Review of the Porous-Fisher-KPP Equation: Mathematical Theory and Applications

The Porous-Fisher-Kolmogorov-Petrovsky-Piscounov (Porous-Fisher-KPP) equation is a nonlinear degenerate parabolic equation that couples the porous medium equation with the classical Fisher-KPP reaction-diffusion model. Because its diffusivity vanishes at zero density, it supports finite-speed propagation and sharp traveling fronts, and it has become a leading framework for biological invasion, tissue growth, and other density-dependent spreading phenomena. This article reviews the mathematical theory of the equation, its traveling-wave and free-boundary formulations, its numerical and asymptotic treatment, and its applications. After setting out the review methodology (Section 2), we develop the mathematical formulation and contrast the porous-medium, Fisher-KPP, and Porous-Fisher-KPP models (Section 3). We then synthesize the traveling-wave literature (Section 4) and the free-boundary literature (Section 5), and survey applications in ecology, cell biology, epidemiology, and tissue engineering (Section 6). Numerical and asymptotic methods (Section 7), recent generalizations (Section 8), and open research directions (Section 9) complete the survey. Throughout, the emphasis is on synthesis and comparison across studies rather than on a study-by-study catalogue, and on the points at which the degeneracy of the diffusion changes the classical Fisher-KPP picture.

Porous-Fisher-KPP equation; Traveling waves; Nonlinear diffusion; Population dynamics; Free boundary; Spreading speed

1. Fisher, R. A. (1937). The wave of advance of advantageous genes. Annals of Eugenics, 7, 355-369. [Google Scholar] [Crossref]

2. Kolmogorov, A. N.; Petrovskii, I. G.; Piskunov, N. S. (1937). A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Byulleten’ Moskovskogo Gosudarstvennogo Universiteta. Matematika, 1, 1-26. [Google Scholar] [Crossref]

3. Aronson, D. G.; Barenblatt, G. I. (1980). Asymptotic behavior of solutions of the one-dimensional porous medium equation. Communications in Mathematical Physics, 70, 115-129. DOI: 10.1007/BF01204333 [Google Scholar] [Crossref]

4. Aronson, D. G. (1986). The porous medium equation. Lecture Notes in Mathematics, 1224, 1-46. DOI: 10.1007/BFb0072687 [Google Scholar] [Crossref]

5. Vazquez, J. L. (2007). The porous medium equation: Mathematical theory. Oxford University Press, 622pp. [Google Scholar] [Crossref]

6. Vazquez, J. L. (1984). Asymptotic behaviour and propagation properties of the one-dimensional flow of gas in a porous medium. Transactions of the American Mathematical Society, 277, 239-253. DOI: 10.2307/1998948 [Google Scholar] [Crossref]

7. Gilding, B. H.; Kersner, R. (2004). Travelling waves in nonlinear diffusion-convection reaction. Birkhauser Verlag, 290pp. DOI: 10.1007/978-3-0348-7964-4 [Google Scholar] [Crossref]

8. Gilding, B. H.; Kersner, R. (1996). The characterization of reaction-convection-diffusion processes by travelling waves. Journal of Differential Equations, 124, 27-79. DOI: 10.1006/jdeq.1996.0002 [Google Scholar] [Crossref]

9. Murray, J. D. (2002). Mathematical biology. I: An introduction. 3rd ed. Springer-Verlag, 551pp. [Google Scholar] [Crossref]

10. Okubo, A. (1980). Diffusion and ecological problems: Mathematical models. Springer-Verlag, 254pp. [Google Scholar] [Crossref]

11. Aronson, D. G. (1969). Regularity properties of flows through porous media. SIAM Journal on Applied Mathematics, 17, 293-313. DOI: 10.1137/0117025 [Google Scholar] [Crossref]

12. Simpson, M. J.; Landman, K. A.; Hughes, B. D. (2006). Animal invasion across a high latitude barrier: Non-Fickian movement and travelling waves. Physica D, 219, 11-22. [Google Scholar] [Crossref]

13. El-Hachem, M.; McCue, S. W.; Jin, W.; Du, Y.; Simpson, M. J. (2019). Revisiting the Fisher-Kolmogorov-Petrovsky-Piskunov equation to interpret the spreading or extinction of initial populations. Bulletin of Mathematical Biology, 81, 4922-4947. DOI: 10.1007/s11538-019-00654-2 [Google Scholar] [Crossref]

14. Diekmann, O. (1978). Thresholds and travelling waves for the geographical spread of an epidemic. Journal of Mathematical Biology, 5, 109-117. [Google Scholar] [Crossref]

