-
Previous Article
Different types of backward bifurcations due to density-dependent treatments
- MBE Home
- This Issue
-
Next Article
Optimal strategies of social distancing and vaccination against seasonal influenza
Chemostats and epidemics: Competition for nutrients/hosts
1. | School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804 |
2. | School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287 |
References:
[1] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size,, J. Math. Biol., 47 (2003), 153.
doi: 10.1007/s00285-003-0207-9. |
[2] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality,, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 175.
doi: 10.3934/dcdsb.2005.5.175. |
[3] |
P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis,, Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1.
doi: 10.3934/dcdsb.2007.8.1. |
[4] |
R. M. Anderson and R. M. May, Coevolution of host and parasites,, Parasitology, 85 (1982), 411.
doi: 10.1111/j.1095-8312.2009.01256.x. |
[5] |
J. Arino, S. S. Pilyugin and G. S. K. Wolkowicz, Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models,, Can. Appl. Math. Q., 11 (2003), 107.
|
[6] |
R. A. Armstrong and R. McGehee, Competitive exclusion,, Amer. Natur., 115 (1980), 151.
doi: 10.1086/283553. |
[7] |
F. B. Bader, Kinetics of double-substrate limited growth,, in, (1982), 1. Google Scholar |
[8] |
M. M. Ballyk, C. C. McCluskey and G. S. K. Wolkowicz, Global analysis of competition for perfectly substituable resources with linear response,, J. Math. Biol., 51 (2005), 458.
doi: 10.1007/s00285-005-0333-7. |
[9] |
M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources,, Math. Biosci., 118 (1993), 127.
doi: 10.1016/0025-5564(93)90050-K. |
[10] |
E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Analysis, 47 (2001), 4107.
doi: 10.1016/S0362-546X(01)00528-4. |
[11] |
F. F. Blackman, Optima and limiting factors,, Ann. Bot. London, 19 (1905), 281. Google Scholar |
[12] |
C. J. Briggs and H. C. J. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations,, Amer. Nat., 145 (1995), 855.
doi: 10.1086/285774. |
[13] |
H.-J. Bremermann and H. R. Thieme, A competition exclusion principle for pathogen virulence,, J. Math. Biol., 27 (1989), 179.
doi: 10.1007/BF00276102. |
[14] |
G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake,, SIAM J. Appl. Math., 45 (1985), 138.
doi: 10.1137/0145006. |
[15] |
V. Capasso, "Mathematical Structures of Epidemic Systems,", Lecture Notes in Biomathematics, 97 (1993).
doi: 10.1007/978-3-540-70514-7. |
[16] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic model,, Math. Biosci., 42 (1978), 43.
doi: 10.1016/0025-5564(78)90006-8. |
[17] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models,, in, (1995), 33. Google Scholar |
[18] |
J. M. Cushing, Two species competition in a periodic environment,, J. Math. Biol., 10 (1980), 385.
doi: 10.1007/BF00276097. |
[19] |
P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach,, J. Math. Biol., 11 (1981), 319.
doi: 10.1007/BF00276900. |
[20] |
O. Diekmann, The many facets of evolutionary dynamics,, J. Biol. Systems, 5 (1997), 325.
doi: 10.1142/S0218339097000205. |
[21] |
O. Diekmann, A beginners guide to adaptive dynamics,, in, 63 (2004), 47.
|
[22] |
O. Diekmann, J. A. P. Heesterbeek and T. Britton, "Mathematical Tools for Understanding Infectious Disease Dynamics,", Princeton Series in Theoretical and Computational Biology, (2013).
|
[23] |
P. W. Ewald and G. De Leo, Alternative transmission modes and the evolution of virulence,, in, (2002), 10.
doi: 10.1017/CBO9780511525728.004. |
[24] |
A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions,, Math. Model. Nat. Phenom., 2 (2007), 55.
doi: 10.1051/mmnp:2008011. |
[25] |
H. I. Freedman and Y. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling,, J. Math. Biol., 31 (1993), 513.
doi: 10.1007/BF00173890. |
[26] |
P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (): 337.
