Transcript Document

ES100:
Community Ecology
8/22/07
What Controls Population Size and
Growth Rate (dN/dt)?
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Density-dependent factors:
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Intra-specific competition
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food
Space
contagious disease
waste production
Interspecific competition
Other species interactions!
Density-independent factors:
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disturbance, environmental conditions
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hurricane
flood
colder than normal winter
Types of Interactions
 Competition
 Predator-Prey
 Mutualism
 Commensalism
Competition
Natural Selection minimizes competition!
Species Interactions
•
How do we model them?
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Start with logistic growth
dN
= r * N (1 –
dt
dN
K
=r*N(
dt
K
N
)
K
-
dN
K-N
=r*N(
)
dt
K
N
)
K
Use this
equation for
2 different
species
Species Interactions
•
Population 1  N1
dN1
K1-N1
= r1 * N1 (
)
dt
K1
•
Population 2  N2
dN2
K2-N2
= r2 * N2 (
)
dt
K2
•
But the growth of one population should have an
effect the size of the other population
Species Interactions
•
New term for interactions
a12  effect of population 2 on population 1
a21  effect of population 1 on population 2
•
Multiply new term by population size
the larger population 2 is, the larger its effect on
population 1 (and vice versa)
a12 * N2
a21 * N1
Competition: Lotka-Volterra Model

If two species are competing, the growth of one
population should reduce the size of the other

Population 1  N1
dN1
K1 - N1 - a12 N2
dt = r1 * N1
K1

Population 2  N2
dN2
K2 - N2 - a21 N1
dt = r2 * N2
K2
Competition

If two species are competing, the growth of one
population should reduce the size of the other
Because this is a negative term, K
is reduced

Population 1  N1
dN1
K1 - N1 - a12 N2
dt = r1 * N1
K1

Population 2  N2
dN2
K2 - N2 - a21 N1
dt = r2 * N2
K2
COMPETITION
Blue Area = Bluejay’s Carrying Capacity
It takes 1squirrel to use the portion of the carrying
capacity occupied by 4 bluejays.
aBS = 4
Interspecific competition regulates bluejay population
 K  N B  4N S
dN B
 rB N B  B
dt
KB




COMPETITION
Green Area = Squirrel’s Carrying Capacity
It takes 4 bluejays to use the portion of the carrying
capacity occupied by 1 squirrel.
aSB =.25
Intraspecific competition regulates squirrel population
 K s  N s  .25N B
dNS
 rs N s 
dt
Ks




Outcomes of Competition Model

Many possible outcomes, depends on the balance of:
 r1 vs r2
 K1 vs K2
 a21
vs a12
 a12
>1
Interspecific competition dominates
population size of species 1
 a12
<1
Intraspecific competition dominates
population size of species 1
a12 is the per capita effect of species 2 on the the pop’n growth
rate of species 1, measured relative to the effect of species 1.
Predator-prey
Predator-Prey Relationships
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Prey defenses: avoid conflict!
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coevolution
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as predator evolves, prey evolves to evade it
warning coloration and mimicry
Camouflage
Predator-Prey
Red = Fox’s Carrying Capacity
It takes 10 rabbits to support 1 fox
aFR =.10
 K F  N F  .10N R 
dNF

 rF N F 
dt
KF


Predator-Prey
Yellow = Rabbits Carrying Capacity
It takes 10 rabbits to support 1 fox
aRF = 10
 K R  N R  10N F
dNR
 rR N R 
dt
KR




Predator-Prey Relationships
•Bottom-up
vs. Top-Down control
•Predators can promote diversity by
keeping competition in check
Predatory-Prey



If it is a predator-prey relationship, then the two
populations have opposite effects on one another
Because this is a negative term, K
is reduced
Prey (N1)
dN1
K1 - N1 - a12 N2
dt = r1 * N1
K1
Predator (N2)
Because this is a positive term, K
is increased
dN2
K2 - N2 + a21 N1
dt = r2 * N2
K2
Mutualism
 Both
species benefit
Mutualism

If it is a mutually beneficial relationship, then the two
populations increase each other’s size

