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REVIEW 5 major objections 4 minor 45 references

On study of cell proliferation and diffusion using nonlinear transforms of heat equation solutions

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that nonlinear transformations of heat equation solutions yield analytic cell-density fields that match scratch assay data across three cell lines.

desk verdict Ansatz I is a nonlinear diffusion equation, not a reaction-diffusion equation, so half the fits don't actually model proliferation; the paper needs major revision but has a salvageable core in Ansatz II. read the letter →

arxiv 2505.23988 v1 pith:4TXBRLMQ submitted 2025-05-29 nlin.CD

Preet Mishra , Shyam Kumar , Sapna Ratan Shah , R.K. Brojen Singh This is my paper · ORCID
classification nlin.CD
keywords Hopf-ColetransformRichardsgrowthfunctioncellproliferationscratchassayHermitepolynomialsfractionalBrownianmotionreaction-diffusionparameterestimation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to show that the spatial and temporal cell-density patterns seen in scratch assays can be reproduced by analytic formulas built from nonlinear transformations of heat equation solutions. It constructs two such transformations using the Richards growth function, represents the heat-equation ingredient with two-variable Hermite polynomials, and fits the resulting expressions to experimental data from three cell lines. The fitted parameters are robust to fractional Brownian noise and agree with previous estimates. If the claim holds, the approach offers a computationally cheap analytic alternative to numerical reaction-diffusion simulations for cell proliferation and migration.

What carries the argument

The engine of the paper is the Hopf-Cole-style nonlinear invertible transformation: a change of variables that maps solutions of the linear heat equation into candidate solutions of a nonlinear growth-diffusion process via cancellation of terms. Two ansatze operationalize this idea for the Richards growth ODE, yielding the explicit formulas in Table II; two-variable Hermite polynomials provide a series representation of the heat solution $h(x,t)$ from the scratch-assay initial condition; and fractional Brownian motion fields supply a noise model for robust parameter estimation.

What would settle it

Numerically solve the intended reaction-diffusion equation $u_t = u_{xx} + F(u)$ with the fitted Richards parameters and the same initial condition, then compare those profiles with the Table II solutions at the same time points; if they diverge appreciably, the transform solutions are not the reaction-diffusion solutions and the claim of describing proliferation plus diffusion collapses.

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Extended reading notes

Core claim

The central discovery is that a cell-density field $u(x,t)$ can be generated by composing a solution $h(x,t)$ of the linear heat equation with a nonlinear growth map, in the spirit of the Hopf-Cole transformation. Two ansatze are derived: Ansatz I sets $u=g(h)$ with $dg/dh=F(g)$, and Ansatz II sets $G(u)=e^t h$ with $\frac{d}{du}\log G(u)=1/F(u)$. For the Richards growth function $F(u)=a u(1-(u/b)^{1/m})$, these produce explicit closed forms (Table II). Using a fourth-order Hermite-polynomial representation of the scratch initial condition, these formulas fit the measured density profiles of the 12505Lu, WM983C, and C8161 cell lines at four time points each, with fitted $m$ near 1 and $b$ around 0.15--0.25, and the parameters remain stable when fractional Brownian noise is added to the residuals.

Load-bearing premise

The load-bearing premise is that the transformed density $u(x,t)$ actually represents a cell population that both grows and diffuses; the paper never derives the partial differential equation that $u$ satisfies, so the proliferation interpretation rests on an assumed Hopf-Cole-style cancellation rather than on a derived growth term.

