A Simplified Model

We begin by making three simplifications to the Leinheiser et al. model (Leinheiser et al. 2024). First, we assume there is no maximum size for oligomers. This approximation is reasonable because concentrations of oligomers larger than size _ℓ _ℓ are small enough that they do not substantially contribute to the sums in (1c) (Leinheiser et al. 2024). Second, we omit the dissociation rate of oligomers. This choice is motivated by the intuition that the growth of oligomers on the membrane is more important than their disassembly. This change will be revisited in the next section. Third, we omit the dynamics characterizing Drp1 and Mff interactions. These variables (M and T) are eliminated because in the Leinheiser et al. model, the concentration of cytosolic Drp1 varies minimally over time, and the limiting factor is the concentration of Mff. As a result, the oligomerization process begins with the pool of Drp1-Mff building blocks (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1$$\end{document}C1) as opposed to the initial interaction of Drp1 and Mff (Fig. 2). The notation for this system is analogous to the previous with the exceptions that the association rate of oligomers with building blocks is simply k, the initial concentration of building blocks is Q, and _f(ℓ ) = a_f(ℓ)=a to shift the discontinuity from _ℓ + 1_ℓ+1 to _ℓ _ℓ. This gives us the following system of equations:

2a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_{1 }&= a\sum _{i = \ell }^\infty i C_i - k\left( C_{1}^{2} + \sum ^{\infty }_{i=1}C_{i}C_{1} \right) , \end{aligned}$$\end{document}C˙1=a∑i=ℓ∞iCi-kC12+∑i=1∞CiC1,

2b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}_{i}&= kC_{1} C_{i-1} - kC_1C_i - f(i) C_i, \quad i \ge 2, \end{aligned}$$\end{document}C˙i=kC1Ci-1-kC1Ci-f(i)Ci,i≥2,

2c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(i)&= {\left\{ \begin{array}{ll} 0, & i < \ell , \\ a, & i \ge \ell , \end{array}\right. } \end{aligned}$$\end{document}f(i)=0,i<ℓ,a,i≥ℓ,

2d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C_1(0)&= Q, \quad C_i(0) = 0, \quad i > 1. \end{aligned}$$\end{document}C1(0)=Q,Ci(0)=0,i>1.

In this system, the total concentration of building blocks, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _{i=1}^\infty i C_i$$\end{document}∑i=1∞iCi, is conserved. Therefore,

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \sum _{i=1}^\infty i C_i = C_1(0) = Q. \end{aligned}$$\end{document}∑i=1∞iCi=C1(0)=Q.

Fig. 2: The reaction schematic for a simplified model of mitochondrial fission. The initial condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1(0) = Q$$\end{document}C1(0)=Q represents an initial cohort of oligomers of size 1. These associate at rate k into oligomers of size 2. Oligomers of size i combine with oligomers of size 1 at rate k to create oligomers of size i + 1i+1. Oligomers of size ℓℓ or larger can initiate fission at rate a

Homogenizing the Simplified Model

System (2) becomes unwieldy as _ℓ _ℓ becomes large because the dimension of the system scales with _ℓ _ℓ. In addition, the steady state solutions include polynomials of degree _ℓ _ℓ which are difficult to estimate numerically (Leinheiser 2023). We can approximate the behavior of this model by converting oligomer size into a continuous variable of a multivariate function. An example of such an approximation can be found in Batkai et al. (2015). We introduce \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (i,t) = C_i(t)$$\end{document}Θ(i,t)=Ci(t) for _i \ge 1_i≥1. Therefore, Equation (2b) is now the dynamics of _Θ (i,t)_Θ(i,t), and Equation (2a) is the left boundary condition. For ease of notation, we will say \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_1(t) = \Theta (1,t) = C$$\end{document}C1(t)=Θ(1,t)=C. As a result, the homogenized simplified model is

3a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= a \int _\ell ^\infty u \Theta (u,t) du - k C^2 - k C \int _1^\infty \Theta (u,t) du, \end{aligned}$$\end{document}C˙=a∫ℓ∞uΘ(u,t)du-kC2-kC∫1∞Θ(u,t)du,

3b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{\partial \Theta }{\partial t}&= - k C \frac{\partial \Theta }{\partial s} - f(s) \Theta (s,t), \quad s > 1, \end{aligned}$$\end{document}∂Θ∂t=-kC∂Θ∂s-f(s)Θ(s,t),s>1,

3c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (1,t)&= C(t). \end{aligned}$$\end{document}Θ(1,t)=C(t).