doi: 10.1137/060654876. |
[27] |
B. S. Goh, Global stability in many species systems,, Amer. Nat., 111 (1977), 135.
doi: 10.1086/283144. |
[28] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Engin., 3 (2006), 513.
doi: 10.3934/mbe.2006.3.513. |
[29] |
H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793.
doi: 10.1090/S0002-9939-08-09341-6. |
[30] |
H. Guo, M. Y. Li and Z. Shuai, Global stability in multigroup epidemic models,, in, 11 (2009), 268.
|
[31] |
W. M. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models for some parasitic infections: Methods for the study of asymptotic behavior,, Comm. Pure Appl. Math., 38 (1985), 733.
doi: 10.1002/cpa.3160380607. |
[32] |
S.-B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760.
doi: 10.1137/0134064. |
[33] |
S.-B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in a continuous culture of micro-organisms,, SIAM J. App. Math., 32 (1977), 366.
doi: 10.1137/0132030. |
[34] |
S.-B. Hsu, A competition model for a seasonally fluctuating nutrient,, J. Math. Biol., 9 (1980), 115.
doi: 10.1007/BF00275917. |
[35] |
S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, Taiwanese J. Math., 9 (2005), 151.
|
[36] |
A. Iggidr, J.-C. Kamgang, G. Sallet and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle,, SIAM J. Appl. Math., 67 (2006), 260.
doi: 10.1137/050643271. |
[37] |
A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays,, Math. Biosci., 209 (2007), 51.
doi: 10.1016/j.mbs.2007.01.008. |
[38] |
V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes,", Yale University Press, (1955). Google Scholar |
[39] |
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75.
doi: 10.1007/s11538-008-9352-z. |
[40] |
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615.
doi: 10.1007/s11538-005-9037-9. |
[41] |
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.
doi: 10.1007/s11538-007-9196-y. |
[42] |
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate,, Math. Med. Biol., 26 (2009), 225.
doi: 10.1093/imammb/dqp009. |
[43] |
A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models,, MMB IMA, 22 (2005), 113. Google Scholar |
[44] |
A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,, Appl. Math. Letters, 15 (2002), 955.
doi: 10.1016/S0893-9659(02)00069-1. |
[45] |
B. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates,, SIAM J. Appl. Math., 59 (1999), 411.
doi: 10.1137/S003613999631100X. |
[46] |
M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.
doi: 10.1137/090779322. |
[47] |
M. Y. Li and Z. Shuai, Global stability problem for coupled systems of differential equations on networks,, J. Differential Eqns., 248 (2010), 1.
doi: 10.1016/j.jde.2009.09.003. |
[48] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays,, J. Math. Anal. Appl., 361 (2010), 38.
doi: 10.1016/j.jmaa.2009.09.017. |
[49] |
X. Lin and J. W.-H. So, Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations,, J. Austral. Math. Soc. Ser. B, 34 (1993), 282.
doi: 10.1017/S0334270000008900. |
[50] |
S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675.
doi: 10.3934/mbe.2010.7.675. |
[51] |
P. Magal, C. C. McCluskey and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109.
doi: 10.1080/00036810903208122. |
[52] |
M. Martcheva, A non-autonomous multi-strain SIS epidemic model,, J. Biol. Dyn., 3 (2009), 235.
doi: 10.1080/17513750802638712. |
[53] |
M. Martcheva, S. S. Pilyugin and R. D. Holt, Subthreshold and superthreshold coexistence of pathogen variants: The impact of host structure,, Math. Biosci., 207 (2007), 58.
doi: 10.1016/j.mbs.2006.09.010. |
[54] |
C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression,, Math. Biosci. Eng., 3 (2006), 603.
doi: 10.3934/mbe.2006.3.603. |
[55] |
C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis,, J. Math. Anal. Appl., 338 (2008), 518.
doi: 10.1016/j.jmaa.2007.05.012. |
[56] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Engin., 6 (2009), 603.
doi: 10.3934/mbe.2009.6.603. |
[57] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. RWA, 11 (2010), 55.
doi: 10.1016/j.nonrwa.2008.10.014. |
[58] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.