Editorial extensions

If this is right

  • If correct, the analytic formulas in Table II can replace numerical solution of reaction-diffusion PDEs when modeling scratch-assay cell dynamics.
  • Recovered values of $m$ near 1 support logistic-like growth for the tested cell lines, consistent with prior biological observations.
  • Robust parameter estimates under fractional Brownian noise suggest the fitting procedure is usable on noisy experimental data.
  • The two ansatze provide alternative temporal behaviors (no modulating factor versus $f(t)=e^t$), which can be matched to different data regimes.
  • The transformation framework extends to other growth ODEs by substituting a different $F(u)$ into the same ansatze.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test would be to derive explicitly the partial differential equation that the transformed density $u(x,t)$ satisfies; if that PDE contains no genuine growth term, the fits are curve fitting rather than reaction-diffusion modeling.
  • The method could be used predictively by fitting parameters to early time points and then forecasting later density profiles, a task not demonstrated in the paper.
  • The same transform recipe might generalize to advection-diffusion or other linear PDEs to model biased cell migration, which the paper does not explore.
  • Because the initial condition is fixed from the $t=0$ data, letting the initial-condition parameters vary in the fitting procedure could either sharpen or break the claimed agreement.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 4 minor

Summary. The paper proposes two nonlinear transformations of heat-equation solutions (Ansatz I and Ansatz II) driven by a Richards growth ODE, claims that the resulting density fields u(x,t) model cell proliferation and diffusion, and fits the parameters m and b to scratch-assay data from three melanoma cell lines (12505Lu, WM983C, C8161). The heat-equation input h(x,t) is approximated by a fourth-order two-variable Hermite polynomial expansion. Parameter estimation is done first by least squares / log-likelihood and then with an additive fractional Brownian motion field. The authors report optimal m and b values, claim agreement with previous growth-rate estimates, and state that the solutions describe the spatiotemporal patterns of the experimental cell densities.

Significance. If the construction were valid, the paper would provide closed-form analytic solutions for reaction-diffusion models with a Richards nonlinearity, which could be useful in cell-biology modeling. The authors make a serious attempt to compare their formulas against real experimental data and to add a noise-robustness analysis. However, the central mathematical claim is not established in the manuscript: the PDE satisfied by u(x,t) is never stated, Ansatz I in fact does not yield a reaction term, and several inconsistencies in the parameter-search ranges and time handling make the reported estimates irreproducible. The in-sample validation does not provide independent confirmation of the model.

major comments (5)
  1. [§Methods, Eqs. (4)-(6) and Table II] Ansatz I does not produce a reaction-diffusion equation with a growth term. Direct differentiation of u = g(h) with dg/dh = F(u) and h_t = h_xx yields u_t = u_xx - [F'(u)/F(u)] u_x^2, which is a nonlinear-diffusion equation with no source term F(u). Thus the fits shown for Ansatz I in Figs. 2-4 do not support the paper's central claim of modeling 'cell proliferation and diffusion'; they are curve fits of a density-dependent diffusion process. The derivation is said to be in the supplementary material, but the claim requires a stated PDE in the main text.
  2. [§Methods, Eqs. (3)-(8)] The PDE satisfied by u(x,t) is never stated for either ansatz. The paper defines the transformation G(u)=g(f(t)h), applies the heat operator formally, and then jumps to the integral relations (6) and (8) and the closed forms in Table II. Without the resulting u_t = ... equation, the interpretation of these formulas as solutions of a reaction-diffusion problem is unverifiable. The derivation is deferred to a supplementary file that is not included in the preprint and cannot be checked.
  3. [§Results, Figs. 2-4 and Eqs. (20)-(21)] The contour plots of the error and log-likelihood surfaces vary m only from 1 to 2, yet the reported optimal values in Eqs. (20)-(21) are m = 0.5, 0.639, 0.54, 0.75, all below 1. The minima shown in the figures are therefore not the reported optima, and the text's statement that 'm values near 1' capture the behavior is inconsistent with the reported estimates. The figures cannot substantiate the claimed 'distinct minimum' of the error surfaces for the parameters finally used.
  4. [§Results, Table II and Eqs. (20)-(21)] The time variable is never reconciled with the experimental times. The data are collected at t = 0, 16, 32, 48 h (or 0, 6, 12, 18 h for C8161), while the Ansatz II solution involves e^t and is plotted only for t ∈ [0,2]. No non-dimensionalization of time is given, and a=1 is fixed without explanation. With t in hours, the term (e^t h)^{-a/m} in Table II would cause the density to saturate almost immediately, contradicting the observed gradual filling of the scratch. The parameter estimates are therefore not reproducible without a clear time-scaling convention.
  5. [§Parameter Estimation, Eqs. (14)-(19) and §Discussion] The claimed confirmation that the solutions 'well describe' the experimental patterns is based on the same least-squares or log-likelihood fit that produced the parameters. The fBM robustness step adds noise and re-estimates the same parameters on the same data, so it does not break the circularity. An independent validation—e.g., held-out time points, model-selection criteria such as AIC (which the Discussion mentions as future work), or comparison against a null diffusion-only model—is required before the descriptive claim can be made.
minor comments (4)
  1. [§Data, Eq. (11)] The initial condition expression is missing parentheses: the intended form appears to be u(x,0) = 1 / (η + α/(1-α) e^{-x^2/(2β^2)}). The current printed formula is ambiguous.
  2. [§Data, text near Eq. (11)] There is a typo: 'otsude' should be 'outside'.
  3. [§Solutions using Hermite polynomials, Eq. (13)] The symbol m is used both for the Hermite truncation order (Eq. 13) and for the Richards nonlinearity exponent throughout the results; this notational collision is confusing and should be resolved.
  4. [§Discussion, limitations paragraph] The justification for truncating the Hermite expansion at order 4 is only 'the higher order terms are O(1/n!)'; this is not a rigorous uniform error bound for the specific initial condition in Eq. (11). A convergence test or an a posteriori error estimate should be reported.

