FW364 Ecological Problem Solving

Class 16: Stage StructureOctober 28, 2013

Continue to make population growth models more realisticby adding in stage structure

Last Class:Introduced stage structure

Objectives for Today:Show how to obtain stage (Leslie) matrices from census dataComplete stage structure exercises

Text (optional reading):Chapter 5

Outline for Today

We considered a stage-structured population of leopard frogs:

Recap from Last Class

We created a box and arrow diagram (i.e., conceptual model) for leopard frogs:

Stage 0 : Eggs

Stage 1 : Tadpole 1

Stage 2 : Tadpole 2

Stage 3 : Adult frogOnly adult stage reproduces

Four stages:

Egg (0) Tadpole 1 Tadpole 201S 12S 23S Frog (3)

3F

33S22S11S

Can remain in a stage for

more than one time step

0 0 0 3F

01S 11S 0 0

0 12S 22S 0

0 0 23S 33S

Recap from Last Class

We created a general Leslie matrix:

0 0 0 1000.3 0.1 0 00 0.1 0.4 00 0 0.05 0.2

And also filled in specific numbers:

We created a box and arrow diagram (i.e., conceptual model) for leopard frogs:

Egg (0) Tadpole 1 Tadpole 201S 12S 23S Frog (3)

3F

33S22S11S

Recap from Last Class

Leopard frog Leslie matrix

01000

00

xNt

=*

0100100

0

xNt+1

0 0 0 1000.3 0.1 0 00 0.1 0.4 00 0 0.05 0.2

Nt+1 = 200Nt = 1000

Saw examples of forecasting population size usinga vector of stage sizes:

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

Stage 0 : Eggs

Stage 1 : Tadpole 1

Stage 2 : Tadpole 2

Stage 3 : Adult frogs

For our leopard frog model:

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

Need to record mortality as wellLet’s look at an example

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

Stage 0 : EggsStage 1 : Tadpole 1Stage 2 : Tadpole 2Stage 3 : Adult frogs

Building a Stage Matrix

Table of census data that could be collected for a frog population across two years

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

Stage 0 : EggsStage 1 : Tadpole 1Stage 2 : Tadpole 2Stage 3 : Adult frogs

Columns for stage of individuals at time t(starting stage-structured population)

Rows for stage of individuals at time t+1(ending stage-structured population,only considering survival, not fecundity)

Number of deaths for each stage at time t

Total number for each stage at time t

Data in the table represent number of individualsmaking each transition, for example:

9 individuals transitioned from stage 0 to stage 12 individuals remained in stage 12 individuals transitioned from stage 1 to stage 2

Building a Stage Matrix

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

To construct the Leslie matrix:Take each number in the table and divide by the total number for each stageResult goes in corresponding survival rate position in the Leslie matrix

E.g., 9 / 30 = 0.3 = 01S2 / 20 = 0.1 = 11S

Leopard frog Leslie matrix

0.3 0.1

Building a Stage Matrix

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

Leopard frog Leslie matrix

To construct the Leslie matrix:Take each number in the table and divide by the total number for each stageResult goes in corresponding survival rate position in the Leslie matrix

E.g., 9 / 30 = 0.3 = 01S2 / 20 = 0.1 = 11S

X X X X

0.3 0.1 0 0

0 0.1 0.4 0

0 0 0.05 0.2

That’s how we get survivals…… now we need fecundities

Building a Stage Matrix

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

Leopard frog Leslie matrix

To obtain fecundities:Determine the reproductive stages

X X X X

0.3 0.1 0 0

0 0.1 0.4 0

0 0 0.05 0.2

For leopard frogs, only adults reproduce

Building a Stage Matrix

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

Leopard frog Leslie matrix

To obtain fecundities:Determine the reproductive stagesCount # stage-0 individuals at time t+1Divide # stage-0 individuals at time t+1 by # adults at time t

0 0 0 X

0.3 0.1 0 0

0 0.1 0.4 0

0 0 0.05 0.2

We’ll say there are 1000 eggs 1000 eggs / 10 adults = 100

For leopard frogs, only adults reproduce

Building a Stage Matrix

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

Leopard frog Leslie matrix

To obtain fecundities:Determine the reproductive stages Count # stage-0 individuals at time t+1Divide # stage-0 individuals at time t+1 by # adults at time t

0 0 0 100

0.3 0.1 0 0

0 0.1 0.4 0

0 0 0.05 0.2

3F = 100

We’ll say there are 1000 eggs 1000 eggs / 10 adults = 100

For leopard frogs, only adults reproduce

Building a Stage Matrix

0 0 0 0

9 2 0 0

0 2 8 0

0 0 1 2

0 1 2 3

0

1

2

3

stage at time t

stage at time t+1

21 16 11 8

30 20 20 10

Deaths

Total

Leopard frog Leslie matrix

0 0 0 100

0.3 0.1 0 0

0 0.1 0.4 0

0 0 0.05 0.2

Note:

I want you to be aware of how a Leslie matrixfor stage-structured data can be obtained,

but I will not test you on how to create a stage-structuredLeslie matrix from census data

Building a Stage Matrix

Age structure is just a special case of stage structurewhere all individuals transition between stages in exactly one time step

The result: no within stage survivals for age structurei.e., the diagonal (except for fecundity) is always 0 for age structure

Age and Stage Structure Summary

0F 1F 2F 3F 4F

0S 0 0 0 0

0 1S 0 0 0

0 0 2S 0 0

0 0 0 3S 0

Age and stage structure are sources of deterministic variationi.e., predictable variation (not stochastic)

(although we can add stochasticity, if desired)

We’ve been making a number of ASSUMPTIONS:

Age structure:Vital rates for individuals (fertilities and survival chances) are related to their ageAmong 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 stageAmong 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 desiredNo environmental or demographic stochasticity could include if desired (Ramas)No density dependence of vital rates could include if desired (Ramas)

Age and Stage Structure Summary

Let’s do somein-class exercises

(note: all of these problems will be posted on the website)

Many insect populations have 4 stages: egg, larvae, pupae, and adult

Gypsy mothlife cycle

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

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

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

Exercise A: Conceptual model with symbols

Egg Larvae Pupae AdultELS 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)

Exercises

Exercise B: Conceptual model with numbers

Egg Larvae Pupae Adult0.3 0.5 0.5

0.2 0.1 0.4

100

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

Exercises Con’t

C. Construct the Leslie matrix for this gypsy moth populationD. 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)?

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

Exercises C: Gypsy moth Leslie matrix

0 0 0 100

0.3 0.2 0 0

0 0.5 0.1 0

0 0 0.5 0.4

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

Exercises D: Forecasting growth

0 0 0 100

0.3 0.2 0 0

0 0.5 0.1 0

0 0 0.5 0.4

2000

200

100

100

10000

640

110

90

* =

Starting with: 2000 eggs 200 larvae 100 pupae 100 adults

Exercises

In the stage-structured Leslie matrix below, there is one unknown transition (labeled with an X):

0 0 0 750.2 X 0 00 0.4 0.3 00 0 0.5 0.4

a. 0b. 1.2c. 0.3d. 0.8

Which of the following could be the value of that transition (more than one answer may be correct)?

Exercises

In the stage-structured Leslie matrix below, there is one unknown transition (labeled with an X):

0 0 0 750.2 X 0 00 0.4 0.3 00 0 0.5 0.4

Which of the following could be the value of that transition (more than one answer may be correct)?

a. 0b. 1.2c. 0.3d. 0.8

Looking Ahead

Next Two Classes:

Metapopulations(i.e., spatial structure)

Lab Tomorrow