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 11 0.0273 12 0.0183 Stage 1 Stage 2 Stage 3 Stage 4 0.0043 0.1132 0.9775 0.9111 0.0736 0.9534 0.0542 .9804 Killer Whale Lefkovich matrix from Brault, S. and H. Caswell (1993) Sardine life table(Sardinops sagax) from Murphy (1967)

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 1st matrix is equal to the number of rows in the 2nd matrix.

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

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

Matrix Multiplication (Population Model): = Answer should be dimension ? Explain Matrix Vector

Matrix Multiplication (Population Model): = N1 * a1,1 + N2* a1,2 N1 * a2,1 + N2* a2,2 Explain Matrix Vector

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”: f3 f4 p1 P2 p3

The general form of an age-structured Leslie Matrix models “Projection Matrix”: f3 f4 p1 P2 p3

(Size class at first reproduction) Age based matrix population model Fecundity fx Survivorship sx Age Class 4 Age Class 3 Age Class 1 Age Class 2 Size class 1 Size class 2 Size class 3 (Size class at first reproduction) Size class 4 Size based matrix models are useful when Demographic parameters, such as, fecundity and survivorship are best expressed as funciions of size and not age Relationship of age and size are not clear Size class 1 Size class 2 Size class 3 Size class 4 f3 f4 s1 s2 s3

The general form of Lefkovitch Matrix model Stage – structured Projection Matrix g1*s1 f3 f4 g2,1*s1 g2,2*s2 g3,1*s1 g3,2*s2 g3,3*s3 g4,3*s3 g4,4*s4

(Size class at first reproduction) The general form of Lefkovitch Matrix model Stage – structured Projection Matrix Size class 1 Size class 2 Size class 3 (Size class at first reproduction) Size class 4 Size class 1 Size class 2 Size class 3 Size class 4 g1*s1 f3 f4 g2,1*s1 g2,2*s2 g3,1*s1 g3,2*s2 g3,3*s3 g4,3*s3 g4,4*s4

(Size class at first reproduction) Stage-based matrix population model Fecundity fx Growth gx and Survivorship sx Size Class 4 Size Class 3 Size Class 1 Size Class 2 Size class 1 Size class 2 Size class 3 (Size class at first reproduction) Size class 4 Size based matrix models are useful when Demographic parameters, such as, fecundity and survivorship are best expressed as funciions of size and not age Relationship of age and size are not clear Size class 1 Size class 2 Size class 3 Size class 4 g1*s1 f3 f4 g2,1*s1 g2,2*s2 g3,1*s1 g3,2*s2 g3,3*s3 g4,3*s3 g4,4*s4

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 t1? Juvenile Adult t0 fx px Njuvenile Nadult fx NJuvenile NAdult px

fx px Njuvenile, t0 Nadult, t0 fx px Njuvenile, t0 Nadult, t0 x = Njuvenile, t1 = 0*(Njuvenile, t0) + fx*(Nadult, t0) Nadult, t1 = px*(Njuvenile, t0) + 0 *(Nadult, t0) fx Juvenile Adult px

Projecting this sample matrix indefinitely will result in the finite population growth rate: λ Age1 Age 2 Age 3 t0 4 0.8 0.5 20 x =

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 f3 f4 p1 P2 p3

Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations Age1 Age 2 Age 3 4 0.8 0.5 λ = 2.00 A 25% decrease in Age 1 survivorship results in a 12% decrease in population growth. Age1 Age 2 Age 3 4 0.6 0.5 λ = 1.76

Elasticity: Proportional sensitivity (of λ ) to matrix element perturbations Age1 Age 2 Age 3 4 0.8 0.5 λ = 2.00 A 25% increase in Age 2 fecundity results in a 9% increase in population growth. Age1 Age 2 Age 3 5 4 0.8 0.5 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