Circularity Check

2 steps flagged · score 6.0 of 10

The central validation claim is an in-sample fit presented as confirmation, and Ansatz I's Richards growth term cancels by construction; partial circularity.

  1. fitted input called prediction [Abstract; Eqs. (14)-(15) and Figs. 2-4 in Results]
    "We estimated the parameters of the Richards function by comparing the model solutions with the experimentally observed data from scratch assays. ... Further, we also confirmed that the spatial-temporal patterns of cell density in the experiments can be well described by the solutions developed using the proposed nonlinear transforms method."

    The parameters Θ={m,b} are obtained by minimizing ‖u_ana(x,t;Θ)−u_obs(x,t)‖ over the very same four-time-point datasets that are then displayed as “fitted analytical solutions”. The abstract's “confirmed ... can be well described” therefore reports the minimum of the fitting objective, not an independent check. Adding an fBM field (Eq. 15) re-estimates Θ,H,σ on the same data and selects the subset nearest the global minimum of the same residual, so the robustness step is also in-sample. Thus the central validation claim reduces to the least-squares fit by construction.

  2. other [Methods, Ansatz I, Eqs. (4)-(6); Table II]
    "By using the relation u=g(h) and the ODE (4) we get u in terms of the heat equation solution h as : ∫ du/F(u)=h (6) where, h(x,t) satisfies ht=hxx."

    Differentiating (6) with h_t=h_xx gives u_t = u_xx − [F'(u)/F(u)] u_x^2. No F(u) source term survives: the Richards growth function appears only in the nonlinear diffusion coefficient. Hence the “growth ODE” input is cancelled by the Hopf-Cole-style substitution, and Ansatz I's u is a solution of a nonlinear diffusion equation, not a reaction-diffusion equation. The paper's claim that these solutions model “cell proliferation and diffusion” is therefore not a derived consequence of the Richards growth law; for Ansatz I the proliferation content is an artifact of how u was defined, i.e., the model's growth term is equivalent to zero by construction.