The conserved quantity is similarly homogenized:Note that Q is finite which implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _1^\infty u \Theta (u,t) du$$\end{document}∫1∞uΘ(u,t)du and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _1^\infty \Theta (u,t)du$$\end{document}∫1∞Θ(u,t)du are bounded. This is biologically sound.

4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C + \int _1^\infty u \Theta (u,t) du = Q. \end{aligned}$$\end{document}C+∫1∞uΘ(u,t)du=Q.

System (3) has two steady states, one is a fission-free equilibrium: where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h: \mathbb {R} \rightarrow \mathbb {R}$$\end{document}h:R→R is any function such that _h(1) = 0_h(1)=0 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _1^{\ell } s , h(s) ds=Q$$\end{document}∫1ℓsh(s)ds=Q. This is a “fission-free" equilibrium since the concentration of building blocks, C, is zero, and the concentration of oligomers large enough to facilitate fission, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \ge \ell $$\end{document}s≥ℓ, is also zero. In other words, all building blocks are caught up in middle sized oligomers and the building process stalls.

5a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C&= 0, \end{aligned}$$\end{document}C=0,

5b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= h(s),&s < \ell , \end{aligned}$$\end{document}Θ(s)=h(s),s<ℓ,

5c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= 0,&s \ge \ell . \end{aligned}$$\end{document}Θ(s)=0,s≥ℓ.

The other steady state is

6a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C&= \frac{- \frac{1}{2}(\ell ^2 + 1) + \sqrt{\frac{1}{4}(\ell ^2 + 1)^2 + 4 \frac{kQ}{a}}}{2\frac{k}{a}}, \end{aligned}$$\end{document}C=-12(ℓ2+1)+14(ℓ2+1)2+4kQa2ka,

6b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= C,&s < \ell , \end{aligned}$$\end{document}Θ(s)=C,s<ℓ,

6c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s)&= C e^{\frac{-a}{k C}(s - \ell )},&s \ge \ell . \end{aligned}$$\end{document}Θ(s)=Ce-akC(s-ℓ),s≥ℓ.

Note that the total amount of fission occurring at this steady state, called the total fission rate, can be calculated with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _\ell ^\infty a \Theta (u)du$$\end{document}∫ℓ∞aΘ(u)du (Leinheiser et al. 2024). The total fission rate of the above equilibrium is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _\ell ^\infty a \Theta (u)du = k C^2$$\end{document}∫ℓ∞aΘ(u)du=kC2.

Discontinuities in the Fission Function and Initial Conditions

In Equation (3b), we choose f(s) to be the same piecewise function as in System (2):This choice of f(s) introduces a discontinuity to Equation (3b). Therefore, we solve the PDE for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s \in [1, \ell ]$$\end{document}s∈[1,ℓ] and use \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (\ell ,t)$$\end{document}Θ(ℓ,t) as the boundary condition for the PDE from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\ell ,\infty )$$\end{document}(ℓ,∞).

7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} f(s) = {\left\{ \begin{array}{ll} 0, \quad s < \ell , \\ a, \quad s \ge \ell . \end{array}\right. } \end{aligned}$$\end{document}f(s)=0,s<ℓ,a,s≥ℓ.

Furthermore, we wish to impose initial conditions analogous to System (2). Recall these initial conditions are chosen to reflect a sudden release of Drp1 from the endoplasmic reticulum, causing an initial impulse of oligomer building material. With the concentrations of Drp1 and Mff removed from the simplified model, we can simulate the release from the endoplasmic reticulum as an initial count of oligomers of size one, with all other sizes initialized to zero.

