Population dynamics of surface-mediated autocatalytic processes

1. [1]

   However, this function must remain bounded between 0 and 1, so that such a growth is not possible

   would imply an exponential growth of ¯Gs(t|x0). However, this function must remain bounded between 0 and 1, so that such a growth is not possible. Even though this equation re- mains valid, it is not suitable for the analysis of the su- percritical regime, and we switch to Eq. ( 18), which in- volves the decaying functions P +(x, t |x0) and S+(t|x0). This...

2. [2]

   Is it possible to get a nonzero steady- state solution? Let us first assume that such a nontrivial solution exists and then inspect the conditions for its existence

   that ¯Gs ≡ 0 is always a solution in the limit t → ∞ . Is it possible to get a nonzero steady- state solution? Let us first assume that such a nontrivial solution exists and then inspect the conditions for its existence. Replacing ¯Gs(t|x0) by its limit ¯Gs(∞| x0) in Eq. ( 18), we get in the leading order ¯Gs(∞| x0) = qc ∫ Γ c dx G+(x, x0) ( 2 ¯Gs(∞| x) − ...

3. [3]

   Given that ¯G1(t|x0) ≡ 0 due to the probability distribution normalization, one might expect that ¯Gs(∞| x0) ≡ 0 for any s

   does not also depend on s. Given that ¯G1(t|x0) ≡ 0 due to the probability distribution normalization, one might expect that ¯Gs(∞| x0) ≡ 0 for any s. We will argue below that this conclusion is valid in the subcritical and critical regimes. In turn, a strictly positive solution ¯Gs(∞| x0) > 0 does exist in the super- critical regime for s < 1, and this s...

4. [4]

   as Gs(∞| x0) = π (x0) + qc ∫ Γ c dx G+(x, x0) [Gs(∞| x)]2, (49) where π (x0) = 1 − qc ∫ Γ c dx G+(x, x0). (50) The integral equation ( 49) has a simple probabilistic interpretation: in the steady-state regime, a particle started from x0 is either absorbed on Γ a with proba- bility π (x0), or branches at a point x ∈ Γ c, creating two identical independent ...

5. [5]

   (58) Since both ¯Gs(∞| x0) and v0(x0) are nonnegative, this equality is only possible if qc/µ 0 > 1

   by v0(x0) and integrate over Γ c that yields, due to the self-adjoint character of the operator A: ( 1 − qc µ 0 ) ∫ Γ c dx0 v0(x0) ¯Gs(∞| x0) = − qc µ 0 ∫ Γ c dx v0(x)[ ¯Gs(∞| x)]2. (58) Since both ¯Gs(∞| x0) and v0(x0) are nonnegative, this equality is only possible if qc/µ 0 > 1. In the marginal case, the left-hand side is zero, implying ¯Gs(∞| x) ≡ 0. ...

6. [6]

   ( 38): qk = − 1 k! lim s→ 0 ∂ k s gs (60) = δk, 1 + qcD C− 0 ∫ Γ c dx u− 0 (x) ∞∫ 0 dt′ eDt′λ − 0 Hk(t′|x), with Hk(t|x) given by Eq

   immediately 12 implies the same asymptotic behavior, Qk(t|x0) ≃ qk u− 0 (x0) C− 0 e− Dtλ − 0 (t → ∞ ), (59) where the prefactor qk is obtained by differentiating the amplitude gs from Eq. ( 38): qk = − 1 k! lim s→ 0 ∂ k s gs (60) = δk, 1 + qcD C− 0 ∫ Γ c dx u− 0 (x) ∞∫ 0 dt′ eDt′λ − 0 Hk(t′|x), with Hk(t|x) given by Eq. ( 24). (ii) In the critical regime, ...

7. [7]

   The linear structure of the PDE allows one to expres- sion its solution in terms of the propagator P − (x, t |x0)

   is linear in Nk(t|x0), whereas the nonlinearity of the boundary condition ( 12b) for the generating function is reflected in the nonlinear structure of the source term Fk(t|x0). The linear structure of the PDE allows one to expres- sion its solution in terms of the propagator P − (x, t |x0). Following the standard technique (see Appendix C), we get Nk(t|x0...