full rationale

The paper does not rely on load-bearing self-citations: the external comparisons to Liu et al. [32] and Falcó et al. [16] are independent, and the Hermite-polynomial representation is standard. The main circularity is in the validation section: m and b are estimated by least-squares on the same scratch-assay spatiotemporal data that are later presented as “well described” by the solutions, so the confirmation is the minimized fitting objective itself rather than an independent prediction. The fBM robustness analysis re-estimates all parameters on the same data and therefore does not break this in-sample loop. A second, construction-level problem affects Ansatz I: differentiating Eq. (6) using h_t=h_xx eliminates the reaction term F(u), leaving u_t=u_xx−(F'/F)u_x^2. Thus the Richards “growth” input cancels by the Hopf-Cole-style substitution, and the Ansatz I fits are nonlinear-diffusion curve fits, not reaction-diffusion predictions of proliferation. Ansatz II does retain a source term F(u), and the external parameter comparison gives independent content, so the circularity is partial rather than total; score 6.

Assumptions & free parameters 8 free parameters · 5 assumptions · 0 invented entities

The model depends on several fitted parameters (a fixed at 1, m, b, initial condition parameters alpha, beta, eta, and noise parameters H, sigma) and on assumptions that the inverse transform exists, the Hermite truncation converges, and residual errors are fBM. The central claim is not derived from a stated reaction-diffusion PDE.

free parameters (8)
  • a = 1
    Growth rate in Richards term set to 1 for all analyses, not estimated. This choice affects the time scale and is entangled with m.
  • m = e.g., 1.207, 0.639, 0.5 for Ansatz 1
    Richards nonlinearity exponent estimated by grid search over m,b. Values vary per cell line and ansatz.
  • b = e.g., 0.15, 0.19, 0.17 for Ansatz 1
    Maximum relative density parameter estimated by grid search.
  • alpha = 0.998, 0.998, 0.9905 per line
    Depth parameter in initial condition fitted to t=0 data.
  • beta = 6.2, 8, 6.2 per line
    Width parameter in initial condition fitted to t=0 data.
  • eta = 10 or 2
    Maximum density offset in initial condition, set per line and ansatz.
  • H = e.g., 0.69, 0.39, 0.574
    Hurst exponent for fBM noise, estimated by minimizing residual error.
  • sigma = e.g., 0.007, 0.007, 0.008
    Noise strength for fBM field, estimated.
assumptions (5)
  • domain assumption The inverse of the growth transform g exists and is closed-form.
    Stated explicitly in Methods: 'A crucial assumption here is that the inverse function of g exists and can be written in closed form.'
  • ad hoc to paper The heat solution h is accurately represented by a 4th-order two-variable Hermite polynomial expansion.
    Methods, eq. (13); the paper claims higher terms are O(1/n!) but gives no error analysis.
  • domain assumption Residuals between model and data are a fractional Brownian motion field with constant H and sigma.
    Methods, eq. (15); used for 'robust' estimation but never validated against the actual residual structure.
  • standard math Hopf-Cole style cancellation conditions reduce the transformation G(u)=g(f h) to the integral relations (6) and (8); detailed derivation is in the supplementary material.
    Methods; the resulting PDE for u is not written in the paper.
  • ad hoc to paper Setting the growth rate a=1 is a valid normalization.
    Results: 'Additionally we keep a=1 fixed for all subsequent analysis.' The time units are hours, so this fixes an arbitrary rate.

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Cite this review

Pith. "Pith review of On study of cell proliferation and diffusion using nonlinear transforms of heat equation solutions." pith.science (2026). https://pith.science/paper/4TXBRLMQ

@misc{pith2026250523988,
  author       = {Pith},
  title        = {Pith review of: On study of cell proliferation and diffusion using nonlinear transforms of heat equation solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4TXBRLMQ}},
  note         = {Machine review of arXiv:2505.23988}
}
read the original abstract