15. Lewis, M. A.; Li, B. (2012). Spreading speed, traveling waves, and minimal domain size in implicit models of kernel dispersal. Journal of Dynamics and Differential Equations, 24, 207-235. DOI: 10.1007/s10884-012-9249-y [Google Scholar] [Crossref]

16. Garcia-Ramos, G.; Villa-Morales, M. (2017). A Fisher-KPP model with degenerate diffusion: Well-posedness and traveling waves. Journal of Differential Equations, 262, 4820-4851. [Google Scholar] [Crossref]

17. Ichida, Y. (2025). Classification of weak and unbounded traveling wave solutions for a Porous-Fisher-KPP equation. Partial Differential Equations and Applications, 6(3). DOI: 10.1007/s42985-025-00335-0 [Google Scholar] [Crossref]

18. Audrito, A.; Vazquez, J. L. (2017). The Fisher-KPP problem with doubly nonlinear diffusion. Journal of Differential Equations, 263, 7600-7650. DOI: 10.1016/j.jde.2017.08.025 [Google Scholar] [Crossref]

19. 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 diffusivity functions in cell biology. Physica D, 398, 1-13. DOI: 10.1016/j.physd.2019.06.005 [Google Scholar] [Crossref]

20. McCue, S. W.; El-Hachem, M.; Simpson, M. J. (2020). Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation. Applied Mathematics Letters, 111, 106918. DOI: 10.1016/j.aml.2020.106918 [Google Scholar] [Crossref]

21. Nadin, G. (2015). How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms? Discrete and Continuous Dynamical Systems - B, 20, 1785-1803. DOI: 10.3934/dcdsb.2015.20.1785 [Google Scholar] [Crossref]

22. Du, Y. (2022). Propagation and reaction-diffusion models with free boundaries. Surveys in Mathematics and its Applications, 17, 1-57. DOI: 10.4171/S1664360722300018 [Google Scholar] [Crossref]

23. Fadai, N. T.; Simpson, M. J. (2020). New travelling wave solutions of the Porous-Fisher model with a moving boundary. Journal of Physics A, 53, 065601. DOI: 10.1088/1751-8121/ab6d3c [Google Scholar] [Crossref]

24. Berestycki, H.; Hamel, F.; Nadirashvili, N. (2005). The speed of propagation for KPP type problems. I: Periodic framework. Journal of the European Mathematical Society, 7, 173-213. DOI: 10.4171/JEMS/26 [Google Scholar] [Crossref]

25. Berestycki, H.; Nirenberg, L. (1992). Travelling fronts in cylinders. Annales de l’Institut Henri Poincare, Analyse Non Lineaire, 9, 497-524. DOI: 10.1016/S0294-1449(16)30229-3 [Google Scholar] [Crossref]

26. Berestycki, H.; Nadin, G.; Perthame, B.; Ryzhik, L. (2009). The non-local Fisher-KPP equation: Travelling waves and steady states. Nonlinearity, 22, 2813-2844. DOI: 10.1088/0951-7715/22/12/002 [Google Scholar] [Crossref]

27. Schaaf, K. W. (1987). Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations. Transactions of the American Mathematical Society, 302, 587-615. DOI: 10.1090/S0002-9947-1987-0891637-2 [Google Scholar] [Crossref]

28. Wang, Z.-C.; Li, W.-T.; Ruan, S. (2007). Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay. Journal of Differential Equations, 233, 210-238. [Google Scholar] [Crossref]

29. Xu, T.; Ji, S.; Mei, M.; Yin, J. (2018). Traveling waves for time-delayed reaction diffusion equations with degenerate diffusion. Journal of Differential Equations, 265, 6079-6115. DOI: 10.1016/j.jde.2018.06.008 [Google Scholar] [Crossref]

30. Xu, T.; Ji, S.; Mei, M.; Yin, J. (2020). Sharp oscillatory traveling waves of structured population dynamics model with degenerate diffusion. Journal of Differential Equations, 269, 10329-10355. DOI: 10.1016/j.jde.2020.06.029 [Google Scholar] [Crossref]

31. Diaz, J. I.; Kamin, S. (2012). Convergence to travelling waves for quasilinear Fisher-KPP type equations. Journal of Mathematical Analysis and Applications, 387, 1019-1033. DOI: 10.1016/j.jmaa.2012.01.018 [Google Scholar] [Crossref]

32. Buenzli, P. R.; Lanaro, M.; Wong, C. S.; McLaughlin, M. P.; Allenby, M. C.; Woodruff, M. A.; Simpson, M. J. (2020). Cell proliferation and migration explain pore bridging dynamics in 3D printed scaffolds of different pore size. PLOS Computational Biology, 16, e1007779. DOI: 10.1371/journal.pcbi.1007779 [Google Scholar] [Crossref]

