Skip to main content
SpringerLink
Log in

Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

When searching for hosts, parasitoids are observed to aggregate in response to chemical signalling cues emitted by plants during host feeding. In this paper we model aggregative parasitoid behaviour in a multi-species host-parasitoid community using a system of reaction-diffusion-chemotaxis equations. The stability properties of the steady-states of the model system are studied using linear stability analysis which highlights the possibility of interesting dynamical behaviour when the chemotactic response is above a certain threshold. We observe quasi-chaotic dynamic heterogeneous spatio-temporal patterns, quasi-stationary heterogeneous patterns and a destabilisation of the steady-states of the system. The generation of heterogeneous spatio-temporal patterns and destabilisation of the steady state are due to parasitoid chemotactic response to hosts. The dynamical behaviour of our system has both mathematical and ecological implications and the concepts of chemotaxis-driven instability and coexistence and ecological change are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alt W., Lauffenburger D.A. (1985). Transient behaviour of a chemotaxis system modelling certain types of tissue inflammation. J. Math. Biol. 24: 691–722

    Google Scholar 

  2. Anderson A.R.A., Chaplain M.A.J. (1998). Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60: 857–899

    Article  MATH  Google Scholar 

  3. Barlow N.D., Beggs J.R., Moller H. (1998). Spread of the wasp parasitoid Sphecophaga vesparum vesparum following its release in New Zealand. N.Z. J. Ecol. 22(2): 205–208

    Google Scholar 

  4. Cameron P.J., Walker G.P. (2002). Field evaluation of Cotesia rubecula. an introduced parasitoid of Pieris rapae in New Zealand. Biol. Control 31(2): 367–374

    Google Scholar 

  5. Chaplain M.A.J., Stuart A.M. (1991). A mathematical model for the diffusion of tumor angiogenesis factor into the surrounding host tissue. IMA J. Math. Appl. Med. Biol. 8: 191–220

    Article  MATH  Google Scholar 

  6. Chaplain M.A.J., Stuart A.M. (1993). A model mechanism for the chemotactic response of endothelial cells to tumor angiogenesis factor. IMA J. Math. Appl. Med. Biol. 10: 149–168

    Article  MATH  Google Scholar 

  7. Chaplain M.A.J., Lolas G. (2005). Mathematical modelling of cancer cell invasion of tissue: the role of the urokinase plasminogen activation system. Math. Model. Methods. Appl. Sci., 15: 1685–1734

    Article  MATH  Google Scholar 

  8. Geervliet J.B.F., Ariens S., Dicke M., Vet L.E.M. (1998). Long-distance assessment of patch profitability through volatile infochemicals by the parasitoids Cotesia glomerata and Cotesia rubecula. Biol. Control 11: 113–121

    Article  Google Scholar 

  9. Geervliet J.B.F., Vreugdenhill A.I., Dicke M., Vet L.E.M. (1998). Learning to discriminate between infochemicals from different plant-host complexes by the parasitoids Cotesia glomerata and Cotesia rubecula. Entomol. Exp. Appl. 86: 241–252

    Article  Google Scholar 

  10. Goldson S.L., Proffitt J.R., McNeill M.R., Baird D.B. (1999). Linear patterns of dispersal and build up of the introduced parasitoid Microctonus hyperode (Hymenoptera: Braconidae) in Canterbury, New Zealand. Bull. Entomol. Res. 89(4): 347–353

    Google Scholar 

  11. Hassell M.P., Wilson H.B. (1997). The dynamics of spatially distributed host-parasitoid systems. In: Tilman, D., Kareiva, P. (eds) Spatial Ecology. Princeton University Press, USA

    Google Scholar 

  12. Höfer T., Sherratt J.A., Maini P.K. (1995). Cellular pattern formation during Dictyostelium aggregation. Phys. D 85: 425–444

    Article  MATH  Google Scholar 

  13. Holt R.D., Hassell M.P. (1993). Environmental heterogeneity and the stability of host-parasitoid interactions. J. Anim. Ecol. 62: 89–100

    Article  Google Scholar 

  14. Keller E.F., Segel L.A. (1970). Initiation of slime mould aggregation viewed as an instability. J. Theor. Biol. 26: 399–415

    Article  Google Scholar 

  15. Keller E.F., Segel L.A. (1971a). Model for Chemotaxis. J. Theor. Biol. 30: 225–234

    Article  Google Scholar 

  16. Keller E.F., Segel L.A. (1971b). Travelling bands of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30: 235–248

    Article  Google Scholar 

  17. Lauffenburger D.A., Aris R., Kennedy C.R. (1984). Travelling bands of chemotactic bacteria in the context of population growth. Bull. Math. Biol. 46: 19–40

    MATH  Google Scholar 

  18. Maini P.K., Myerscough M.R., Winters K.H., Murray J.D. (1991). Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation. Bull. Math. Biol. 53(5): 701–719

    MATH  Google Scholar 

  19. McDougall S.R., Anderson A.R.A., Chaplain M.A.J., Sherratt J.A. (2002). Mathematical modelling of flow through vascular networks: implications for tumor-induced angiogenesis and chemotherapy strategies. Bull. Math. Biol. 64: 673–702