8. [8]

   in terms of products of lower-order moments, one can resort to the induction argument, i.e., one can assume that Eq. (

9. [9]

   , k − 1 and then check it for k

   holds for j = 1, 2, . . . , k − 1 and then check it for k. This assumption implies the long-time behavior Fk(t|x0) ≃ (u− 0 (x0)C− 0 )2e− 2Dtλ − 0 k− 1∑ j=1 (k j ) njnk− j, (74) so that the Laplace transform of Fk(t|x0) does not have a pole at p0 = − Dλ − 0 . As a consequence, the long-time asymptotic behavior of the right-hand side of Eq. (

10. [10]

    ( 73) by setting nk = 1 + qcD C− 0 ∫ Γ c dx u− 0 (x) ∞∫ 0 dt′ eDt′λ − 0 Fk(t′|x)

    is Nk(t|x0) ≃ C− 0 u− 0 (x0)e− Dtλ − 0 + qcDu− 0 (x0)e− Dtλ − 0 ∫ Γ c dx u− 0 (x) ∞∫ 0 dt′eDt′λ − 0 Fk(t′|x), 14 that is reduced to Eq. ( 73) by setting nk = 1 + qcD C− 0 ∫ Γ c dx u− 0 (x) ∞∫ 0 dt′ eDt′λ − 0 Fk(t′|x). (75) As in Sec. IV B, the inconvenience of this representa- tion is that one needs to know Fk(t′|x) for all t′ > 0. If the long-time relati...

11. [11]

    Since N2(t|R) rapidly decreases, its accurate estimation from Monte Carlo simulations requires too many realizations (see dispersion of crosses at long times)

    with the approximate form (76) for n2 provides an accurate approximation. Since N2(t|R) rapidly decreases, its accurate estimation from Monte Carlo simulations requires too many realizations (see dispersion of crosses at long times). One practical consequence of the asymptotic relation (

12. [12]

    is that the coefficient of variation, i.e., the ratio be- tween the standard deviation and the mean, γ(t|x0) = √ N2(t|x0) − N 2 1 (t|x0) N1(t|x0) ∝ eDtλ − 0 / 2, (77) diverges as t → ∞ . In other words, fluctuations become more and more relevant in the subcritical regime so that rare realizations of the stochastic process N (t) can sig- nificantly affect its m...

13. [13]

    More explicitly, we have N2(t|x0) ≃ u− 0 (x0)C− 0 ( 1 + 2Dt qcq0 ) , (78) with exponentially decaying correction terms

    yields t in the leading order. More explicitly, we have N2(t|x0) ≃ u− 0 (x0)C− 0 ( 1 + 2Dt qcq0 ) , (78) with exponentially decaying correction terms. In the Laplace domain, this behavior means that the product of two functions with a simple pole p0 = 0 yields a func- tion with a double pole p0 = 0. Repeating this procedure iteratively, one gets Nk(t|x0) ...

14. [14]

    34); and (c) supercritical (qc = 1

    91); (b) critical ( qc = qcrit c ≈ 4. 34); and (c) supercritical (qc = 1. 1 qcrit c ≈ 4. 78), with qcrit c = 1/ (R ln(L/R )). Solid blue line presents the exact result obtained via a numerical solu - tion of the integral equation ( 70) as described in Appendix G; crosses show the empirical mean of N 2(t) obtained over 10 6 random realizations (see Appendi...

15. [15]

    implies Fk(t|x0) ∝ ekDt|λ − 0 | and this function determines the asymptotic behavior of Nk(t|x0). Using the Laplace transform and the above argument for the poles, we get the long-time behavior Nk(t|x0) ≃ qcDekDt|λ − 0 | ∫ Γ c dx (k− 1∑ j=1 (k j ) nj(x)nk− j(x) ) × ∞∫ 0 dt′e− kDt′|λ − 0 | P − (x, t ′|x0). (81) The dependence on the starting point x0 is th...

16. [16]

    The time derivative of Eq

    or ( 26), or by solving the initial-value problem ( 27) with k = 0. The time derivative of Eq. ( 22) yields an integral equation for the probability density: J0(t|x0) = Ja(t|x0) + 2qcD ∫ Γ c dx t∫ 0 dt′P +(x, t ′|x0) × Q0(t − t′|x)J0(t − t′|x), (87) 16 where Ja(t|x0) = qaD ∫ Γ a dx P +(x, t |x0) (88) is the flux of particles onto the absorbing boundary Γ a...

17. [17]

    In addition, this relation identified T +(x0) as a natural lower bound for the mean extinction time

    still depends on the probability Q0(t|x), the spatial dependence of the mean Ex0 {T0} is now explicitly captured via the Green’s func- tion D ˜P +(x, 0|x0). In addition, this relation identified T +(x0) as a natural lower bound for the mean extinction time. Figure 7 shows the probability density J0(t|x0) for the population dynamics in the circular annulus ...

