Akademik Veri Yönetim Sistemi
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, cilt.34, sa.09, ss.1649-1700, 2024 (SCI-Expanded)
In this paper, we study stability, bifurcation and spikes of positive stationary solutions of the following parabolic-elliptic chemotaxis system with singular sensitivity and logistic source: {u(t) = u(xx) - chi(u/v v(x))(x) + u(a - bu), 0 < x < L, t > 0, 0 = v(xx) - mu v+nu u, 0 < x < L, t > 0, u(x)(t,0) = u(x) (t, L) = v(x)(t, 0) = v(x) (t, L) = 0 t > 0, where chi, a, b, mu, nu are positive constants. Among others, we prove there are chi* > 0 and and {chi(k)*} subset of [chi*, infinity) (chi* is an element of{chi(k)*}) such that the constant solution (a/b, nu/mu a/b) of system is locally stable when 0 < chi < chi* and is unstable when chi > chi*, and under some generic condition, for each k >= 1, a (local) branch of nonconstant stationary solutions of system bifurcates from (a/b, nu/mu a/b) when chi passes through chi(k)*, and global extension of the local bifurcation branch is obtained. We also prove that any sequence of nonconstant positive stationary solutions {(u(center dot; chi(n)); v(center dot; chi(n)))g of system with chi = chi(n)(-> infinity) develops spikes at any x* satisfying lim inf(n ->infinity) u(x*; chi(n)) > a/b. Some numerical analysis is carried out. It is observed numerically that the local bifurcation branch bifurcating from (a/b, nu/mu a/b) when chi passes through chi* can be extended to chi = infinity and the stationary solutions on this global bifurcation extension are locally stable when chi >> 1 and develop spikes as chi -> infinity.