FW364 Ecological Problem Solving Class 16: Stage Structure October 28, 2013

Outline for Today Continue to make population growth models more realistic by adding in stage structure Last Class: Introduced stage structure Objectives for Today: Show how to obtain stage (Leslie) matrices from census data Complete stage structure exercises Text (optional reading): Chapter 5

We considered a stage-structured population of leopard frogs: Recap from Last Class We considered a stage-structured population of leopard frogs: Stage 0 : Eggs Four stages: Stage 1 : Tadpole 1 Can remain in a stage for more than one time step Stage 2 : Tadpole 2 Only adult stage reproduces Stage 3 : Adult frog We created a box and arrow diagram (i.e., conceptual model) for leopard frogs: 11S 22S 33S 01S 12S 23S Egg (0) Tadpole 1 Tadpole 2 Frog (3) 3F

Recap from Last Class 11S 22S 33S We created a general Leslie matrix: And also filled in specific numbers: 3F 01S 11S 12S 22S 23S 33S 100 0.3 0.1 0.4 0.05 0.2  We created a box and arrow diagram (i.e., conceptual model) for leopard frogs: 11S 22S 33S 01S 12S 23S Egg (0) Tadpole 1 Tadpole 2 Frog (3) 3F

Recap from Last Class Saw examples of forecasting population size using a vector of stage sizes: Leopard frog Leslie matrix xNt xNt+1 100 0.3 0.1 0.4 0.05 0.2 1000 100 = * Nt = 1000 Nt+1 = 200 One item we did not address: How do we get data to build a stage structured Leslie matrix?

Building a Stage Matrix How do we get data to build a stage structured Leslie matrix? First, need to decide on stages for a population For our leopard frog model: Stage 0 : Eggs Stage 1 : Tadpole 1 Stage 2 : Tadpole 2 Stage 3 : Adult frogs Then follow individuals through at least two time steps (time t to t+1) Can be tricky for some organisms! Record stage of all individuals at each time step Let’s look at an example Need to record mortality as well

Building a Stage Matrix stage at time t 1 2 3 1 2 3 9 2 8 1 Table of census data that could be collected for a frog population across two years stage at time t+1 Deaths 21 16 11 8 Total 30 20 10 Stage 0 : Eggs Stage 1 : Tadpole 1 Stage 2 : Tadpole 2 Stage 3 : Adult frogs

Building a Stage Matrix stage at time t Columns for stage of individuals at time t (starting stage-structured population) 1 2 3 1 2 3 9 2 8 1 Rows for stage of individuals at time t+1 (ending stage-structured population, only considering survival, not fecundity) stage at time t+1 Deaths 21 16 11 8 Number of deaths for each stage at time t Total 30 20 10 Total number for each stage at time t Data in the table represent number of individuals making each transition, for example: 9 individuals transitioned from stage 0 to stage 1 2 individuals remained in stage 1 2 individuals transitioned from stage 1 to stage 2 Stage 0 : Eggs Stage 1 : Tadpole 1 Stage 2 : Tadpole 2 Stage 3 : Adult frogs

Building a Stage Matrix Leopard frog Leslie matrix stage at time t Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 stage at time t+1 0.3 0.1 Deaths 21 16 11 8 Total 30 20 10 To construct the Leslie matrix: Take each number in the table and divide by the total number for each stage Result goes in corresponding survival rate position in the Leslie matrix E.g., 9 / 30 = 0.3 = 01S 2 / 20 = 0.1 = 11S

Building a Stage Matrix stage at time t Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 X 0.3 0.1 0.4 0.05 0.2 stage at time t+1 Deaths 21 16 11 8 Total 30 20 10 To construct the Leslie matrix: Take each number in the table and divide by the total number for each stage Result goes in corresponding survival rate position in the Leslie matrix E.g., 9 / 30 = 0.3 = 01S 2 / 20 = 0.1 = 11S That’s how we get survivals… … now we need fecundities

Building a Stage Matrix Leopard frog Leslie matrix stage at time t Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 X 0.3 0.1 0.4 0.05 0.2 stage at time t+1 Deaths 21 16 11 8 Total 30 20 10 To obtain fecundities: Determine the reproductive stages  For leopard frogs, only adults reproduce

Building a Stage Matrix Leopard frog Leslie matrix stage at time t Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 X 0.3 0.1 0.4 0.05 0.2 stage at time t+1 Deaths 21 16 11 8 Total 30 20 10 To obtain fecundities: Determine the reproductive stages Count # stage-0 individuals at time t+1 Divide # stage-0 individuals at time t+1 by # adults at time t  For leopard frogs, only adults reproduce  We’ll say there are 1000 eggs  1000 eggs / 10 adults = 100

Building a Stage Matrix Leopard frog Leslie matrix stage at time t Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 100 0.3 0.1 0.4 0.05 0.2 stage at time t+1 Deaths 21 16 11 8 Total 30 20 10 To obtain fecundities: Determine the reproductive stages Count # stage-0 individuals at time t+1 Divide # stage-0 individuals at time t+1 by # adults at time t  For leopard frogs, only adults reproduce  We’ll say there are 1000 eggs  1000 eggs / 10 adults = 100  3F = 100

