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Chapter 3��Bifurcations

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Definition

  • If variation of a parameter changes the qualitative behaviour of the solution, we call it a bifurcation.

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  • To understand a mathematical model properly, it is important to know when and how a bifurcation occurs.
  • We will introduce four common bifurcations, namely bifurcations that occur at equilibria.

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Saddle-Node Bifurcation

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Bifurcation Diagram

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Transcritical Bifurcation

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Bifurcation Diagram

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Pitchfork Bifurcation

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Bifurcation Diagram

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Hopf Bifurcation

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Bifurcation Diagram

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Mathematics behind Hopf bifurcation

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Example: The Spruce Budworm Model

  • To describe the outbreak of budworm populations in Canadian forests. Large budworm populations are able to defoliate and finally kill the balsam fir tree.

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  • The budworm population is susceptible to predation by birds.

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  • The model in its simplest, nondimensionalized form is given as

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Saddle-Node Bifurcation

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Types of specific solutions other than equilibria

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A high-dimensional challenging example in stoichiometry: LKE Model

Algal population

(measured by carbon)

Daphnia population

(measured by carbon)

Loladze et al. “Stoichiometry in Producer-Grazer Systems: Linking Energy Flow with Element Cycling”, BMB, Vol. 62, pp 1137-1162 (2000).

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Holling Type II

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Now we start to perform stability and bifurcation analysis according to

the varying parameter K (representing the light intensity).

Li, X., Wang, H. and Kuang, Y., 2011. Global analysis of a stoichiometric producer–grazer model with Holling type functional responses. Journal of Mathematical Biology, 63(5), pp.901-932.

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light

light

light

light

(no limit cycle, the one

plotted is only for proof)

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(no limit cycle, the one

plotted is only for proof)

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Summary

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Li, X., Wang, H. and Kuang, Y., 2011. Global analysis of a stoichiometric producer–grazer model with Holling type functional responses. Journal of Mathematical Biology, 63(5), pp.901-932.

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For the complete analysis with all varying parameters, see the paper:

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Xie, T., Yang, X., Li, X. and Wang, H., 2018. Complete global and bifurcation analysis of a stoichiometric predator–prey model. Journal of Dynamics and Differential Equations, 30(2), pp.447-472.

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  • Do exercises at the end of 4 Bifurcation in the optional textbook

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  • Lab 1: on Septebmer 24th, XPPAUT for plotting bifurcation diagrams

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  • Lab 2: on September 29th, Matlab Tutorial, simulations for sample models (discrete-time models, ODE models)

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  • Lab 3: on October 8th, Parameter estimation, model selection, and sensitivity analysis

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  • Lab 4: on October 29th. Building Mathematical Models from Experimental Data

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  • Exam: on October 15th, in class time and in lecture classroom, all contents taught before the exam

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