Relaxation Oscillations in a System with Delays Modeling the Predator--Prey Problem

S. A. Kashchenko

Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 1, pp. 52-98.

Résumé

A new asymptotic method for investigating complex relaxation oscillations of a system with delay was offered. Applying it, we can reduce the problem of predator-prey system dynamics to problem of one-dimensional maps analysis. Some conclusions of biological nature based on the asymptotic analysis were made.

Détail

Keywords: delay differential equation, large parameter, asymptotic, periodic solution.

Bibliographie

[1] Yang Kuang, Delay Differential Equations. With Applications in Population Dynamics, Academic Press, 1993 | MR

[2] E. M. Wright, “A non-linear differential equation”, J. Reine Angew. Math., 194:1–4 (1955), 66–87 | MR | Zbl

[3] S. Kakutani, L. Markus, “On the non-linear difference-differential equation $y(t)=(a-by(t-\tau))y(t)$ contributions to the theory of non-linear oscillations”, Ann. Math. Stud., IV, Princeton University Press, Princeton, 1958, 1–18 | MR

[4] S. A. Kaschenko, “K voprosu ob otsenke v prostranstve parametrov oblasti globalnoi ustoichivosti uravneniya Khatchinsona”, Nelineinye kolebaniya v zadachakh ekologii, YarGU, Yaroslavl, 1985, 55–62

[5] S. A. Kaschenko, “Asimptotika periodicheskogo resheniya obobschennogo uravneniya Khatchinsona”, Issledovaniya po ustoichivosti i teorii kolebanii, YarGU, Yaroslavl, 1981, 64–85

[6] G. S. Jones, “The existence of periodic solutions of $f(x)=-\alpha f(x-1)[1+f(x)]$”, J. Math. Anal. and Appl., 5 (1962), 435–450 | DOI | MR | Zbl

[7] S. A. Kaschenko, “O periodicheskikh resheniyakh uravneniya $x(t)=-lx(t-1)[1+x(t)]$”, Issledovaniya po ustoychivosti i teorii kolebaniy, YarGU, Yaroslavl, 1978, 110–117 (in Russian)

[8] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Extremal dynamics of the generalized Hutchinson equation”, Computational Mathematics and Mathematical Physics, 49:1 (2009), 71–83 | DOI | MR | Zbl

[9] Yu. S. Kolesov, “Matematicheskiye modeli ekologii”, Issledovaniya po ustoychivosti i teorii kolebaniy, Yaroslavl, 1979, 3–40 (in Russian) | MR | Zbl

[10] Yu. S. Kolesov, D. I. Shvitra, Avtokolebaniya v sistemakh s zapazdyvaniyem, Mokslas, Vil'nyus, 1979, 146 pp. (in Russian) | MR | Zbl

[11] Yu. S. Kolesov, E. P. Kubyshkin, “Dvukhchastotnyy podkhod k zadache “khishchnik–zhertva””, Issledovaniya po ustoychivosti i teorii kolebaniy, Yaroslavl, 1979, 111–121 (in Russian) | MR | Zbl

[12] S. D. Glyzin, “O stabiliziruyushchey roli neodnorodnogo soprotivleniya vneshney sredy v zadache “khishchnik–zhertva””, Issledovaniya po ustoychivosti i teorii kolebaniy, Yaroslavl, 1982, 126–-129 (in Russian)

[13] A. A. Zakharov, Yu. S. Kolesov, A. N. Spokoynov, N. B. Fedotov, “Teoreticheskoye ob"yasneniye desyatiletnego tsikla kolebaniy chislennosti mlekopitayushchikh v Kanade i Yakutii”, Issledovaniya po ustoychivosti i teorii kolebaniy, Yaroslavl, 1980, 79–131 (in Russian) | MR

[14] Yu. S. Kolesov, E. P. Kubyshkin, “Chislennoye issledovaniye odnoy sistemy differentsial'no-raznostnykh uravneniy, modeliruyushchey zadachu “khishchnik–zhertva””, Faktory raznoobraziya v matematicheskoy ekologii i populyatsionnoy genetike, Pushchino, 1980, 54–62 (in Russian) | MR | Zbl

[15] A. A. Zakharov, “Chislennyye issledovaniya sistemy uravneniy Kolesova, modeliruyushchikh zadachu “khishchnik–zhertva” s uchetom davleniya khishchnika na zhertvu i yego migratsii za granitsu areala obitaniya”, Differentsial'nyye uravneniya i ikh primeneniye, 29 (1981), 9–26 (in Russian) | Zbl

[16] S. D. Glyzin, “A registration of age groups for the Hutchinson's equation”, Modeling and Analysis of Information Systems, 14:3 (2007), 29–42 (in Russian) | MR

[17] S. A. Kaschenko, “Issledovanie metodami bolshogo parametra sistemy nelineinykh differentsialno-raznostnykh uravnenii, modeliruyuschei zadachu «khischnik–zhertva»”, Dokl. AN SSSR, 266:4 (1982), 792–795 | MR

[18] S. A. Kaschenko, “Biologicheskoe ob'yasnenie nekotorykh zakonov funktsionirovaniya prosteishikh ekosistem v ekstremalnykh sluchayakh”, Issledovaniya po ustoichivosti i teorii kolebanii, Yaroslavl, 1982, 85–103

[19] S. A. Kaschenko, “Periodicheskie resheniya sistemy nelineinykh uravnenii s zapazdyvaniyami, modeliruyuschikh zadachu «khischnik–zhertva»”, Issledovaniya po ustoichivosti i teorii kolebanii, Yaroslavl, 1981, 136–143

[20] S. A. Kaschenko, Statsionarnyye rezhimy v zadache “khishchnik–zhertva”, Preprint No 84.54, In-t matematiki AN USSR, Kiyev, 1984, 58 pp. (in Russian)

[21] S. A. Kaschenko, “Asymptotic of solutions of generalized Hutchinson's equation”, Modeling and Analysis of Information Systems, 19:3 (2012), 32–62 (in Russian)

[22] A. N. Sharkovskii, Yu. L. Maistrenko, E. Yu. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, 1993, 372 pp. | MR | MR