Cell proliferation and diffusion can be modeled through reaction-diffusion systems describing the space-time evolution of a density variable. In this work, we present non-linear transformations of heat equation solutions to model cellular growth and diffusion using a Richards growth function. The solutions are obtained by using two-variable Hermite Polynomials. We estimated the parameters of the Richards function by comparing the model solutions with the experimentally observed data from scratch assays. To check robustness of the parameters estimated, we used a fractional Brownian Motion (fBM) field type noise with given Hurst exponent (H) for minimizing residual errors. We found that the parameters can be robustly estimated and match with previous estimations. Further, we also confirmed that the spatial-temporal patterns of cell density in the experiments can be well described by the solutions developed using the proposed nonlinear transforms method.

Figures

Figures reproduced from arXiv: 2505.23988 by the authors.

Figure 1
Figure 1. A single realization of σWH(x, t):= the fBM fields with different Hurst exponents (H). (a)H=0.3, (b)H=0.8 For all the of the figures the σ = 0.006 Parameter Estimation We do the parameter estimation in two steps. In first step we assume the residual error to be given by the expression: u obs(x, t) = u ana(x, t; Θ) + ǫ(x, t) (14) where, ǫ are the residuals(error) to be calculated from the given model and the data and… view at source ↗
Figure 2
Figure 2. Residual Error, log-likelihood surfaces and estimation of p [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Residual Error, log-likelihood surfaces and estimation of p [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Residual Error, log-likelihood surfaces and estimation of p [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Parameters estimated from the stochastic analysis using [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: Surfaces from optimal parameters for Ansatz 1. Param [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: Surfaces from optimal parameters for Ansatz 2. Param [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

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Works this paper leans on

45 extracted references · 35 canonical work pages

  1. [1]

    Derive analytical expressions for space-time dependent cell de nsity variableu(x,t ) from trans- formations of heat equation solutions using the growth ODE of Richa rds model

    work page

  2. [2]

    Apply the analytical solutions u(x,t ) using two-variable Hermite polynomials to fit experi- mental cell density data from a scratch assay, estimating parame tersm andb in the Richards Growth term

    work page

  3. [3]

    The paper is organized as follows : In Model and Methods section we d escribe in details the methods required for all the 3 objectives as described above

    Evaluate the robustness and accuracy of the proposed solutio ns through parameter estimation, comparing the performance of different ansatzes through stoch astic analysis. The paper is organized as follows : In Model and Methods section we d escribe in details the methods required for all the 3 objectives as described above. In Results se ction we describe...

    work page

  4. [4]

    W., Ottino-L¨ offler, B., & Kardar, M

    Swartz, D. W., Ottino-L¨ offler, B., & Kardar, M. (2022). Se ascape origin of Richards growth. Physical Review E, 105(1), 014417. https://doi.org/10.1103/PhysRevE.105.014417

    work page doi:10.1103/physreve.105.014417 2022

  5. [5]

    We first compute for a fix parameter set Θ the norm across each space point(i.e., per time ti): ǫti = /radicaltp /radicalvertex /radicalvertex √ M∑ xj =1 (ǫti xj )2 (16) where, (ǫti xj ) := (uana(xj,t i; Θ) +σW H (xj,t i)) −uobs(xj,t i) (17)

    work page

  6. [6]

    RESULTS Table II: Summary table of solutions(Richards Model)

    Then we find the total residual error as the norm across all ti: Error = /radicaltp /radicalvertex /radicalvertex √ N∑ i=1 (ǫti )2 (18) This total residual Error is minimized to get the optimal parameter s et Θ,H,σ . RESULTS Table II: Summary table of solutions(Richards Model). Ansatz I Ansatz II u(x, t) = 1[ b− 1 m +e− ah(x,t) m ] m u(x, t) = 1[ b− 1 m +[...

    work page

  7. [7]

    Estimation of the parameters Θ = {m,b } using Euclidean norm of the residuals(error) and using the maximum of the log-likelihood surface as described by assum ing the form ( 14)

    work page

  8. [8]

    Estimation using residual error and log-likelihood surfac es First we take up ( 14), where we assume a Gaussian distribution of the residual Errors

    Estimation of the same parameters Θ = {m,b },H,σ where the parameters H is the Hurst exponent and noise strength σ by addition of a fBM field to the analytical solution and subsequent computation of the error norm as described by assum ing the form ( 15). Estimation using residual error and log-likelihood surfac es First we take up ( 14), where we assume a...