18. [18]

    paradoxical

    1 qcrit c ≈ 4. 78, dotted red line), with qcrit c = 1 / (R ln(L/R )). The function J0(t|R) was obtained at short times by solv- ing numerically the integral equation ( 89); at longer times, J0(t|R) was calculated directly from a finite-difference ap- proximation of the derivative of Q0(t|R). Symbol present the empirical probability densities in the first two...

    2024

19. [19]

    Moreover, as the PDE problem ( E1) is linear, one can write its solution as N A 1 (t|x0) = ∫ A dx P − (x, t |x0)

    for N1(t|x0) reveals the only difference in the initial condition. Moreover, as the PDE problem ( E1) is linear, one can write its solution as N A 1 (t|x0) = ∫ A dx P − (x, t |x0). (E2) As the mean population size in any subset A of the do- main can be obtained by integrating the propagator over A, P − (x, t |x0)dx can be interpreted as the mean popu- lati...

20. [20]

    Principal Steklov eigenmode As discussed in the main text, the principal eigenvalue µ 0 of the Steklov problem (

21. [21]

    As the principal eigenfunction is rotationally invariant, one can search it as v0(x) = A(ln(|x|/R ) + B), with two unknown constants A and B

    determines the condition for the critical regime. As the principal eigenfunction is rotationally invariant, one can search it as v0(x) = A(ln(|x|/R ) + B), with two unknown constants A and B. Two boundary conditions fix the eigenvalue, µ 0 = 1 R[1/ (qaL) + ln(L/R )] , (F1) and the constant B so that v0(x) = A ( ln(L/ |x|) + 1 qaL ) . (F2) 22 The remaining ...

22. [22]

    Principal Laplacian eigenmode In this section, we summarize the standard formulas to compute the principal eigenvalue λ − 0 and the associated eigenfunction u− 0 (x) by solving Eqs. ( 28). While this computation can be easily adjusted to get other Lapla- cian eigenmodes [ 58–60], we focus on the principal eigen- mode that determines the long-time behavior...

23. [23]

    (F8d) In the following, we treat separately three regimes

    by replacing qc by − qc, and for the principal eigenpair {λ 0, u 0(x)} associated to qc = 0 and satisfying − ∆ uk = λ kuk in Ω , (F8a) ∂nuk = 0 on Γ c, (F8b) ∂nuk + qauk = 0 on Γ a, (F8c) ∂nuk = 0 on Γ r. (F8d) In the following, we treat separately three regimes. Subcritical regime In the subcritical regime ( qc < µ 0), the principal eigen- value λ − 0 is...

24. [24]

    Integrating u− 0 (x) over Ω, we also get C− 0 = πA ( (L2 − R2)(1/ 2 + 1/ (qaL)) − R2 ln(L/R ) )

    and ( 52) for the Laplacian and Steklov eigenmodes are identical, so that the principal Laplacian eigenfunc- tion u− 0 is actually proportional to the principal Steklov eigenfunction v0 (we recall that these eigenfunctions have different normalizations): u− 0 (x) = A ( ln(L/ |x|) + 1 qaL ) , (F16) where the normalization constant A is obtained from 1 = ∫ Ω...

25. [25]

    To avoid technical details, we focus on the Laplace-transformed propagator, for which the computation is simpler: ˜P − (x, p |x0) = ∞∫ 0 dt e− pt P − (x, t |x0)

    The propagator Since the Laplacian eigenmodes are known for the cir- cular annulus, the propagator P − (x, t |x0) can be found through the spectral expansion ( 31). To avoid technical details, we focus on the Laplace-transformed propagator, for which the computation is simpler: ˜P − (x, p |x0) = ∞∫ 0 dt e− pt P − (x, t |x0). (F25) Moreover, as our setting...

26. [26]

    Mean population size For the circular annulus, an explicit computation of the mean population size N1(t|x0) is possible in the Laplace domain. For this purpose, it is sufficient to integrate the Laplace-transformed propagator ˜P − (x, p |x0) over Ω that yields ˜N1(p|r0) = 1 p ( 1 + qc vL(r0) − qa vR(r0) W ) , (F32) where vL(r0), vR(r0) and W are given by Eq...

27. [27]

    Long-time limit of the generating function Since the annulus is rotationally invariant, the depen- dence of the generating function Gs(t|x0) on x0 is re- duced to the radial coordinate r0 = |x0|. As a conse- quence, the Green’s function G+(x, x0) in the integral equation ( 45) is averaged over the angular coordinate, yielding g(r0) = qc ∫ Γ c dx G+(x, x0)...

