Transcript Density dependent mortality.

Upcoming Seminars:
• EECB seminars – 4:00 Thurs in OSN 102
– Thurs Feb 12: Mark Lindberg (U. A.
Fairbanks) “Patterns and rates of dispersal in
avian populations: Is scale important?”
Outline
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Introduction to population ecology
Spatial structure
Plant interactions and density dependence
Age and size structure
Plant demography
Population growth models and parameters
Life tables
Survivorship curves
Reading Assignments
1. Textbook chapter 4 and 5
2. Radford et al. 2002. Austral Ecology
27:258-358.
3. Supplemental (not required)
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Allcock and Hik 2004. Oecologia 138:231-241.
Silvertown and Lovett-Doust 1993.
Introduction to plant population biology.
Blackwell Scientific Publications, London.
Population Biology
Population: collection of individuals of the
same species living in the same area
Population structure: spatial, age, size
Population biology tries to explain origin of
structure types, how they interact, and how
they change with time.
Spatial Structure
Patterns of distribution: random, dispersed,
clumped.
Patterns affected by biological and abiotic
interactions. Test for randomness
mean:variance ratio, Poisson analysis
Spatial Structure
Why does pattern matter?
Spatial Structure
Why does pattern matter?
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Interpret causes of patterns
Stratification
Appropriate sampling regimes (density,
frequency, non-quadrat)
Affects interactions
Plant interactions
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Space affects population biology in two
ways:
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“neighborhood” – area of genetic or ecological
influence
Density – number of plants per unit area.
Affects resource competition.
Density influences growth, survival, fitness.
Law of Constant Yield
Biomass/unit area increases with density, then
levels off and becomes independent of
density.
Y=wmN(1 + aN)-1
Y= Yield
wm=max potential biomass/plant
N=density
A=area necessary to achieve wm
Law of Constant Yield
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At high density Y is constant and
proportional to wma-1 and w=Y/N.
Plant size is inversely proportional to
density : w=wm(1+aN)-1
Generalization: to allow for changing curves
at high density (some species DECLINE in
yield) replace –1 exponent with “-b”
Competition-Density Effect
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w=wm(1+aN)-b describes variation in weight
with density at a given moment in time.
Parameters vary during growth and with
environmental conditions.
Competition also leads to reduction in N
over time (self-thinning)
“-3/2 power law”
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Self-thinning: smaller individuals die,
reducing density as plant size and
competition increases. Density dependent
mortality.
w=cN-k and log w = log c – k log N
-k = slope of “self thinning line” (boundary)
log c = constant between 3.5 and 5
-k is usually –3/2
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Dense populations reach boundary line
before sparse ones
Slope of w:N constant across very different
plant groups
Controversy about “law” but there is a
geometrical explanation (Yoda 1963)
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Plant weight proportional to volume (L3), plant
sits on area (L2).
When plant occupies all available space, ratio
of weight to area CANNOT exceed 3:2
Age and Size Structure
Size distribution of a given age rarely normal.
Plants usually display highly skewed frequency
distribution ( L shaped curve)
Skewed size distribution
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Two causes:
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Growth rate is normally distributed, and faster
growing plants change normal size distribution
to skewed.
Larger plants suppress smaller ones
(asymmetric competition).
Self-thinning in even aged stand can return
normal size/age distribution in time.
Age and Size Structure
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Maturity affected by size
Size affected by environmental conditions
and intraspecific competition
Age-based models of populations often not
appropriate for plants…use stage (or size)
based
Modular growth
• Plants have indeterminate growth
• Plants grow by adding modules (roots,
stems, leaves, clones)
• Genet = one genetic individual (e.g.
aspen grove)
• Ramet = clonally produced part of a
plant (may be essentially independent)
Demography
Study of changes in population size and
structure over time.
Nt+1=Nt + B – D + I – E
Nt+1/Nt = Finite rate of increase = λ
when λ = 1 population is stable
When λ < 1 population is shrinking
When λ> 1 population is growing
Modeling populations:
unrestricted growth
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Compare population at time t to population at
time t+1 (difference equation)
Nt+1=RNt+Nt or Nt+1=λNt
λ=R+1 and R=geometric rate of increase
For arbitrary time step: NT=N0λT
For instantaneous growth (continuous time;
differential equation)
dN/dt=rN(t) and N(T)/N(0)=erT
So N(T)=N(0)erT
Modeling populations:
unrestricted growth
N
Time
Unrestricted growth
What sorts of populations exhibit exponential or
geometric growth?
1. Unrestricted resources
2. No competition or other limitations
Unrestricted growth
What sorts of populations exhibit exponential or
geometric growth?
1. Unrestricted resources
2. No competition or other limitations
Invasive species, expanding populations
Exponential decline: constant mortality rate > birth rate
Density dependent growth
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Biological factors interact to produce a negative
feedback between N and R.
Examples:
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Resources decrease (are used up)
Available space is filled
Interference (agression etc) may increase
More efficient predation as prey density increases
Emmigration or dispersal increases
Immigration decreases
Density dependent growth
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Can model density dependent growth with
logistic equation:
dN/dt=rN((K-N)/N)
K= population at equilibrium carrying capacity
At K population growth rate is zero
Density dependent growth
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As intrinsic rate of increase goes up, behaviour
of model changes:
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Carrying capacity: one equilibrium value for N
Stable limit cycles: N oscillates among several values
As R increases, number of values in cycle doubles (for
2.1<R<2.57)
Eventually (R>2.57) dynamics are CHAOTIC. Not
random, highly density dependent, but unpredictable.
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Time lags can also create cycles:
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Resource availability changes with time and
population size (eg herbivores and food source)
dN/dt=rN((K-N(t-T))/K)
Classic example: lynx and hares…
Characteristics of populations
Ecologists use life tables and fecundity
schedules to organize demographic data:
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Age or stage-specific survivorship, birth rates, death
rates, reproductive value etc.
– Can be based on cohorts (cohort life table) or
age/stage classes (static life table)
– May contain the following parameters:
Age/stage, number surviving (Nx), survivorship (lx=Nx/N1),
mortality (dx=(lx-1-lx)), mortality rate (qx=dx/lx), fecundity
(bx= offspring per individual), reproductive value (Vx =
bx+ Σ(lx+I/lx)bx+I)
Survivorship Curves
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Plots of number of survivors (log scale) versus
age/stage.
Three basic shapes: different life histories.
Type I
Type II
Type III
Survivorship
What do the shapes of the survivorship curves
mean? Examples?
Type I
Type II
Type III
Importance
Importance
Demography affects current distributions, historical
range shifts/spread, gene frequencies, and
population structures.
Population dynamics important for commercial
species: yield, growth, survival, etc.
Use population models to create management plans
for both endangered and invasive species
Herbivory (eg stock production) can affect population
parameters of range species: Riginos and
Hoffman (2003). Journal of Applied Ecology
40:615-625.
Lab: life cycle diagrams, matrix models, life tables,
and their applications for management
Next lecture: metapopulations, life history strategies,
allocation.