8a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (1,0)&= C(0) = Q, \end{aligned}$$\end{document}Θ(1,0)=C(0)=Q,

8b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Theta (s,0)&= 0, \quad s > 1. \end{aligned}$$\end{document}Θ(s,0)=0,s>1.

This choice introduces another discontinuity which begins at (1, 0) and propagates through the solution. Therefore, any solution with these initial conditions would solve System (3) in the weak sense. Note, the PDE holds on either side of the discontinuity caused by the initial conditions.

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^(t)$$\end{document}s∗(t) be the size where the discontinuity propagated from the initial conditions appears. We take _Θ (s,t)_Θ(s,t) to be left continuous and therefore \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Theta (s^,t) = Q$$\end{document}Θ(s∗,t)=Q when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^* < \ell $$\end{document}s∗<ℓ. Furthermore _Θ (s,t) = 0_Θ(s,t)=0 for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s > s^*$$\end{document}s>s∗. We use the Leibniz rule, Equation (3a), and Equation (3b) to differentiate Equation (4):

9a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} Q&= C + \int _1^{\infty } u \Theta (u,t)du = C + \int _1^{s^*} u \Theta (u,t)du, \end{aligned}$$\end{document}Q=C+∫1∞uΘ(u,t)du=C+∫1∗uΘ(u,t)du,

9b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0&= \dot{C} + \int _1^{s^*} u \frac{\partial \Theta }{\partial t} du + s^* \Theta (s^*, t )\frac{ds^*}{dt}, \end{aligned}$$\end{document}0=C˙+∫1∗u∂Θ∂tdu+s∗Θ(s∗,t)ds∗dt,

9c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0&= - k C^2 - k C \int _1^{s^*} \Theta (u,t) du - k C \int _1^{s^*} u \frac{ \partial \Theta }{\partial u} du + s^* Q \frac{ds^*}{dt}, \end{aligned}$$\end{document}0=-kC2-kC∫1∗Θ(u,t)du-kC∫1∗u∂Θ∂udu+s∗Qds∗dt,

9d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0&= - kC^2 -kC \int _1^{s^*} \Theta (u,t) du \end{aligned}$$\end{document}0=-kC2-kC∫1∗Θ(u,t)du

9e\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\quad - k C \left( s^* \Theta (s^*, t) - C - \int _1^{s^*} \Theta (u,t) du \right) + s^* Q \frac{ds^*}{dt}, \quad \text {(Int. by parts)}, \end{aligned}$$\end{document}-kCs∗Θ(s∗,t)-C-∫1∗Θ(u,t)du+s∗Qds∗dt,(Int. by parts),

9f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} 0&= -k C s^* Q + s^* Q \frac{d s^*}{dt}, \end{aligned}$$\end{document}0=-kCs∗Q+s∗Qds∗dt,

9g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \frac{ds^*}{dt}&= k C. \end{aligned}$$\end{document}ds∗dt=kC.

We further define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^*$$\end{document}t∗ using integration:

10a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s^*(t^*) - s^*(0)&= \ell - 1 \end{aligned}$$\end{document}s∗(t∗)-s∗(0)=ℓ-1

10b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&= \int _{0}^{t^*} \frac{ds^*}{d\xi } d\xi = \int _{0}^{t^*} k C(\xi ) d \xi \end{aligned}$$\end{document}=∫0∗ds∗dξdξ=∫0∗kC(ξ)dξ

Intuitively, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ is the time where the discontinuity created by the initial conditions crosses the discontinuity at size _ℓ _ℓ. We note there is no guarantee such a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ exists.

Due to the nonlocal interactions in the PDE boundary condition and velocity of the traveling wave, we defineFor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^(t) < \ell $$\end{document}s∗(t)<ℓ, the impulse created by the initial conditions has only traveled to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^ < \ell $$\end{document}s∗<ℓ, soThen we can use the Leibniz rule to write,

11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n(t) = \int _1^\infty \Theta (u,t) du. \end{aligned}$$\end{document}n(t)=∫1∞Θ(u,t)du.

