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Existence of nontrivial steady states for populations structured with respect to space and a continuous trait
Qualitative analysis and travelling wave solutions for the SI model with vertical transmission
1. | UMR CNRS 5251, I.M.B. and INRIA Bordeaux Sud-ouest Anubis, case 36, UFR Sciences et Modélisation, Université Victor Segalen Bordeaux 2, 3 ter, place de la Victoire - 33076 Bordeaux cedex |
2. | Institut de Mathématiques de Bordeaux, UMR CNRS 5251, INRIA Bordeaux sud-ouest, EPI Anubis, UFR Sciences de la Vie, Université Victor Segalen Bordeaux 2, 3 ter Place de la Victoire, 33076 Bordeaux, France |
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford Univ. Press, (1991). Google Scholar |
[2] |
A. Arapostathis, M. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems,, Comm. Part. Diff. Eq., 24 (1999), 1555.
|
[3] |
F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer, (2000).
|
[4] |
S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases,", Biomathematics volume 23, (1993).
|
[5] |
V. Capasso, "Mathematical Structures of Epidemic Systems,", Lecture Notes in Biomathematics volume 97, (1993).
|
[6] |
Z. Q. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic system,, J. Diff. Eq., 139 (1997), 261.
|
[7] |
O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,", Wiley, (2000).
|
[8] |
R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. Google Scholar |
[9] |
W. E. Fitzgibbon and M. Langlais, A diffusive S.I.S. model describing the propagation of F.I.V.,, Communications of Applied Analysis, 7 (2003), 387.
|
[10] |
W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains,, p. 115-164, (1936), 115.
|
[11] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of Feline Leukemia Virus (FeLV) through a highly heterogeneous spatial domain,, SIAM J. Math. Analysis, 33 (2001), 570.
|
[12] |
B. S. Goh, Global stability in a class of predator-prey models,, Bull. Math. Biol., 40 (1978), 525.
|
[13] |
J. Hale, "Asymptotic Behavior of Dissipation Systems,", American Mathematical Society, (1988).
|
[14] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Universit\'e d'Etat Moscou, (1937), 1. Google Scholar |
[15] |
J. J. Morgan, Boundedness and decay results for reaction diffusion systems,, SIAM J. Math. Anal., 20 (1990), 1128.
|
[16] |
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", Springer-Verlag, (2003).
|
[17] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in, (2007), 99.
|
[18] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology,'', Chapman & Hall/CRC, (2009), 293. Google Scholar |
[19] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd, edition, (1994).
|
[20] |
H. R. Thieme, "Mathematics in Population Biology,", Princeton Univ. Press, (2003).
|
[21] |
A. Volpert, Vit. Volpert and Vl. Volpert, "Travelling Wave Solutions of Parabolic Systems,", Monographs, (1994).
|
show all references
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford Univ. Press, (1991). Google Scholar |
[2] |
A. Arapostathis, M. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems,, Comm. Part. Diff. Eq., 24 (1999), 1555.
|
[3] |
F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology,", Springer, (2000).
|
[4] |
S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases,", Biomathematics volume 23, (1993).
|
[5] |
V. Capasso, "Mathematical Structures of Epidemic Systems,", Lecture Notes in Biomathematics volume 97, (1993).
|
[6] |
Z. Q. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic system,, J. Diff. Eq., 139 (1997), 261.
|
[7] |
O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation,", Wiley, (2000).
|
[8] |
R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. Google Scholar |
[9] |
W. E. Fitzgibbon and M. Langlais, A diffusive S.I.S. model describing the propagation of F.I.V.,, Communications of Applied Analysis, 7 (2003), 387.
|
[10] |
W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains,, p. 115-164, (1936), 115.
|
[11] |
W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of Feline Leukemia Virus (FeLV) through a highly heterogeneous spatial domain,, SIAM J. Math. Analysis, 33 (2001), 570.
|
[12] |
B. S. Goh, Global stability in a class of predator-prey models,, Bull. Math. Biol., 40 (1978), 525.
|
[13] |
J. Hale, "Asymptotic Behavior of Dissipation Systems,", American Mathematical Society, (1988).
|
[14] |
A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique,, Bulletin Universit\'e d'Etat Moscou, (1937), 1. Google Scholar |
[15] |
J. J. Morgan, Boundedness and decay results for reaction diffusion systems,, SIAM J. Math. Anal., 20 (1990), 1128.
|
[16] |
J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", Springer-Verlag, (2003).
|
[17] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in, (2007), 99.
|
[18] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology,'', Chapman & Hall/CRC, (2009), 293. Google Scholar |
[19] |
J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd, edition, (1994).
|
[20] |
H. R. Thieme, "Mathematics in Population Biology,", Princeton Univ. Press, (2003).
|
[21] |
A. Volpert, Vit. Volpert and Vl. Volpert, "Travelling Wave Solutions of Parabolic Systems,", Monographs, (1994).
|
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