On nonstandard chemotactic dynamics with logistic growth induced by a modified complex Ginzburg–Landau equation

José Luis López
2021 Studies in applied mathematics (Cambridge)  
In this paper, we derive a variant of the classical Keller-Segel model of chemotaxis incorporating a growth term of logistic type for the cell population 𝑛(𝑡, 𝑥), say 𝜈𝑛(1 − 𝑛) with 𝜈 > 0, and a nonstandard chemical productiondegradation mechanism involving first-and secondorder derivatives of the logarithm of the cell density, say 𝑓 𝜆𝑎𝑏 (𝑛, 𝑛 𝑥 , 𝑛 𝑥𝑥 ) = 𝜆𝑛 + 𝑎 𝑛 𝑥𝑥 𝑛 + 𝑏 𝑛 2 𝑥 𝑛 2 with 𝜆, 𝑎, 𝑏 ∈ ℝ, via the (𝑛, 𝑆)-hydrodynamical system associated with a modified Ginzburg-Landau equation
more » ... ing the evolution of the complex wavefunction 𝜓 = √ 𝑛 𝑒 𝑖𝑆 . In a chemotactic context, 𝑆(𝑡, 𝑥) will play the role of the concentration of chemical substance. Then, after carry- ing out a detailed analysis of the modulational stability of uniform-in-space plane waves, dark soliton-shaped traveling wave densities of the former system are constructed from solitary wave solutions of the latter.
doi:10.1111/sapm.12440 fatcat:bw7x4p6vjveflf7ikghetyu74u