HW 1 Agestructured populations Stagestructure populations Life cycle diagrams Projection matrices Context Sea Turtle Conservation But first background Population Structure ID: 759418
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Slide1
Ecology 8310Population (and Community) Ecology
HW #1Age-structured populationsStage-structure populations Life cycle diagramsProjection matrices
Slide2Context: Sea
Turtle Conservation(But first … background)
Slide3PopulationStructure:
From vianica.com
Slide4Life CycleDiagram:
Age 1
Age 2
Age 0
Age 3
Age-based approach. What now?
Slide5More transitions?
Life CycleDiagram:
Age 1
Age 2
Age 0
Age 3
Dead
Are we done?
Slide6Add values?Pij = Per capita transition from group j to i
Group 2, Age 1
Group 3, Age 2
Group 1, Age 0
Group 4, Age 3
Dead
Now what?
P
21
P
14
P
32
P
43
P
13
1-P
21
1-
P
43
1-
P
32
1-
P
54
Life Cycle
Diagram:
Slide7Project from time t to time t+1….
Group 2, Age 1
Group 3, Age 2
Group 1, Age 0
Group 4, Age 3
Dead
P
21
P
14
P
32
P
43
P
13
1-P
21
1-
P
43
1-
P
32
1-
P
54
Projections:
Slide8nx,t = abundance (or density) of class x at time t.So, given that we know n1,t, n2,t, ….And all of the transitions (Pij's)…… What is n1,t+1, n2,t+1, n3,t+1, … ?
Projections:
Slide9n2,t+1 = ?? = P21 x n1,t
Group 2, Age 1
Group 3, Age 2
Group 1, Age 0
Group 4, Age 3
Dead
P
21
P
14
P
32
P
43
P
13
Projections:
Slide10n1,t+1 = ?? = (P14 x n4,t) + (P13 x n3,t)
Group 2, Age 1
Group 3, Age 2
Group 1, Age 0
Group 4, Age 3
Dead
Project what?
P
21
P
14
P
32
P
43
P
13
Projections:
Slide11Is there a way to write this out more formally
(e.g., as in geometric growth model)?
Slide12Matrix algebra:
n
is a vector of abundances for the groups;
A is a matrix of transitionsNote similarity to:
Slide13Matrix
algebra
:
For our age-based approach
Slide14Matrix
algebra
:
Slide15Our age-basedexample:
Group 2, Age 1
Group 3, Age 2
Group 1, Age 0
Group 4, Age 3
P
21
P
14
P
32
P
43
P
13
Slide16A simplerexample:
Group 2
Group 3
Group 1
P
21
P
32
P
13
P
12
Slide17Simpleexample:
What is
n
t+1
?
Slide18Simple
example
:
Slide19Group 2
Group 3
Group 1
P
21
P
32
P
13
P
12
Simple
example
:
Slide20Time:1234567n1,t1000609036108103n2,t060036542265n3,t00300182711Nt1006090126108157179
Simple
example
:
Slide21n1/N1.000.670.710.330.69n2/N01.000.290.500.14n3/N000.3300.170.17Annual growth rate=(Nt+1/Nt).601.501.400.861.451.14
Time:
1
2
3
4
5
6
n
1
100
0
60
90
36
108
n
2
0
60
0
36
54
22
n
3
0
0
30
0
18
27
N
100
60
90
126
108
157
Slide22Let's plot this…
Slide23Dynamics:
What about a longer timescale?
Slide24Dynamics
:
Are the age classes growing at similar rates?
Slide25Dynamics
:
Thus, the composition is constant…
Slide26Constant
proportions through
time = Stable Age Distribution (SAD)
If no growth (Nt=Nt+1), then:SAD Stationary Age DistributionSAD is the same as the “survivorship curve” … (return later)
Age
structure:
Slide27Dynamics
:
If
A
constant,
then SAD, and
Geometric growth
N
t
+1
/
N
t
=
l
N
t
=N
0
l
t
Here,
l
=1.17
Slide28How do we obtain a survivorship schedule from our transition matrix,
A
?
Slide29p(x) = Probability of surviving from age x to age x+1 (same as the “survival” elements in age-based transition matrix: e.g. p(0)=P21).l(x) = Probability of surviving from age 0 to age xl(x) = Pp(x) ; e.g., l(2)=p(0)p(1)
Survivorship
schedule:
Slide30“Group”Age, xPx+2,x+1=p(x)l(x)100.61.0210.50.6320.00.3430.00.0
Survivorshipschedule:
Recall:
Slide31Survivorship
curves:
Age specific survival?
Slide32Back to the question:
The age distribution should mirror the survivorship schedule.
Does it?
Slide33Survivorshipcurves:
Does the age distribution match the survivorship curve?
“Group”Age, xl(x)Stable A.D.Rescaled AD101.00.581.0210.60.300.52320.30.130.22
Why not?
Slide34The population increases 17% each year
So what was the original size of each cohort? And how does that affect SAD?
Survivorship
curves
:
Slide35Population Growth!How can we adjust for growth?
l(x)Stable A.D.Adjusted by growthRescaled 1.00.58=0.58/1.1721.00.60.30=0.30/1.170.60.30.130.130.3
Survivorship
curves
:
Slide36Survivorshipcurves:
Static Method: count individuals at time t in each age class and then estimate l(x) as n(
x,t
)/n(0,t
)
Caveat: assumes each cohort started with same n(0)!
Cohort
Method: follow a cohort through time and then estimate l(x) as n(
x,t+x
)/n(0)
Slide37ReproductiveValue:
Contribution of an individual to future population growth
Depends on:
Future reproduction
Pr
(surviving) to realize it
Timing (e.g., how soon – so your kids can start reproducing)
Slide38How can we calculate it?Directly estimate it from transition matrix (requires math)Simulate itPut 1 individual in a stageProject Compare future N to what you get when you put the 1 individual in a different stage
Reproductive
Value
:
Slide39Group (Age class)N (t=25)Reproductive Value1 (0)341.02 (1)671.93 (2)882.6
Reproductive
Value
:
Slide40ReproductiveValue:
Always increase up to maturation (why?)May continue to increase after maturationEventually it declines (why?)Why might this be useful for turtle conservation policy?
From vianica.com
Slide41Slide42Non-age based approachesDensity dependenceOther forms of non-constant AHow you obtain fecundity and survival data (and use it to get A)Issues related to timing of the projection vs. birth pulsesSensitivities and elasticitiesHow you obtain the SAD and RV's (right and right eigenvectors) and l (dominant eigenvalue)
Issues we've ignored:
Slide43Age-structured:
Stage-structured:
Age 1
Age 2
Age 0
Age 3
Stage 2
Stage 3
Stage 1
Stage 4
How will these models differ?
Generalizing
t
he approach:
Slide44Age-structured:
Stage-structured:
Age 1
Age 2
Age 0
Age 3
Stage 2
Stage 3
Stage 1
Stage 4
Slide45To do:
Go back through the previous results for age-structure and think about how they will change for stage-structured populations.
Read Vonesh and de la Cruz (carefully and deeply) for discussion next time.
We'll also go into more detail about the analysis of these types of models.