A&S 320: Mathematical Modeling in Biology

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A&S 320: Mathematical Modeling in Biology David Murrugarra Department of Mathematics, University of Kentucky http://www.

1 A&S 320: Mathematical Modeling in Biology David Murrugarra Department of Mathematics, University of Kentucky Spring 2016 David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

2 Difference Equations If there is a matrix A such that x 1 = Ax 0, x 2 = Ax 1, and, in general, x k+1 = Ax k for k = 0, 1, 2,... (1) then Eq 1 is called a linear difference equation (or recurrent relation). A subject of interest to demographers is the movement of populations or group of people from one region to another. David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

3 Difference Equations The simple model here considers the changes in the population of a certain city and its surrounding suburbs over a period of years. Fix an initial year say 2016 and denote the population of the city and suburbs of that year by r 0 and s 0, respectively. Let x 0 be the population vector [ ] r0 city population in x 0 = suburb population in 2016 s 0 For 2017 and subsequent years, denote the populations of the city and suburbs by the by the vectors [ ] [ ] r1 r2 x 1 =, x 2 =,... s 1 s 2 David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

4 Difference Equations Our goal is to describe mathematically how these vectors might be related. Suppose demographic studies show that each year about 5% of the city s population moves to the suburbs (95% remains in the city), while 3% of the suburban population move to the city (and 97% remains in the suburbs) City Suburbs Figure: Migration between city and suburbs. David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

5 Difference Equations After a year, the original r 0 persons in the city are now distributed between city and suburbs as [ ] [ ].95r0.95 remain in the city = r.05r move to suburb (2) The s 0 persons in the suburbs in 2016 are distributed 1 year later as [ ] [ ] 0.03s move to city = s 0.97s 0 (3) remain in suburb Vectors in Eq 2 and Eq 3 account for all population in David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

6 Difference Equations Thus (ignoring births, deaths, migration into and out of the city/suburban region) That is, x 1 = [ r1 s 1 ] [ ] [ ] = r 0 + s.05 0 =.97 where M is the migration matrix determined by [ ] M = [ ] [ ] r0 s 0 x 1 = Mx 0 (4) Eq 4 describes how the population changes from 2016 to David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

7 Difference Equations If the migration percentages remain constant, then the change from 2017 to 2018 is given by x 2 = Mx 1 and similarly from 2018 to 2019 and subsequent years. In general, x k+1 = Mx k for k = 0, 1, 2,... (5) The sequence of vectors {x 0, x 1, x 2,... } describes the population of the city/suburban region over a period of years. David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

8 Difference Equations The annual migration between these two parts of the metropolitan region is governed by the migration matrix, [ ] M = Compute the population of the region for the years 2017 and 2018, given that population in 2016 was 600,000 in the city and 400,000 in the suburbs. [ ] 600, 000 x 0 = 400, 000 David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

9 Stochastic Matrix Definition Consider a nonnegative vector x = is called a probability vector. [ r0 s 0 ] such that r 0 + s 0 = 1. Then x Definition A stochastic matrix is a square matrix whose columns are probability vectors. M = [.95 ] David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

10 Markov Chains Definition A Markov Chain is a sequence of probability vectors x 0, x 1, x 2,..., together with a stochastic matrix P such that x 1 = Px 0, x 2 = Px 1, and, in general, x k+1 = Px k for k = 0, 1, 2,... (6) Eq 6 is called a first order linear difference equation. David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

11 Markov Chains Suppose demographic studies show that each year about 5% of the city s population moves to the suburbs (95% remains in the city), while 3% of the suburban population move to the city (and 97% remains in the suburbs) City Suburbs Figure: Migration between city and suburbs. David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

12 Markov Chains The annual migration between these two parts of the metropolitan region is governed by the migration matrix, [ ] M = Compute the distribution of population of the region just described for the years 2017 and 2018, given that population in 2016 was 600,000 in the city and 400,000 in the suburbs. [ ] 600, 000 x 0 = 400, 000 David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

13 Markov Chains Initial distribution of the population x 0 = [ ] The following equation describes how the population distribution changes from 2016 to [ ] [ ] x 1 = Mx 0 = = and the change from 2017 to 2018 is given by [ ] [ ] x 2 = Mx 1 = = [ ] [ ] (7) (8) David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

14 Predicting the Distant Future What happens to the system as time passes? [ ] [ ] [ ] x 3 = Mx 2 = = [ ] [ ] [ ] x 4 = Mx 3 = = [ ] [ ] [ ] x 5 = Mx 4 = = David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

15 Steady State Vectors Definition If P is a stochastic matrix, then a steady state vector (or equilibrium vector) for P is a probability vector q such that [ ].6.2 Let P = and q =.4.8 Pq = q [ ] 0.3. Is q a steady-state vector for P? 0.7 David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

16 Steady State Vectors Definition If P is a stochastic matrix, then a steady state vector (or equilibrium vector) for P is a probability vector q such that [ ].6.2 Let P = and q =.4.8 No! Because, Pq = Pq = q [ ] 0.3. Is q a steady-state vector for P? 0.7 [ ] [ ] = [ ] 0.32 q 0.68 David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

17 Steady State Vectors Definition If P is a stochastic matrix, then we say that P is regular if some matrix power P k contains only strictly positive entries. M = [.95 ] David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

18 Steady State Vectors Theorem If P is an n n regular stochastic matrix, then P has a unique steady state vector q. Further, if x 0 is any initial state and x k+1 = Px k for k = 0, 1, 2,... then the Markov Chain {x k } converges to q as k. David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

19 Steady State Vectors [ ] Let P =. Find a steady state vector for P David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

20 Steady State Vectors [ ] [ ] 3/ The probability vector q = = is a steady state vector for 5/ the population migration matrix M because, [ ] [ ] [ ] q = Mq = = David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

21 Practice Problems 1 Suppose the residents of a metropolitan region move according to the probabilities in the migration matrix, [ ] M = and a resident is chosen at random. Then the state of a vector for a certain year may be interpreted as giving the probabilities that the person is a city resident or a suburban resident at that time. [ 1 (a) Suppose the person chosen is a city resident now, so that x 0 =. 0] What is the likelihood that the person will live in the suburbs next year? (b) What is the likelihood that the person will be living in the suburbs in two years? 2 What percentage of the population will live in the suburbs after many years? David Murrugarra (University of Kentucky) A&S 320: Section Spring / 21

TMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013

TMA Calculus 3. Lecture 21, April 3. Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 TMA4115 - Calculus 3 Lecture 21, April 3 Toke Meier Carlsen Norwegian University of Science and Technology Spring 2013 www.ntnu.no TMA4115 - Calculus 3, Lecture 21 Review of last week s lecture Last week