recently studied by Gu, Lou and Zhou \cite{GLZ}. In this paper we consider the time-periodic case and study the long time behavior of the solutions. We show that a vanishing-spreading dichotomy result holds when $β$ is small; a vanishing-transition-virtual spreading trichotomy result holds when $β$ is a medium-sized function; all solutions vanish when $β$ is large. Here the partition of $β(t)$ is much more complicated than the case when $β$ is a real number, since it depends not only on the "size" $\barβ:= \frac{1}{T}\int_0^T β(t) dt$ of $β(t)$ but also on its "shape" $\tildeβ(t) := β(t) - \barβ$.

math.AP↗