Show all 45 references
  1. [9]

    Mathema tical Modeling of Cell Growth via Inverse Problem and Computational Approach

    Andrusyak, I.; Brodyak, O.; Pukach, P.; Vovk, M. Mathema tical Modeling of Cell Growth via Inverse Problem and Computational Approach. Computation 12, 26 (20 24)

  2. [10]

    (1937) A study of the diffusion equation with increase in the amount of substance,” and its application to a biologi cal problem

    Kolmogorov A.N., Petrovskii I., and Piskunov N. (1937) A study of the diffusion equation with increase in the amount of substance,” and its application to a biologi cal problem. In V. M. Tikhomirov, editor, Selected Works of A. N. Kolmogorov I, pages 248–270. Kluwer 1 991, Tran...

    1937

  3. [11]

    to the t = 0 data provided in the sheets[ 31]. Solutions using Hermite polynomials The initial conditions taken in the above section cannot be used in the heat kernel to obtain closed form solutions hence we take the functional approximation route b y representing solutions th...

  4. [12]

    Fisher, R. A. (1937). The wave of advance of advantageous genes. Annals of eugenics, 7(4), 355-369

    1937

  5. [13]

    Brunet, ´E., Derrida, B. (1999). Microscopic models of traveling wav e equations. Computer Physics Communications, 121, 376-381. https://doi.org/10.1016/S0010-4655(99)00358-6

    1999 doi

  6. [14]

    Panja, D. (2004). Effects of fluctuations on propagating f ronts. Physics Reports, 393(2), 87-174. https://doi.org/10.1016/j.physrep.2003.12.001

    2004 doi

  7. [15]

    Canosa, J. (1973). On a nonlinear diffusion equation desc ribing population growth. IBM Journal of Research and Development, 17(4), 307-313. https://doi.org/10.1147/rd.174.0307

    1973 doi

  8. [16]

    Gazdag, J., & Canosa, J. (1974). Numerical solution of Fi sher’s equation. Journal of Applied Proba- bility, 11(3), 445-457. https://doi.org/10.2307/3212689

    1974 doi

  9. [17]

    W., Jin, W., Moroney, T

    McCue, S. W., Jin, W., Moroney, T. J., Lo, K. Y., Chou, S. E. , & Simpson, M. J. (2019). Hole- closing model reveals exponents for nonlinear degenerate d iffusivity functions in cell biology. Physica D: Nonlinear Phenomena, 398, 130-140. https://doi.org/10.1016/j.physd.2019.06.005

    2019 doi

  10. [18]

    Q., Landman, K

    Cai, A. Q., Landman, K. A., & Hughes, B. D. (2007). Multi- scale modeling of a wound-healing cell migration assay. Journal of Theoretic al Biology, 245(3), 576- 594.https://doi.org/10.1016/j.jtbi.2006.10.024

    2007 doi

  11. [19]

    Scribner, E., & Fathallah-Shaykh, H. M. (2017). Single cell mathematical model suc- cessfully replicates key features of GBM: go-or-grow is not necessary. PLoS One, 12(1), e0169434.https://doi.org/10.1371/journal.pone.0169434

    2017 doi

  12. [20]

    E. M. Rutter, T. L. Stepien, B. J. Anderies, J. D. Plasenc ia, E. C. Woolf, A. C. Scheck, G. H. Turner, Q. Liu, D. Frakes, V. Kodibagkar, Y. Kuang, M. C. Pr eul, and E. J. Kostelich, Mathematical analysis of glioma growth in a murine model, Sc i. Rep., 7 (2017), p. 2508, https...

