Understanding Structured Populations and Life Histories: A Demographic Analysis

2. Characterizing (structured) populations How fast does a population increase/decrease? (r, lambda) What will the population size be next year? (management) • Need to summarize Births & Deaths (age structure) of a population • Cohort Life Tables • follow one group from birth until the last one dies • Static Life Table • census population for abundance in each age/stage combined with estimates of survival and reproductive output by age/stage • Limitations? • Must be able to age/stage each organism (uncertainty?)

3. Survivorship curves (age-specific survival) Which type(s) of survivorship suggest a need for a structured model?

4. Structured demographic models • Translate variation in individuals (measureable) to population dynamics (less measurable) • Structured around life-history schedules of different species or populations • Demography (vital rates by age, sex, stage, size, etc.) captures both the dynamics & structure of populations • Widely adopted to evaluate species vulnerability (Population Viability Analysis, late 1990’s )

5. Life history components (esp. reproduction & lifespan) • Maturity- age at 1st reproduction • Mosquito (14 days), Desert tortoise (25-30 years) • Parity- # of episodes for reproduction • Sockeye salmon (1), White footed mouse (4-12) • Semelparity vs. Iteroparity (annual, perennial) • Fecundity- # offspring/episode • Elephant (1), Western Toad (8,000-15,000) • Aging/Senescence- survival/life span - Fruit fly (35 d), Blue Whale (80-90 y)

6. Not all combinations are possible... (aka tradeoffs) Log 2 eggs x 2 / yr 20d parental care Lifespan 2-3yr Log 1 egg/ ~2yr 9 mo. parental care Lifespan >60yr

7. Life history of Cascades frogs • 4 life stages: • Embryonic: lasts 1-3 weeks, • 0-80 (~60)% survival • Larval (tadpole): lasts 8-12 weeks • ~50% survival • Juvenile (metamorph): lasts 2-3 years • (10-40??)% annual survival • Adult: lasts 8-20+ years, 300-700 offspring/yr • 70-80% annual survival Rana cascade

8. Life history diagram for Cascades frogs Vital rates, also called transition probabilities 300-700/yr Embryos Adults Larvae Juveniles x 0.6 0.5 1 - x 0.7-0.8 0.1-0.4 4-7 yr 2-3 yr 1 yr

9. Life history of elephants • 5 life stages: • Yearling: 80% survival, lasts 1 year • Pre-reproductive: 98% survival, lasts 15 year • Early reproductive: 98% survival, lasts 5 years, 0.08 offspring/yr • Middle reproductive: 95% survival, lasts 25 years, 0.3 offspring/yr • Post-reproductive: 80% survival, 5-25 years

10. 0.8 0.95 0.98 0.98 0.8 Life history diagram for elephants 0.1/yr 0.08/yr Post-reproductive Middle age Pre-reproductive Early Repro Yearling x x x 1- x 1- x 1- x 1 yr 5 yr 25 yr 5-25 yr 15 yr

11. Other ways to depict life cycle Age-based Stage-based Size-based

12. Life histories • Bubble diagrams summarize average life history events • with fixed time steps (survival per week, year, decade) • Result of natural selection • Organisms exist to maximize lifetime reproductive success • Represent successful ways of allocating limited resources to carry out various functions of living organisms • Survival, growth, reproduction

13. Questions we can answer with age/stage structured population models: • How much harvest of a population can occur while still have less than an X% chance of extinction? • What life history stages should conservation (or eradication) efforts be focused on to achieve the biggest change in population size/growth rate? • How many (sub)populations of a species need be preserved to ensure reasonable protection from periodic local extinctions and infrequent catastrophic events? • Which life-history strategies are more or less vulnerable to anthropogenic impacts and extinction? • Is it worth the effort to try and recover a particular population, or is it so likely to go extinct that limited resources should be invested elsewhere? • and hundreds more!

