Structured PVA 1 Vital rates (Processes that contribute to change - PowerPoint PPT Presentation

Structured PVA 1

Vital rates (Processes that contribute to change in population size) Birth and death rates Growth rate Fecundity Vital rates often depend on age and size 2

Survival rate depends on age Hydra 3

Plant fecundity depends on size Ln(number of seeds) Plant size 4

Types of PVA’s Count based: simple -- all individuals are the same (age, size, etc.) Structured (demographic): different vital rates for different classes of individuals 5

Structured (demographic) models Age-structured - use data on each age group 6

Structured (demographic) models Age-structured - use data on each age group Stage structured - used data on size or stage groups Adults > 40 cm Juveniles 20 < x < 40 cm Tadpoles < 20 cm 0 25 50 75 100 0 12.5 25.0 37.5 50.0 Individuals Individuals 7

Building a stage structured model Understand your species Decide how many stages to include 8

Building a stage structured model (for loggerhead sea turtles) 9

Building a stage structured model (for loggerhead sea turtles) 10

mating near shore foraging open ocean nesting on beaches 11

How many stages to include? Biological Intuition - stages should differ in vital rates from other stages What the data will allow - balance accuracy of more stages with amount of available data 12

For turtle PVA we use 5 stages Hatchlings (and eggs): first year Small juveniles: 1-7 years Large juveniles: 8-15 years Subadults 16-21 years (mostly non-breeding) Mature adults 22-55 years, breeding 13

Life-cycle diagram Small juveniles Nestlings 14

Life-cycle diagram Transition rate Stage Large juveniles Small Subadults juveniles Nestlings Adults 15

Life-cycle diagram Large juveniles Small Subadults juveniles Nestlings Adults 16

Building a stage structured model Understand your species Decide how many stages to include Gather data 17

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Turtle data Small Large Mature Nestlings Subadults juveniles juveniles adults Marked in year 1 1000 1000 1000 1000 1000 Recaptured in 0 703 657 682 809 same class Recaptured in 675 47 19 61 - next larger class Eggs/female/year 0 0 0 4.665 61.896 20

Building a stage structured model Understand your species Decide how many stages to include Gather data Calculate transition rates Fractions surviving but not growing Fractions surviving and growing Number of female offspring per year and female 21

Life-cycle diagram Large juveniles Small Subadults juveniles Nestlings Adults 22

Turtle data Small Large Mature Nestlings Subadults juveniles juveniles adults Marked in year 1 1000 1000 1000 1000 1000 Recaptured in 0 703 657 682 809 same class Recaptured in 675 47 19 61 - next larger class Eggs/female/year 0 0 0 4.665 61.896 23

Life-cycle diagram Large juveniles Small Subadults juveniles 0.675 Nestlings Adults 24

Life-cycle diagram Large juveniles Small Subadults juveniles 0.675 Nestlings Adults 25

Turtle data Small Large Mature Nestlings Subadults juveniles juveniles adults Marked in year 1 1000 1000 1000 1000 1000 Recaptured in 0 703 657 682 809 same class Recaptured in 675 47 19 61 - next larger class Eggs/female/year 0 0 0 4.665 61.896 26

Life-cycle diagram 0.703 Large juveniles Small Subadults juveniles Nestlings Adults 27

Life-cycle diagram 0.657 0.703 0.682 Large juveniles 0.047 0.019 Small Subadults juveniles 4.665 0.061 0.675 Nestlings Adults 0.809 61.896 0.703 28

Building a stage structured model Understand your species Decide how many stages to include Gather data Calculate transition rates Make model 29

Population (Projection) matrix The projection matrix is the summary of all transition probabilities (all vital rates) 30

Population (Projection) matrix Number of new turtles (size class 1) produces by F i an average individual of size i per year Fraction of size i turtles surviving and STAYING in S i the same size class per year Fraction of size i turtles surviving and GROWING G i to size class i+1 per year 31

A generic projection matrix Size this year 1 2 3 4 5 1   S 1 F 2 F 3 F 4 F 5 Size 2 G 1 S 2 0 0 0   next   3 G 2 S 3 0 0 0   year   4 G 3 S 4 0 0 0   5 G 4 S 5 0 0 0 F i new S i surviving G i advancing 32

