FW364 Ecological Problem Solving

FW364 Ecological Problem Solving Class 15: Stage Structure October 23, 2013

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

Stage Structure Introduction Age structure Distribution of individuals in each age class in a population Stage structure Distribution of individuals in defined stages in a population Works well when: We cannot determine age of individuals, but we can define stages Stage is important for ecology Works well when: We can determine the exact age of individuals Age is important for ecology

Stage Structure Introduction Types of stages Physiological e.g., stages of pregnancy and/or lactation Morphological e.g., changes in defensive traits Size e.g., 0-10mm, 11-20mm fish Developmental e.g., tadpole, frog e.g., larva, pupa, adult …or any combination (often the case)

Stage Structure Introduction When stage can be more important to ecology than age: Example (from text): Forest trees Trees can be defined in stages: seeds, saplings, and canopy trees  Stages are important for ecology • Saplings cannot become canopy trees until a canopy opening occurs • Because canopy openings typically occur by chance (i.e., sporadically), • the age of a tree is not a good predictor of its likelihood of growing to the canopy (and reaching reproductive maturity) • Stage structured model better than age for many trees

Modeling Stage Structure Work our way into a stage structured model by building conceptual diagrams: Case 1: Simple age-structured model Case 2: Composite age class model (type of stage model) Case 3: Stage-structured model

Case 1: Age Structure Leslie matrix for age-structured population Assume 0F = 0

Case 1: Age Structure Leslie matrix for age-structured population Assume 0F = 0 Not going to assume 1F = 2F = 3F… With age structure, there are limits to the types of transitions allowed: Individuals can only: Move into next higher age class e.g., Age 1  Age 2 Produce offspring e.g., Age 2  Age 0 With stage structure, more transitions are possible Some plant species can fill every stage transition in a Leslie matrix!

Case 1: Age Structure Leslie matrix for age-structured population Assume 0F = 0 Not going to assume 1F = 2F = 3F… Let’s build a conceptual model Can use box and arrow diagrams to represent age/stage transitions 0S 1S Age 1 Age 2 Age 0 2F Conceptually similar to stock and flow models: Boxes are age (stage) classes (like “stocks”) Arrows are transitions between classes (like “flows”) NOTE: we are NOT doing / assuming “mass balance” or T = S/F

Case 1: Age Structure Leslie matrix for age-structured population Assume 0F = 0 Not going to assume 1F = 2F = 3F… Let’s build a box and arrow model that corresponds to the Leslie matrix above How many boxes (i.e., age classes) will the model have?  5 age classes How many arrows (i.e., transitions) will the model have?  8 transitions 0S 1S 2S 3S Age 1 Age 2 Age 3 Age 0 Age 4 1F 2F 3F 4F

Case 2: Composite Age Class In some situations, it might be advantageous to pool multiple age classes into a single age class (often pool oldest classes) e.g., the honeyeater 3+ age class We are justified in pooling age classes when vital rates for classes do not vary For many organisms (mammals, birds), - Individuals do not grow after reaching maturity - Fecundity and survival may not be very different after reaching maturity  Can pool age classes into a composite age class to simplify modeling e.g., Pool age 3 individuals and older into class for age 3+ age 0 age 1 age 2 age 3+

Case 2: Composite Age Class In some situations, it might be advantageous to pool multiple age classes into a single age class (often pool oldest classes) e.g., the honeyeater 3+ age class We are justified in pooling age classes when vital rates for classes do not vary age 0 age 1 age 2 age 3+ • Composite age classes (e.g., the honeyeater age 3+ class) • are not really true age classes • Rather, composite age classes are a type of stage class • Let’s build a conceptual model with a composite age class

Case 2: Composite Age Class How would our conceptual model below change if we pooled ages 3 and 4? • We would have: Fecundity for Age 3+ and age 1 and 2 Box for Age 3+ and age 0, 1, and 2 Survival within Age 3+ and other between-class survivals 0S 0S 1S 1S 2S 2S 3S 3+S Age 1 Age 1 Age 2 Age 2 Age 3+ Age 3 Age 0 Age 0 Age 4 1F 1F “Retention” in age 3+ 2F 2F 3F 3+F 4F

Case 2: Composite Age Class 3+S There are two possible transitions to the age 3+ class: - From age 2 to age 3+ (2S) - Age 3+ remaining in age 3+ (3+S) To make the transitions more explicit, we’ll use a new notation for survivals: 33S 01S 0S 12S 1S 23S 2S Age 1 Age 1 Age 2 Age 2 Age 3+ Age 3+ Age 0 Age 0 1F 1F 2F 2F 3+F 3+F

Case 2: Composite Age Class 3+S With the new notation: 01S : Survival from age 0 to age 1 12S : Survival from age 1 to age 2  NOT survival for age 12! 23S : Survival from age 2 to age 3 NOT survival for age 23! 33S : Survival within age 3+  NOT survival for age 33! Flexible notation for stages; also may help with remembering transitions There are two possible transitions to the age 3+ class: - From age 2 to age 3+ (2S) - Age 3+ remaining in age 3+ (3+S) To make the transitions more explicit, we’ll use a new notation for survivals: 33S 01S 0S 12S 1S 23S 2S Age 1 Age 1 Age 2 Age 2 Age 3+ Age 3+ Age 0 Age 0 1F 1F 2F 2F 3+F 3+F

