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Population models with singular equilibrium. (English) Zbl 1116.92049

Summary: A class of models of biological populations and communities with a singular equilibrium at the origin is analyzed; it is shown that these models can possess a dynamical regime of deterministic extinction, which is crucially important from the biological standpoint. This regime corresponds to the presence of a family of homoclinics to the origin, so-called elliptic sectors. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. An algorithmic approach to analyze system behavior with parameter changes is presented. The developed methods and algorithm are applied to existing mathematical models of biological systems. In particular, we analyze a model of anticancer treatment with oncolytic viruses, a parasite-host interaction model, and a model of Chagas’ disease.

MSC:

92D25 Population dynamics (general)
37N25 Dynamical systems in biology
35B99 Qualitative properties of solutions to partial differential equations
92-08 Computational methods for problems pertaining to biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
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[1] Lotka, A. J., Elements of physical biology (1925), Williams & Wilkins: Williams & Wilkins Baltimore · JFM 51.0416.06
[2] Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem R Accad Naz dei Lincei Ser VI, 2 (1926) · JFM 52.0450.06
[3] Kermack, W. O.; McKendrick, A. G., A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London Series A, Containing Papers of a Mathematical and Physical Character, 115, 772, 700 (1927) · JFM 53.0517.01
[4] Holling, C. S., The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Canadian Entomologist, 91, 293 (1959)
[5] Holling, C. S., Some characteristics of simple types of predation and parasitism, Canadian Entomologist, 91, 385 (1959)
[6] Arditi, R.; Ginzburg, L. R., Coupling in Predator Prey Dynamics - Ratio-Dependence, Journal of Theoretical Biology, 139, 3, 311 (1989)
[7] Bazykin, A. D., Nonlinear dynamics of interacting population (1998), World Scientific · Zbl 0605.92015
[8] Hassell, M. P.; Varley, G. C., New inductive population model for insect parasites and its bearing on biological control, Nature, 223, 1133 (1969)
[9] DeAngelis, D. L.; Goldstein, R. A.; O’Neill, R. V., A model for trophic interaction, Ecology, 56, 881 (1975)
[10] Beddington, J. R., Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 51, 331 (1975)
[11] Abrams, P. A.; Ginzburg, L. R., The nature of predation: prey dependent, ratio dependent or neither?, Trends in Ecology & Evolution, 15, 8, 337 (2000)
[12] Akcakaya, H. R.; Arditi, R.; Ginzburg, L. R., Ratio-dependent predation – an abstraction that works, Ecology, 76, 3, 995 (1995)
[13] Berezovskaya, F.; Karev, G.; Arditi, R., Parametric analysis of the ratio-dependent predator-prey model, Journal of Mathematical Biology, 43, 3, 221 (2001) · Zbl 0995.92043
[14] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, Journal of Mathematical Biology, 42, 6, 489 (2001) · Zbl 0984.92035
[15] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bulletin of Mathematical Biology, 61, 1, 19 (1999) · Zbl 1323.92173
[16] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36, 4, 389 (1998) · Zbl 0895.92032
[17] Tang, Y. L.; Zhang, W. N., Heteroclinic bifurcation in a ratio-dependent predator-prey system, Journal of Mathematical Biology, 50, 6, 699 (2005) · Zbl 1067.92050
[18] Xiao, D. M.; Ruan, S. G., Global dynamics of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 43, 3, 268 (2001) · Zbl 1007.34031
[19] Diekmann, O.; Heesterbeek, J. A.P., Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (2000), John Wiley: John Wiley New York · Zbl 0997.92505
[20] McCallum, H.; Barlow, N.; Hone, J., How should pathogen transmission be modelled?, Trends in Ecology & Evolution, 16, 6, 295 (2001)
[21] May, R. M.; Anderson, R. M., Transmission dynamics of HIV-infection, Nature, 326, 6109, 137 (1987)
[22] Hethcote, H. W., The mathematics of infectious diseases, Siam Review, 42, 4, 599 (2000) · Zbl 0993.92033
[23] Busenberg, S.; Cooke, K., Vertically Transmitted Diseases (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0512.92017
[24] Menalorca, J.; Hethcote, H. W., Dynamic-models of infectious-diseases as regulators of population sizes, Journal of Mathematical Biology, 30, 7, 693 (1992) · Zbl 0748.92012
