×

Branching random walks with several sources. (English) Zbl 1409.92216

Summary: A continuous-time branching random walk on multidimensional lattices with a finite number of branching sources of three types leads to explicit conditions for the exponential growth of the total number of particles. These conditions are expressed in terms of the spectral characteristics of the operator describing the mean number of particles both at an arbitrary point and on the entire lattice.

MSC:

92D25 Population dynamics (general)
92C17 Cell movement (chemotaxis, etc.)
60G50 Sums of independent random variables; random walks
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akhmerov , R.R. , Kamenskiĭ , M.I. , and Potapov , A.S. ( 1992 ). Measures of Noncompactness and Condensing Operators, 55 of Operator Theory: Advances and Applications . Basel : Birkhäuser Verlag .
[2] Albeverio , S. and Bogachev , L. ( 2000 ). Branching random walk in a catalytic medium. I. Basic equations . Positivity , 4 ( 1 ): 41 - 100 . · Zbl 0953.60079
[3] Albeverio , S. , Bogachev , L. , and Yarovaya , E. ( 1998 ). Asymptotics of branching symmetric random walk on the lattice with a single source . Comptes-rendus de l’Académie des Sciences, Paris, séries I , 326 : 975 - 980 . · Zbl 0917.60080
[4] Bogachev , L.V. and Yarovaya , E.B. ( 1998a ). A limit theorem for a supercritical branching random walk on Z^d with a single source . Uspehi Matematicheskih Nauk , 53 : 229 - 230 . · Zbl 0940.60042
[5] Bogachev , L.V. and Yarovaya , E.B. ( 1998b ). Moment analysis of a branching random walk on a lattice with a single source . Doklady Akademii Nauk , 363 : 439 - 442 . · Zbl 0963.60083
[6] Fedotov , S. and Iomin , A. ( 2008 ). Probabilistic approach to a proliferation and migration dichotomy in tumor cell invasion . Physical Review, E 77 : 031911 1 - 10 .
[7] Gohberg , I.C. and Kreĭn , M.G. ( 1957 ). Fundamental aspects of defect numbers, root numbers and indexes of linear operators . Uspehi Matematicheskih Nauk , 12 : 43 - 118 . · Zbl 0088.32101
[8] Golitsyna , A.G. and Molchanov , S.A. ( 1987 ). A multidimensional model of rare random scatterers . Doklady Akademii Nauk SSSR , 294 : 1302 - 1306 . · Zbl 0662.60068
[9] Kato , T. ( 1966 ). Perturbation Theory for Linear Operators . New York : Springer-Verlag . · Zbl 0148.12601
[10] Reed , M. and Simon , B. ( 1978 ). Methods of Modern Mathematical Physics. IV. Analysis of Operators . New York : Academic Press . · Zbl 0401.47001
[11] Robertson , A.P. and Robertson , W.J. ( 1964 ). Topological Vector spaces. Cambridge Tracts in Mathematics and Mathematical Physics, 53 . New York : Cambridge University Press . · Zbl 0123.30202
[12] Schur , I. ( 1911 ). Bemerkungen zur Theorie der beschränkten Bilinearformen mit unebdlich vielen Veränderlichen . The Journal für die reine und angewandte Mathematik , 140 : 1 - 28 . · JFM 42.0367.01
[13] Shubin , M.A. ( 1985 ). Pseudodifference operators and their Green function . Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya , 49 : 652 - 671 . · Zbl 0574.39006
[14] Vatutin , V. and Topchii , V. ( 2004 ). Limit theorem for critical catalytic branching random walks . Teoriya Veroyatnostei i ee Primeneniya , 49 : 463 - 484 .
[15] Vatutin , V.A. , Topchiĭ , V.A. , and Yarovaya , E.B. ( 2003 ). Catalytic branching random walks and queueing systems with a random number of independent servers . Teoriya Ĭmovīrnosti ta Matematichnoĭ Statistiki , 69 : 1 - 15 . · Zbl 1097.60068
[16] Yarovaya , E.B. ( 2007 ). Branching Random Walks in Nonhomogeneous Medium . Moscow : MSU .
[17] Yarovaya , E.B. ( 2010 ). Critera for exponential growth of the numbers of particles in models of branching random walks . Teoriya Veroyatnostei i ee Primeneniya , 55 ( 4 ): 705 - 731 .
[18] *Communicated by Peter Jagers and Christine Jacob.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.