ON A MACROPHAGE AND TUMOR CELL CHEMOTAXIS SYSTEM WITH BOTH PARACRINE AND AUTOCRINE LOOPS

Li Xie, Shigui Ruan

Research output: Contribution to journalArticlepeer-review

Abstract

In the paper, we consider a homogeneous Neumann initial-boundary value problem (IBVP) for the following two-species and two-stimuli chemotaxis model with both paracrine and autocrine loops: {equation presented}, where u(t, x) and w(t, x) denote the density of macrophages and tumor cells at time t and location x ∈ Ω, respectively, v(t, x) and z(t, x) represent the concentration of colony stimulating factor 1 (CSF-1) secreted by the tumor cells and epidermal growth factor (EGF) secreted by macrophages at time t and location x ∈ Ω, respectively. Ω ⊂ Rnis a bounded region with smooth boundary, τi≥0(i = 1, 2), Di(s) ≥ di(s + 1)mi-1with parameters mi≥1 (i = 1, 2) and Sj(s)≤ (s + 1)qjwith parameters qj>, 0 (j = 1, 2, 3). For the case without autocrine loop (i.e., S3(w) = 0), it is shown that when qj≤ 1 (j = 1, 2), if one of qjis smaller than one or one of miis larger than one, then the IBVP has a global classical solution which is uniformly bounded. Moreover, when m1= m2= q1= q2= 1, an inequality involving the product d1d2and the product of the two species' initial mass is obtained which guarantees the existence of global bounded classical solutions. More specifically, it allows one of dito be small or one of the species initial mass to be large. For the case with autocrine loop (i.e S3(w) ≠ 0), similar results hold only if q3<, 1. If q3= 1, solutions to the IBVP exist globally only when d2is suitably large or the mass of species w is suitably small.

Original languageEnglish (US)
Pages (from-to)1447-1479
Number of pages33
JournalCommunications on Pure and Applied Analysis
Volume21
Issue number4
DOIs
StatePublished - Apr 2022
Externally publishedYes

Keywords

  • Chemotaxis model
  • global existence
  • macrophage
  • paracrine and autocrine loops
  • tumor cell

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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