    2017 doi

  13. [21]

    L., Rutter, E

    Stepien, T. L., Rutter, E. M., & Kuang, Y. (2015). A data- motivated density-dependent diffu- sion model of in vitro glioblastoma growth. Mathematical Bi osciences & Engineering, 12(6), 1157- 1172.https://doi.org/10.3934/mbe.2015.12.1157

    2015 doi

  14. [22]

    J., Baker, R

    Warne, D. J., Baker, R. E., & Simpson, M. J. (2019). Using experimental data and information criteria to guide model selection for reaction–diffusion pr oblems in mathematical biology. Bulletin of 17 Mathematical Biology, 81(6), 1760-1804. https://doi.org/10.1007/s11538-019-00589-x

    2019 doi

  15. [23]

    M., & Baker, R

    Falc´ o, C., Crossley, R. M., & Baker, R. E. (2024). Travelling waves in a minimal go-or-grow model of cell invasion. Applied Mathematics Letters, 158, 109209. https://doi.org/10.1016/j.aml.2024.109209

    2024

  16. [24]

    J., Carrillo, J

    Falc´ o, C., Cohen, D. J., Carrillo, J. A., & Baker, R. E. ( 2024). Quantifying cell cycle regulation by tissue crowding. Biophysical Journal. https://doi.org/10.1016/j.bpj.2024.05.003

    2024 doi

  17. [25]

    Communications on Pure and Applied Mathematics 3(3):201-230 https://doi.org/10.1002/cpa.3160030302

    Hopf, E.(1950) The Partial differential equation ut+uux = µuxx. Communications on Pure and Applied Mathematics 3(3):201-230 https://doi.org/10.1002/cpa.3160030302

    1950 doi

  18. [26]

    Cole, J. D. (1951). On a quasi-linear parabolic equatio n occur- ring in aerodynamics. Quarterly of Applied Mathematics, 9( 3), 225–236. https://www.ams.org/journals/qam/1951-09-03/S0033-569X-1951-42889-X/

    1951

  19. [27]

    An, J., Henderson, C., & Ryzhik, L. (2023). Quantitativ e steepness, semi-FKPP re- actions, and pushmi-pullyu fronts. Archive for Rational Me chanics and Analysis, 247(5), 88.https://doi.org/10.1007/s00205-023-01924-2

    2023 doi

  20. [28]

    Anco, S., Bluman, G., Wolf, T. (2008). Invertible mappi ngs of nonlinear PDEs to lin- ear PDEs through admitted conservation laws. Acta Applican dae Mathematicae, 101, 21- 38.https://doi.org/10.1007/s10440-008-9205-7

    2008 doi

  21. [29]

    Kumei, S., Bluman, G. W. (1982). When nonlinear differen tial equations are equiva- lent to linear differential equations. SIAM Journal on Appli ed Mathematics, 42(5), 1157- 1173.https://doi.org/10.1137/0142079

    1982 doi

  22. [30]

    Cheviakov, A., Zhao, P. (2024). Integrability and Othe r Analytical Properties of Nonlinear PDE Sys- tems. In: Analytical Properties of Nonlinear Partial Differ ential Equations. CMS/CAIMS Books in Mathematics, vol 10. Springer

    2024

  23. [31]

    W., Newell, G

    Montroll, E. W., Newell, G. F. (1952). Unsteady-State S eparation Performance of Cascades. I. Journal of Applied Physics, 23(2), 184-194. https://doi.org/10.1063/1.1702171

    1952 doi

  24. [32]

    Kenkre, V. M. (2004). Results from variants of the Fishe r equation in the study of epi- demics and bacteria. Physica A: Statistical Mechanics and i ts Applications, 342(1-2), 242- 248.https://doi.org/10.1016/j.physa.2004.04.084

