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
Single species population
growth models
Matrix population models
4 x 4 size-structured matrix
(also called Lefkovitch matrix)
é P11
êP
A2 = ê 21
ê0
ê
ë0
0
0
P22
0
P32
P33
0
P43
F4 ù
0 úú
0ú
ú
P44 û
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 prereproductive sizes (maturity at age
four).
**additional complexities like shrinking
or moving more than one class back
or forward is easy to incorporate
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
Matrix models put impacts in context
Simple (deterministic):
Adult #’s
45%
650
10%
85%
10%
15%
1%
7%
deterministic
λ for each
30 years
Population grows (or shrinks)
exponentially as a function of the
combination of fixed vital rates
Matrix models put impacts in context
More realistic (stochastic simulation):
Sc
Long summer
Fa
Se
Drought year
Sj
Survey yr
1
Se 0.89
Sl
0.08
Sj
0.15
Sa 0.08
Sc 0.47
Fa 498
Adult #’s
Sa
Sl
30 years
2
0.86
0.07
0.15
0.2
0.6
711
3
0.66
0.02
0.15
0.09
0.48
884
9
0.97
0.11
0.15
0.05
0.27
509
not est = fixed
Population varies from year to
year as a function of a randomly
drawn matrix
Matrix models put impacts in context
0
1000
100
30 years
prob.
survival
fecundity
0
survival
0
survival
100
prob.
prob.
0
100
Adult #’s
survival
prob.
0
prob.
prob.
More realistic (stochastic simulation):
0
survival
100
100
Population varies from year to
year as a function of the
combination of randomly drawn
vital rates
Matrix models put impacts in context
0
1000
100
30 years
prob.
survival
fecundity
0
survival
0
survival
100
prob.
prob.
0
100
Adult #’s
survival
prob.
0
prob.
prob.
Simulation-based
stochastic
model:
More
realistic (stochastic
simulation):
0
survival
100
100
Stochastic projections
Issues to consider:
1. 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
Stochastic projections
Issues to consider:
-Demographic stochasticity:
OUTPUTS: Stochastic lambda,
extinction probability CDF
*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)
Life cycle models put impacts in context
0
survival
fecundity
1000
100
prob.
0
100
Adult #’s
survival
prob.
0
prob.
prob.
More realistic (stochastic simulation):
Simulation-based stochastic model:
survival
100
prob.
0
survival
100
prob.
0
30 years
0
survival
100
Stochastic lambda (λG) =
Geometric mean λ
Arithmetic lambda >> λG (esp. with high var)
λG = 0.98
λG = 0.91
λG = 0.96
From Morris & Doak Ch. 2
Life cycle models put impacts in context
0
survival
fecundity
1000
100
30 years
prob.
0
100
prob.
survival
prob.
0
Adult #’s
prob.
More realistic (stochastic simulation):
Simulation-based stochastic model:
survival
100
Quasi-extinction threshold
survival
100
0
survival
100
Cumulative Pr(Extinction)
# times population went extinct in each year
(x thousands of simulations)
P (extinction)
0
prob.
prob.
0
30 years