doi: 10.1016/j.nonrwa.2009.11.005. |
[59] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. Engin., 7 (2010), 837.
doi: 10.3934/mbe.2010.7.837. |
[60] |
J. Mena-Lorca, J. X. Velasco-Hernandez and C. Castillo-Chavez, Density-dependent dynamics and superinfection in an epidemic model,, IMA J. Math. Appl. Med. Biol., 16 (1999), 307. Google Scholar |
[61] |
J. A. J. Metz, S. D. Mylius and O. Diekmann, When does evolution optimise?, Evolutionary Ecology Research, 10 (2008), 629. Google Scholar |
[62] |
J. Prüss, L. Pujo-Menjouet and G. F. Webb, Analysis of a model for the dynamics of prions,, Discr. Contin. Dyn. Syst. B, 6 (2006), 225.
|
[63] |
J. Roughgarden, "Theory of Population Genetics and Evolutionary Ecology: An Introduction,", Macmillan, (1979). Google Scholar |
[64] |
S. Ruan and X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling,, SIAM J. Appl. Math., 58 (1998), 170.
doi: 10.1137/S0036139996299248. |
[65] |
T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields,, Math. Biosci. Eng., 8 (2011), 827.
doi: 10.3934/mbe.2011.8.827. |
[66] |
H. L. Smith, Competitive coexistence in an oscillating chemostat,, SIAM J. Appl. Math., 40 (1981), 498.
doi: 10.1137/0140042. |
[67] |
H. L. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (1995).
doi: 10.1017/CBO9780511530043. |
[68] |
S. Tennenbaum, T. G. Kassem, S. Roudenko and C. Castillo-Chavez, The role of transactional sex in spreading HIV in Nigeria,, in, 410 (2006), 367.
doi: 10.1090/conm/410/07737. |
[69] |
H. R. Thieme, "Mathematics in Population Biology,", Princeton Series in Theoretical and Computational Biology, (2003).
|
[70] |
H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators,, J. Differential Eqns., 250 (2011), 3772.
doi: 10.1016/j.jde.2011.01.007. |
[71] |
H. R. Thieme, Pathogen competition and coexistence and the evolution of virulence,, in, (2007), 123.
|
[72] |
V. Volterra, "Leçons sur la Théorie Mathématique de la Lutte pour la Vie,", Gauthier-Villars, (1931). Google Scholar |
[73] |
E. B. Wilson and J. Worcester, The law of mass action in epidemiology, Part I,, Proc. Nat. Acad. Sci., 31 (1945), 24.
doi: 10.1073/pnas.31.9.294. |
[74] |
G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat,, Math. Biosci., 93 (1989), 249.
doi: 10.1016/0025-5564(89)90025-4. |
[75] |
G. S. K. Wolkowicz, M. M. Ballyk and S. P. Daoussis, Interaction in a chemostat: Introduction of a competitor can promote greater diversity,, Rocky Mountain Journal of Mathematics, 25 (1995), 515.
doi: 10.1216/rmjm/1181072300. |
[76] |
G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates,, SIAM J. Appl. Math., 52 (1992), 222.
doi: 10.1137/0152012. |
[77] |
G. S. K. Wolkowicz and H. Xia, Global asymptotic behavior of a chemostat model with discrete delay,, SIAM J. Appl. Math., 57 (1997), 1019.
doi: 10.1137/S0036139995287314. |
[78] |
G. S. K. Wolkowicz and X.-Q. Zhao, $N$-species competition in a periodic chemostat,, Differential Integral Equations, 11 (1998), 465.
|
show all references
References:
[1] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion and coexistence for pathogens in an epidemic model with variable population size,, J. Math. Biol., 47 (2003), 153.
doi: 10.1007/s00285-003-0207-9. |
[2] |
A. S. Ackleh and L. J. S. Allen, Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality,, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 175.
doi: 10.3934/dcdsb.2005.5.175. |
[3] |
P. Adda, J. L. Dimi, A. Iggidr, J. C. Kamgang, G. Sallet and J. J. Tewa, General models of host-parasite systems. Global analysis,, Disc. Cont. Dyn. Syst. Ser. B, 8 (2007), 1.
