HW 1 Recreate Deterministic Matrix from Literature Quick overview Speciessystem ElasticitySensitivity which classes Issues Matrix population models 4 x 4 size structured matrix also called Lefkovitch matrix ID: 917880
Download Presentation The PPT/PDF document "Single species population growth models" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
Single species population growth models
Slide2HW 1: Recreate Deterministic Matrix from Literature
Quick overviewSpecies/systemElasticity/Sensitivity (which classes?)Issues?
Slide3Matrix population models
Slide44 x 4
size-structured matrix(also called Lefkovitch matrix)
P
ij
=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
Slide5What 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
Slide6What 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
Slide7Matrix models put impacts in context
Simple (
deterministic
):
650
85%
7%
15%
10%
45%
30 years
Adult #
’
s
10%
1%
Population grows (or shrinks) exponentially as a function of the combination of
fixed
vital rates
λ
= lambda
ω
= stable age/stage
S = sensitivity matrix
E = elasticity matrix
N
t
Slide8More realistic (
stochastic
simulation):
30 years
Adult #
’
s
Population varies from year to year as a function of
a
randomly drawn
matrix
Matrix models put impacts in context
Survey
yr
1 2 3 9
S
e
0.89 0.86 0.66 0.97
S
l
0.08 0.07 0.02 0.11
S
j
*
0.15
0.15
0.15
0.15
S
a
0.08 0.2 0.09 0.05
S
c
0.47 0.6 0.48 0.27
F
a
498 711 884 509
Drought year
Long summer
Slide9More realistic (
stochastic
simulation):
30 years
Adult #
’
s
Population varies from year to year as a function of the combination of
randomly drawn
vital rates
Matrix models put impacts in context
Slide10Simulation-based stochastic model:
30 years
Adult #
’
s
More realistic (
stochastic
simulation):
Matrix models put impacts in context
Slide11Stochastic projections
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
Issues to consider:
Slide122. Additional
structure-Demographic
stochasticity
?
*Important @ Small
population sizes
-Monte Carlo sims of individual fate given distributions of vital rates (quasi-extinction is easier…)
-Density-dependence in specific vital rates?
-vital
rate function
of density in pop (Nt) or specific stage (
N
it
)
(very difficult
to parameterize
)
-Correlation structure? -within years (common), across years (cross-correlation, harder)
-Quasi
-extinction threshold?
-minimum ‘viable’ level (below which model is unreliable & pop unlikely to recover)
-Starting Pop Size?
Stochastic projections
Slide13Can no longer calculate deterministi
c properties of a single matrix (λ, w, S, E)
Stochastic Lambda (
λ
s
)
-
can be calculated from stochastic outputs -estimated by approximation (more on this Thurs…)
Extinction Probability (CDF) -how does extinction probability change with time simulated in future?
Population Size at t (
N
t
)
-describe variation in est. pop size @ specific points in future
Sensitivity/Elasticity
-need a new approach (Wed class)
Stochastic projections
Outputs:
Slide14Simulation-based stochastic model:
30 years
Adult #
’
s
Life cycle models put impacts in context
More realistic (
stochastic
simulation):
Stochastic lambda (
λ
G
) =
Geometric mean
λ
λ
G
= 0.98
λ
G
= 0.91
λ
G
= 0.96
Arithmetic lambda >>
λ
G
(esp. with high
var
)