# American Institute of Mathematical Sciences

December  2011, 4(6): 1599-1610. doi: 10.3934/dcdss.2011.4.1599

## Novel dynamics of a simple Daphnia-microparasite model with dose-dependent infection

 1 Department of Mathematics, College of Medicine, Third Military Medical University, Chongqing, 400038 2 School of Mathematics and Statistical Sciences, Arizona State University, Tempe, AZ 85281

Received  March 2009 Revised  September 2009 Published  December 2010

Many experiments reveal that Daphnia and its microparasite populations vary strongly in density and typically go through pronounced cycles. To better understand such dynamics, we formulate a simple two dimensional autonomous ordinary differential equation model for Daphnia magna-microparasite infection with dose-dependent infection. This model has a basic parasite production number $R_0=0$, yet its dynamics is much richer than that of the classical mathematical models for host-parasite interactions. In particular, Hopf bifurcation, stable limit cycle, homoclinic and heteroclinic orbit can be produced with suitable parameter values. The model indicates that intermediate levels of parasite virulence or host growth rate generate more complex infection dynamics.
Citation: Kaifa Wang, Yang Kuang. Novel dynamics of a simple Daphnia-microparasite model with dose-dependent infection. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1599-1610. doi: 10.3934/dcdss.2011.4.1599
##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases I, Nature (London), 280 (1979), 361-367. doi: 10.1038/280361a0. [2] R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Phil. Tran. R. Soc. Lond. B, 291 (1981), 451-524. doi: 10.1098/rstb.1981.0005. [3] D. Ebert, Infectivity, multiple infections, and the genetic correlation between withinhost growth and parasite virulence: A reply to Hochberg, Evolution, 52 (1998), 1869-1871. doi: 10.2307/2411360. [4] D. Ebert, "Ecology, Epidemiology, and Evolution of Parasitism in Daphnia," [Internet]. Bethesda (MD): National Library of Medicine (US), National Center for Biotechnology Information. Available from: http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=Books, 2005. [5] D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: Experimental epidemiology with Daphnia and six microparasites, Am. Nat., 156 (2000), 459-477. doi: 10.1086/303404. [6] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209. doi: 10.1007/PL00008847. [7] J. Guckenheimer and P. Holmes, "Nonlinear Oscillation, Dynamixal Systems and Bifurcations of Vector Fields," Springer-Verlag. 1983. [8] T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol., 46 (2003), 17-30. doi: 10.1007/s00285-002-0165-7. [9] T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Math. Biosci. Eng., 2 (2005), 743-751. [10] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [11] W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956. [12] L. Perko, "Differential Equations and Dynamical Systems," Springer, New York, 1996. [13] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. B, 269 (2002), 271-279. doi: 10.1098/rspb.2001.1816. [14] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [15] P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [16] H. Wang, K. Dunning, J. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete Continuous Dynam. Systems - B, 12 (2009), 481-493. [17] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [18] Y. Yu, J.J. Nieto, A. Torres and K. Wang, A viral infection model with a nonlinear infection rate, Bound. Value Probl., (2009), doi:10.1155/2009/958016.

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##### References:
 [1] R. M. Anderson and R. M. May, Population biology of infectious diseases I, Nature (London), 280 (1979), 361-367. doi: 10.1038/280361a0. [2] R. M. Anderson and R. M. May, The population dynamics of microparasites and their invertebrate hosts, Phil. Tran. R. Soc. Lond. B, 291 (1981), 451-524. doi: 10.1098/rstb.1981.0005. [3] D. Ebert, Infectivity, multiple infections, and the genetic correlation between withinhost growth and parasite virulence: A reply to Hochberg, Evolution, 52 (1998), 1869-1871. doi: 10.2307/2411360. [4] D. Ebert, "Ecology, Epidemiology, and Evolution of Parasitism in Daphnia," [Internet]. Bethesda (MD): National Library of Medicine (US), National Center for Biotechnology Information. Available from: http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=Books, 2005. [5] D. Ebert, M. Lipsitch and K. L. Mangin, The effect of parasites on host population density and extinction: Experimental epidemiology with Daphnia and six microparasites, Am. Nat., 156 (2000), 459-477. doi: 10.1086/303404. [6] D. Ebert, C. D. Zschokke-Rohringer and H. J. Carius, Dose effects and density-dependent regulation of two microparasites of Daphnia magna, Oecologia, 122 (2000), 200-209. doi: 10.1007/PL00008847. [7] J. Guckenheimer and P. Holmes, "Nonlinear Oscillation, Dynamixal Systems and Bifurcations of Vector Fields," Springer-Verlag. 1983. [8] T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations, J. Math. Biol., 46 (2003), 17-30. doi: 10.1007/s00285-002-0165-7. [9] T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Math. Biosci. Eng., 2 (2005), 743-751. [10] A. Korobeinikov, Global properties of basic virus dynamics models, Bull. Math. Biol., 66 (2004), 879-883. doi: 10.1016/j.bulm.2004.02.001. [11] W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956. [12] L. Perko, "Differential Equations and Dynamical Systems," Springer, New York, 1996. [13] R. R. Regoes, D. Ebert and S. Bonhoeffer, Dose-dependent infection rates of parasites produce the Allee effect in epidemiology, Proc. R. Soc. Lond. B, 269 (2002), 271-279. doi: 10.1098/rspb.2001.1816. [14] S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X. [15] P. van den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [16] H. Wang, K. Dunning, J. J. Elser and Y. Kuang, Daphnia species invasion, competitive exclusion, and chaotic coexistence, Discrete Continuous Dynam. Systems - B, 12 (2009), 481-493. [17] D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025. [18] Y. Yu, J.J. Nieto, A. Torres and K. Wang, A viral infection model with a nonlinear infection rate, Bound. Value Probl., (2009), doi:10.1155/2009/958016.
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