12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} n(t) = \int _1^{s^*} \Theta (u,t)du. \end{aligned}$$\end{document}n(t)=∫1∗Θ(u,t)du.

13\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}(t)&= \int _1^{s^*} \frac{\partial \Theta (u,t)}{\partial t} du + \frac{ds^*}{dt} \Theta (s^*,t) \nonumber \\&= - kC \int _1^{s^*} \frac{\partial \Theta (u,t)}{\partial s}du + k C Q \nonumber \\&= k C^2. \end{aligned}$$\end{document}n˙(t)=∫1∗∂Θ(u,t)∂tdu+ds∗dtΘ(s∗,t)=-kC∫1∗∂Θ(u,t)∂sdu+kCQ=kC2.

A Stalled Wave

For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^(t) < \ell $$\end{document}s∗(t)<ℓ, System (3), with Equation (13), and initial conditions (8) create a new ODE system. Note for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^(t) < \ell $$\end{document}s∗(t)<ℓ, the first integral term in Equation (3a) is 0. Then This system can be solved for C and plugged into \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _0^{t^} k C(\xi ) d\xi = \ell - 1$$\end{document}∫0∗kC(ξ)dξ=ℓ-1 to solve for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗. However, there is no guarantee a solution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ exists. If there is no solution, the wave defined by Equation (3b) never reaches size _ℓ _ℓ, and the method of characteristics produces a stalled wave (Fig. 3). With this stalled wave, the impulse created by the initial conditions never reaches size _ℓ _ℓ, implying we approach the fission-free steady state. The following theorem proves there is no solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ when _ℓ > 3_ℓ>3.

14a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{C}&= -k C^2 - k C n, \end{aligned}$$\end{document}C˙=-kC2-kCn,

14b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{n}&= k C^2, \end{aligned}$$\end{document}n˙=kC2,

14c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(0)&= Q, \quad n(0) = 0. \end{aligned}$$\end{document}C(0)=Q,n(0)=0.

Fig. 3: A graph of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^(t)$$\end{document}s∗(t) in the homogenized simplified model with parameters ℓ = 15ℓ=15, b = 0.02b=0.02, k = 1k=1, and Q = 12Q=12. Note \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^$$\end{document}s∗ does not travel far in the size direction and never reaches the red dotted line of ℓ = 15ℓ=15

Theorem 1

If _k > 0_k>0, _Q > 0_Q>0, and C(t) is a solution to System (14), then

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^\infty k C(\xi ) d\xi < 2. \end{aligned}$$\end{document}∫0∞kC(ξ)dξ<2.

Proof

Let C(t) and n(t) be solutions to System (14). Note that if _C(t) = 0_C(t)=0 for any t, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C}(t) = 0$$\end{document}C˙(t)=0, and therefore C(t) cannot cross the t-axis. Since _C(0) = Q > 0_C(0)=Q>0, _C(t) > 0_C(t)>0 for _t \ge 0_t≥0. Since \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{n} = k C(t)^2$$\end{document}n˙=kC(t)2, n(t) is increasing. We seek to define new ODEs with closed-form solutions that are guaranteed to be larger than C(t) for all t. Note that _k > 0_k>0 implies \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C} = -k C^2 - k C n \le -k C^2$$\end{document}C˙=-kC2-kCn≤-kC2 for _t \ge 0_t≥0. We can use this fact to define dynamics on a new ODE that has a derivative always larger than C(t) (but still negative).

Define A(t) to be the solution to

15a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{A}&= -k A^2, \end{aligned}$$\end{document}A˙=-kA2,

15b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} A(0)&= Q. \end{aligned}$$\end{document}A(0)=Q.

