Matrix Population Models Life tables Intro to matrix

Matrix Population Models • Life tables • Intro to matrix multiplication • Examples of age and stage structured models • Elasticity analysis

Life Tables and Matrices: Accounting demographic parameters Age lx mx 2 1. 000 0. 5147 3 0. 6703 1. 3618 4 0. 4493 1. 6819 5 0. 3102 1. 8816 6 0. 2019 2. 0257 7 0. 1353 2. 1358 8 0. 0907 2. 2347 9 0. 0608 2. 2686 10 0. 0408 2. 2686 11 0. 0273 2. 2686 12 0. 0183 2. 2686 Sardine life table(Sardinops sagax) from Murphy (1967) Stage 1 Stage 2 Stage 3 Stage 4 0 0. 0043 0. 1132 0 0. 9775 0. 9111 0 0 0. 0736 0. 9534 0 0. 0542 . 9804 Killer Whale Lefkovich matrix from Brault, S. and H. Caswell (1993)

Matrix multiplication Scalar Multiplication - each element in a matrix is multiplied by a constant.

Matrix multiplication Multiply rows times columns. You can only multiply if the number of columns in the 1 st matrix is equal to the number of rows in the 2 nd matrix.

Matrix multiplication Multiply rows times columns. You can only multiply if the number of columns in the 1 st matrix is equal to the number of rows in the 2 nd matrix. They must match. Dimensions: 2 x 3 3 x 2 The dimensions of your answer.

*Answer should be dimension ? 2 x 2 0(4) + (-1)(-2) 0(-3) + (-1)(5) 1(4) + 0(-2) 1(-3) +0(5)

Matrix Multiplication (Population Model): a 1, 1 a 1, 2 a 2, 1 a 2, 2 Answer should be dimension ? x N 1 N 2 =

Matrix Multiplication (Population Model): a 1, 1 a 1, 2 a 2, 1 a 2, 2 x N 1 * a 1, 1 + N 2* a 1, 2 N 1 * a 2, 1 + N 2* a 2, 2 N 1 N 2 =

Introduction to Matrix Models • Vital rates describe the development of individuals through their life cycle (Caswell 1989) • Vital rates are : birth, growth, development, reproductive, mortality rates • The response of these rates to the environment determines: – population dynamics in ecological time – the evolution of life histories in evolutionary time

The general form of an age-structured Leslie Matrix models “Projection Matrix”: 0 p 1 0 0 P 2 0 f 3 0 0 p 3 f 4 0 0 0

The general form of an age-structured Leslie Matrix models “Projection Matrix”: Age 1 Age 2 Age 3 Age 4 0 p 1 0 0 Age 2 Age 3 Age 4 0 0 P 2 0 f 3 0 0 p 3 f 4 0 0 0

Age based matrix population model • Fecundity fx • Survivorship sx Age Class 4 Age Class 3 Age Class 1 Size class 2 Size class 3 Size class 4 Age Class 2 Size class 1 Size class 2 0 s 0 1 2 Size class 3 (Size class at first reproduction) f 3 0 0 s 3 Size class 4 f 4 0 0 0

The general form of Lefkovitch Matrix model Stage – structured Projection Matrix g *s 0 1 1 2, 1 1 3, 1 1 0 g *s 0 2, 2 2 3, 2 2 f 3 0 g *s 3, 3 3 4, 3 3 f 4 0 0 g *s 4, 4 4

The general form of Lefkovitch Matrix model Stage – structured Projection Matrix Size class 1 Size class 2 Size class 3 Size class 4 Size class 1 Size class 2 g *s 0 0 g *s 0 1 1 2, 1 1 3, 1 1 2, 2 2 3, 2 2 Size class 3 (Size class at first reproduction) f 3 0 g *s 3, 3 3 4, 3 3 Size class 4 f 4 0 0 g *s 4, 4 4

Stage-based matrix population model Size Class 4 Size Class 3 Size Class 1 Size Class 2 • Fecundity fx • Growth gx and Survivorship sx Size class 1 Size class 2 Size class 3 Size class 4 Size class 1 Size class 2 g *s 0 0 g *s 0 1 1 2, 1 1 3, 1 1 2, 2 2 3, 2 2 Size class 3 (Size class at first reproduction) f 3 0 g *s 3, 3 3 4, 3 3 Size class 4 f 4 0 0 g *s 4, 4 4

Some of the utility of matrix population models • Population projection – deterministic and stochastic • Elasticity Analysis – Conservation – Management • Meta population dynamics

Caswell • Types of model variability

Population projection Example: What is the population at t 1? Juvenile Adult t 0 0 fx Njuvenile px 0 Nadult fx NJuvenile NAdult px

0 fx px 0 x Njuvenile, t 0 Nadult, t 0 = Njuvenile, t 1 = 0*(Njuvenile, t 0) + fx*(Nadult, t 0) Nadult, t 1 = px*(Njuvenile, t 0) + 0 *(Nadult, t 0) fx Juvenile Adult px

Projecting this sample matrix indefinitely will result in the finite population growth rate: λ Age 1 Age 2 Age 3 t 0 0 4 4 20 0. 8 0 0. 5 0 x 20 20 =

Look at the y-axis • λ is on the natural log scale…

Stable age distribution

Stable age distribution. Expected in a static environment…

Population trajectories with process and observation error 0 p 1 0 0 P 2 0 f 3 0 0 p 3 f 4 0 0 0

Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations Age 1 Age 2 Age 3 0 4 4 0. 8 0 0. 5 0 Age 1 Age 2 Age 3 0 4 4 0. 6 0 0. 5 0 λ = 2. 00 A 25% decrease in Age 1 survivorship results in a 12% decrease in population growth. λ = 1. 76

Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations Age 1 Age 2 Age 3 0 4 4 0. 8 0 0. 5 0 Age 1 Age 2 Age 3 0 5 4 0. 8 0 0. 5 0 λ = 2. 00 A 25% increase in Age 2 fecundity results in a 9% increase in population growth. λ = 2. 18

Brault and Caswell •

Elasticity • A type of “perturbation” analysis • The elasticity eij indicates the relative impact on of a modification of the value of the parameter aij • Scaled, therefore The elasticity is independent on the metric of the parameter aij and

Stage-specific survival and reproduction • G, P, F

Stage-specific survival and reproduction • G, P, F

Stage-specific survival and reproduction • Initialize matrix A