P.A. Kuznetsov. Analytic diffusion waves in a nonlinear parabolic "predator–prey" model ... P. 158-167

We consider a system of two nonlinear degenerate parabolic equations that are nonlinear generalizations of the Fisher–Kolmogorov–Petrovskii–Piskunov equation. This system is the basis for the predator–prey mathematical model. Its interesting peculiarity is that it has solutions of the diffusion (heat, filtration) wave type propagating over a zero background with a finite velocity. This peculiarity is a consequence of nonlinear degeneracy. We consider the problem of constructing a diffusion wave of the system that has a known law of front motion. A theorem of existence and uniqueness of a piecewise analytic solution is proved. The proof is constructive: we find a solution in the form of power series and give recursive formulas for the coefficients. The local convergence is proved by the majorant method. The obtained results follow the tradition of Academician A. F. Sidorov’s scientific school to use the power series method to solve degenerate parabolic problems. Note that similar studies were previously conducted for single equations, as well as for reaction–diffusion systems that were significantly simpler in structure than the one mentioned above. The increased complexity makes it impossible to automatically transfer the earlier results to the case under consideration and affects both the construction of the solution and the proof of convergence. The convergence is local, but the obtained exact solutions of traveling wave type can illustrate the behavior of the solution outside the convergence domain. In order to construct the solution, we reduce the original problem to the Cauchy problem for a system of ordinary differential equations. This system is integrated in quadratures, and its solutions are written explicitly. The obtained formulas may be used to verify numerical calculations.

Keywords: nonlinear degenerate parabolic system, predator–prey model, diffusion wave, existence theorem, power series, majorant method, exact solutions

Received March 9, 2022

Revised March 24, 2022

Accepted March 27, 2022

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-07-00407 A) and jointly by the Russian Foundation for Basic Research and the Government of the Irkutsk oblast (project no. 20-41-385002).

Pavel Alexandrovich Kuznetsov, Cand. Sci. (Phys.-Math.), Matrosov Institute for System Dynamics and Control Theory SB RAS, Irkutsk, 664033 Russia, e-mail: kuznetsov@icc.ru

REFERENCES

1.   Perthame B. Parabolic equations in biology: Growth, reaction, movement and diffusion. Cham: Springer, 2015, 199 p. ISBN: 9783319195001 .

2.   Fisher R.A. The wave of advance of advantageous genes. Ann. Eugenics, 1937, vol. 7, no. 4, pp. 353–369. doi: 10.1111/j.1469-1809.1937.tb02153.x 

3.   Kolmogorov A.N., Petrovskii I.G., Piskunov N.S. Studies of the diffusion with the increasing quantity of the substance; its application to a biological problem. Bull. Moscow Univ. Math. Mech., 1937, vol. 1, no. 6, pp. 1–26.

4.   Achouri T., Ayadi M., Habbal A., Yahyaoui B. Numerical analysis for the two-dimensional Fisher–Kolmogorov–Petrovski–Piskunov equation with mixed boundary condition. J. Appl. Math. Comp., 2021, no. 1, pp. 1–26. doi: 10.1007/s12190-021-01679-7 

5.   Aleshin S.V., Glyzin S.D., Kaschenko S.A. Fisher–Kolmogorov–Petrovskii–Piscounov equation with delay. Modeling and Analysis of Information Systems, 2015, vol. 22, no. 2, pp. 304–321 (in Russian). doi: 10.18255/1818-1015-2015-2-304-321 

6.   Viguerie A., Veneziani A., Lorenzo G., Baroli D., Aretz-Nellesen N., Patton A., Yankeelov T.E., Reali A., Hughes T.J.R., Auricchio F. Diffusion–reaction compartmental models formulated in a continuum mechanics framework: application to COVID-19, mathematical analysis, and numerical study. Comput. Mech., 2020, vol. 66, no. 5, pp. 1131–1152. doi: 10.1007/s00466-020-01888-0 

7.   Murray J.D. Mathematical biology II: Spatial models and biomedical applications. Ser. Interdisciplinary Appl. Math., vol. 18. NY: Springer, 2003, 837 p. doi: 10.1007/b98869 

8.   Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Blow-up in quasilinear parabolic equations. Berlin: Walter de Gruyter, 1995. 554 p. doi: 10.1515/9783110889864 . Original Russian text published in Samarskii A.A., Galaktionov V.A., Kurdyumov S.P., Mikhailov A.P. Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Moscow: Nauka Publ., 1987, 480 p.

9.   Zel’dovich Ya.B., Raizer Yu.P. Physics of shock waves and high-temperature hydrodynamic phenomena. NY: Dover Publ., 2002, 944 p. ISBN: 0486420027 . Original Russian text published in Zel’dovich Ya.B. Raizer Yu.P. Fizika udarnykh voln vysokotemperaturnykh gidrodinamicheskikh yavlenii, Moscow: Fizmatlit Publ., 1966, 632 p.

10.   Barenblatt G., Entov V., Ryzhik V. Theory of fluid flows through natural rocks. Dordrecht: Springer Science+Business Media, 1990, 396 p. ISBN: 978-90-481-4042-8 . Original Russian text published in Barenblatt G.I., Entov V.N., Ryzhik V.M. Dvizhenie zhidkostei i gazov v prirodnykh plastakh, Moscow: Nedra Publ., 1984, 211 p.