Since C(t) and A(t) have the same initial condition and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C}(t) \le \dot{A}(t)$$\end{document}C˙(t)≤A˙(t) for _t \ge 0_t≥0, we know _C(t) łe A(t)_C(t)≤A(t) for _t \ge 0_t≥0. The solution is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(t) = \frac{Q}{k Q t + 1}$$\end{document}A(t)=QkQt+1. Thus,This A(t) is one of the closed-form solutions we sought. However, the integral of A(t) is unbounded. Therefore, we define yet another ODE with a closed-form solution that is guaranteed to be larger than C(t) for t greater than some non-zero time. Fix time _α > 0_α>0. Since n(t) is increasing, we know _n(α ) > 0_n(α)>0 for any _α > 0_α>0. This allows us to define B(t) to be the solution to

16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(t) \le \frac{Q}{k Q t + 1}, \quad t \ge 0. \end{aligned}$$\end{document}C(t)≤QkQt+1,t≥0.

17a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{B}&= -k B n(\alpha ), \end{aligned}$$\end{document}B˙=-kBn(α),

17b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B(\alpha )&= \frac{Q}{k Q\alpha + 1}. \end{aligned}$$\end{document}B(α)=QkQα+1.

Since n(t) is increasing, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{C}(t) \le \dot{B}(t)$$\end{document}C˙(t)≤B˙(t) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge \alpha $$\end{document}t≥α. Then because \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(\alpha ) = A(\alpha ) \ge C(\alpha )$$\end{document}B(α)=A(α)≥C(α), we know _C(t) łe B(t)_C(t)≤B(t) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge \alpha $$\end{document}t≥α. The solution is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B(t) = Q e^{k n(\alpha )(\alpha - t)}/(k Q \alpha + 1)$$\end{document}B(t)=Qekn(α)(α-t)/(kQα+1). Thus,Thus, we have two equations which we can integrate to bound the integral of kC(t). The first, A(t), is a bound on C(t) which we cannot use for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow \infty $$\end{document}t→∞ because its integral is unbounded. So, we use bound A(t) until time _α _α. At _t = α _t=α we can use the fact that _n(α ) > 0_n(α)>0 to define bound B(t) which we use for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \ge \alpha $$\end{document}t≥α.

18\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} C(t) \le \frac{Q e^{k n(\alpha )(\alpha - t)}}{k Q \alpha + 1}, \quad t \ge \alpha . \end{aligned}$$\end{document}C(t)≤Qekn(α)(α-t)kQα+1,t≥α.

By Equations (16) and (18), and the continuity of the integral, we getSince _n(α )_n(α) appears in the denominator of this bound, we need an underestimate of _n(α )_n(α). We use a similar strategy, defining a new ODE with a closed-form solution whose solution will always be smaller than n(t). Then we can use this new solution evaluated at _α _α in the bound of the integral of kC(t).

19\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^\infty k C(\xi ) d\xi&\le \int _0^\alpha \frac{k Q}{k Q \xi + 1} d\xi + \int _\alpha ^\infty \frac{k Q e^{k n(\alpha )(\alpha - \xi )}}{k Q \alpha + 1} d\xi \nonumber \\&= \ln (k Q \alpha + 1) + \frac{Q}{(k Q \alpha + 1) n(\alpha )}. \end{aligned}$$\end{document}∫0∞kC(ξ)dξ≤∫0αkQkQξ+1dξ+∫α∞kQekn(α)(α-ξ)kQα+1dξ=ln(kQα+1)+Q(kQα+1)n(α).

Define F(t) and g(t) to be the solutions to the ODE system

20a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{F}&= -k Q F - k F g, \end{aligned}$$\end{document}F˙=-kQF-kFg,

20b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \dot{g}&= k F^2, \end{aligned}$$\end{document}g˙=kF2,

20c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(0)&= Q, \quad g(0) = 0. \end{aligned}$$\end{document}F(0)=Q,g(0)=0.