doi: 10.3934/dcdsb.2007.8.1. |
[4] |
R. M. Anderson and R. M. May, Coevolution of host and parasites,, Parasitology, 85 (1982), 411.
doi: 10.1111/j.1095-8312.2009.01256.x. |
[5] |
J. Arino, S. S. Pilyugin and G. S. K. Wolkowicz, Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models,, Can. Appl. Math. Q., 11 (2003), 107.
|
[6] |
R. A. Armstrong and R. McGehee, Competitive exclusion,, Amer. Natur., 115 (1980), 151.
doi: 10.1086/283553. |
[7] |
F. B. Bader, Kinetics of double-substrate limited growth,, in, (1982), 1. Google Scholar |
[8] |
M. M. Ballyk, C. C. McCluskey and G. S. K. Wolkowicz, Global analysis of competition for perfectly substituable resources with linear response,, J. Math. Biol., 51 (2005), 458.
doi: 10.1007/s00285-005-0333-7. |
[9] |
M. M. Ballyk and G. S. K. Wolkowicz, Exploitative competition in the chemostat for two perfectly substitutable resources,, Math. Biosci., 118 (1993), 127.
doi: 10.1016/0025-5564(93)90050-K. |
[10] |
E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay,, Nonlinear Analysis, 47 (2001), 4107.
doi: 10.1016/S0362-546X(01)00528-4. |
[11] |
F. F. Blackman, Optima and limiting factors,, Ann. Bot. London, 19 (1905), 281. Google Scholar |
[12] |
C. J. Briggs and H. C. J. Godfray, The dynamics of insect-pathogen interactions in stage-structured populations,, Amer. Nat., 145 (1995), 855.
doi: 10.1086/285774. |
[13] |
H.-J. Bremermann and H. R. Thieme, A competition exclusion principle for pathogen virulence,, J. Math. Biol., 27 (1989), 179.
doi: 10.1007/BF00276102. |
[14] |
G. J. Butler and G. S. K. Wolkowicz, A mathematical model of the chemostat with a general class of functions describing nutrient uptake,, SIAM J. Appl. Math., 45 (1985), 138.
doi: 10.1137/0145006. |
[15] |
V. Capasso, "Mathematical Structures of Epidemic Systems,", Lecture Notes in Biomathematics, 97 (1993).
doi: 10.1007/978-3-540-70514-7. |
[16] |
V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic model,, Math. Biosci., 42 (1978), 43.
doi: 10.1016/0025-5564(78)90006-8. |
[17] |
C. Castillo-Chavez and H. R. Thieme, Asymptotically autonomous epidemic models,, in, (1995), 33. Google Scholar |
[18] |
J. M. Cushing, Two species competition in a periodic environment,, J. Math. Biol., 10 (1980), 385.
doi: 10.1007/BF00276097. |
[19] |
P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach,, J. Math. Biol., 11 (1981), 319.
doi: 10.1007/BF00276900. |
[20] |
O. Diekmann, The many facets of evolutionary dynamics,, J. Biol. Systems, 5 (1997), 325.
doi: 10.1142/S0218339097000205. |
[21] |
O. Diekmann, A beginners guide to adaptive dynamics,, in, 63 (2004), 47.
|
[22] |
O. Diekmann, J. A. P. Heesterbeek and T. Britton, "Mathematical Tools for Understanding Infectious Disease Dynamics,", Princeton Series in Theoretical and Computational Biology, (2013).
|
[23] |
P. W. Ewald and G. De Leo, Alternative transmission modes and the evolution of virulence,, in, (2002), 10.
doi: 10.1017/CBO9780511525728.004. |
[24] |
A. Fall, A. Iggidr, G. Sallet and J. J. Tewa, Epidemiological models and Lyapunov functions,, Math. Model. Nat. Phenom., 2 (2007), 55.
doi: 10.1051/mmnp:2008011. |
[25] |
H. I. Freedman and Y. Xu, Models of competition in the chemostat with instantaneous and delayed nutrient recycling,, J. Math. Biol., 31 (1993), 513.
