2016/12/03 by Rachidi B. Salako, Wenxian Shen, Salako, Rachidi B. +1 · 3 citations
Mathematics · Biochemistry, Genetics and Molecular Biology · Medicine · #Mathematical Biology Tumor Growth #Gene Regulatory Network Analysis #Mathematical and Theoretical Epidemiology and Ecology Models
paper · pdf · doi:10.48550/arxiv.1612.00924
We consider the following chemotaxis systems \begincasesut=Δu-χ1∇(u∇ v1)+χ2∇(u∇ v2)+u(a-bu), x∈\mathbb RN,tgt;0,
0=(Δ-λ1I)v1+μ1u, x∈\mathbb RN,tgt;0,
0=(Δ-λ2I)v2+μ2u, in x∈\mathbb RN, tgt;0,
u(⋅,0)=u0, x∈\mathbb RN,\endcaseswhere χi, λi, μi, i=1,2 and a, b are positive constant real numbers and N is a positive integer. Under some conditions on the parameters, we prove the global existence and boundedness of classical solutions (u(x,t;u0),v1(x,t;u0),v2(x,t;u0)) for nonnegative, bounded, and uniformly continuous initials u0(x). Next, we show that, for every strictly positive initial u0(x),limt→∞[‖u(⋅,t;u0)-(a)/(b)‖∞+‖λ1v1(⋅,t;u0)-(a)/(b)μ1‖∞+‖λ2v2(⋅,t;u0)-(a)/(b)μ2‖∞]=0. Finally, we explore the spreading properties of the global solutions and prove that there are two positive numbers 0c^*+(χ1,μ1,λ1,χ2,μ2,λ2). Furthermore we show thatlim(χ1,χ2)→(0,0)c^*-(χ1,μ1,λ1,χ2,μ2,λ2)=lim(χ1,χ2)→(0,0)c^*+(χ1,μ1,λ1,χ2,μ2,λ2)=2√(a).