[25] Gao, L. Q.; Hethcote, H. W., Disease transmission models with density-dependent demographics, Journal of Mathematical Biology, 30, 7, 717 (1992) · Zbl 0774.92018
[26] Busenberg, S.; Vargas, C., Modeling Chagas’ desease: variable population size and demographic implications, (Arino, O.; Axelrod, D.; Kimmel, M., Mathematical Population Dynamics (1991), Marcel Dekker: Marcel Dekker New York), 283 · Zbl 0744.92028
[27] Hwang, T. W.; Kuang, Y., Deterministic extinction effect of parasites on host populations, Journal of Mathematical Biology, 46, 1, 17 (2003) · Zbl 1015.92042
[28] Berezovsky, F.; Karev, G.; Song, B.; Castillo-Chavez, C., A simple epidemic model with surprising dynamics, Mathematical Biosciences and Engineering, 2, 1, 133 (2005) · Zbl 1061.92052
[29] Hwang, T. W.; Kuang, Y., Host extinction dynamics in a simple parasite-host interaction model, Mathematical Biosciences and Engineering, 2, 4, 743 (2005) · Zbl 1108.34045
[30] de Jong, M. C.M.; Diekmann, O.; Heesterbeek, H., How does transmission of infection depend on population size?, (Mollison, D., Epidemic Models: Their Structure and Relation to Data (1995), Cambridge University Press), 84 · Zbl 0850.92042
[31] Novozhilov, A. S.; Berezovskaya, F. S.; Koonin, E. V.; Karev, G. P., Mathematical modeling of tumor therapy with oncolytic viruses: Regimes with complete tumor elimination within the framework of deterministic models, Biology Direct, 1, 1, 6 (2006)
[32] de Pillis, L. G.; Radunskaya, A. E.; Wiseman, C. L., A validated mathematical model of cell-mediated immune response to tumor growth, Cancer Research, 65, 7950 (2005)
[33] Andronov, A. A.; Leontovich, E. A.; Gordon, I. I.; Maier, A. G., Qualitative Theory of Second-order Dynamic Systems (1973), John Wiley & Sons, Translation · Zbl 0282.34022
[34] F.S. Berezovskaya, About asymptotics of trajectories of a system of two differential equations, Dep. in the All-Union Information Center (USSR), No 3447-76, (1976) 17 p. (in Russian).; F.S. Berezovskaya, About asymptotics of trajectories of a system of two differential equations, Dep. in the All-Union Information Center (USSR), No 3447-76, (1976) 17 p. (in Russian).
[35] Berezovskaya, F. S., Topological normal form for a system of two differential equations, Russian Mathematical Surveys, 33, 2, 227 (1978)
[36] Berezovskaya, F. S., The main topological part of plane vector fields with fixed Newton diagram, (Le, D. T.; Saito, K.; Teissier, B., Proceedings on Singularity Theory (1995), Word Scientific: Word Scientific Singapore, New Jersey, London, Hong Kong), 55 · Zbl 0944.32036
[37] Berezovskaya, F. S.; Medvedeva, N. B., A complicated singular point of “center-focus” type and the Newton diagram, Selecta Mathematica Formely Sovietica, 13, 1 (1994) · Zbl 0792.58032
[38] A.D. Bruno, Power Geometry in Algebraic and Differential equations, IMPRINT: Amsterdam; New York: Elsevier, 2000.; A.D. Bruno, Power Geometry in Algebraic and Differential equations, IMPRINT: Amsterdam; New York: Elsevier, 2000.
[39] Dumortier, F., Singularities of vector fields in the plane, Journal of Differential equations, 23, 53 (1977) · Zbl 0346.58002
[40] Parato, K. A.; Senger, D.; Forsyth, P. A.; Bell, J. C., Recent progress in the battle between oncolytic viruses and tumours, Nature Reviews. Cancer, 5, 12, 965 (2005)
[41] Wodarz, D., Viruses as antitumor weapons: defining conditions for tumor remission, Cancer Research, 61, 8, 3501 (2001)
[42] Harrison, D.; Sauthoff, H.; Heitner, S.; Jagirdar, J.; Rom, W. N.; Hay, J. G., Wild-type adenovirus decreases tumor xenograft growth, but despite viral persistence complete tumor responses are rarely achieved-deletion of the viral E1b-19-kD gene increases the viral oncolytic effect, Human Gene Therapy, 12, 10, 1323 (2001)
[43] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Rich dynamics of a ratio-dependent one-prey two-predators model, Journal of Mathematical Biology, 43, 5, 377 (2001) · Zbl 1007.34054
[44] Hsu, S. B.; Hwang, T. W.; Kuang, Y., A ratio-dependent food chain model and its applications to biological control, Mathematical Biosciences, 181, 1, 55 (2003) · Zbl 1036.92033
[45] Arnold, V. I.; Ilyashenko, Yu. S., Ordinary Differential Equations, 1. Encyclopedia of Mathematical Science, Vol. 1 (1994), Springer
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