doi: 10.1007/BF00173890. |
[26] |
P. Georgescu and Y.-H. Hsieh, Global stability for a virus dynamics model with nonlinear incidence of infection and removal,, SIAM J. Appl. Math., 67 (): 337.
doi: 10.1137/060654876. |
[27] |
B. S. Goh, Global stability in many species systems,, Amer. Nat., 111 (1977), 135.
doi: 10.1086/283144. |
[28] |
H. Guo and M. Y. Li, Global dynamics of a staged progression model for infectious diseases,, Math. Biosci. Engin., 3 (2006), 513.
doi: 10.3934/mbe.2006.3.513. |
[29] |
H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions,, Proc. Amer. Math. Soc., 136 (2008), 2793.
doi: 10.1090/S0002-9939-08-09341-6. |
[30] |
H. Guo, M. Y. Li and Z. Shuai, Global stability in multigroup epidemic models,, in, 11 (2009), 268.
|
[31] |
W. M. Hirsch, H. Hanisch and J.-P. Gabriel, Differential equation models for some parasitic infections: Methods for the study of asymptotic behavior,, Comm. Pure Appl. Math., 38 (1985), 733.
doi: 10.1002/cpa.3160380607. |
[32] |
S.-B. Hsu, Limiting behavior for competing species,, SIAM J. Appl. Math., 34 (1978), 760.
doi: 10.1137/0134064. |
[33] |
S.-B. Hsu, S. P. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in a continuous culture of micro-organisms,, SIAM J. App. Math., 32 (1977), 366.
doi: 10.1137/0132030. |
[34] |
S.-B. Hsu, A competition model for a seasonally fluctuating nutrient,, J. Math. Biol., 9 (1980), 115.
doi: 10.1007/BF00275917. |
[35] |
S.-B. Hsu, A survey of constructing Lyapunov functions for mathematical models in population biology,, Taiwanese J. Math., 9 (2005), 151.
|
[36] |
A. Iggidr, J.-C. Kamgang, G. Sallet and J.-J. Tewa, Global analysis of new malaria intrahost models with a competitive exclusion principle,, SIAM J. Appl. Math., 67 (2006), 260.
doi: 10.1137/050643271. |
[37] |
A. Iggidr, J. Mbang and G. Sallet, Stability analysis of within-host parasite models with delays,, Math. Biosci., 209 (2007), 51.
doi: 10.1016/j.mbs.2007.01.008. |
[38] |
V. S. Ivlev, "Experimental Ecology of the Feeding of Fishes,", Yale University Press, (1955). Google Scholar |
[39] |
A. Korobeinikov, Lyapunov functions and global properties for SEIR and SEIS epidemic models,, Math. Med. Biol., 21 (2004), 75.
doi: 10.1007/s11538-008-9352-z. |
[40] |
A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission,, Bull. Math. Biol., 68 (2006), 615.
doi: 10.1007/s11538-005-9037-9. |
[41] |
A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence,, Bull. Math. Biol., 69 (2007), 1871.
doi: 10.1007/s11538-007-9196-y. |
[42] |
A. Korobeinikov, Global asymptotic properties of virus dynamics models with dose-dependent parasite reproduction and virulence and non-linear incidence rate,, Math. Med. Biol., 26 (2009), 225.
doi: 10.1093/imammb/dqp009. |
[43] |
A. Korobeinikov and P. K. Maini, Nonlinear incidence and stability of infectious disease models,, MMB IMA, 22 (2005), 113. Google Scholar |
[44] |
A. Korobeinikov and G. C. Wake, Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,, Appl. Math. Letters, 15 (2002), 955.
doi: 10.1016/S0893-9659(02)00069-1. |
[45] |
B. Li, Global asymptotic behavior of the chemostat: General response functions and different removal rates,, SIAM J. Appl. Math., 59 (1999), 411.
doi: 10.1137/S003613999631100X. |
[46] |
M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434.
doi: 10.1137/090779322. |
[47] |
M. Y. Li and Z. Shuai, Global stability problem for coupled systems of differential equations on networks,, J. Differential Eqns., 248 (2010), 1.
