Skip to main content
SpringerLink
Log in
Book cover

SEAMS School on Dynamical Systems and Bifurcation Analysis

DySBA 2018: Dynamical Systems, Bifurcation Analysis and Applications pp 117–132Cite as

Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology

  • Conference paper
  • First Online:

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 295))

Abstract

Bifurcation analysis is a powerful technique for investigating the dynamical behaviours of nonlinear systems. While this approach has been employed extensively in analysing ordinary-differential equations and other deterministic models, the use of bifurcation analysis in studying the dynamics of partial differential equations (PDE) is yet limited. This chapter illustrates the progress on how numerical bifurcation and stability analyses can be used in understanding the overall dynamics of a PDE system under consideration. By considering an ecological example of competitive system with environmental suitability and spatial diffusion terms, distinct behaviours of the model e.g. alternative stable states, multi-species coexistence and extinction phenomena are demonstrated as interspecific competition and dispersal strength change. Further investigation reveals the existence of several threshold values in ecologically-relevant parameters corresponding to distinct bifurcations (e.g. saddle-node and transcritical), which lead to different stability properties of PDE solution branches.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Mohd, M.H.B.: Modelling the presence-absence of multiple species. Ph.D. Thesis, University of Canterbury, New Zealand (2016)

    Google Scholar 

  2. Shmida, A.V.I., Wilson, M.V.: Biological determinants of species diversity. J. Biogeogr. 1–20 (1985)

    Google Scholar 

  3. Pearson, R.G., Dawson, T.P.: Predicting the impacts of climate change on the distribution of species: are bioclimate envelope models useful? Global Ecol. Biogeogr. 12(5), 361–371 (2003)

    Article  Google Scholar 

  4. Soberón, J.: Grinnellian and Eltonian niches and geographic distributions of species. Ecol. Lett. 10(12), 1115–1123 (2007)

    Article  Google Scholar 

  5. Kearney, M., Porter, W.: Mechanistic niche modelling: combining physiological and spatial data to predict species-ranges. Ecol. Lett. 12(4), 334–350 (2009)

    Article  Google Scholar 

  6. Wisz, M.S., Pottier, J., Kissling, W.D., Pellissier, L., Lenoir, J., Damgaard, C.F., Dormann, C.F., Forchhammer, M.C., Grytnes, J.A., Guisan, A.: The role of biotic interactions in shaping distributions and realised assemblages of species: implications for species distribution modelling. Biol. Rev. 88(1), 15–30 (2013)

    Article  Google Scholar 

  7. Case, T.J., Holt, R.D., McPeek, M.A., Keitt, T.H.: The community context of species’ borders: ecological and evolutionary perspectives. Oikos 108(1), 28–46 (2005)

    Article  Google Scholar 

  8. Gilman, S.E., Urban, M.C., Tewksbury, J., Gilchrist, G.W., Holt, R.D.: A framework for community interactions under climate change. Trends Ecol. Evol. 25(6), 325–331 (2010)

    Article  Google Scholar 

  9. Darwin, C.: On the Origins of Species by Means of Natural Selection. John Murray (1859)

    Google Scholar 

  10. Grinnell, J.: The niche-relationships of the California thrasher. Auk 34(4), 427–433 (1917)

    Article  Google Scholar 

  11. Gause, G.F.: Experimental studies on the struggle for existence. J. Exp. Biol. 9(4), 389–402 (1932)

    Google Scholar 

  12. Gause, G.F.: The Struggle for Existence. Dover Publications (1934)

    Google Scholar 

  13. Hutchinson, G.E.: Concluding Remarks: Cold Spring Harbor Symposium on Quantitative Biology, vol. 22. Cold Spring Harbor Laboratory Press, pp. 415–427 (1957)

    Google Scholar 

  14. Lotka, A.J.: Elements of Physical Biology. Williams and Wilkins (1925)

    Google Scholar 

  15. Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)

    Article  Google Scholar 

  16. Cosner, C.: Reaction-diffusion-advection models for the effects and evolution of dispersal. Discret. Contin. Dyn. Syst. A 34(5), 1701–1745 (2014)

    Article  MathSciNet  Google Scholar 

  17. Holmes, E.E., Lewis, M.A., Banks, J.E., Veit, R.R.: Partial differential equations in ecology: spatial interactions and population dynamics. Ecology 75(1), 17–29 (1994)

    Article  Google Scholar 

  18. MacLean, W.P., Holt, R.D.: Distributional patterns in St. Sroix Sphaerodactylus lizards: the taxon cycle in action. Biotropica 11(3), 189–195 (1979)

    Article  Google Scholar 

  19. Roughgarden, J.: Theory of population genetics and evolutionary ecology: an introduction. Macmillan Publishing (1979)

    Google Scholar 

  20. Ermentrout, B.: Simulating, analyzing, and animating dynamical systems: a guide to XPPAUT for researchers and students. SIAM (2002)

    Google Scholar 

  21. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press (2001)

