Mathematical model of ideal free distribution in the predator--prey system

P. A. Zelenchuk ; V. G. Tsybulin

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 2, pp. 237-249.

Résumé

We consider a system of reaction–diffusion–advection equations which describes the evolution of spatial distributions of antagonistic populations under directed migration. The concept of an ideal free distribution (IFD) for a predator–prey system is introduced. We find conditions on parameters under which there exist explicit stationary solutions with nonzero densities of both species. The numerical approach with staggered grids is used to analyze solutions in case of violation of the conditions on the coefficients that provide the IFD. We construct asymptotic expansions for an inhomogeneous one-dimensional area and present the results of a computational experiment in the case of violation of the IFD conditions.

Détail

Keywords: mathematical ecology, predator–prey system.
Mots-clés : reaction–diffusion–advection equations

Bibliographie

[1] Bazykin A. D., Nelineinaya dinamika vzaimodeistvuyuschikh populyatsii, Inst. komp. issl., Izhevsk, 2003

[2] Budyanskii A. V., Tsibulin V. G., “Modelirovanie mnogofaktornogo taksisa v sisteme «khischnik—zhertva»”, Biofizika, 64:2 (2019), 343–349 | DOI

[3] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vysshaya shkola, M., 1990 | MR

[4] Zelenchuk P. A., Tsibulin V. G., “Idealnoe svobodnoe raspredelenie v modeli «khischnik—zhertva» pri mnogofaktornom taksise”, Biofizika, 66:3 (2021), 546–554 | DOI

[5] Myurrei Dzh. D., Matematicheskaya biologiya, v. 1, Inst. komp. issl., M.—Izhevsk, 2011

[6] Naife A., Vvedenie v metody vozmuschenii, Mir, M., 1984

[7] Riznichenko G. Yu., Matematicheskoe modelirovanie biologicheskikh protsessov. Modeli v biofizike i ekologii, Yurait, M., 2020

[8] Trubetskov D. I., “Fenomen matematicheskoi modeli Lotki—Volterry i skhodnykh s nei”, Izv. vuzov. Prikl. nelin. dinam., 19:2 (2011), 69–88

[9] Tyutyunov Yu. V., Titova L. I., “Ot Lotki—Volterra k Arditi—Ginzburgu: 90 let evolyutsii troficheskikh funktsii”, Zhurn. obsch. biol., 79:6 (2018), 428–448 | DOI

[10] Kha T. D., Tsibulin V. G., “Uravneniya diffuzii—reaktsii—advektsii dlya sistemy khischnik—zhertva v geterogennoi srede”, Komp. issl. i model., 13:6 (2021), 1161–1176

[11] Kha T. D., Tsibulin V. G., “Multistabilnost dlya matematicheskoi modeli dinamiki khischnikov i zhertv na neodnorodnom areale”, Sovrem. mat. Fundam. napravl., 68, no. 3, 2022, 509–521 | MR

[12] Tsibulin V. G., Kha T. D., Zelenchuk P. A., “Nelineinaya dinamika sistemy khischnik—zhertva na neodnorodnom areale i stsenarii lokalnogo vzaimodeistviya vidov”, Izv. vuzov. Prikl. nelin. dinam., 29:5 (2021), 751–764

[13] Averill I., Lou Y., Munther D., “On several conjectures from evolution of dispersal”, J. Biol. Dyn., 6:2 (2012), 117–130 | DOI | MR | Zbl

[14] Budyansky A. V., Frischmuth K., Tsybulin V. G., “Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat”, Discrete Contin. Dyn. Syst., 24 (2019), 547–561 | MR | Zbl

[15] Cantrell R. S., Cosner C., “Evolutionary stability of ideal free dispersal under spatial heterogeneity and time periodicity”, Math. Biosci., 305 (2018), 71–76 | DOI | MR | Zbl

[16] Cantrell R. S., Cosner C., DeAngelis D. L., Padron V., “The ideal free distribution as an evolutionarily stable strategy”, J. Biol. Dyn., 1:3 (2007), 249–271 | DOI | MR | Zbl

[17] Cantrell R. S., Cosner C., Lou Y., “Evolution of dispersal and the ideal free distribution”, Math. Biosci. Engrg., 7:1 (2010), 17–36 | DOI | MR | Zbl

[18] Cantrell R. S., Cosner C., Martinez S., Torres N., “On a competitive system with ideal free dispersal”, J. Differ. Equ., 265 (2018), 3464–3493 | DOI | MR

[19] Cosner C., “Reaction–diffusion–advection models for the effects and evolution of dispersal”, Discrete Contin. Dyn. Syst. Ser. A, 34:5 (2014), 1701–745 | DOI | MR

[20] Cosner C., Cantrell R., Spatial ecology via reaction–diffusion equations, John Wiley Sons, Chichester, 2003 | MR | Zbl

[21] Cressman R., Garay G., Křivan V., “Ideal free distributions, evolutionary games, and population dynamics in multiple-species environments”, American Naturalist, 164:4 (2004), 473–489 | DOI

[22] Fretwell S. D., Lucas H. L., “On territorial behavior and other factors influencing habitat distribution in birds”, Acta Biotheoretica, 19 (1970), 16–36 | DOI

[23] Ha T. D., Tsybulin V. G., Zelenchuk P. A., How to model the local interaction in the predator–prey system at slow diffusion in a heterogeneous environment?, Ecol. Complexity, 52 (2022), 101026 | DOI

[24] Kacelnik A., Krebs J. R., Bernstein C., “The ideal free distribution and predator–prey populations”, Trends Ecol. Evol., 7 (1992), 50–55 | DOI

[25] Kim K., Choi W., “Local dynamics and coexistence of predator–prey model with directional dispersal of predator”, Math. Biosci. Engrg, 17 (2020), 6737–6755 | DOI | MR | Zbl

[26] Schwinning S., Rosenzweig M. L., “Periodic oscillations in an ideal-free predator–prey distribution”, OIKOS, 59 (1990), 85–91 | DOI

[27] Tyutyunov Y. V., Zagrebneva A. D., Azovsky A. I., “Spatiotemporal pattern formation in a prey–predator system: The case study of short-term interactions between diatom microalgae and microcrustaceans”, Mathematics, 8:7 (2020), 1065–1079 | DOI