doi: 10.1016/j.jde.2009.09.003. |
[48] |
M. Y. Li, Z. Shuai and C. Wang, Global stability of multi-group epidemic models with distributed delays,, J. Math. Anal. Appl., 361 (2010), 38.
doi: 10.1016/j.jmaa.2009.09.017. |
[49] |
X. Lin and J. W.-H. So, Global stability of the endemic equilibrium and uniform persistence in epidemic models with subpopulations,, J. Austral. Math. Soc. Ser. B, 34 (1993), 282.
doi: 10.1017/S0334270000008900. |
[50] |
S. Liu and L. Wang, Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy,, Math. Biosci. Eng., 7 (2010), 675.
doi: 10.3934/mbe.2010.7.675. |
[51] |
P. Magal, C. C. McCluskey and G. F. Webb, Liapunov functional and global asymptotic stability for an infection-age model,, Applicable Analysis, 89 (2010), 1109.
doi: 10.1080/00036810903208122. |
[52] |
M. Martcheva, A non-autonomous multi-strain SIS epidemic model,, J. Biol. Dyn., 3 (2009), 235.
doi: 10.1080/17513750802638712. |
[53] |
M. Martcheva, S. S. Pilyugin and R. D. Holt, Subthreshold and superthreshold coexistence of pathogen variants: The impact of host structure,, Math. Biosci., 207 (2007), 58.
doi: 10.1016/j.mbs.2006.09.010. |
[54] |
C. C. McCluskey, Lyapunov functions for tuberculosis models with fast and slow progression,, Math. Biosci. Eng., 3 (2006), 603.
doi: 10.3934/mbe.2006.3.603. |
[55] |
C. C. McCluskey, Global stability for a class of mass action systems allowing for latency in tuberculosis,, J. Math. Anal. Appl., 338 (2008), 518.
doi: 10.1016/j.jmaa.2007.05.012. |
[56] |
C. C. McCluskey, Global stability for an SEIR epidemiological model with varying infectivity and infinite delay,, Math. Biosci. Engin., 6 (2009), 603.
doi: 10.3934/mbe.2009.6.603. |
[57] |
C. C. McCluskey, Complete global stability for an SIR epidemic model with delay-distributed or discrete,, Nonlinear Anal. RWA, 11 (2010), 55.
doi: 10.1016/j.nonrwa.2008.10.014. |
[58] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and nonlinear incidence,, Nonlinear Anal. RWA, 11 (2010), 3106.
doi: 10.1016/j.nonrwa.2009.11.005. |
[59] |
C. C. McCluskey, Global stability for an SIR epidemic model with delay and general nonlinear incidence,, Math. Biosci. Engin., 7 (2010), 837.
doi: 10.3934/mbe.2010.7.837. |
[60] |
J. Mena-Lorca, J. X. Velasco-Hernandez and C. Castillo-Chavez, Density-dependent dynamics and superinfection in an epidemic model,, IMA J. Math. Appl. Med. Biol., 16 (1999), 307. Google Scholar |
[61] |
J. A. J. Metz, S. D. Mylius and O. Diekmann, When does evolution optimise?, Evolutionary Ecology Research, 10 (2008), 629. Google Scholar |
[62] |
J. Prüss, L. Pujo-Menjouet and G. F. Webb, Analysis of a model for the dynamics of prions,, Discr. Contin. Dyn. Syst. B, 6 (2006), 225.
|
[63] |
J. Roughgarden, "Theory of Population Genetics and Evolutionary Ecology: An Introduction,", Macmillan, (1979). Google Scholar |
[64] |
S. Ruan and X.-Z. He, Global stability in chemostat-type competition models with nutrient recycling,, SIAM J. Appl. Math., 58 (1998), 170.
doi: 10.1137/S0036139996299248. |
[65] |
T. Sari and F. Mazenc, Global dynamics of the chemostat with different removal rates and variable yields,, Math. Biosci. Eng., 8 (2011), 827.
doi: 10.3934/mbe.2011.8.827. |
[66] |
H. L. Smith, Competitive coexistence in an oscillating chemostat,, SIAM J. Appl. Math., 40 (1981), 498.
doi: 10.1137/0140042. |
[67] |
H. L. Smith and P. Waltman, "The Theory of the Chemostat: Dynamics of Microbial Competition,", Cambridge Studies in Mathematical Biology, 13 (1995).
doi: 10.1017/CBO9780511530043. |
[68] |
S. Tennenbaum, T. G. Kassem, S. Roudenko and C. Castillo-Chavez, The role of transactional sex in spreading HIV in Nigeria,, in, 410 (2006), 367.
doi: 10.1090/conm/410/07737. |
[69] |
H. R. Thieme, "Mathematics in Population Biology,", Princeton Series in Theoretical and Computational Biology, (2003).
|
[70] |
H. R. Thieme, Global stability of the endemic equilibrium in infinite dimension: Lyapunov functions and positive operators,, J. Differential Eqns., 250 (2011), 3772.
doi: 10.1016/j.jde.2011.01.007. |
[71] |
H. R. Thieme, Pathogen competition and coexistence and the evolution of virulence,, in, (2007), 123.
|
[72] |
V. Volterra, "Leçons sur la Théorie Mathématique de la Lutte pour la Vie,", Gauthier-Villars, (1931). Google Scholar |
[73] |
E. B. Wilson and J. Worcester, The law of mass action in epidemiology, Part I,, Proc. Nat. Acad. Sci., 31 (1945), 24.
doi: 10.1073/pnas.31.9.294. |
[74] |
G. S. K. Wolkowicz, Successful invasion of a food web in a chemostat,, Math. Biosci., 93 (1989), 249.
doi: 10.1016/0025-5564(89)90025-4. |
[75] |
G. S. K. Wolkowicz, M. M. Ballyk and S. P. Daoussis, Interaction in a chemostat: Introduction of a competitor can promote greater diversity,, Rocky Mountain Journal of Mathematics, 25 (1995), 515.
doi: 10.1216/rmjm/1181072300. |
[76] |
G. S. K. Wolkowicz and Z. Lu, Global dynamics of a mathematical model of competition in the chemostat: General response functions and differential death rates,, SIAM J. Appl. Math., 52 (1992), 222.
doi: 10.1137/0152012. |
[77] |
G. S. K. Wolkowicz and H. Xia, Global asymptotic behavior of a chemostat model with discrete delay,, SIAM J. Appl. Math., 57 (1997), 1019.
doi: 10.1137/S0036139995287314. |
[78] |
G. S. K. Wolkowicz and X.-Q. Zhao, $N$-species competition in a periodic chemostat,, Differential Integral Equations, 11 (1998), 465.
|
[1] |
Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228 |
[2] |
Robert Stephen Cantrell, King-Yeung Lam. Competitive exclusion in phytoplankton communities in a eutrophic water column. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020361 |
[3] |
Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020 doi: 10.3934/jcd.2021006 |
[4] |
Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020378 |
[5] |
Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3657-3682. doi: 10.3934/dcds.2020051 |
[6] |
Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161 |
[7] |
Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 |
[8] |
Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084 |
[9] |
Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003 |
[10] |
Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077 |
[11] |
Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020288 |
[12] |
Yunfeng Geng, Xiaoying Wang, Frithjof Lutscher. Coexistence of competing consumers on a single resource in a hybrid model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 269-297. doi: 10.3934/dcdsb.2020140 |
[13] |
Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118 |
[14] |
Yancong Xu, Lijun Wei, Xiaoyu Jiang, Zirui Zhu. Complex dynamics of a SIRS epidemic model with the influence of hospital bed number. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021016 |
[15] |
Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304 |
[16] |
Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020104 |
[17] |
Matthieu Alfaro, Isabeau Birindelli. Evolution equations involving nonlinear truncated Laplacian operators. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3057-3073. doi: 10.3934/dcds.2020046 |
[18] |
Bimal Mandal, Aditi Kar Gangopadhyay. A note on generalization of bent boolean functions. Advances in Mathematics of Communications, 2021, 15 (2) : 329-346. doi: 10.3934/amc.2020069 |
[19] |
Andreas Koutsogiannis. Multiple ergodic averages for tempered functions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1177-1205. doi: 10.3934/dcds.2020314 |
[20] |
Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]