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

‘A size-based projection matrix model and elasticity analysis of red abalone, Haliotis rufescens, in northern California. ’ Robert Leaf and Laura Rogers-Bennett rleaf@mlml. calstate. edu Moss Landing Marine Laboratories U. C. Bodega Marine Laboratory This work was funded by Sea. Grant Traineeship # R/CZ-69 PD TR

Abalone harvest in California Adapted from Karpov et al. 2000

Recreational abalone fishery • North of San Francisco Bay • Only red abalone may be taken • 7 inch limit (178 mm) • Free-dive only (no SCUBA) • 3 red abalone per day • Maximum take: 24 abalone during a calendar year

What are the most effective conservation measures for red abalone? • “Outplanting” or seeding juveniles – Increase juvenile survivorship • Marine Protected Areas – Increase adult survivorship (>178 mm) and eliminate incidental mortality • New Size Limits – Are current limits set appropriately?

Size based matrix population model • Fecundity fx • Growth gx • Survivorship sx Size class 1 Size class 2 Size class 3 Size class 4 Size Class 3 Size Class 1 Size Class 2 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

Elasticity Example if ei, j = 0. 5 10% change in the value of a parameter (ai, j) will result in a 5% increase in the population growth rate.

Annual growth CDFG Tagging study 1971 – 1978 (Schultz and De. Martini unpublished) • Red abalone tagged at the five sites (n = 5997) • 41. 5 to 222 mm SL • 845 recaptured at one year intervals plus or minus 30 days (335 to 395 days) • Growth was normalized to one year. Pt. Cabrillo (N. and S. ) Mendocino county Van Damme SP Pt. Arena Sonoma county Ft. Ross

Growth transition matrix - gx Size class (mm) t 1 Size class (mm) t 0 0 to 25 25. 1 to 50 50. 1 to 75 75. 1 to 100. 1 to 125. 1 to 150. 1 to 178. 1 to 200 > 200. 1 0 to 25 24 0 0 0 0 25. 1 to 50 15 1 0 0 0 0 50. 1 to 75 1 1 1 0 0 0 75. 1 to 100 0 0 8 16 0 0 0 100. 1 to 125 0 0 1 37 28 0 0 125. 1 to 150 0 4 63 71 0 0 0 150. 1 to 178 0 0 2 76 322 10 0 178. 1 to 200 0 0 0 29 150 3 > 200. 1 0 0 0 0 2 22 2 10 57 93 147 351 162 Number of indivuals 40 in size class at t 0 25

Growth transition frequencies - gx Size class (mm) t 1 Size class (mm) t 0 0 to 25 25. 1 to 50 50. 1 to 75 75. 1 to 100. 1 to 125. 1 to 150. 1 to 178. 1 to 200 > 200. 1 0 to 25 0. 600 0 0 0 0 25. 1 to 50 0. 375 0. 500 0 0 0 50. 1 to 75 0. 025 0. 500 0. 100 0 0 0 75. 1 to 100 0 0 0. 800 0. 281 0 0 0 100. 1 to 125 0 0 0. 100 0. 649 0. 301 0 0 125. 1 to 150 0 0. 070 0. 677 0. 483 0 0 0 150. 1 to 178 0 0 0. 022 0. 517 0. 917 0. 062 0 178. 1 to 200 0 0 0 0. 083 0. 926 0. 120 > 200. 1 0 0 0 0. 012 0. 880

Size based matrix population model • Fecundity fx • Growth gx • Survivorship sx Size class 1 Size class 2 Size class 3 Size class 4 Size Class 3 Size Class 1 Size Class 2 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

Individual tag number Annual Survival Estimates Julian Day Model Name Number of individuals Annual Survival Estimate Standard Error ‘cryptic’ (< 100 mm) 179 0. 525 0. 0506 ‘emergent’ (≥ 100 mm) 567 0. 691 0. 0240