Recall, _Q \ge C(t)_Q≥C(t) for _t \ge 0_t≥0. So \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\dot{F}(t) \le \dot{C}(t)$$\end{document}F˙(t)≤C˙(t) and thus _F(t) łe C(t)_F(t)≤C(t) for _t \ge 0_t≥0. So, _g(t) łe n(t)_g(t)≤n(t) for _t \ge 0_t≥0. The solutions for F(t) and g(t) are

21a\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} F(t)&= \sqrt{2} Q \sqrt{1 - \tanh ^2(\sqrt{2} k Q t + \text {arccosh}(-\sqrt{2}))}, \end{aligned}$$\end{document}F(t)=2Q1-tanh2(2kQt+arccosh(-2)),

21b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} g(t)&= -Q + \sqrt{2} Q \tanh (\sqrt{2} k Q t + \text {arccosh}(- \sqrt{2})). \end{aligned}$$\end{document}g(t)=-Q+2Qtanh(2kQt+arccosh(-2)).

Define \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ^* = k Q \alpha $$\end{document}α∗=kQα. By (19) and (21b):Parameters k and Q affect the value of _α _α where the bound is minimized, but they do not effect the minimum value. Recall the bound holds for any _α > 0_α>0 and therefore any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ^* > 0$$\end{document}α∗>0. One can calculate that when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha ^* = 1.65$$\end{document}α∗=1.65 the bound is equal to 1.89571 < 2_1.89571<2. Thus,□ _□

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\int _0^\infty k C(\xi ) d\xi \le \ln (k Q \alpha + 1) + \frac{Q}{(k Q \alpha + 1) n(\alpha )} \\&\le \ln (k Q \alpha + 1) + \frac{Q}{(k Q \alpha + 1) (-Q + \sqrt{2} Q \tanh (\sqrt{2} k Q \alpha + \text {arccosh}(- \sqrt{2})))} \\&= \ln (\alpha ^* + 1) + \frac{1}{(\alpha ^* + 1) (-1 + \sqrt{2} \tanh (\sqrt{2} \alpha ^* + \text {arccosh}(- \sqrt{2})))}. \end{aligned}$$\end{document}∫0∞kC(ξ)dξ≤ln(kQα+1)+Q(kQα+1)n(α)≤ln(kQα+1)+Q(kQα+1)(-Q+2Qtanh(2kQα+arccosh(-2)))=ln(α∗+1)+1(α∗+1)(-1+2tanh(2α∗+arccosh(-2))).

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \int _0^\infty k C(\xi ) d\xi&< 2. \end{aligned}$$\end{document}∫0∞kC(ξ)dξ<2.

By this theorem, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int _0^{t^} k C(\xi ) d\xi = \ell -1$$\end{document}∫0∗kC(ξ)dξ=ℓ-1 has no solution for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t^$$\end{document}t∗ when _ℓ > 3_ℓ>3. The theorem makes use of the chosen initial conditions for the simplified model. These initial conditions were chosen to be analogous to the Leinheiser et al. model, but we note that different initial conditions may escape the basin of attraction of the fission-free, stalled-wave equilibrium. Such initial conditions could be investigated in future work.

Based on the above theorem, we conjectured that a stalled wave exists in the Leinheiser et al. model when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_- = 0$$\end{document}k-=0. Indeed, numerical simulations with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_-=0$$\end{document}k-=0 confirm that without dissociation of some kind, the Leinheiser et al. model also exhibits a stalled-wave equilibrium where fission does not occur (Fig. 4).

Fig. 4: A numerical simulation of the Leinheiser et al. model with parameters \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k+ = 5$$\end{document}k+=5, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_- = 0$$\end{document}k-=0, ℓ = 25ℓ=25, a = 5a=5, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_1 = 1$$\end{document}k1=1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{-1} = 0.02$$\end{document}k-1=0.02, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {T} = 20$$\end{document}T=20, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {M} = 15$$\end{document}M=15, γ = 0.01γ=0.01, and N = 30N=30. The left figure shows there is no fission with these parameters. The right figure shows snapshots of the size distribution interpolated between integer sizes at various time steps. The concentrations of sizes beyond s = 8s=8 are omitted, as they are all 0. The middle sizes of oligomers stall as early as t = 0.35t=0.35, which is consistent with the findings from the simplified model_