    Article  Google Scholar 

  20. Murray J.D. (2002). Mathematical Biology. I: An Introduction. II: Spatial models and Biomedical Applications. Springer, New York

    Google Scholar 

  21. Pearce I.G., Chaplain M.A.J., Schofield P.G., Anderson A.R.A., Hubbard S.F. (2006). Modelling the spatio-temporal dynamics of multi-species host-parasitoid interactions: heterogeneous patterns and ecological implications. J. Theor. Biol. 241: 876–886

    Google Scholar 

  22. Petrovsikii S.V., Malchow H. (1999). A minimal model of pattern formation in a predator-prey system. Math. Comp. Model. 29(8): 49–63

    Article  Google Scholar 

  23. Petrovsikii S.V., Malchow H. (2001). Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics. Theor. Popul. Biol. 59: 157–174

    Article  Google Scholar 

  24. Preedy, K., Schofield, P.G., Chaplain, M.A.J., Hubbard, S.F.: Disease induced dynamics in host-parasitoid systems: Chaos and coexistence. R. Soc. Interface, doi: 10.1098/rsif.2006.0184 (2006)

  25. Rohani P., Miramontes O. (1996). Chaos or quasiperiodicity in laboratory insect populations? J. Anim. Ecol. 65(6): 847–849

    Article  Google Scholar 

  26. Savill N.J., Rohani P., Hogeweg P. (1997). Self-reinforcing spatial patterns enslave evolution in a host-parasitoid system. J. Theor. Biol. 188: 11–20

    Article  Google Scholar 

  27. Schofield P.G., Chaplain M.A.J., Hubbard S.F. (2002). Mathematical modelling of host-parasitoid systems: effects of chemically mediated parasitoid foraging strategies on within- and between-generation spatio-temporal dynamics. J. Theor. Biol. 214: 31–47

    Article  Google Scholar 

  28. Schofield P.G., Chaplain M.A.J., Hubbard S.F. (2005). Dynamic heterogeneous spatio-temporal pattern formation in host-parasitoid systems with synchronised generations. J. Math. Biol. 50: 559–583

    Article  MATH  Google Scholar 

  29. Schofield P.G., Chaplain M.A.J., Hubbard S.F. (2005). Evolution of searching and life history characteristics in individual-based models of hostf́bparasitoidf́bmicrobe associations. J. Theor. Biol. 237: 1–16

    Article  Google Scholar 

  30. Sherratt J.A. (1994). Chemotaxis and chemokinesis in eukaryotic cells: the Keller–Segel equations as an approximation to a detailed model. Bull. Math. Biol. 56: 129–146

    MATH  Google Scholar 

  31. Sherratt J.A., Lewis M.A., Fowler A.C. (1995). Ecological chaos in the wake of invasion. Proc. Natl. Acad. Sci. USA 92: 2524–2528

    Article  MATH  Google Scholar 

  32. Sherratt J.A., Eagan B.T., Lewis M.A. (1997). Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality? Phil. Trans. R. Soc. Lond. B 352: 21–38

    Article  Google Scholar 

  33. Sherratt J.A. (2003). Periodic travelling wave selection by Dirichlet boundary conditions in oscillatory reaction-diffusion systems. SIAM. J. Appl. Math. 63(5): 1520–1538

    Article  MATH  Google Scholar 

  34. Tyson R.C., Lubkin S.R., Murray J.D. (1999). Model and analysis of chemotactic patterns in a liquid medium. J. Math. Biol. 38: 359–375

    Article  MATH  Google Scholar 

  35. Roosjen M., Pumarino L., Dicke M., Poecke R.M.P. (2003). Attraction of the specialist parasitoid Cotesia rubecula to Arabidopsis thaliana infested by host or non-host herbivore species. Entomol. Exp. Appl. 107: 229–236

    Article  Google Scholar 

  36. Vos M., Berrocal S.M., Karamaouna F., Hemerik L., Vet L.E.M. (2001). Plant-mediated indirect effects and the persistence of parasitoid-herbivore communities. Ecol. Lett. 4: 38–45

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Division of Mathematics, University of Dundee, Dundee, DD1 4HN, UK

    Ian G. Pearce, Mark A. J. Chaplain & Alexander R. A. Anderson

  2. College of Life Sciences, University of Dundee, Dundee, DD1 4HN, UK

    Pietà G. Schofield & Stephen F. Hubbard

Authors
  1. Ian G. Pearce
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mark A. J. Chaplain
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pietà G. Schofield
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Alexander R. A. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Stephen F. Hubbard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ian G. Pearce.

Additional information

I. G. Pearce gratefully acknowledges the financial support of the NERC.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pearce, I.G., Chaplain, M.A.J., Schofield, P.G. et al. Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems. J. Math. Biol. 55, 365–388 (2007). https://doi.org/10.1007/s00285-007-0088-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-007-0088-4

Keywords

Search

Navigation