Analysis of a population model with advection and an autocatalytic-type growth. (English) Zbl 1505.92166
Summary: This paper analyzes a population model with time-dependent advection and an autocatalytic-type growth. As opposed to a logistic growth where the rate of growth of the population decreases from the onset, an autocatalytic growth has a point of inflection where the rate of growth switches from an increasing trend to a decreasing trend. Employing the idea of Painlevé property, we show that a variety of exact traveling wave solutions can be obtained for this model depending on the choice of the advection term. In particular, due to situations in resource distribution or environmental variations, if the advection is represented as a decaying function in time or an oscillating function in time, we are able to find exact solutions with interesting behavior. We also carry out a computational study of the model using an exponentially upwinded numerical scheme and illustrate the movement of the solutions and their characteristics pictorially.
MSC:
| 92D25 | Population dynamics (general) |
| 35C07 | Traveling wave solutions |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
Painlevé property; traveling wave solutions; standing wave; exponentially upwinded numerical schemeReferences:
| [1] | Ablowitz, M. J. and Clarkson, P. A., Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform (Cambridge University Press, Cambridge, 1991). · Zbl 0762.35001 |
| [2] | Alzaleq, L., A Klein-Gordon Equation Revisited: New Solutions and A Computational Method (Washington State University, 2016). |
| [3] | L. Alzaleq and V. Manoranjan, Analysis of the Fisher-KPP equation with a time-dependent Allee effect, IOP SciNotes (2020), https://doi.org/10.1088/2633-1357/ab99cc. |
| [4] | Beltrami, E., Mathematics for Dynamic Modeling (Academic Press, San Diego, CA, 1987). · Zbl 0625.58012 |
| [5] | Bhrawy, A. H. and Alghamdi, M. A., Approximate solutions of Fisher’s type equations with variable coefficients, Abstract Appl. Anal.2013 (2013), Article ID: 176730, 10 pp. · Zbl 1297.65126 |
| [6] | Boman, B. M., Guetter, A., Boman, R. M. and Runquist, O. A., Autocatalytic tissue polymerization reactionmechanism in colorectal cancer development and growth, Cancers12(2) (2020) 460, https://doi.org/10.3390/cancers12020460. |
| [7] | X. Y. Chen, Numerical methods for the Burgers-Fisher equation, University of Aeronautics and Astronautics, China (2007). |
| [8] | Feng, Q. H. and Zheng, B., Traveling wave solutions for the fifth-order Sawada-Kotera equation and the general Gardner equation by \(\left(\right.\frac{ G^\prime}{ G}\left)\right.\)-expansion method, WSEAS Trans. Math.9(3) (2010) 171-180. |
| [9] | Fisher, R. A., The wave of advance of advantageous genes, Ann. Hum. Genet.7(4) (1937) 355-369. · JFM 63.1111.04 |
| [10] | Gazdag, J. and Canosa, J., Numerical solution of Fisher’s equation, J. Appl. Probab.11(3) (1974) 445-457. · Zbl 0288.65055 |
| [11] | Ginzburg, L. R., The theory of population dynamics: I. Back to first principles, J. Theor. Biol.122(4) (1986) 385-399. |
| [12] | Hammond, J. F. and Bortz, D. M., Analytical solutions to Fisher’s equation with time-variable coefficients, Appl. Math. Comput.218(6) (2011) 2497-2508. · Zbl 1239.35165 |
| [13] | Harko, T. and Mak, M. K., Exact travelling wave solutions of non-linear reaction-convection-diffusion equations-An Abel equation based approach, J. Math. Phys.56 (2015) 111501. · Zbl 1328.35096 |
| [14] | Harko, T. and Mak, M. K., Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach, Math. Biosci. Eng.12 (2015) 41-69. · Zbl 1316.35070 |
