Asymptotic analysis of a bi-monomeric nonlinear Becker-Döring system

To provide a mechanistic explanation of sustained then damped oscillations observed in a depolymerisation experiment, a bi-monomeric variant of the seminal Becker-Döring system has been proposed in Doumic et al (2019 J. Theor. Biol. 480 241-61). When all reaction rates are constant, the equations are the following: d v d t = − v w + v ∑ j = 2 ∞ c j , d w d t = v w − w ∑ j = 1 ∞ c j , d c j d t = J j − 1 − J j , j ⩾ 1 , J j = w c j − v c j + 1 , j ⩾ 1 , J 0 = 0 , where v and w are two distinct unit species, and c_i represents the concentration of clusters containing i units. We study in detail the mechanisms leading to such oscillations and characterise the different phases of the dynamics, from the initial high-amplitude oscillations to the progressive damping leading to the convergence towards the unique positive stationary solution. We give quantitative approximations for the main quantities of interest: period of the oscillations, size of the damping (corresponding to a loss of energy), number of oscillations characterising each phase. We illustrate these results by numerical simulation, in line with the theoretical results, and provide numerical methods to solve the system.