Incorporation of annual survival rates into growth transition frequencies Size class (mm) 0 to 25 25 to 50 50 to 75 75 to 100 to 125 to 150 to 178 to 200 > 200 0 to 25 0. 600 0 0 0 0 25 to 50 0. 375 0. 500 0 0 0 50 to 75 0. 025 0. 500 0. 100 0 0 0 75 to 100 0 0 0. 800 0. 281 0 0 0 100 to 125 0 0 0. 100 0. 649 0. 301 0 0 125 to 150 0 0. 070 0. 677 0. 483 0 0 0 150 to 178 0 0 0. 022 0. 517 0. 917 0. 062 0 178 to 200 0 0 0 0. 083 0. 926 0. 120 > 200 0 0 0 0. 012 0. 880 x 0. 525 y -1 x 0. 691 y -1

Incorporation of annual survival rates into growth transition frequencies Size class (mm) 0 to 25 25. 1 to 50 50. 1 to 75 75. 1 to 100. 1 to 125. 1 to 150. 1 to 178. 1 to 200 > 200. 1 0 to 25 0. 315 0 0 0 0 25. 1 to 50 0. 197 0. 263 0 0 0 0 50. 1 to 75 0. 013 0. 263 0. 053 0 0 0 75. 1 to 100 0 0 0. 421 0. 148 0 0 0 100. 1 to 125 0 0 0. 053 0. 341 0. 158 0 0 125. 1 to 150 0 0. 037 0. 356 0. 334 0 0 0 150. 1 to 178 0 0 0. 011 0. 357 0. 643 0. 043 0 178. 1 to 200 0 0 0 0. 057 0. 640 0. 083 > 200. 1 0 0 0 0. 009 0. 608

Size based matrix population model • Fecundity fx • Growth gx • Survivorship sx Size class 1 Size class 2 Size class 3 Size class 4 Size Class 3 Size Class 1 Size Class 2 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

Fecundity Rogers-Bennett et al. 2004

Incorporate fecundities (number of eggs) and solve for first year survivorship assuming a stable population growth rate. Size class (mm) 25. 1 to 50 50. 1 to 75 75. 1 to 100. 1 to 125. 1 to 150. 1 to 178. 1 to 200 3. 8 * 105 x P 0 5 * 105 113 58 * 10 x Px 0 P 0 > 200. 1 25. 1 to 50 0. 263 0 0 0. 5 * 105 x P 0 141 * 105 x P 0 50. 1 to 75 0. 263 0. 053 0 0 0 75. 1 to 100 0 0. 421 0. 148 0 0 0 100. 1 to 125 0 0. 053 0. 341 0. 158 0 0 125. 1 to 150 0 0 0. 037 0. 356 0. 334 0 0 0 150. 1 to 178 0 0. 011 0. 357 0. 643 0. 043 0 178. 1 to 200 0 0 0. 057 0. 640 0. 083 > 200. 1 0 0 0 0. 009 0. 608 So, P 0 = 1. 5 x 10 -6, approximately 1 individual in 665, 000 survives to the beginning of their second year

Elasticity results Current size limit 178 mm

What are the most effective conservation measures for red abalone? • “Outplanting” or seeding juveniles – Increase juvenile survivorship • Marine Protected Areas – Increase adult survivorship (>178 mm) and eliminate incidental mortality • New Size Limits – Are current limits set appropriately?

Elasticity results Current size limit 178 mm

What are the most effective conservation measures for red abalone? • “Outplanting” or seeding juveniles – Increase juvenile survivorship • Marine Protected Areas – Increase adult survivorship (>178 mm) and eliminate incidental mortality • New Size Limits – Are current limits set appropriately?

Elasticity results Current size limit 178 mm

What are the most effective conservation measures for red abalone? • “Outplanting” or seeding juveniles – Increase juvenile survivorship • Marine Protected Areas – Increase adult survivorship (>178 mm) and eliminate incidental mortality • New Size Limits – Are current limits set appropriately?

Elasticity results Current size limit 178 mm

Conclusions • Matrix models can be potentially effective in analysis of proposed management strategies. • Static vital rates