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)

Single species population growth models

Matrix population models

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

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

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

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

Matrix models put impacts in context More realistic (stochastic simulation): Long summer Adult #’s Drought year 30 years Survey yr 1 2 3 9 Se 0.89 0.86 0.66 0.97 Sl 0.08 0.07 0.02 0.11 Sj 0.15 0.15 0.15 0.15 Sa 0.08 0.2 0.09 0.05 Sc 0.47 0.6 0.48 0.27 Fa 498 711 884 509 Population varies from year to year as a function of a randomly drawn matrix not est = fixed 8 8

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

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

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

Stochastic projections Issues to consider: OUTPUTS: Stochastic lambda, extinction probability CDF -Demographic stochasticity: *Important @ 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

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

From Morris & Doak Ch. 2

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