Case 2: Composite Age Class 33S • In-Class Challenge: • What would a Leslie matrix look like with a composite age class? • See if you can figure out the (general) Leslie matrix for the model above 01S 12S Some hints / guiding questions: How many rows and columns will the matrix have? Where will 33S go?  Talking through the transitions for each matrix element will help 23S Age 1 Age 2 Age 3+ Age 0 1F 2F 3+F

Case 2: Composite Age Class 33S Leslie matrix with composite age class: How many rows and columns? Where does 33S go? 01S 12S 23S Age 1 Age 2 Age 3+ Age 0 1F 2F 3+F

Case 2: Composite Age Class 33S Leslie matrix with composite age class: Key Features: New notation for survivals 33S goes on the diagonal All “within class” rates go on diagonal Fecundities do not look very different 01S 12S 23S Age 1 Age 2 Age 3+ Age 0 1F 2F 3+F  We can now take this general form and build a full stage-structure model

Case 3: Stage Structure Stage structure example: Leopard Frogs – 4 stages Tadpoles and frogs can remain in a stage for more than one time step:  We’ll have transitions within Stages 2, 3, and 4 Stage 1: Eggs Stage 2: Tadpole 1 Stage 3: Tadpole 2 (tadpole with legs, but still aquatic) Stage 4: Frog (adult; only mature stage)

Case 3: Stage Structure Leopard frog conceptual model: 22S 11S 33S And corresponding Leslie matrix: Key Features: Fecundity for frogs only (3F) Within stage survivals on the diagonal: 11S, 22S, 33S Between stage survivals on sub-diagonal: 01S, 12S, 23S All other transitions are zeros (not possible for leopard frogs) 01S 12S 23S Tadpole 1 Tadpole 2 Frog (3) Egg (0) 3F

Stage Structure – Forecasting Growth As we did for age structure, we can forecast stage-structured population growth using: - Stage structured Leslie matrix - Vector of stage sizes (xNt) (number of individuals, N, at stage class, x, at time, t) Leopard frog Leslie matrix xNt Each element is the number of individuals in each stage Let’s do some examples

Stage Structure – Forecasting Growth Let’s start with a population of solely 1000 stage 1 tadpoles: xNt Stage 0 : Eggs Stage 1 : Tadpole 1 Stage 2 : Tadpole 2 Stage 3 : Adult We need to put some numbers in the Leopard frog Leslie matrix

Stage Structure – Forecasting Growth In-Class Exercise: Forecasting to time t+1 xNt+1 xNt Leopard frog Leslie matrix Nt = 1000 Nt+1 = ?

Stage Structure – Forecasting Growth In-Class Exercise: Forecasting to time t+1 xNt+1 xNt Leopard frog Leslie matrix Nt = 1000 Nt+1 = 200 80% population decline

Stage Structure – Forecasting Growth Forecasting to time t+2 xNt+2 xNt+1 Leopard frog Leslie matrix Nt+1 = 200 Nt+2 = 65 68% population decline

Stage Structure – Forecasting Growth Forecasting to time t+3 xNt+3 xNt+2 Leopard frog Leslie matrix Nt+2 = 65 Nt+3 = 526 709% population increase

Stage Structure – Forecasting Growth Forecasting to time t+4 xNt+4 xNt+3 Leopard frog Leslie matrix Nt+3 = 526 Nt+4 = 561 If we kept going, we’d find out the population starts steadily shrinking!  Try on your own 7% population increase

Stage Structure – Forecasting Growth Let’s look a bit closer at the realism of the leopard frog Leslie matrix Leopard frog Leslie matrix • These were just • made-up data… • … let’s consider the ecological plausibility Would we expect 11S (0.1) to be less than 22S (0.4)? Means “retention” of stage 1 tadpoles as stage 1 is less than “retention” of stage 2 tadpoles as stage 2 (i.e., individuals stay in stage 2 tadpole longer than stage 1 tadpole)  Depends on ecology we don’t know (but we could research)

Stage Structure – Forecasting Growth Let’s look a bit closer at the realism of the leopard frog Leslie matrix Leopard frog Leslie matrix • These were just • made-up data… • … let’s consider the ecological plausibility Would we expect 12S (0.1) to be greater than 23S (0.05)? Means survival from stage 1 tadpole to stage 2 tadpole is greater than survival of stage 2 tadpole to adults  Maybe not too likely; survival is typically better at later stages… … although for the frog example, 23S represents an important transition from water to land, which could involve high mortality

Stage Structure – Forecasting Growth Let’s look a bit closer at the realism of the leopard frog Leslie matrix Leopard frog Leslie matrix • These were just • made-up data… • … let’s consider the ecological plausibility • Are there other general restrictions on the components of a stage-structured Leslie matrix? • All survival rates must be less than 1

Stage Structure – Forecasting Growth Let’s look a bit closer at the realism of the leopard frog Leslie matrix Leopard frog Leslie matrix • These were just • made-up data… • … let’s consider the ecological plausibility • Are there other general restrictions on the components of a stage-structured Leslie matrix? • All survival rates must be less than 1 • Survival column sums must also be less than 1! • e.g., 11S + 12S + 13S < 1  0.1 + 0.1 + 0 < 1

Stage Structure – Forecasting Growth Let’s look a bit closer at the realism of the leopard frog Leslie matrix Leopard frog Leslie matrix • These were just • made-up data… • … let’s consider the ecological plausibility General point: It’s good to think about the ecological plausibility of any Leslie matrix you build… … and more generally, any model you build!

Looking Ahead Next Class: Continue with stage structure Practice stage structure exercises

FW364 Ecological Problem Solving