On a parabolic–elliptic chemotactic system with non-constant chemotactic sensitivity

We study a parabolic–elliptic chemotactic system describing the evolution of a population’s density “uu” and a chemoattractant’s concentration “vv”. The system considers a non-constant chemotactic sensitivity given by “χ(N−u)χ(N−u)”, for N≥0N≥0, and a source term of logistic type “λu(1−u)λu(1−u)”. The existence of global bounded classical solutions is proved for any χ>0χ>0, N≥0N≥0 and λ≥0λ≥0. By using a comparison argument we analyze the stability of the constant steady state u=1u=1, v=1v=1, for a range of parameters. –For N>1N>1 and Nλ>2χNλ>2χ, any positive and bounded solution converges to the steady state.–For N≤1N≤1 the steady state is locally asymptotically stable and for χN<λχN<λ, the steady state is globally asymptotically stable.