Because this is a positive term, K
is increased

Because this is a positive term, K
is increased
Population 1  N1
dN1
K1 - N1 + a12 N2
= r1 * N1
dt
K1
 ti
Population 2  N2
dN2
K2 - N2 + a21 N1
dt = r2 * N2
K2
Commensalism
 One
species benefits, the other is unaffected
Commensalism
If the relationship is commensalistic, one species
benefits (the commensal) and the other is unaffected
 Population 1  N1

Because this is a positive term, K
is increased
dN1
K1 - N1 + a12 N2
=
r
*
N
1
1
dt
K1

Population 2  N2
dN2
dt = r2 * N2
Because there is no a21 term, K is
unchanged
K2 - N 2
K2
Assumptions of Lotka-Volterra
Models
 All
assumptions of logistic growth model… plus:
 Interaction
coefficients, carrying capacities, and
intrinsic growth rates are constant.
Summary of Interaction Equations:
Competition: (- , -)
Predator/Prey: (+, -)
Mutualism:
(+, +)
Commensalism: (+, 0)
 K1  N1 ? a12 N 2 
dN1

 r1 N1 
dt
K1


 K 2  N 2 ? a21 N1 
dN2

 r2 N 2 
dt
K2


Test you knowledge!
What type of relationship– what equation to use?

A coati eats tree fruit.

Your dog has a flea

You use a fast bicyclist to “draft” off of
Problems with Simple Logistic Growth
1.
Births and deaths not separated
-you might want to look at these processes separately
-predation may have no effect on birth rate
2.
3.
Carrying capacity is an arbitrary, set value
No age structure
1. Separate Births and Deaths
dN
= Births - Deaths
dt
Births = b*N
Deaths = d*N
Births and deaths may be density dependent
1. Separate Births and Deaths
dN
= Births - Deaths
dt
Births = b*N
Example:
Births = b*N(1- N )
K
Deaths = d*N
Deaths = db+a21N2
Births rate may be density dependent
Death rate may be dominated by predator effects
2. Refine Carrying Capacity
If the population is a herbivore, K may depend
on the population of plants
Kherbivore= Nplant
dNH
NH
dt = rH * NH (1 – NP )
Remaining Problems
 Age
Structure
 Space:
animals rely on different parts of landscape
for different parts of their life cycle
 Individuality:
Populations are collections of
individuals, not lumped pools
General Notes on Using Models
 How
complex should model be? K.I.S.S.
 Identify research needs:
 Build
model structure
 Test model to see what it is most sensitive to
 Do research to find values of unknown parameters
 If
build a model that accurately predicts dynamics,
it can be used as a management tool.
 Look
critically at assumptions!
Community Dynamics
Community: a group of populations (both plants and animals)
that live together in a defined region.
Trophic Cascade
predator/
tertiary consumer
Eagles
4th trophic level
predator/
secondary consumer
Foxes
3rd trophic level
herbivore/
primary consumer
Mice
2nd trophic level
autotroph/
primary producer
Plants
1st trophic level
How would we Model the Fox
Population?
 K F  N F  aFM N M  aFE N E 
dNF

 rF N F 
dt
KF


Why not include the effect of the plant population?
What if foxes had a competitor?
Trophic Cascade
if eagles go
extinct, what
could happen
to…
foxes?
mice?
plants?
Eagles
4th trophic level
Foxes
3rd trophic level
Mice
2nd trophic level
Plants
1st trophic level
Trophic Cascade
If a new predator
on mice is
introduced, what
could happen to…
mice?
plants?
foxes?
eagles?
Eagles
4th trophic level
Foxes
3rd trophic level
Mice
2nd trophic level
Plants
1st trophic level
Trophic Cascade
If drought caused
a dip in plant
production, what
would happen to…
mice?
foxes?
eagles?
Eagles
4th trophic level
Foxes
3rd trophic level
Mice
2nd trophic level
Plants
1st trophic level
Simplified Temperate Forest Food Web
What happens to when it’s a WEB instead of a CHAIN?
Eagle
Wolf
Deer
Oak seedling
Fox
Shrews
Rabbit
Caterpillars
Grasses
Herbs
In long term, balance is restored
Food Web doesn’t account
for Keystone Species
Otters eat sea urchins
Kelp provides otter habitat
Sea urchins eat kelp
Summary

Modeling Species Interactions
Competition
 Predator-prey
 Mutualism
 Commensalism


Community Dynamics
Food Webs
Keystone Species