33. He, Q. (2024). Porous medium type reaction-diffusion equation: Large time behaviors and regularity of free boundary. Journal of Functional Analysis, 287, 110643. DOI: 10.1016/j.jfa.2024.110643 [Google Scholar] [Crossref]

34. Calvo, J.; Campos, J.; Caselles, V.; Sanchez, O.; Soler, J. (2015). Flux-saturated porous media equations and applications. ESAIM: Proceedings and Surveys, 50, 1-19. DOI: 10.4171/EMSS/11 [Google Scholar] [Crossref]

35. Zhan, H. (2017). The nonlinear diffusion equation of the ideal barotropic gas through a porous medium. Open Mathematics, 15, 895-906. DOI: 10.1515/math-2017-0074 [Google Scholar] [Crossref]

36. Ughi, M. (1986). A degenerate parabolic equation modelling the spread of an epidemic. Annali di Matematica Pura ed Applicata, 142, 313-328. DOI: 10.1007/BF01769226 [Google Scholar] [Crossref]

37. Du, Y.; Li, W.-T. (2015). The completely degenerate Fisher-KPP equation with free boundary. Journal of Differential Equations, 259, 5825-5853. [Google Scholar] [Crossref]

38. Li, W.-T.; Wang, Z.-C. (2020). Nonlocal dispersal systems with applications to reaction-diffusion and evolution equations. World Scientific Publishing, 354pp. [Google Scholar] [Crossref]

39. Lewis, M. A.; Schmitz, A. (2023). On the dynamics of invasion: Travelling waves and the impact of spatial structure. Bulletin of Mathematical Biology, 85, 96. DOI: 10.1007/s11538-023-01202-z [Google Scholar] [Crossref]

40. Goriely, A. (2019). Multidisciplinary applications of nonlinear science. Cambridge University Press, 1-17. [Google Scholar] [Crossref]

41. Oelschlager, K. (1990). Large systems of interacting particles and the porous medium equation. Journal of Differential Equations, 88, 294-346. DOI: 10.1016/0022-0396(90)90101-T [Google Scholar] [Crossref]

42. Mansour, M. B. A. (2010). Traveling wave solutions for the extended Fisher/KPP equation. Reports on Mathematical Physics, 66, 375-383. DOI: 10.1016/S0034-4877(10)80009-6 [Google Scholar] [Crossref]

43. Paudel, L. P.; Iaia, J. (2013). Traveling wave solutions of the porous medium equation with degenerate interfaces. Nonlinear Analysis: Theory, Methods & Applications, 81, 110-129. DOI: 10.1016/j.na.2012.10.016 [Google Scholar] [Crossref]

44. Simpson, M. J.; McCue, S. W. (2024). Traveling waves for free boundary models of tissue growth. Annual Review of Fluid Mechanics, 56, 279-307. [Google Scholar] [Crossref]

45. Lui, R. F. (2000). Traveling waves of Fisher-KPP type in heterogeneous environments. Journal of Differential Equations, 166, 173-218. [Google Scholar] [Crossref]

46. Norris, J. R. (1997). The Brownian motion and Brownian local time of a one-dimensional diffusing particle. Cambridge University Press, 417pp. [Google Scholar] [Crossref]

47. Mei, M. (2006). Asymptotic stability of traveling waves for quasilinear parabolic systems. Journal of Differential Equations, 227, 355-388. [Google Scholar] [Crossref]

48. Filipov, A. F. (1988). Differential equations with discontinuous righthand sides. Kluwer Academic Publishers, 304pp. [Google Scholar] [Crossref]

49. Volkov, E. A. (2019). Numerical methods for the porous medium equation with source terms. Journal of Computational Physics, 399, 108923. [Google Scholar] [Crossref]

50. Sun, Y.; Zhang, W. (2024). Numerical analysis for porous medium equation with nonlinear reaction and absorption. Mathematics and Computers in Simulation, 224, 48-64. [Google Scholar] [Crossref]

• Interplay of Students’ Emotional Intelligence and Attitude toward Mathematics on Performance in Grade 10 Algebra
• Numerical Simulation of Fitzhugh-Nagumo Dynamics Using a Finite Difference-Based Method of Lines
• Fixed Point Theorem in Controlled Metric Spaces
• Usage of Moving Average to Heart Rate, Blood Pressure and Blood Sugar
• Exploring Algebraic Topology and Homotopy Theory: Methods, Empirical Data, and Numerical Examples