arXiv subjects

R. B. Salako

Publications and source records attributed to R. B. Salako.

4 recordsLinked to original sources

arXiv2024

Global dynamics of a two-stage structured diffusive population model in time-periodic and spatially heterogeneous environments

This work examines the global dynamics of classical solutions of a two-stage (juvenile-adult) reaction-diffusion population model in time-periodic and spatially heterogeneous environments. It is shown that the sign of the principal eigenvalue $λ_*$ of the time-periodic linearized system at the trivial solution completely determines the persistence of the species. Moreover, when $λ_*>0$, there is at least one time-periodic positive entire solution. A fairly general sufficient condition ensuring the uniqueness and global stability of the positive time-periodic solution is obtained. In particular, classical solutions eventually stabilize at the unique time-periodic positive solutions if either each subgroup's intra-stage growth and inter-stage competition rates are proportional, or the environment is temporally homogeneous and both subgroups diffuse slowly. In the later scenario, the asymptotic profile of steady states with respect to small diffusion rates is established.

math.AP↗

arXiv2019

On traveling wave solutions in full parabolic Keller-Segel chemotaxis systems with logistic source

This paper is concerned with traveling wave solutions of the following full parabolic Keller-Segel chemotaxis system with logistic source, \begin{equation} \begin{cases} u_t=Δu -χ\nabla\cdot(u\nabla v)+u(a-bu),\quad x\in\mathbb{R}^N \cr τv_t=Δv-λv +μu,\quad x\in \mathbb{R}^N, \end{cases}(1) \end{equation} where $χ, μ,λ,a,$ and $b$ are positive numbers, and $τ\ge 0$. Among others, it is proved that if $b>2χμ$ and $τ\geq \frac{1}{2}(1-\fracλ{a})_{+} ,$ then for every $c\ge 2\sqrt{a}$, (1) has a traveling wave solution $(u,v)(t,x)=(U^{τ,c}(x\cdotξ-ct),V^{τ,c}(x\cdotξ-ct))$ ($\forall\, ξ\in\mathbb{R}^N$) connecting the two constant steady states $(0,0)$ and $(\frac{a}{b},\fracμλ\frac{a}{b})$, and there is no such solutions with speed $c$ less than $2\sqrt{a}$, which improves considerably the results established in \cite{SaSh3}, and shows that (1) has a minimal wave speed $c_0^*=2\sqrt a$, which is independent of the chemotaxis.

math.AP↗

arXiv2018

Traveling waves of a full parabolic attraction-repulsion chemotaxis systems with logistic sources

In this paper, we study traveling wave solutions of the chemotaxis systems \begin{equation} \begin{cases} u_{t}=Δu -χ_1\nabla( u\nabla v_1)+χ_2 \nabla(u\nabla v_2 )+ u(a -b u), \qquad \ x\in\mathbb{R} \\ τ\partial_tv_1=(Δ- λ_1 I)v_1+ μ_1 u, \qquad \ x\in\mathbb{R}, \\ τ\partial v_2=(Δ- λ_2 I)v_2+ μ_2 u, \qquad \ \ x\in\mathbb{R}, \end{cases} (0.1) \end{equation} where $τ>0,χ_{i}> 0,λ_i> 0,\ μ_i>0$ ($i=1,2$) and $\ a>0,\ b> 0$ are constants, and $N$ is a positive integer. Under some appropriate conditions on the parameters, we show that there exist two positive constant $ 0<c^{*}(τ,χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)<c^{**}(τ,χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)$ such that for every $c^{*}(τ,χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)\leq c<c^{**}(τ,χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)$, $(0.1)$ has a traveling wave solution $(u,v_1,v_2)(x,t)=(U,V_1,V_2)(x-ct)$ connecting $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ and $(0,0,0)$ satisfying $$ \lim_{z\to \infty}\frac{U(z)}{e^{-μz}}=1, $$ where $μ\in (0,\sqrt a)$ is such that $c=c_μ:=μ+\frac{a}μ$. Moreover, $$ \lim_{(χ_1,χ_2)\to (0^+,0^+))}c^{**}(τ,χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)=\infty$$ and $$\lim_{(χ_1,χ_2)\to (0^+,0^+))}c^{*}(τ,χ_1,μ_1,λ_1,χ_2,μ_2,λ_2)= c_{\tildeμ^*}, $$ where $\tildeμ^*={\min\{\sqrt{a}, \sqrt{\frac{λ_1+τa}{(1-τ)_{+}}},\sqrt{\frac{λ_2+τa}{(1-τ)_{+}}}\}}$. We also show that $(0.1)$ has no traveling wave solution connecting $(\frac{a}{b},\frac{aμ_1}{bλ_1},\frac{aμ_2}{bλ_2})$ and $(0,0,0)$ with speed $c<2\sqrt{a}$.

math.AP↗

arXiv2018

Parabolic-elliptic chemotaxis model with space-time dependent logistic sources on $\mathbb{R}^N$. III. Transition fronts

The current work is the third of a series of three papers devoted to the study of asymptotic dynamics in the space-time dependent logistic source chemotaxis system, $$ \begin{cases} \partial_tu=Δu-χ\nabla\cdot(u\nabla v)+u(a(x,t)-b(x,t)u),\quad x\in R^N,\cr 0=Δv-λv+μu ,\quad x\in R^N, \end{cases} (0.1) $$ where $N\ge 1$ is a positive integer, $χ, λ$ and $μ$ are positive constants, the functions $a(x,t)$ and $b(x,t)$ are positive and bounded. In the first of the series, we studied the phenomena of persistence, and the asymptotic spreading for solutions. In the second of the series, we investigate the existence, uniqueness and stability of strictly positive entire solutions. In the current part of the series, we discuss the existence of transition front solutions of (0.1) connecting $(0,0)$ and $(u^*(t),v^*(t))$ in the case of space homogeneous logistic source. We show that for every $χ>0$ with $χμ\big(1+\frac{\sup_{t\in R}a(t)}{\inf_{t\in R}a(t)}\big)<\inf_{t\in R}b(t)$, there is a positive constant $c^{*}_χ$ such that for every $\underline{c}>c^{*}_χ$ and every unit vector $ξ$, (0.1) has a transition front solution of the form $(u(x,t),v(x,t))=(U(x\cdotξ-C(t),t),V(x\cdotξ-C(t),t))$ satisfying that $C'(t)=\frac{a(t)+κ^2}κ$ for some number $κ>0$, $\liminf_{t-s\to\infty}\frac{C(t)-C(s)}{t-s}=\underline{c}$, and$$\lim_{x\to-\infty}\sup_{t\in R}|U(x,t)-u^*(t)|=0 \quad \text{and}\quad \lim_{x\to\infty}\sup_{t\in R}|\frac{U(x,t)}{e^{-κx}}-1|=0.$$Furthermore, we prove that there is no transition front solution $(u(x,t),v(x,t))=(U(x\cdotξ-C(t),t),V(x\cdotξ-C(t),t))$ of (0.1) connecting $(0,0)$ and $(u^*(t),v^*(t))$ with least mean speed less than $2\sqrt{\underline{a}}$, where $\underline{a}=\liminf_{t-s\to\infty}\frac{1}{t-s}\int_{s}^{t}a(τ)dτ$.

math.AP↗