14. Single species population growth models

15. Matrix population models

16. Life history diagram for Cascades frogs Vital rates, also called transition probabilities 300-700/yr Embryos Adults Larvae Juveniles x 0.6 0.5 1 - x 0.7-0.8 0.1-0.4 4-7 yr 2-3 yr 1 yr

17. a13 1 2 3 a21 a32 a33 Basic matrix construction aijtransition prob. to row i from column j (per timestep) Columns = j (from) Rows = i (to)

18. a13 1 2 3 a21 a32 a33 Basic matrix construction aijtransition prob. to row i from column j (per timestep) Columns = j (from) Top row = reproduction (F) Rows = i (to) • Must be ‘square’ • Zeros for impossible transitions • Proportions (0-1), except? Off-Diagonal = move forward Diagonal = remain in stage

19. Basic matrix construction a13 1 2 3 a21 a32 a33 Matrix (A) x population vector (nt) = population vector (nt+1) If there were only one class, what would this look like? x Underlying this model is an assumption of EXPONENTIAL growth

20. 3 x 3 age-structured matrix(also called Leslie matrix) P=probability of surviving from one age to the next F=fecundity of individuals at each age In this case, there are two pre-reproductive years (maturity at age three) *only need one subscript here because indivds must move ages each time step See any inconsistency here?

21. 4 x 4 size-structured matrix(also called Lefkovitch matrix) Pij=probability of growing from one size to the next or remaining the same size (need subscripts to denote new possibilities) F=fecundity of individuals at each size In this case, there are three pre-reproductive sizes (maturity at age four). **additional complexities like shrinking or moving more than one class back or forward is easy to incorporate

22. x nt per ha Average matrix Matrix multiplication a13 a12 a11 Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) Three stages (1,2,3+ yrs), with reproduction at each stage. 1 2 3 a21 a32 a33 1 2 3 1 2 3

23. Matrix multiplication 1 2 3 1 x 2 3 nt How do we calculate the number present in each class next year (nt+1)? =

24. Matrix multiplication nt x n(1) n(3) n(4) n(5) n(6) n(7) n(8) n(2)

25. n(1) n(3) n(4) n(5) n(6) n(7) n(8) n(2) Matrix multiplication x nt Is this population declining/increasing/stable?

26. n(1) n(3) n(4) n(5) n(6) n(7) n(8) n(2) Matrix multiplication x nt = sum (n2) / sum(n1) What’s the proportional change in the population from one time to the next? = lambda (λ) (0.639)

27. Matrix multiplication x Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Is this population declining/increasing/stable? Dominant eigenvalue(1) *stable environment or average matrix* Asymptototic measure of geometric, density-independent population growth rate Is the population at a stable distribution of stages/ages? Dominant eigenvector (w)

28. What to do with a deterministic matrix? Fixed environment assumption is unrealistic. BUT… can evaluate the relative performance of different management/conservation options can use the framework to conduct ‘thought experiments’ not possible in natural contexts can ask whether the results of a short-term experiment/study affecting survival/reproduction could influence population dynamics

29. A short detour for an example

30. moderate UV-B low UV-B high UV-B % survival % survival % survival -UV +UV -UV +UV -UV +UV How does egg mortality change across a natural gradient of UV-B exposure?

32. dispersal UV ?? 10,000 eggs 6,000 larvae 3,000 larvae adults (4-10 years) eggs (1-3 weeks) larvae (8 weeks-3+ years) 100 juveniles

33. What to do with a deterministic matrix? Fixed environment assumption is unrealistic. BUT… can evaluate the relative performance of different management/conservation options can use the framework to conduct ‘thought experiments’ not possible in natural contexts can ask whether the results of a short-term experiment/study affecting survival/reproduction could influence population dynamics *can evaluate the relative sensitivity of  to different vital rates

34. Reproductive value Prop in j @ stable age constant Matrix element sensitivities x Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Sensitivity of 1 to element aij is Sij

35. Matrix element sensitivities x Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix • Sensitivity analysis of a deterministic matrix (by brute force): • Vary vital rates individually (small change vs. biologically realistic range?) • Re-calculate deterministic 1 • Plot change in each rate versus change in 1

36. X X X Matrix element sensitivities x Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix Sensitivity analysis of a deterministic matrix:

37. Matrix element sensitivities x Semipalmated sandpiper data from Hitchcock & Gatto-Trevor (1997) nt Average matrix • Relative sensitivities (comparable across vital rates) called Elasticities • proportional change of vital rates compared to proportional change in lambda • Why does might this matter?