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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.

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