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Matrix Population Models

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Presentation on theme: "Matrix Population Models"— Presentation transcript:

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

2 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)

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

4 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.

5 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.

6 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)

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

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

9 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

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

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

12 (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

13 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

14 (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

15 (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

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

17 Caswell Types of model variability

18 Population projection Example: What is the population at t1?
Juvenile Adult t0 fx px Njuvenile Nadult fx NJuvenile NAdult px

19 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

20 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 =

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

22 Stable age distribution

23 Stable age distribution. Expected in a static environment…

24 Population trajectories with process and observation error
f3 f4 p1 P2 p3

25 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

26 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

27 Brault and Caswell

28 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

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

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

31 Stage-specific survival and reproduction
Initialize matrix A

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

33 Abalone harvest in California
Suseptible to overfishing: Slow growth – reach fishery size of 7” or 178 mm in years Shallow subtidal distribution predominately in intertidal to 7 to 10 meters demand for their meat and shells Spatial and temporal Serial depletion of abalone stocks have been documented in California Commercial fishery closed: White abalone are at 0.1 to 1.0% of historical abundances, pink abalone at less than 0.01% of historical abundances, black abalone at ~1.0% of historical abundances Adapted from Karpov et al. 2000

34 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 Recreational and commercial abalone fisheries in southern California were closed in 1997 Red abalone attain the largest sizes 7 inch limit is estimated to take 12 to 13 years to attain in northern california, I should mention that the ages of very large individuals has not been validated Free dive only fishery, Karpov et al., reports a refuge by depth Reduction in the daily and yearly bag limits bag limit occurred in 2001

35 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?

36 (Size class at first reproduction)
Size based matrix population model Fecundity fx Growth gx 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

37 Elasticity Example if ei,j = 0. 5
Elasticity Example if ei,j = % change in the value of a parameter (ai,j) will result in a 5% increase in the population growth rate. For conservation of a population emphasis should be place on preserving or increasing transitions with larg elasticities.

38 Annual growth CDFG Tagging study 1971 – 1978 (Schultz and DeMartini 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. Mendocino county Pt. Cabrillo (N. and S.) Van Damme SP Pt. Arena Sonoma county We derived annaul growth and survival estimates from tagging data at five locations Growth data from a total of 845 abalone recaptured at one year intervals plus or minus 30 days (335 to 395 days) were used in our analysis of growth transition rates. Growth was normalized for all individuals to one year by determining the daily growth rate and then standardizing to 365 days. We used 27 abalone, ranging in size at the time of tagging from mm, from South Cabrillo Cove. Growth data was obtained from red abalone ranging in size at the time of stocking from 5 to 29 mm (Rogers-Bennett and Pearse 1998). Ft. Ross

39 Growth transition matrix - gx
Size class (mm) t0 0 to 25 25.1 to 50 50.1 to 75 75.1 to 100 100.1 to 125 125.1 to 150 150.1 to 178 178.1 to 200 > 200.1 24 25.1 to 50 15 1 50.1 to 75 75.1 to 100 8 16 100.1 to 125 37 28 125.1 to 150 4 63 71 150.1 to 178 2 76 322 10 178.1 to 200 29 3 22 Size class (mm) t1 Number of indivuals in size class at t0 40 2 10 57 93 147 351 162 25

40 Growth transition frequencies - gx
Size class (mm) t0 0 to 25 25.1 to 50 50.1 to 75 75.1 to 100 100.1 to 125 125.1 to 150 150.1 to 178 178.1 to 200 > 200.1 0.600 25.1 to 50 0.375 0.500 50.1 to 75 0.025 0.100 75.1 to 100 0.800 0.281 100.1 to 125 0.649 0.301 125.1 to 150 0.070 0.677 0.483 150.1 to 178 0.022 0.517 0.917 0.062 178.1 to 200 0.083 0.926 0.120 0.012 0.880 Size class (mm) t1

