Asymptotic limits for a nonlinear integro-differential equation modelling leukocytes’ rolling on arterial walls

We consider a nonlinear integro-differential model describing z, the position of the cell center on the real line presented in Grec et al (2018 J. Theor. Biol. 452 35–46). We introduce a new ɛ-scaling and we prove rigorously the asymptotics when ɛ goes to zero. We show that this scaling characterizes the long-time behavior of the solutions of our problem in the cinematic regime (i.e. the velocity z ˙ tends to a limit). The convergence results are first given when ψ, the elastic energy associated to linkages, is convex and regular (the second order derivative of ψ is bounded). In the absence of blood flow, when ψ, is quadratic, we compute the final position z ∞ to which we prove that z tends. We then build a rigorous mathematical framework for ψ being convex but only Lipschitz. We extend convergence results with respect to ɛ to the case when ψ′ admits a finite number of jumps. In the last part, we show that in the constant force case [see model 3 in Grec et al (2018 J. Theor. Biol. 452 35–46), i.e. ψ is the absolute value)] we solve explicitly the problem and recover the above asymptotic results.