    2004 doi

  25. [33]

    Giuggioli, L., Kenkre, V. M. (2003). Analytic solution s of a nonlinear convective equation in population dynamics. Physica D: Nonlinear Phen omena, 183(3-4), 245- 259.https://doi.org/10.1016/S0167-2789(03)00176-3

    2003 doi

  26. [34]

    J., Zeppetella, A

    Ablowitz, M. J., Zeppetella, A. (1979). Explicit solut ions of Fisher’s equation for a special wave speed. Bulletin of Mathematical Biology, 41(6), 835-840. https://doi.org/10.1007/BF02462380

    1979 doi

  27. [35]

    Cherniha, R. M. (1998). New non-Lie ans¨ atze and exact s olutions of nonlinear reaction- diffusion-convection equations. Journal of Physics A: Math ematical and General, 31(40), 8179. https://doi.org/10.1088/0305-4470/31/40/012

    1998 doi

  28. [36]

    A., Mansfield, E

    Clarkson, P. A., Mansfield, E. L. (1994). Symmetry reduc tions and exact solutions of a class of nonlinear heat equations. Physica D: Nonlinear Phenomena, 70(3), 250- 288.https://doi.org/10.1016/0167-2789(94)90017-5

    1994 doi

  29. [37]

    Hood, S. (2000). On direct, implicit reductions of a non linear diffusion equation with an arbitrary function-generalizations of Clarkson’s and Kruskal’s met hod. IMA journal of applied mathematics, 64(3), 223-244. https://doi.org/10.1093/imamat/64.3.223

    2000 doi

  30. [38]

    T., McCue, S

    Vittadello, S. T., McCue, S. W., Gunasingh, G., Haass, N . K., & Simpson, M. J. (2018). Mathematical models for cell migration with real-time cell cycle dynamic s. Biophysical Journal, 114(5), 1241-1253. url{https://doi.org/10.1016/j.bpj.2017.12.041}

    2018 doi

  31. [39]

    https://www.cell.com/cms/10.1016/j.bpj.2017.12.041/attachment/1a1897ab-08fe-41fe-92b1-97c4e3c02344/mmc 2.xls

    2017 doi

  32. [40]

    K., Cohen, D

    Liu, Y., Suh, K., Maini, P. K., Cohen, D. J., & Baker, R. E. (2024). Parameter identifiability and model selection for partial differential equation models of cell invasion. Journal of the Royal Society Interface, 21(212), 20230607. https://doi.org/10.1098/rsif.2023.0607

    2024

  33. [41]

    Nelder, J. A. (1961). The fitting of a generalization of t he logistic curve. Biometrics, 17(1), 89- 110.https://doi.org/10.2307/2527498

    1961 doi

  34. [42]

    Richards, F. J. (1959). A flexible growth function for em pirical use. Journal of experimental Botany, 10(2), 290-301. https://doi.org/10.1093/jxb/10.2.290

    1959 doi

  35. [43]

    (1943) Introduction to the Mathema tical Theory of the Conduction of Heat in 18 Solid Bodies

    (1943) Carslaw H.S. (1943) Introduction to the Mathema tical Theory of the Conduction of Heat in 18 Solid Bodies. Dover, New York.(Second Edition)

    1943

  36. [44]

    Dattoli, G., Chiccoli, C., Lorenzutta, S., Maino, G., & Torre, A. (1994). Theory of generalized Hermite polynomials. Computers & Mathematics with Applications, 28(4), 71- 83.https://doi.org/10.1016/0898-1221(94)00128-6

    1994 doi

  37. [45]

    A., & Viens, F

    Tindel, S., Tudor, C. A., & Viens, F. (2003). Stochastic evolution equations with fractional Brownian motion. Probability Theory and Re lated Fields, 127, 186- 204.https://doi.org/10.1007/s00440-003-0282-2

    2003 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.