41 (Size class at first reproduction)
Size based matrix population model Fecundity fx Growth gx 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

42 Annual Survival Estimates
Individual tag number GOF tests variance inflation factor Capture-Mark-Recapture (CMR) Julian Day Annual Model Name Number of individuals Survival Estimate Standard Error ‘cryptic’ (< 100 mm) ‘emergent’ (≥ 100 mm)

43 Incorporation of annual survival rates into growth transition frequencies
Size class (mm) 0 to 25 25 to 50 50 to 75 75 to 100 100 to 125 125 to 150 150 to 178 178 to 200 > 200 0.600 25 to 50 0.375 0.500 50 to 75 0.025 0.100 75 to 100 0.800 0.281 100 to 125 0.649 0.301 125 to 150 0.070 0.677 0.483 150 to 178 0.022 0.517 0.917 0.062 178 to 200 0.083 0.926 0.120 0.012 0.880 x y x y -1

44 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 100 100.1 to 125 125.1 to 150 150.1 to 178 178.1 to 200 > 200.1 0.315 25.1 to 50 0.197 0.263 50.1 to 75 0.013 0.053 75.1 to 100 0.421 0.148 100.1 to 125 0.341 0.158 125.1 to 150 0.037 0.356 0.334 150.1 to 178 0.011 0.357 0.643 0.043 178.1 to 200 0.057 0.640 0.083 0.009 0.608

45 (Size class at first reproduction)
Size based matrix population model Fecundity fx Growth gx 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

46 Fecundity Rogers-Bennett et al. 2004

47 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 100 100.1 to 125 125.1 to 150 150.1 to 178 178.1 to 200 > 200.1 25.1 to 50 0.263 0.5 * 105 x P0 3.8 * 105 58 * 105 141 * 105 50.1 to 75 0.053 75.1 to 100 0.421 0.148 100.1 to 125 0.341 0.158 125.1 to 150 0.037 0.356 0.334 150.1 to 178 0.011 0.357 0.643 0.043 178.1 to 200 0.057 0.640 0.083 0.009 0.608 113 * 105 x P0 So, P0 = 1.5 x 10-6, approximately 1 individual in 665,000 survives to the beginning of their second year

48 Elasticity results Current size limit 178 mm
The proportional elasticity of the red abalone matrix suggests that the matrix element composed of vital rates of remaining in and surviving in the 7th size class ( mm) had the largest elasticity (0.254) and therefore the most influence on population growth rate λ of the model Survival made up 88% of the elasticity value while reproduction comprised 12%. In general, elasticities for survival (px) were much greater than for reproduction and first year survivorship (fx)

49 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?

50 Elasticity results Current size limit 178 mm
The proportional elasticity of the red abalone matrix suggests that the matrix element composed of vital rates of remaining in and surviving in the 7th size class ( mm) had the largest elasticity (0.254) and therefore the most influence on population growth rate λ of the model Survival made up 88% of the elasticity value while reproduction comprised 12%. In general, elasticities for survival (px) were much greater than for reproduction and first year survivorship (fx)

51 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?

52 Elasticity results Current size limit 178 mm
The proportional elasticity of the red abalone matrix suggests that the matrix element composed of vital rates of remaining in and surviving in the 7th size class ( mm) had the largest elasticity (0.254) and therefore the most influence on population growth rate λ of the model Survival made up 88% of the elasticity value while reproduction comprised 12%. In general, elasticities for survival (px) were much greater than for reproduction and first year survivorship (fx)

53 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?

54 Elasticity results Current size limit 178 mm
The proportional elasticity of the red abalone matrix suggests that the matrix element composed of vital rates of remaining in and surviving in the 7th size class ( mm) had the largest elasticity (0.254) and therefore the most influence on population growth rate λ of the model Survival made up 88% of the elasticity value while reproduction comprised 12%. In general, elasticities for survival (px) were much greater than for reproduction and first year survivorship (fx)

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

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