what is an individual counting...
TRANSCRIPT
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Act II: Single Species Dynamics
Scene I: Life, Death, and Life Histories
(the chapter from hell)
Population
β’ Number of individuals in an area
β’ Density
β’ ππ‘βππ = ππππ€ + π + π + π + π
What is an individual
Unitary Modular
Final size Determinate Indeterminate
Predictable shape Predictable Less predictable
Form of growth Unit Modular
Examples Human Tree, corals, hydroids
Senescence Predictable Less predictable
Ecological individual Genet Ramet or Genet
Counting Individuals
β’ Modular organisms
β Ramets, genets, biomass
β’ Unitary organisms
β Census
β Index
β Population estimate
Births and deaths
β’ Damned difficult
β’ Births β Hard to quantify
β 50% of zygotes in rabbits abort
β 30% of human zygotes make it full term
β’ Deaths β Rarely find dead individuals
β Small plants and animals may get consumed completely
Iteroparity and semelparity
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Climate and reproductive strategies
β’ Seasonal climates have a greater proportion of semelparous or seasonal iteroparous species
β’ Seasonality may be induced by precipitation
Examples of diversity in life cycles
Semelparous/Overlapping generations Long-lived
semelparous species
Long-lived (we hope) iteroparous species
Annuals
lx = ax/a0
dx = lx β lx+1
qx = dx/lx*1/days
β’ dx can be summed
β’ qx gives intensity in a particular period
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Seed Banks
Static vs. Cohort Life Tables
β’ Cohort
β Strengths
β Disadvantages
β’ Static
β Strengths
β Disadvantages
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Basic reproductive rate
β’ π 0 = ππ₯ππ₯
β’ When non-overlapping generations R = R0
β’ π1 = π0π
β’ π2 = π1π = π0π π = π0π 2
β’ π3 = π2π π = π0π π π = π0π 3
β’ So, generally, ππ‘ = ?
β’ And π 0 = π π‘ -> take log of both sides gives:
Intrinsic rate of increase
β’ If Ln R = r
β’ Then π =ln π 0
π
β’ When population are overlapping then use
β’ π =ln π 0
ππ where Tc is the cohort generation
time
β’ ππ = π₯ππ₯ππ₯
π 0
Stable Age Distribution
Life history
β’ Lifetime pattern of growth, differentiation, storage, and reproduction
p.s. donβt Google βorchid seedβ = w.t.h.!
Two majors elements
β’ Optimization
β’ Bet-hedging
β’ Trade-offs
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Allocation of limited resources
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Clutch size vs. litter size
β’ In general, if more produced then smaller produced
Survivorship and reproduction
β’ When survivorship is low, reproduction is high
β’ Age of maturity
β’ Semelparity
r-K spectrum
β’ Developed by MacArthur and Wilson 1967 in the Theory of Island Biogeography
β’ r species: maximize reproduction
β’ K species: maximize competitive edge
β’ Make a table that includes habitat, propagule size, time to maturity, clutch size, parental care, semelparity
β’ Support
Phenotypic Plasticity
β’ P = G
β’ P = G + E
β’ P = G + E + G*E
β’ Plasticity is the aspect of the phenotype of the phenotype that is determined by the environment
β’ Are phenotypic responses optimal?
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Constraints
β’ Phylogenetic
β’ Physics
β’ Example: Allometric relationships
β At the cell
β At the organismal
Phylogenetic effects
β’ The comparative method β Harvey and Pagel 1991
More on single species dynamics
Leslie Matrix
L =
F0 F1 F2 F3
S0 0 0 0
0 S1 0 0
0 0 S2 0
Γ©
Γ«
ΓͺΓͺΓͺΓͺΓͺ
ΓΉ
Γ»
ΓΊΓΊΓΊΓΊΓΊ
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Elements of Leslie Matrix (L)
Fx β Age-specific Fecundity Γ age-specific survival
Sx βAge-specific Survival
Fx = Sx mx+1
How does the Leslie Matrix estimate Population Growth?
