Stationary states in population dynamics with migration and distributed offspring

A. A. Davydov ; Kh. A. Khachatryan

Contemporary Mathematics. Fundamental Directions, Contemporary Mathematics. Fundamental Directions, Tome 69 (2023) no. 4, pp. 578-587.

Résumé

For an integral equation whose solutions provide stationary states of a population distributed in an arithmetic space, we find the conditions for the existence of its solution and conditions under which this equation has no more than one solution.

Détail

Keywords: population dynamics, stationary state, distributed offspring, integral equation.
Mots-clés : migration

Bibliographie

[1] Arabadzhyan L. G., “Ob odnom integralnom uravnenii teorii perenosa v neodnorodnoi srede”, Diff. uravn., 23:9 (1987), 1618–1622 | MR | Zbl

[2] Belyakov A. O., Davydov A. A., “Optimizatsiya effektivnosti tsiklicheskogo ispolzovaniya vozobnovlyaemogo resursa”, Tr. IMM UrO RAN, 22, no. 2, 2016, 38–46

[3] Davydov A. A., Danchenko V. I., Zvyagin M. Yu., “Suschestvovanie i edinstvennost statsionarnogo raspredeleniya biologicheskogo soobschestva”, Tr. MIAN, 267, 2009, 46–55 | Zbl

[4] Davydov A. A., Danchenko V. I., Nikitin A. A., “Ob integralnom uravnenii dlya statsionarnykh raspredelenii biologicheskikh soobschestv”, Problemy dinamicheskogo upravleniya, MAKS Press, M., 2010, 15–29

[5] Danchenko V. I., Rubai R. V., “Ob odnom integralnom uravnenii statsionarnogo raspredeleniya biologicheskikh sistem”, Sovrem. mat. Fundam. napravl., 36, 2010, 50–60 | Zbl

[6] Nikolaev M. V., Dikman U., Nikitin A. A., “Primenenie spetsialnykh funktsionalnykh prostranstv k issledovaniyu nelineinykh integralnykh uravnenii, voznikayuschikh v ravnovesnoi prostranstvennoi logisticheskoi dinamike”, Dokl. RAN, 499 (2021), 35–39 | DOI | Zbl

[7] Nikolaev M. V., Nikitin A. A., “O suschestvovanii i edinstvennosti resheniya odnogo nelineinogo integralnogo uravneniya”, Dokl. RAN, 488 (2019), 595–598 | DOI | Zbl

[8] Rudin U., Funktsionalnyi analiz, Mir, M., 1975 | MR

[9] Sergeev A. G., Khachatryan Kh. A., “O razreshimosti odnogo klassa nelineinykh integralnykh uravnenii v zadache rasprostraneniya epidemii”, Tr. Mosk. mat. ob-va, 80, no. 1, 2019, 113–131 | Zbl

[10] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1981 | MR

[11] Belyakov A. O., Davydov A. A., “Efficiency optimization for the cyclic use of a renewable resource”, Proc. Steklov Inst. Math., 299, suppl. 1 (2017), 14–21 | DOI | MR

[12] Belyakov A. O., Davydov A. A., Veliov V. M., “Optimal cyclic exploitation of renewable resources”, J. Dyn. Control Syst., 21:3 (2015), 475–494 | DOI | MR | Zbl

[13] Davydov A. A., “Existence of optimal stationary states of exploited populations with diffusion”, Proc. Steklov Inst. Math., 310 (2020), 124–130 | DOI | MR | Zbl

[14] Davydov A. A., Danchenko V. I., Zvyagin M. Yu., “Existence and uniqueness of a stationary distribution of a biological community”, Proc. Steklov Inst. Math., 267 (2009), 40–49 | DOI | MR | Zbl

[15] Davydov A. A., Platov A. S., “Optimal stationary solution in forest management model by accounting intra-species competition”, Mosc. Math. J., 12:2 (2012), 269–273 | DOI | MR | Zbl

[16] Dieckmann U., Law R., “Relaxation projections and the method of moments”, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge University Press, Cambridge, 2000, 412–455 | DOI

[17] Diekmann O., “Threshold and travelling waves for the geographical spread of infection”, J. Math. Biol., 6 (1978), 109–130 | DOI | MR | Zbl

[18] Diekmann O., Gyllenberg M., Metz J. A. J., “Steady-state analysis of structured population models”, Theor. Popul. Biol., 63 (2003), 309–338 | DOI | Zbl

[19] Fisher R. A., “The wave of advance of advantageous genes”, Ann. Eugenics, 7:4 (1937), 353–369

[20] Khachatryan Kh. A., Petrosyan H. S., “On solvability of a class of multidimensional integral equations in the mathematical theory of geographic distribution of an epidemic”, J. Contemp. Math. Anal., 56:5 (2021), 143–157 | DOI | MR | Zbl

[21] Kolmogorov A. N., Petrovskii I. G., Piskunov N. S., “A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem”, Bull. Moscow Univ. Math. Mech., 1 (1937), 1–25 | MR | Zbl

[22] Law R., Dieckmann U., “Moment approximations of individual-based models”, The Geometry of Ecological Interactions: Simplifying Spatial Complexity, Cambridge University Press, Cambridge, 2000, 252–270 | DOI

[23] Nikolaev M. V., Dieckmann U., Nikitin A. A., “Application of special function spaces to the study of nonlinear integral equations arising in equilibrium spatial logistic dynamics”, Dokl. Math., 104:1 (2021), 188–192 | DOI | MR | Zbl

[24] Malthus T., An essay on the principle of population, St. Paul's Church-Yard, London, 1798

[25] McKendrick A. G., “Applications of mathematics to medical problems”, Proc. Edinb. Math. Soc., 44:1 (1926), 98–130 | MR

[26] Verhulst P. F., “Notice sur la loi que la population poursuit dans son accroissement”, Corr. Math. Phys., 10 (1838), 113–121

[27] Von Foerster H., “Some remarks on changing populations”, The Kinetics of Cellular Proliferation, Grune and Stratton, New York, 1959, 382–407

[28] Yengibarian N. B., “Renewal equation on the whole line”, Stoch. Process Appl., 85:2 (2000), 237–247 | DOI | MR | Zbl