scholarly journals Asymptotic spreading speed for the weak competition system with a free boundary

2019 ◽  
Vol 39 (9) ◽  
pp. 5223-5262 ◽  
Author(s):  
Zhiguo Wang ◽  
◽  
Hua Nie ◽  
Yihong Du ◽  
Keyword(s):  
Free Boundary ◽  
Spreading Speed ◽  
Competition System ◽  
Asymptotic Spreading

An evolutional free-boundary problem of a reaction–diffusion–advection system

2017 ◽  
Vol 147 (3) ◽  
pp. 615-648 ◽  
Author(s):  
Ling Zhou ◽  
Shan Zhang ◽  
Zuhan Liu
Keyword(s):  
Free Boundary ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Ecological Model ◽  
Reaction Diffusion ◽  
Spreading Speed ◽  
Heterogeneous Environment ◽  
Long Run ◽  
Strong Competition ◽  
Asymptotic Spreading

In this paper we consider a system of reaction–diffusion–advection equations with a free boundary, which arises in a competition ecological model in heterogeneous environment. The evolution of the free-boundary problem is discussed, which is an extension of the results of Du and Lin (Discrete Contin. Dynam. Syst. B19 (2014), 3105–3132). Precisely, when u is an inferior competitor, we prove that (u, v) → (0, V) as t→∞. When u is a superior competitor, we prove that a spreading–vanishing dichotomy holds, namely, as t→∞, either h(t)→∞ and (u, v) → (U, 0), or limt→∞h(t) < ∞ and (u, v) → (0, V). Moreover, in a weak competition case, we prove that two competing species coexist in the long run, while in a strong competition case, two species spatially segregate as the competition rates become large. Furthermore, when spreading occurs, we obtain some rough estimates of the asymptotic spreading speed.

scholarly journals ASYMPTOTIC SPREADING IN A COMPETITION SYSTEM WITH NONLOCAL DISPERSAL

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Xiaoming Yang ◽  
◽  
Guo Lin ◽  
Jianing Yang ◽  
Keyword(s):  
Nonlocal Dispersal ◽  
Competition System ◽  
Asymptotic Spreading

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 Spreading speed revisited: Analysis of a free boundary model

2012 ◽  
Vol 7 (4) ◽  
pp. 583-603 ◽  
Author(s):  
Gary Bunting ◽  
◽  
Yihong Du ◽  
Krzysztof Krakowski
Keyword(s):  
Free Boundary ◽  
Spreading Speed ◽  
Boundary Model
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