    Google Scholar 

  22. Gotelli, N.J.: A Primer of Ecology. Sinauer Associates Incorporated (1995)

    Google Scholar 

  23. Chesson, P.: Mechanisms of maintenance of species diversity. Annu. Rev. Ecol. Syst. 31, 343–358 (2000)

    Article  Google Scholar 

  24. Case, T.J.: An Illustrated Guide to Theoretical Ecology. Oxford University Press (2000)

    Google Scholar 

  25. Colwell, R.K., Rahbek, C., Gotelli, N.J.: The mid-domain effect and species richness patterns: what have we learned so far? Am. Nat. 163(3), E1–E23 (2004)

    Article  Google Scholar 

  26. Colwell, R.K., Lees, D.C.: The mid-domain effect: geometric constraints on the geography of species richness. Trends Ecol. Evol. 15(2), 70–76 (2000)

    Article  Google Scholar 

  27. McCain, C.M.: The mid-domain effect applied to elevational gradients: species richness of small mammals in Costa Rica. J. Biogeogr. 31(1), 19–31 (2004)

    Article  MathSciNet  Google Scholar 

  28. McCain, C.M.: Elevational gradients in diversity of small mammals. Ecology 86(2), 366–372 (2005)

    Article  Google Scholar 

  29. Mohd M.H.B.: Modelling biotic interactions, dispersal effects and the stability of multi-species community compositions. In: AIP Conference Proceedings, vol. 1974, p. 020079. AIP Publishing (2018)

    Google Scholar 

  30. Higham, D.J., Higham, N.J.: MATLAB guide. SIAM (2016)

    Google Scholar 

  31. Kooi, B.W., Boer, M.P., Kooijman, S.A.L.M.: Resistance of a food chain to invasion by a top predator. Math. Biosci. 157(1), 217–236 (1999)

    Article  MathSciNet  Google Scholar 

  32. Van Voorn, G.A.K., Kooi, B.W.: Smoking epidemic eradication in a eco-epidemiological dynamical model. Ecol. Complex. 14, 180–189 (2013)

    Article  Google Scholar 

  33. Saputra, K.V.I., Van Veen, L., Quispel, G.R.W.: The saddle-node-transcritical bifurcation in a population model with constant rate harvesting. Dyn. Contin. Discret. Impuls. Syst. Ser. B Math. Anal. 14, 233–250 (2010)

    Article  MathSciNet  Google Scholar 

  34. Sinclair, A.R.E., Fryxell, J.M., Caughley, G.: Wildlife Ecology. Wiley, Conservation and Management (2006)

    Google Scholar 

  35. Short, J., Bradshaw, S.D., Giles, J., Prince, R.I.T., Wilson, G.R.: Reintroduction of macropods (Marsupialia: Macropodoidea) in Australia a review. Biol. Conserv. 62(3), 189–204 (1992)

    Article  Google Scholar 

  36. Dytham, C.: Evolved dispersal strategies at range margins. Proc. R. Soc. Lond. B Biol. Sci. 276(1661), 1407–1413 (2009)

    Article  Google Scholar 

  37. Davis, A.J., Jenkinson, L.S., Lawton, J.H., Shorrocks, B., Wood, S.: Making mistakes when predicting shifts in species range in response to global warming. Nature 391(6669), 783–786 (1998)

    Article  Google Scholar 

  38. Davis, A.J., Lawton, J.H., Shorrocks, B., Jenkinson, L.S.: Individualistic species responses invalidate simple physiological models of community dynamics under global environmental change. J. Anim. Ecol. 67(4), 600–612 (1998)

    Article  Google Scholar 

  39. Pulliam, H.R.: On the relationship between niche and distribution. Ecol. Lett. 3(4), 349–361 (2000)

    Article  Google Scholar 

  40. Amarasekare, P., Nisbet, R.M.: Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species. Am. Nat. 158(6), 572–584 (2001)

    Article  Google Scholar 

  41. Lei, G., Hanski, I.: Spatial dynamics of two competing specialist parasitoids in a host metapopulation. J. Anim. Ecol. 67(3), 422–433 (1998)

    Article  Google Scholar 

Download references

Acknowledgements

Mohd Hafiz Mohd is supported by the Universiti Sains Malaysia (USM) Fundamental Research Grant Scheme (FRGS) No. 203/PMATHS/6711645.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia

    Mohd Hafiz Mohd

Authors
  1. Mohd Hafiz Mohd
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohd Hafiz Mohd .

Editor information

Editors and Affiliations

  1. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Mohd Hafiz Mohd

  2. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Norazrizal Aswad Abdul Rahman

  3. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Nur Nadiah Abd Hamid

  4. School of Mathematical Sciences, Universiti Sains Malaysia, USM, Penang, Malaysia

    Yazariah Mohd Yatim

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Singapore Pte Ltd.

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Mohd, M.H. (2019). Numerical Bifurcation and Stability Analyses of Partial Differential Equations with Applications to Competitive System in Ecology. In: Mohd, M., Abdul Rahman, N., Abd Hamid, N., Mohd Yatim, Y. (eds) Dynamical Systems, Bifurcation Analysis and Applications. DySBA 2018. Springer Proceedings in Mathematics & Statistics, vol 295. Springer, Singapore. https://doi.org/10.1007/978-981-32-9832-3_7

Download citation

Publish with us

Policies and ethics

Search

Navigation