28. [28]

    In turn, one has G1(t|x0) = 1 due to the normalization of probabilities in any regime. Appendix G: Numerical solution of integral equations In this Appendix, we present a numerical scheme for solving integral equations ( 25, 26) to determine the prob- abilities Qk(t|x0), as well as the generating function Gs(t|x0), in the case of the circular annulus with...

29. [29]

    To see this point, we inspect the auxiliary function f (t) = ∫ Γ c dx0 ∫ Γ c dx P (x, t |x0) (G2) in the general setting

    Short-time behavior of the kernel Let us first discuss the weak singularity of the kernel P (R, t ′|R). To see this point, we inspect the auxiliary function f (t) = ∫ Γ c dx0 ∫ Γ c dx P (x, t |x0) (G2) in the general setting. We assume that Γ a and Γ c are disconnected regions of the boundary (as in the case of the circular annulus). As a consequence, Γ a ...

30. [30]

    These quantities can be found either in terms of spectral expan- sions over the Laplacian eigenfunctions, or via the inverse Laplace transform

    Numerical computation of the kernel To proceed, we need to evaluate S(t|R) and p(t). These quantities can be found either in terms of spectral expan- sions over the Laplacian eigenfunctions, or via the inverse Laplace transform. In the Laplace domain, we can use Eq. ( F28) to get ˜P (R, p |R) and Eq. ( F32) to get ˜S(p|R), by setting qc = 0. We employ the...

31. [31]

    ( G6) as ¯Q0(nδ) ≈ S(nδ|R) + n∑ j=0 wj v((n − j)δ), (G10) where wj are suitable weights

    Discretization Choosing a small timestep δ = t/n and equally spaced grid, we discretize Eq. ( G6) as ¯Q0(nδ) ≈ S(nδ|R) + n∑ j=0 wj v((n − j)δ), (G10) where wj are suitable weights. In a basic quadrature, we use piecewise-constant ap- proximations for functions p(t′) and v(t − t′) on each interval (δj, δ (j + 1)), so that tj+1∫ tj dt′ √ t′ p(t′) v(t − t′) ...

32. [32]

    Solutions for k > 0 In the same way, one can discretize Eq. (

33. [33]

    We get then Qk(nδ) ≈ w0Qk(nδ) ( 2Q0(nδ) − 1 ) + fn, (G17) where fn = S(nδ|R)δk, 1 +w0v0(nδ)+ n∑ j=1 wj v((n− j)δ)

    with k > 0 as Qk(nδ) ≈ S(nδ|R)δk, 1 + n∑ j=0 wj v((n − j)δ), (G15) with v(t) = Qk(t) ( 2Q0(t) − 1 ) + v0(t), (G16a) v0(t) = n− 1∑ i=1 Qi(t)Qn− i(t), (G16b) where we wrote separately the term 2 Q0(t)Qk(t) from Hk(t|x). We get then Qk(nδ) ≈ w0Qk(nδ) ( 2Q0(nδ) − 1 ) + fn, (G17) where fn = S(nδ|R)δk, 1 +w0v0(nδ)+ n∑ j=1 wj v((n− j)δ). (G18) As a consequence, ...

34. [34]

    In contrast to the generating function, the moments Nk(t|x0), as well as the kernel P − (x, t |x0) in Eq

    Computation of the moments The same technique can be used to compute the mo- ments Nk(t|x0) by solving the integral equation ( 70). In contrast to the generating function, the moments Nk(t|x0), as well as the kernel P − (x, t |x0) in Eq. ( 70) grow exponentially in the supercritical regime. This issue can be easily resolved by rescaling the moment Nk(t|x0...

35. [35]

    (G22) In this way, the integral equation ( G21) does not contain any exponentially growing functions in the supercritical regime

    as ¯Nk(t|x0) = ¯S− (t|x0) (G21) + qcD ∫ Γ c dx t∫ 0 dt′ ¯P − (x, t ′|x0) ¯Fk(t − t′|x), where ¯S− (t|x0) = S− (t|x0)e− kDt|λ − 0 |, ¯P − (x, t |x0) = P − (x, t |x0)e− kDt|λ − 0 | and ¯Fk(t|x) = Fk(t|x)e− kDt|λ − 0 | = k− 1∑ j=1 (k j ) ¯Nj(t|x) ¯Nk− j(t|x). (G22) In this way, the integral equation ( G21) does not contain any exponentially growing functions...

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