Propagation dynamics of a nonlocal dispersal Fisher-KPP equation in a time-periodic shifting habitat

2020 ◽  
Vol 268 (6) ◽  
pp. 2852-2885 ◽  
Author(s):  
Guo-Bao Zhang ◽  
Xiao-Qiang Zhao
Keyword(s):  
Nonlocal Dispersal ◽  
Kpp Equation ◽  
Propagation Dynamics ◽  
Time Periodic

Asymptotic behaviour of solutions of Fisher–KPP equation with free boundaries in time-periodic environment

2019 ◽  
Vol 31 (3) ◽  
pp. 423-449
Author(s):  
JINGJING CAI ◽  
LI XU
Keyword(s):  
Free Boundary ◽  
Asymptotic Behaviour ◽  
Chemical Species ◽  
Free Boundaries ◽  
Periodic Functions ◽  
Kpp Equation ◽  
Periodic Environment ◽  
A Free Boundary Problem ◽  
Time Periodic ◽  
Asymptotic Spreading

We study a free boundary problem of the form: ut = uxx + f(t, u) (g(t) < x < h(t)) with free boundary conditions h′(t) = −ux(t, h(t)) – α(t) and g′(t) = −ux(t, g(t)) + β(t), where β(t) and α(t) are positive T-periodic functions, f(t, u) is a Fisher–KPP type of nonlinearity and T-periodic in t. This problem can be used to describe the spreading of a biological or chemical species in time-periodic environment, where free boundaries represent the spreading fronts of the species. We study the asymptotic behaviour of bounded solutions. There are two T-periodic functions α0(t) and α*(t; β) with 0 < α0 < α* which play key roles in the dynamics. More precisely, (i) in case 0 < β< α0 and 0 < α < α*, we obtain a trichotomy result: (i-1) spreading, that is, h(t) – g(t) → +∞ and u(t, ⋅ + ct) → 1 with $c\in (-\overline{l},\overline{r})$, where $ \overline{l}:=\frac{1}{T}\int_{0}^{T}l(s)ds$, $\overline{r}:=\frac{1}{T}\int_{0}^{T}r(s)ds$, the T-periodic functions −l(t) and r(t) are the asymptotic spreading speeds of g(t) and h(t) respectively (furthermore, r(t) > 0 > −l(t) when 0 < β < α < α0; r(t) = 0 > −l(t) when 0 < β < α = α0; $0 \gt \overline{r} \gt -\overline{l}$ when 0 < β < α0 < α < α*); (i-2) vanishing, that is, $\lim\limits_{t \to \mathcal {T}}h(t) = \lim\limits_{t \to \mathcal {T}}g(t)$ and $\lim\limits_{t \to \mathcal {T}}\max\limits_{g(t)\leq x\leq h(t)} u(t,x)=0$, where $\mathcal {T}$ is some positive constant; (i-3) transition, that is, g(t) → −∞, h(t) → −∞, $0<\lim\limits_{t \to \infty}[h(t)-g(t)] \lt +\infty$ and u(t, ⋅) → V(t, ⋅), where V is a T-periodic solution with compact support. (ii) in case β ≥ α0 or α ≥ α*, vanishing happens for any solution.

scholarly journals The generalised principal eigenvalue of time-periodic nonlocal dispersal operators and applications

2020 ◽  
Vol 269 (6) ◽  
pp. 4960-4997
Author(s):  
Yuan-Hang Su ◽  
Wan-Tong Li ◽  
Yuan Lou ◽  
Fei-Ying Yang
Keyword(s):  
Principal Eigenvalue ◽  
Nonlocal Dispersal ◽  
Time Periodic

scholarly journals Spreading speeds and traveling waves for space-time periodic nonlocal dispersal cooperative systems

2019 ◽  
Vol 18 (1) ◽  
pp. 361-396 ◽  
Author(s):  
Xiongxiong Bao ◽  
◽  
Wenxian Shen ◽  
Zhongwei Shen ◽  
◽  
...  
Keyword(s):  
Traveling Waves ◽  
Cooperative Systems ◽  
Space Time ◽  
Spreading Speeds ◽  
Nonlocal Dispersal ◽  
Time Periodic

Entire solutions in the Fisher-KPP equation with nonlocal dispersal

2010 ◽  
Vol 11 (4) ◽  
pp. 2302-2313 ◽  
Author(s):  
Wan-Tong Li ◽  
Yu-Juan Sun ◽  
Zhi-Cheng Wang
Keyword(s):  
Entire Solutions ◽  
Nonlocal Dispersal ◽  
Kpp Equation
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