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Leslie MatricesModelling Age Structured Populations with EigenvaluesMatthew Roughanmatthew.roughan@adelaide.edu.auSchool of Mathematical SciencesUniversity of AdelaideMarch 20, 2014Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20141 / 21
Maths as an ArtMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20142 / 21
Maths as an ArtMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20143 / 21
Maths as an ArtEngineers and Scientist see Maths as a toolIIILike a hammer, you get it out when you need it, and put it away whenyou don’tYou don’t think too hard about how to use a hammer, you just hitthings with itSome people build better hammers, but that’s their problem, not mineI see Maths more like an artIIIIts a living corpus of workIf you are going to use it, you need to understand the loose edgesEveryone who uses Maths should be making it betterMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20144 / 21
Population ModelsThere are lots of models of populations:Exponential growthLogistic growthLotka-Volterra (predator-prey)Stochastic models: birth and death processesMost of them assume the population is homogeneous, but real populationshave structure, e.g.,Male/femaleGeographyDifferent agesMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20145 / 21
Ageing populationsThe distribution of ages mattersdeath rate can change with agebirth rate can change with agehttp://amrita.vlab.co.in/?sub 3&brch 65&sim 183&cnt 1Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20146 / 21
Example 1: Australian DemographicsGovernments need to predict populations in different age categories inorder to plan:Schools (how many children will there be?)Pensions (how many retired people will there be?)Australia has an “ageing” population.Proportion of population over 15.http://demographics.treasury.gov.au/content/ download/australiasdemographic challenges/html/adc-04.aspMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20147 / 21
Example 2: Australian Teachers“Australia’s Teachers: Australia’s Future”, Chapter 5, pp.53–64,DEST, Committee for the Review of Teaching and Teacher Education, 2003, ISBN 1877032 80 8.Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20148 / 21
Example 3: Weed KillersImagine you want to control a weed (or other pest) and you have twochoices of weedicide1is extremely effective, but only kills mature plants2is less effective, but kills germinating seedswhich is better?Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 20149 / 21
The ModelAge Classesageing012k 1birthAge specific survival rate governs ageing, from class i to i 1.Age specific fecundity (per capita birth rate) governs births, but allbirths start in age category 0Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201410 / 21
TerminologyEach time step, from t t 1, individuals age and potential die, and/orgive birth:survival rate: si is the proportion of individuals from Age Class i thatsurvive to i 1.fecundity: fi is the proportion of individuals from Age Class i whogive birth to new individuals in Age Class 0.population: at time step t is kept in the vector nt .The above often only makes sense if we model female populations (asmales don’t give birth).Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201411 / 21
The Leslie Matrix: DefinitionThe equation for one time step of the model as f0 f1 f2 f3. . . fk 1 .0nt 1 (0) s0 0 0 0 nt 1 (1) 0 s1 0 0.0 0 0 s2 0.0 . . 0 0 0. .0nt 1 (k 1)0 0 0 . . . sk 20 nt (0) nt (1) . . nt (k 1) or more succinctly asnt 1 Lntwhere L is called the Leslie Matrix.Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201412 / 21
The Leslie Matrix EquationSimple extrapolation of the equationnt 1 Lntfrom the first time step, where the population is n0 givesnt Lt n0so we can calculate future populations, just by taking powers of the Lesliematrix.Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201413 / 21
Let’s PlayLoginUsername:Password:Open Internet Explorer (not Firefox), and go to the following URL:http://bandicoot.maths.adelaide.edu.au/Leslie matrix/leslie.cgiMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201414 / 21
What you should seeMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201415 / 21
ResultsMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201416 / 21
What we should seeThe model parameters (survival rate, and fecundity) play a big role indetermining whether the population lives or dies.The starting population isn’t so important.IIGrowth or decay aren’t determined by starting populations.The final proportions of each Age Class don’t depend on the startingproportionsIn many cases it’s quite hard to guess whether a population will growor die.Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201417 / 21
What you may have noticedThe calculator also reports two extra results:IIThe first eigenvalue, which we will denote λ1Its corresponding eigenvectorYou may have noticedIIGrowth and decay are linked to the eigenvalue:If λ1 1 you get growthIf λ1 1 you get decayThe final proportions of each Age Class match the eigenvectorMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201418 / 21
Eigenvalues and EigenvectorsDefinition: Take a square n n matrix A, then a non-zero vector inx IR n is called an eigenvector if and only if it satisfiesAx λxfor some scalar λ, which is called an eigenvalue of A.x is said to be the eigenvector corresponding to λ.Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201419 / 21
Why does it work?The other session will talk some more about eigenvalues, but theapproximate view here isnt Lt n0 ' γ λt1 x1for large t, where λ1 is the largest eigenvalue of L, and x1 is itscorresponding eigenvector.Matthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201420 / 21
ConclusionModelling is all about tractable vs realism tradeoffsMaths models for growth are somewhat limitedIneed to account for ageThe Leslie model provides a very simple way to do soMathematical analysis can be used to understand its behaviourBut the Leslie model still has limitationsIIIno migrationit’s linearonly one speciesMatthew Roughan ( School of Mathematical Sciences UniversityLeslieofMatricesAdelaide [3mm] )March 20, 201421 / 21
Matthew Roughan ( School of Mathematical Sciences University of Adelaide [3mm] )Leslie Matrices March 20, 2014 18 / 21. Eigenvalues and Eigenvectors De nition: Take a square n n matrix A, then a non-zero vector in x 2IRn i
