1. Presentation topics Agent/individual based models
Joyce, Gavia, Tamara, Kathleen
Host-pathogen models / SIR
Kari, Shir Yi
Lotka-Volterra & functional response
Dennis
Metapopulation models
Rylee
Occupancy models
Kristen
Dates
Feb. 22
Mar. 1
Mar. 8
Mar. 15/22
Mar. 29
Lily- wants to do island biogeography (species-area relationships)

2. Single species population growth models

3. Matrix population models

4. 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 4

5. 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

6. What we’ve covered so far:
Translating life histories into stage/age/size -based matrices
Understanding matrix elements (survival and fecundity rates)
Basic matrix multiplication in fixed environments
Deterministic matrix evaluation (1 , stable stage/age)
Initial framework for sensitivity analysis
Next:
Incorporating demographic & environmental stochasticity 6

7. deterministic λ for each
Matrix models put impacts in context
Simple (deterministic):
650
85% 7%
15%
10%
45%
Adult #’s
deterministic λ for each
10%
30 years
Population grows (or shrinks) exponentially as a function of the combination of fixed vital rates 1% 7 7

8. Matrix models put impacts in context
More realistic (stochastic simulation):
Long summer
Adult #’s
Drought year
30 years
Survey yr Se Sl Sj Sa Sc Fa
Population varies from year to year as a function of a randomly drawn matrix
not est = fixed 8 8

9. Matrix models put impacts in context
More realistic (stochastic simulation):
Adult #’s
30 years
Population varies from year to year as a function of the combination of randomly drawn vital rates 9 9

10. Simulation-based stochastic model:
Matrix models put impacts in context
More realistic (stochastic simulation):
Simulation-based stochastic model:
Adult #’s
30 years 10 10

11. Stochastic projections
Issues to consider:
Form of stochasticity: in matrix or vital rates?
-Environmental stochasticity:
Series of fixed matrices (as opposed to mean matrix)
-random = env. conditions ‘independent’ (no autocorrelation*)
-preserves within year correlations among vital rates (whether you can estimate them or not)
Vary individual vital rates each timestep
-separate from sampling variation
-draw vital rates from distribution describing variation
(Lognormal, beta, etc.)
*Either can be mechanistic: vital rates affected by periodic conditions (ENSO, flood recurrence, etc.)  probabilistic draw 11

12. Stochastic projections
Issues to consider:
OUTPUTS: Stochastic lambda, extinction probability CDF
-Demographic stochasticity:
Small population sizes
-Monte Carlo sims of individual fate given distributions of vital rates (quasi-extinction is easier…)
-Quasi-extinction threshold?
-minimum ‘viable’ level
(below which model is unreliable & pop unlikely to recover)
-Density-dependence in specific vital rates
-vital rate function of density in pop (Nt) or specific stage (Nit)
(very difficult to parameterize)
-Correlation structure?
-within years (common), across years (cross-correlation, harder) 12

13. Life cycle models put impacts in context
More realistic (stochastic simulation):
Simulation-based stochastic model:
Adult #’s
30 years
Arithmetic lambda always higher, and geometric smaller (more so when more variable)
*Take home, When more variable pop growth (lambda) the slower the population grows
λG = 0.98
Stochastic lambda (λG) =
Geometric mean λ
λG = 0.91
λG = 0.96
Arithmetic lambda >> λG (esp. with high var) 13 13

14. From Morris & Doak Ch. 2

15. Life cycle models put impacts in context
More realistic (stochastic simulation):
Simulation-based stochastic model:
Adult #’s
30 years
Quasi-extinction threshold
P (extinction)
Cumulative Pr(Extinction)
# times population went extinct in each year
(x thousands of simulations)
30 years 15 15