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UCLA - LIFESCI 30B - Class Notes - Week 2

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UCLA - LIFESCI 30B - Class Notes - Week 2

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School: University of California - Los Angeles
Department: Life Science
Course: Mathematics for Life Scientists
Professor: Jukka Keranen
Term: Winter 2019
Tags:
Name: LS30B WEEK2
Description: Lecture notes
Uploaded: 01/20/2019
48 Pages 44 Views 35 Unlocks
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Unformatted text preview: * 14th Jan 19 b lower Lec 4 Chaps: An Introduction The Hastings Equation Ecosystem with three speues : grass X Sheep wolvesz Suppose the model is logistic growth saturating consumption of grans per sheep Pag God Y - d, Y - inz per capita bhith roke proportional to grasi ea constant per capita death rate schuro.bing consumption of sheep per nolla Z' = ( ( 02.Z - d27 per capita birth rate proportoval to sheep eate constant per capita death rate 110 01-5, 6-3, 22-01, b2 + 2, d.-N.4, 12*0. DL 10 + population i inte 2 1 150 200 250 300 350 40 Each cycle is different!Trajectory in state space Wolves population drop rapidly wolne Yee reverer less sheep wolves eat more Sheep grass recover Dynamics D the Hastings. How did we go here?) 2 *lf b is low, sheey would reach grass s turatio quickly The System has a point attracy mirease b 2 < b < 3) + Hopf bifurcation MAAN IVVVVVV time The system now has a limit cycle atractor bifurcation Further increase b (2< b < 3) period doubling Ya e MMMMM Lime more complex Olymastic limit yde attractorFinally increase bi to 3 period becomes L The system exhibits aperiodic dynamics in spite of the rough amalitative similarity between the 'cycles the system never repeats exactly never approaches exact repetition 4 4 The system now has a chaotic attraer

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