Building a Stage Matrix stage at time t Leopard frog Leslie matrix 1 2 3 1 2 3 9 2 8 1 100 0.3 0.1 0.4 0.05 0.2 stage at time t+1 Deaths 21 16 11 8 Total 30 20 10 Note: I want you to be aware of how a Leslie matrix for stage-structured data can be obtained, but I will not test you on how to create a stage-structured Leslie matrix from census data

Age and Stage Structure Summary Age structure is just a special case of stage structure where all individuals transition between stages in exactly one time step The result: no within stage survivals for age structure i.e., the diagonal (except for fecundity) is always 0 for age structure 0F 1F 2F 3F 4F 0S 1S 2S 3S Age and stage structure are sources of deterministic variation i.e., predictable variation (not stochastic) (although we can add stochasticity, if desired)

Age and Stage Structure Summary We’ve been making a number of ASSUMPTIONS: Age structure: Vital rates for individuals (fertilities and survival chances) are related to their age Among individuals of the same age, there is little variation in the vital rates (relative to variation between ages) Stage structure: Vital rates for individuals (fertilities and survival chances) are related to their stage Among individuals of the same stage, there is little variation in the vital rates (relative to variation between stages) For both age and stage structure: Working in closed populations  could include immigration or emigration if desired No environmental or demographic stochasticity  could include if desired (Ramas) No density dependence of vital rates  could include if desired (Ramas)

(note: all of these problems will be posted on the website) Let’s do some in-class exercises (note: all of these problems will be posted on the website)

Many insect populations have 4 stages: egg, larvae, pupae, and adult Exercises Many insect populations have 4 stages: egg, larvae, pupae, and adult Gypsy moth life cycle http://www.youtube.com/watch?v=N1FJjS7K_7I http://www.cbsnews.com/video/watch/?id=2921074n

Exercises Consider a gypsy moth population with the following vital rates: Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die (i.e., no eggs ever stay an egg after one time step) Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce) Exercises Draw a box and arrow diagrams that illustrate: A. All of these stages and transitions conceptually (just use symbols for S and F; use E, L, P, and A for stage notations) B. All of these stages and transitions using numbers from above

Exercises ELS LPS PAS LLS PPS AAS AF Consider a gypsy moth population with the following vital rates: Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die (i.e., no eggs ever stay an egg after one time step) Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce) Exercise A: Conceptual model with symbols Egg Larvae Pupae Adult ELS LPS PAS LLS PPS AAS AF

Exercises Consider a gypsy moth population with the following vital rates: Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die (i.e., no eggs ever stay an egg after one time step) Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce) Exercise B: Conceptual model with numbers Egg Larvae Pupae Adult 0.3 0.5 0.2 0.1 0.4 100

Exercises Exercises Con’t Consider a gypsy moth population with the following vital rates: Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die (i.e., no eggs ever stay an egg after one time step) Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce) Exercises Con’t C. Construct the Leslie matrix for this gypsy moth population D. If there are currently 2000 eggs, 200 larvae, 100 pupae, and 100 adults, how many individuals of each stage will there be next year (i.e., in the next time step)?

Exercises Exercises C: Gypsy moth Leslie matrix Consider a gypsy moth population with the following vital rates: Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die (i.e., no eggs ever stay an egg after one time step) Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce) Exercises C: Gypsy moth Leslie matrix 100 0.3 0.2 0.5 0.1 0.4

Exercises Exercises D: Forecasting growth Consider a gypsy moth population with the following vital rates: Eggs have a 0.3 probability of hatching; if eggs do not hatch, they die (i.e., no eggs ever stay an egg after one time step) Larvae and pupae have a 0.2 and 0.1 probability, respectively, of surviving and staying in the same stage, and both have a 0.5 probability of surviving and moving to the next stage Adults have a 0.4 probability of surviving and have a per capita fecundity of 100 (larvae and pupae do not reproduce) Exercises D: Forecasting growth Starting with: 2000 eggs 200 larvae 100 pupae 100 adults 100 0.3 0.2 0.5 0.1 0.4 2000 200 100 10000 640 110 90 * =

Exercises In the stage-structured Leslie matrix below, there is one unknown transition (labeled with an X): 75 0.2 X 0.4 0.3 0.5 Which of the following could be the value of that transition (more than one answer may be correct)? a. 0 b. 1.2 c. 0.3 d. 0.8

Exercises In the stage-structured Leslie matrix below, there is one unknown transition (labeled with an X): 75 0.2 X 0.4 0.3 0.5 Which of the following could be the value of that transition (more than one answer may be correct)? a. 0 b. 1.2 c. 0.3 d. 0.8

(i.e., spatial structure) Looking Ahead Next Two Classes: Metapopulations (i.e., spatial structure) Lab Tomorrow