11.   Sidorov A.F. Izbrannye trudy: Matematika. Mekhanika [Selected Works: Mathematics and Mechanics]. Moscow: Fizmatlit Publ., 2001, 576 p. ISBN: 5-9221-0103-X .

12.   Bautin S.P. Analiticheskaya teplovaya volna [The analytical heat wave]. Moscow: Fizmatlit Publ., 2003, 88 p. ISBN: 5-9221-0443-8 .

13.   Kazakov A.L., Lempert A.A. Analytical and numerical studies of the boundary value problem of a nonlinear filtration with degeneration. Vychisl. Tekhnol., 2012, vol. 17, no. 1, pp. 57–68 (in Russian).

14.   Kazakov A.L. On exact solutions to a heat wave propagation boundary-value problem for a nonlinear heat equation. Sib. Elektron. Mat. Izv., 2019, vol. 16, pp. 1057–1068 (in Russian). doi: 10.33048/semi.2019.16.073 

15.   Filimonov M.Yu., Korzunin L.G., Sidorov A.F. Approximate methods for solving nonlinear initial boundary-value problems based on special construction of series. Russ. J. Numer. Anal. Math. Modelling, 1993, vol. 8, no. 2, pp. 101–125. doi: 10.1515/rnam.1993.8.2.101 

16.   Kazakov A.L., Kuznetsov P.A. On the analytic solutions of a special boundary value problem for a nonlinear heat equation in polar coordinates. J. Appl. Indust. Math., 2018, vol. 12, no. 2, pp. 255–263. doi: 10.1134/S1990478918020060 

17.   Kazakov A.L., Kuznetsov P.A., Lempert A.A. Analytical solutions to the singular problem for a system of nonlinear parabolic equations of the reaction-diffusion type. Symmetry, 2020, vol. 12, no. 6, art. no. 999. doi: 10.3390/sym12060999 

18.   Kazakov A.L., Kuznetsov P.A., Spevak L.F. Construction of solutions to the boundary value problem with singularity for a nonlinear parabolic system. Sib. Zh. Ind. Mat., 2021, vol. 24, no. 4, pp. 64–78 (in Russian). doi: 10.33048/SIBJIM.2021.24.405 

19.   Vasin V.V., Akimova E.N., Miniakhmetova A.F. Iterative Newton type algorithms and its applications to inverse gravimetry problem. Vestnik YuUrGU. Ser. Mat. Model. Progr., 2013, vol. 6, no. 3, pp. 26–37 (in Russian).

20.   Korotkii A.I., Starodubtseva Yu.V. Modelirovanie pryamykh i obratnykh granichnykh zadach dlya statsionarnykh modelei teplomassoperenosa [Modeling of direct and inverse boundary value problems for stationary heat and mass transfer models]. Ekaterinburg: Ural. Univ. Publ., 2015, 168 p.

21.   Kovrizhnykh O.O. On construction of an asymptotic solution to the degenerate nonlinear parabolic equation. Comput. Math. Math. Phys., 2003, vol. 43, no. 10, pp. 1430–1436.

22.   Filimonov M.Yu. Representation of solutions of boundary value problems for nonlinear evolution equations by special series with recurrently caculated coefficients. J. Physics: Conference Series, 2019, vol. 1268, art. no. 012071, 6 p. doi: 10.1088/1742-6596/1268/1/012071 

23. Kazakov A.L.,  Kuznetsov P.A.,  Spevak L.F. On a degenerate boundary value problem for the porous medium equation in spherical coordinates. {\it Trudy Inst. Mat. i Mekh. UrO RAN}, 2014, vol. 20, no. 1, pp. 119-129.

24.   Kazakov A.L., Orlov Sv.S., Orlov S.S. Construction and study of exact solutions to a nonlinear heat equation. Sib. Math. J., 2018, vol. 59, no. 3, pp. 427–441. doi: 10.1134/S0037446618030060 

25.   Arnold V.I. Ordinary differential equations. Berlin; Heidelberg: Springer, 1992, 338 p. ISBN: 978-3-540-34563-3 . Original Russian text published in Arnol’d V.I. Obyknovennye differentsial’nye uravneniya, Moscow: MTsNMO Publ., 2012, 344 p.

26.   Vasin V.V., Sidorov A.F. Some methods of approximate solution of differential and integral equations. Russian Math. (Iz. VUZ), 1983, vol. 27, no. 7, pp. 14–33.

27.   Kedrin V.S., Arguchintsev A.V., Dobrinets I.M. Mechanisms of polymorphic systematization of bioecological data within the BaikalIntelli platform for organizing computational models of population dynamics. Journal of Physics: Conference Series, 2021, vol. 1847, art. no. 012029. doi: 10.1088/1742-6596/1847/1/012029 

Cite this article as: P.A. Kuznetsov. Analytic diffusion waves in a nonlinear parabolic “predator–prey” model. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2022, vol. 28, no. 2, pp. 158–167.