| [15] | He, J. H. and Wu, X. H., Exp-function method for nonlinear wave equations,Chaos Solitons Fractals30(3) (2006) 700-708. · Zbl 1141.35448 |
| [16] | Hirota, R., Exact solution of the Korteweg-De Vries equation for multiple collisions of solitons,Phys. Rev. Lett.27(18) (1971) 1192-1194. · Zbl 1168.35423 |
| [17] | Kavanagh, A. J. and Richards, O. W., The autocatalytic growth-curve, Am. Nat.68(714) (1934) 54-59. |
| [18] | Li, X. and Wang, M., A sub-ODE method for finding exact solutions of a generalized KdV-mKdV equation with high-order nonlinear terms,Phys. Lett. A361(1-2) (2007) 115-118. · Zbl 1170.35085 |
| [19] | Mancas, S. C. and Rosu, H. C., Integrable Abel equations and Vein’s Abel equation,Math. Methods Appl. Sci.39 (2016) 1376-1387. · Zbl 1338.34009 |
| [20] | Manoranjan, V. S. and Gomez, M. O., Alternating direction implicit method with exponential upwinding,Comput. Math. Appl.30(11) (1995) 47-58. · Zbl 0835.65098 |
| [21] | Marciniak-Czochra, A., Karch, G. and Suzuki, K., Instability of Turing patterns in reaction-diffusion-ODE systems,J. Math. Biol.74(3) (2017) 583-618. · Zbl 1356.35043 |
| [22] | Mickens, R. E., A best finite-difference scheme for the fisher equation,Numer. Methods Part. Differ. Equ.10(5) (1994) 581-585. · Zbl 0810.65131 |
| [23] | Mickens, R. E. and Gumel, A. B., Construction and analysis of a non-standard finite difference scheme for the Burgers-Fisher equation, J. Sound Vib.257(4) (2002) 791-797. · Zbl 1237.65095 |
| [24] | Murray, J. D., Mathematical Biology \(:\) I. An Introduction (Springer Verlag, New York, 2002). · Zbl 1006.92001 |
| [25] | Öğün, A. and Kart, C., Exact solutions of Fisher and generalized Fisher equations with variable coefficients, Acta Math. Appl. Sinica Engl. Ser.23(4) (2007) 563-568. · Zbl 1137.35423 |
| [26] | Roessler, J. and Hüssner, H., Numerical solution of the 1+2 dimensional Fisher’s equation by finite elements and the Galerkin method, Math. Comput. Model.25(3) (1997) 57-67. · Zbl 0885.65105 |
| [27] | Şahin, A., Dağ, İ. and Saka, B., A B-spline algorithm for the numerical solution of Fisher’s equation,Kybernetes37(2) (2008) 326-342. · Zbl 1177.41008 |
| [28] | Schuster, P., What is special about autocatalysis? Monatshefte für Chemie-Chemical Monthly, 150(5) (2019) 763-775, https://doi.org/10.1007/s00706-019-02437-z. |
| [29] | Sirendaoreji, S. and Sun, J., Auxiliary equation method for solving nonlinear partial differential equations, Phys. Lett. A309 (5-6) (2003) 387-396. · Zbl 1011.35035 |
| [30] | Turchin, P., Does population ecology have general laws?Oikos94(1) (2001) 17-26. |
| [31] | Wang, M. L., Exact solution for a compound KdV-Burgers equations, Phys. Lett. A213(5-6) (1996) 279-287. · Zbl 0972.35526 |
| [32] | Weiss, J., The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative, J. Math. Phys.24(6) (1983) 1405-1413. · Zbl 0531.35069 |
| [33] | Weiss, J., Tabor, M. and Carnevale, G., The Painlevé property for partial differential equations, J. Math. Phys.24(3) (1983) 522-526. · Zbl 0514.35083 |
| [34] | Yang, X.-F., Deng, Z.-C. and Wei, Y., A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Differ. Equ.117 (2015) 117-134. · Zbl 1422.35153 |
| [35] | Zhao, H. and Bazant, M. Z., Population dynamics of driven autocatalytic reactive mixtures,Phys. Rev. E100(1) (2019) 012144. |
| [36] | Zhao, T., Li, C., Zang, Z. and Wu, Y., Chebyshev-Legendre pseudo-spectral method for the generalised Burgers-Fisher equation, Appl. Math. Model.36(3) (2012) 1046-1056. · Zbl 1243.65126 |
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