Nt+1 = LΒ΄Nt
Population Projection
Nt+1 =
F0 F1 F2 F3
S0 0 0 0
0 S1 0 0
0 0 S2 0
Γ©
Γ«
ΓͺΓͺΓͺΓͺΓͺ
ΓΉ
Γ»
ΓΊΓΊΓΊΓΊΓΊ
Β΄Nt
Population Projection
N0,t+1
N1.t+1
N2,t+1
N3,t+1
Γ©
Γ«
ΓͺΓͺΓͺΓͺΓͺ
ΓΉ
Γ»
ΓΊΓΊΓΊΓΊΓΊ
=
F0 F1 F2 F3
S0 0 0 0
0 S1 0 0
0 0 S2 0
Γ©
Γ«
ΓͺΓͺΓͺΓͺΓͺ
ΓΉ
Γ»
ΓΊΓΊΓΊΓΊΓΊ
Β΄
N0,t
N1,t
N2,t
N3,t
Γ©
Γ«
ΓͺΓͺΓͺΓͺΓͺ
ΓΉ
Γ»
ΓΊΓΊΓΊΓΊΓΊ
Assumptions
β’ Individuals can be aged reliably
β’ No age-effects in vital rates
β’ Vital rates are constant
β Constant environment
β No density dependence
β stochastic Leslie Matrices possible
β’ Sex ratio at birth is 1:1
β i.e., male and female vital rates are congruent
Advantages of Leslie Matrix
β’ Stable-age distribution not assumed
β’ Sensitivity analyses β
β can identify main age-specific vital rates that affect abundance and age structure
β’ Modify the analyses to include density-dependence
β’ Derive finite rate of population change (Ξ») and SAD
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Disadvantage of Leslie Matrix
β’ See assumptions
β’ Age data may not be available
β can use stage-based Lefkovitch Matrix
β’ Fecundity data may not be available for all ages
EigenAnalysis of L
β’ Eigenvalues β β dominant = population growth rate
β’ asymptotic growth rate at Stable Age Distribution
β’ Stable Age Structure β right eigenvector
β’ Reproductive Value β left eigenvector
Other Statistics β’ Sensitivities
β how Ξ» varies with a change in matrix elements β’ absolute changes in matrix elements
β’ Elasticities β how Ξ» varies with a change in a vital rate holding
other rates constant
β
β’ Damping ratio β rate population approaches equilibrium - SAD
r =
l1
l2
Act II Scene II
β’ Darwin
β’ Competition
February 22, 1809
Happy Darwin Day
Darwinβs Childhood
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Darwinβs Education Darwinβs Education
Voyage of the Beagle Voyage of the Beagle
Voyage of the Beagle Voyage of the Beagle
β’ Immensity of Time
β Charles Lyell
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Voyage of the Beagle
β’ Thomas Malthus
β Struggle for existence
Voyage of the Beagle
Back home Back home
Back home Development of natural selection
β’ Deep time
β’ Struggle for existence
β’ Genealogical connection with past creatures
β’ Isolation
β’ Variation is important
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Alfred Russell Wallace Origin of Species
Other stuff Darwin did
β’ Expression of emotion
β’ Human history
β’ Action of earthworms
β’ Barnacles
β’ Pollination
Darwinβs children
Up to now
β’ How organisms interact with the environment
β’ What has shaped the organisms (natural selection + constraints)
β’ Simple populations
Intraspecific Competition
β’ Exploitation
β’ Interference
β Includes mates
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So, increased density =
β’ Increased mortality
β’ Decreased fertility
β’ Caveats
β Group selection
Density
β’ Individuals per unit area
β’ Individuals per unit resource
β’ Caveats
β Group size
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Carrying Capacity
β’ b = d
What does this mean for K?
Sigmoidal Growth Curve Sigmoidal Growth Curve
β’ Growth rates along the curve
β’ Caveat
β Beware the asymptote!
β Competition requires
β What else happens to populations
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Models
β’ What is a model?
β’ All models are wrong
β’ But some are useful
Discrete breeding seasons
β’ R is the fundamental net reproductive rate
β’ ππ‘+1 = ππ‘π
β’ ππ‘ = π0π π‘
β’ Assumes R is constant
Discrete with competition
β’ At a small population size the effect of competition is small so
β’ππ‘
ππ‘+1=
1
π
β’ To add competition
β’ ππ‘+1 =ππ‘π
1+(π β1)ππ‘
πΎ
but, if (R-1)/K=a then
β’ ππ‘+1 =ππ‘π
1+πππ‘ what is the effect of a on Nt+1 as
a increases
Time lags
β’ Models presented so far represent populations that respond instantaneously to competition
β’ A time lag implies that the population responds to previous competition β Time lags can be -1 generations or more β Time lags induce oscillations β Dampening is the loss of oscillations
β’ These models can be called time-dependent or autoregressive models
β’ The effect of neighbors on a data point is called autocorrelation (spatial and temporal.. Or both)
Time lag
β’ ππ‘+1 =ππ‘π
1+πππ‘β1
β’ ππ‘+1 =ππ‘π
1+ πππ‘π
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Implications
β’ When R is large populations are likely to oscillate without extrinsic forces
Continuous breeding
β’ππ
ππ‘= ππ
β’ r = ln(R)
β’ Note that this is not the population size but what?
β’ With intraspecific competition
β’ππ
ππ‘= ππ
πΎβπ
πΎ
β’ Logistic equation (Verhulst 1838)
β’ Why logistic?
Competition: Effects on individuals
β’ Body size: body size decreases but skewness increases with increasing density
Asymmetry in competition Asymmetry in intraspecific
competition
β’IS NATURAL SELECTION
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Territoriality
β’ Resources (usually space) is defended β’ Costs
β Defense, moving
β’ Benefits β Mates β Predictable resources
β’ When does territoriality break down? β Unpredictable resources β Superabundant resources β Simply put c > b
Self-thinning