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

  • Mohd Hafiz MohdEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (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.

Keywords

Partial differential equations Numerical bifurcation Stability analysis Ecological modelling 

Notes

Acknowledgements

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

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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversiti Sains MalaysiaPenangMalaysia

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