Population (Projection) matrix Number of new turtles (size class 1) produces by F i an average individual of size i per year Fraction of size i turtles surviving and STAYING in S i the same size class per year Fraction of size i turtles surviving and GROWING G i to size class i+1 per year Note that since S and G are fractions surviving. They are between 0 and 1. 33

Projection matrix for loggerhead sea turtles Size this year 1 2 3 4 5 1   0 0 0 4 . 665 61 . 896 Size 2 0 . 675 0 . 703 0 0 0   next   3 0 0 . 047 0 . 657 0 0   year   4 0 0 0 . 019 0 . 682 0   5 0 0 0 0 . 061 0 . 8091 34

Life-cycle diagram 0.657 0.703 0.682 Large juveniles 0.047 0.019 Small Subadults juveniles 4.665 0.061 0.675 Nestlings Adults 0.809 61.896 0.703 35

recall count based method N t = λ N t − 1 36

Structured model N t = PN t − 1 37

Stage distribution vector a column showing the number (or density) of individuals in each stage   Nestlings 23 . 85 Small juveniles 64 . 78     Large juveniles 10 . 33   Subadults   0 . 73   Adults 0 . 31 100.00 Total density 38

Stable stage (or age or size) distribution distribution of individuals among stages that won’t change over time (if population size changes at a constant rate) Example: 100% of individuals in stage 1 is not stable – the next year there will be individuals in other stages 39

Stable stage (or age or size) distribution distribution of individuals among stages that won’t change over time (if population size changes at a constant rate) Example: 100% of individuals in stage 1 is not stable – the next year there will be individuals in other stages Stage distribution will converge to the stable stage distribution over time 40

N t = PN t − 1 41

N t − 1 N t P       0 0 0 4 . 665 61 . 896 23 . 85 0 . 675 0 . 703 0 0 0 64 . 78             ? = 0 0 . 047 0 . 657 0 0 10 . 33             0 0 0 . 019 0 . 682 0 0 . 73       0 0 0 0 . 061 0 . 8091 0 . 31 Use matrix algebra..... 42

N t − 1 N t P       22 . 59 0 0 0 4 . 665 61 . 896 23 . 85 61 . 64 0 . 675 0 . 703 0 0 0 64 . 78             9 . 83 0 0 . 047 0 . 657 0 0 10 . 33 =             0 . 69 0 0 0 . 019 0 . 682 0 0 . 73       0 . 30 0 0 0 0 . 061 0 . 8091 0 . 31 43

Eggs Juveniles # Large juveniles Subadults Adults Time 44

Same graph as last slide, but changing Eggs scale on y-axis Juveniles # Large juveniles Subadults Adults Time 45

Eggs Stable stage distribution Juveniles Large juveniles Subadults Freq Adults Time 46

N t − 1 N t P       22 . 59 0 0 0 4 . 665 61 . 896 23 . 85 61 . 64 0 . 675 0 . 703 0 0 0 64 . 78             9 . 83 0 0 . 047 0 . 657 0 0 10 . 33 =             0 . 69 0 0 0 . 019 0 . 682 0 0 . 73       0 . 30 0 0 0 0 . 061 0 . 8091 0 . 31 How do we know if population is growing or shrinking? 47

Recall that: N t λ = N t − 1 48

N t − 1 N t P       22 . 59 0 0 0 4 . 665 61 . 896 23 . 85 61 . 64 0 . 675 0 . 703 0 0 0 64 . 78             9 . 83 0 0 . 047 0 . 657 0 0 10 . 33 =             0 . 69 0 0 0 . 019 0 . 682 0 0 . 73       0 . 30 0 0 0 0 . 061 0 . 8091 0 . 31 95.05 100.0 95.05/100 = 0.9505 = � 49

lambda Time 50

� again In a count based model N t = λ N t − 1 In a structured model N t = PN t − 1 P is playing the same role as the count based � . 51

� again In a count based model N t = λ N t − 1 In a structured model N t = PN t − 1 P is playing the same role as the count based � . The information in P can be summarized by a matrix � (dominant eigenvalue) 52

In structured models, change in N is still called � but can be measured in two ways N t /N t − 1 Summarize the information P as a single number, the dominant eigenvalue � . 53

In structured models, change in N is still called � but can be This only will be constant if the population is at the stable stage distribution, variable N t /N t − 1 until then Summarize the information P as a single number, the dominant eigenvalue � . 54