ANDRONOV-HOPF S BIFURCATION IN A DYNAMIC MODEL OF CELL POPULATION

Transcription

1 Mathematical Moelig ANDRONOV-HOPF S BIFURCATION IN A DYNAMIC MODEL OF CELL POPULATION JuG Nekhozhia VA Sobolev Samara Natioal Reearch Uiverity Samara Ruia Abtract The mathematical moel of the ifferetial cacae with a ymmetrical iviio of tem cell i coiere i the aer The uig of the Routh- Hurwitz criterio a the Aroov-Hof theorem [] allow to fi the coitio of bifurcatio cycle i the moel uer coieratio a the heomeo of "oft lo of tability" wa tate Keywor: bifurcatio of the cycle tability yamic moel cell oulatio Citatio: Nekhozhia JuG Sobolev VA Aroov-Hof bifurcatio i a yamic moel of cell oulatio CEUR Workho Proceeig 6; 638: DOI: 887/ Itrouctio Symmetric iviio of tem cell i a key way to icreae the umber of cell i evelomet a regeeratio of tiue after ijury However thi ca lea to ucotrolle growth of maligat tumor The moelig of the emergece a growth of maligat tumor i certaily oe of the mai irectio of mathematical moelig i biology I orer to eamie the role of ymmetric cell iviio a mathematical moel of cell oulatio wa coiere The aer eamie variou imeio of the ytem that ecribe thi moel The aim of thi work i to fi the miimum value of the umber of equatio i the ytem at which tability i lot Aalytically the coucte aalyi ha allowe to etablih the lo of tability whe it i a low-imeioal roblem a to cofirm thi fact by umerical eerimet Moel ecritio Dyamic moel of cell oulatio ca be rereete by the followig ytem of oriary ifferetial equatio []: Iformatio Techology a Naotechology ITNT-6 636

2 Mathematical Moelig Nekhozhia JuG Sobolev VA Iformatio Techology a Naotechology ITNT i i i i i i i i I thi ytem: rereet the umber of tem cell i with i = - the umber of cell i the i comartmet at the metatable tate a i the umber of mature cell + rereet the amout of a cytokie which regulate the ize of the tem cell comartmet through a egative-feeback rereet growth rate of tem cell i with i = i the umber of ivie cell er ay λ i with i = rereet the ee of eath of the relevat cell i a oitive arameter The roblem coit i fiig the miimum value of at which the ytem` tability i lot Whe = the teay tate with oitive cooriate i aymtotically table So let tart with = The tuy of the tability of the ytem whe = I thi cae the ytem ha three equatio Thi ytem look a follow: Ivetigate the tability of thi ytem Firt we fi the teay tate of the ytem from equatio: There are two equilibrium oitio oe of which i the trivial equilibrium that i utable Therefore let coier a o-zero equilibrium oitio It look like thi: ; ; To eamie the tability of the teay tate coier the matri of the liearizatio of at the teay tate It ha the form:

3 Mathematical Moelig Nekhozhia JuG Sobolev VA Iformatio Techology a Naotechology ITNT Comoe the characteritic equatio of thi matri For thi let calculate the etermiat a equate it to zero: Or 3 4 Where The Hurwitz matri for the characteritic equatio i Accorig to the criterio of Routh-Hurwitz the characteritic olyomial ha root with egative real art a hece teay tate of ytem i table if = α > = αβ γ > 3 = γ >

4 Mathematical Moelig Nekhozhia JuG Sobolev VA However the characteritic equatio ha urely imagiary root whe Iee if λ = ± iω are the root of equatio 4 the ubtitutig i 4 we get: 3 : i i 3 : i i Aig thee two equatio we get Subtract the eco equatio from the firt The a therefore O the other ha So Coier the cae whe the eco orer mior of the Hurwitz matri i equal to zero a fi uch value of the arameter uer which it i mae ; Hece at thi value of the arameter the characteritic equatio 4 ha urely imagiary root It i eay to verify that it will atify to all coitio of the Aroov- Hof bifurcatio theorem Thi mea that for > there are table olutio at = the equilibrium oitio i the ceter a if < there are a utable olutio a a table limit cycle Eamle Let The correoig teay tate i 4 64 ; ; A the critical value 587 Iformatio Techology a Naotechology ITNT-6 639

5 Mathematical Moelig Nekhozhia JuG Sobolev VA ; i A o-trivial equilibrium ha the form: ; ; 8 The trajectory i reete i the followig figure: Fig The trajectory for = 587 Now coier the cae whe 5 The teay tate ha the form: ; 88359; 8 The trajectory i reete i the followig figure: Fig The trajectory for = 5 I thi cae the teay tate i utable Now coier the cae whe The teay tate ha the form: 34343; ; 8 The trajectory i reete i the followig figure: Iformatio Techology a Naotechology ITNT-6 64

6 Mathematical Moelig Nekhozhia JuG Sobolev VA Fig 3 The trajectory for = I thi cae the teay tate i aymtotically table Cocluio It i how i the aer that for = the Aroov-Hof bifurcatio take lace i the yamical moel uer coieratio The heomeo of "oft lo of tability" wa tate i thi moel ee alo [3-5] Ackowlegemet Thi work i uorte i art by the Ruia Fouatio for Baic Reearch grat a the Miitry of Eucatio a Sciece of the Ruia Feeratio uer the Cometitivee Ehacemet Program of Samara Uiverity 3 Referece Mare JE McCracke M The Hof bifurcatio a it alicatio NY: Sriger- Verlag 976 Sachez-Taltavull D Alarco T Robute of ifferetiatio cacae with ymmetric tem cell iviio J R Soc Iterface 4; : 464 DOI: 98/rif464 3 Shcheakia E Caar a black wa i moel of a 3-D autocatalator Joural of Phyic: Coferece Serie 5; : Shcheakia E Korotkova O Coitio for caar eloio i a emicouctor otical amlifier Joural of the Otical Society of America B: Otical Phyic ; 88: Shcheakia E Korotkova O Caar eloio i chemical a otical ytem Dicrete a Cotiuou Dyamical Sytem - Serie B 3; 8: Iformatio Techology a Naotechology ITNT-6 64