American Institute of Mathematical Sciences

Advanced
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

Kaifa Wang 1, and  Yang Kuang 2,

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.
Keywords: Hopf bifurcation, Daphnia magna-microparasite model, limit cycle, heteroclinic orbit, dose-dependent infection, homoclinic orbit.
Mathematics Subject Classification: Primary: 92D25; Secondary: 34C6.
Citation: Kaifa Wang, Yang Kuang. Novel dynamics of a simple Daphnia-microparasite model with dose-dependent infection. Discrete & 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. doi: 10.1038/280361a0. Google Scholar

[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. doi: 10.1098/rstb.1981.0005. Google Scholar

[3]

D. Ebert, Infectivity, multiple infections, and the genetic correlation between withinhost growth and parasite virulence: A reply to Hochberg,, Evolution, 52 (1998), 1869. doi: 10.2307/2411360. Google Scholar

[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: \url{http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=Books}, (2005). Google Scholar

[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. doi: 10.1086/303404. Google Scholar

[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. doi: 10.1007/PL00008847. Google Scholar

[7]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillation, Dynamixal Systems and Bifurcations of Vector Fields,", Springer-Verlag. 1983., (1983). Google Scholar

[8]

T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations,, J. Math. Biol., 46 (2003), 17. doi: 10.1007/s00285-002-0165-7. Google Scholar

[9]

T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model,, Math. Biosci. Eng., 2 (2005), 743. Google Scholar

[10]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879. doi: 10.1016/j.bulm.2004.02.001. Google Scholar

[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. doi: 10.1007/BF00276956. Google Scholar

[12]

L. Perko, "Differential Equations and Dynamical Systems,", Springer, (1996). Google Scholar

[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. doi: 10.1098/rspb.2001.1816. Google Scholar

[14]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate,, J. Differential Equations, 188 (2003), 135. doi: 10.1016/S0022-0396(02)00089-X. Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[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. Google Scholar

[17]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate,, Math. Biosci., 208 (2007), 419. doi: 10.1016/j.mbs.2006.09.025. Google Scholar

[18]

Y. Yu, J.J. Nieto, A. Torres and K. Wang, A viral infection model with a nonlinear infection rate,, Bound. Value Probl., (2009). Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, Population biology of infectious diseases I,, Nature (London), 280 (1979), 361. doi: 10.1038/280361a0. Google Scholar

[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. doi: 10.1098/rstb.1981.0005. Google Scholar

[3]

D. Ebert, Infectivity, multiple infections, and the genetic correlation between withinhost growth and parasite virulence: A reply to Hochberg,, Evolution, 52 (1998), 1869. doi: 10.2307/2411360. Google Scholar

[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: \url{http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?db=Books}, (2005). Google Scholar

[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. doi: 10.1086/303404. Google Scholar

[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. doi: 10.1007/PL00008847. Google Scholar

[7]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillation, Dynamixal Systems and Bifurcations of Vector Fields,", Springer-Verlag. 1983., (1983). Google Scholar

[8]

T. W. Hwang and Y. Kuang, Deterministic extinction effect of parasites on host populations,, J. Math. Biol., 46 (2003), 17. doi: 10.1007/s00285-002-0165-7. Google Scholar

[9]

T. W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model,, Math. Biosci. Eng., 2 (2005), 743. Google Scholar

[10]

A. Korobeinikov, Global properties of basic virus dynamics models,, Bull. Math. Biol., 66 (2004), 879. doi: 10.1016/j.bulm.2004.02.001. Google Scholar

[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. doi: 10.1007/BF00276956. Google Scholar

[12]

L. Perko, "Differential Equations and Dynamical Systems,", Springer, (1996). Google Scholar

[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. doi: 10.1098/rspb.2001.1816. Google Scholar

[14]

S. Ruan and W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate,, J. Differential Equations, 188 (2003), 135. doi: 10.1016/S0022-0396(02)00089-X. Google Scholar

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[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. Google Scholar

[17]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate,, Math. Biosci., 208 (2007), 419. doi: 10.1016/j.mbs.2006.09.025. Google Scholar

[18]

Y. Yu, J.J. Nieto, A. Torres and K. Wang, A viral infection model with a nonlinear infection rate,, Bound. Value Probl., (2009). Google Scholar

[1]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293
[2]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[3]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[4]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[5]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883
[6]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485
[7]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351
[8]

Matthias Rumberger. Lyapunov exponents on the orbit space. Discrete & Continuous Dynamical Systems - A, 2001, 7 (1) : 91-113. doi: 10.3934/dcds.2001.7.91
[9]

Stefano Galatolo. Orbit complexity and data compression. Discrete & Continuous Dynamical Systems - A, 2001, 7 (3) : 477-486. doi: 10.3934/dcds.2001.7.477
[10]

Shiqiu Liu, Frédérique Oggier. On applications of orbit codes to storage. Advances in Mathematics of Communications, 2016, 10 (1) : 113-130. doi: 10.3934/amc.2016.10.113
[11]

Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162
[12]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[13]

Rui Dilão, András Volford. Excitability in a model with a saddle-node homoclinic bifurcation. Discrete & Continuous Dynamical Systems - B, 2004, 4 (2) : 419-434. doi: 10.3934/dcdsb.2004.4.419
[14]

Heide Gluesing-Luerssen, Katherine Morrison, Carolyn Troha. Cyclic orbit codes and stabilizer subfields. Advances in Mathematics of Communications, 2015, 9 (2) : 177-197. doi: 10.3934/amc.2015.9.177
[15]

Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221
[16]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[17]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846
[18]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[19]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[20]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

2018 Impact Factor: 0.545

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences