Lecture 6 - Stochastic population models

Slide 1

Slide 1 text

Stochasticity

Slide 2

Slide 2 text

Today’s topics 1 Introduction 2 Geometric Growth 3 Logistic growth

Slide 3

Slide 3 text

Random Variables A random variable is a variable whose value can’t be predicted with certainty. Introduction Geometric Growth Logistic growth 3 / 17

Slide 4

Slide 4 text

Random Variables A random variable is a variable whose value can’t be predicted with certainty. Examples? • Weather • Our own behavior • Population size Introduction Geometric Growth Logistic growth 3 / 17

Slide 5

Slide 5 text

Probability distributions A random variable (X) can be described by a probability distribution. Introduction Geometric Growth Logistic growth 4 / 17

Slide 6

Slide 6 text

Probability distributions A random variable (X) can be described by a probability distribution. There are many types of probability distributions • Normal (or Gaussian) • Poisson • Binomial • Multinomial • etc. . . Introduction Geometric Growth Logistic growth 4 / 17

Slide 7

Slide 7 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X Introduction Geometric Growth Logistic growth 5 / 17

Slide 8

Slide 8 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q Introduction Geometric Growth Logistic growth 5 / 17

Slide 9

Slide 9 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 10

Slide 10 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 11

Slide 11 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 12

Slide 12 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 13

Slide 13 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 14

Slide 14 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q q q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 15

Slide 15 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q q q q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 16

Slide 16 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q q q q q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 17

Slide 17 text

Normal (Gaussian) Distribution X ∼ Normal(µ = 0, σ2 = 1) −3 −2 −1 0 1 2 3 X q q q q q q q q q q Introduction Geometric Growth Logistic growth 5 / 17

Slide 18

Slide 18 text

Normal (Gaussian) Distribution −2 −1 0 1 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 X Relative probability µ=0, σ2=0.6 µ=0, σ2=0.4 µ=0, σ2=0.2 Introduction Geometric Growth Logistic growth 6 / 17

Slide 19

Slide 19 text

A purely stochastic model Nt ∼ Normal(µ = 50, σ2 = 1) qq q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qqq q q q q q q q q q q q q q q q q q q q qq q q q q qq q q 0 20 40 60 80 100 48 49 50 51 52 Time Population size (N) Introduction Geometric Growth Logistic growth 7 / 17

Slide 20

Slide 20 text

Two important types of stochasticity Environmental stochasticity • Random variation in weather, habitat, etc. . . among years Introduction Geometric Growth Logistic growth 8 / 17

Slide 21

Slide 21 text

Two important types of stochasticity Environmental stochasticity • Random variation in weather, habitat, etc. . . among years Demographic stochasticity • Random variation in the number of births and deaths among years Introduction Geometric Growth Logistic growth 8 / 17

Slide 22

Slide 22 text

Geometric growth with environmental stochasticity Nt+1 = Nt + Nt r + Xt where Xt ∼ Normal(0, σ2 e ) Introduction Geometric Growth Logistic growth 9 / 17

Slide 23

Slide 23 text

Geometric growth with environmental stochasticity Nt+1 = Nt + Nt r + Xt where Xt ∼ Normal(0, σ2 e ) R code: r <- 0.1 sigma.e <- 10 for(t in 2:nYears) { X[t-1] <- rnorm(n=1, mean=0, sd=sigma.e) N[t] <- N[t-1] + N[t-1]*r + X[t-1] } Introduction Geometric Growth Logistic growth 9 / 17

Slide 24

Slide 24 text

Example N0 = 100, r = 0.1, µ = 0, σ2 e = 100 q q q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 0 200 400 600 800 1000 Time Population size (N) Introduction Geometric Growth Logistic growth 10 / 17

Slide 25

Slide 25 text

Slide 26

Slide 26 text

Slide 27

Slide 27 text

Slide 28

Slide 28 text

Slide 29

Slide 29 text

Slide 30

Slide 30 text

Slide 31

Slide 31 text

Slide 32

Slide 32 text

Slide 33

Slide 33 text

Slide 34

Slide 34 text

Example N0 = 100, r = 0.1, µ = 0, σ2 e = 10000 q q q q q q q q q q q q q q q q q q 0 5 10 15 20 0 200 400 600 800 1000 Time Population size (N) Introduction Geometric Growth Logistic growth 11 / 17

Slide 35

Slide 35 text

Example N0 = 100, r = 0.1, µ = 0, σ2 e = 10000 q q q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 0 200 400 600 800 1000 Time Population size (N) Introduction Geometric Growth Logistic growth 11 / 17

Slide 36

Slide 36 text

Slide 37

Slide 37 text

Example N0 = 100, r = 0.1, µ = 0, σ2 e = 10000 q q q q q q q q q q q q 0 5 10 15 20 0 200 400 600 800 1000 Time Population size (N) Introduction Geometric Growth Logistic growth 11 / 17

Slide 38

Slide 38 text

Slide 39

Slide 39 text

Slide 40

Slide 40 text

Example N0 = 100, r = 0.1, µ = 0, σ2 e = 10000 q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 0 200 400 600 800 1000 Time Population size (N) Introduction Geometric Growth Logistic growth 11 / 17

Slide 41

Slide 41 text

Slide 42

Slide 42 text

Slide 43

Slide 43 text

Example N0 = 100, r = 0.1, µ = 0, σ2 e = 10000 q q q q q q q q q 0 5 10 15 20 0 200 400 600 800 1000 Time Population size (N) Introduction Geometric Growth Logistic growth 11 / 17

Slide 44

Slide 44 text

Geometric growth with demographic stochasticity Nt+1 = Nt + Nt rt Introduction Geometric Growth Logistic growth 12 / 17

Slide 45

Slide 45 text

Geometric growth with demographic stochasticity Nt+1 = Nt + Nt rt where rt ∼ Normal(¯ r, σ2 d ) Introduction Geometric Growth Logistic growth 12 / 17