ES100:

Community Ecology

8/22/07

What Controls Population Size and Growth Rate (dN/dt)?

• Density-dependent factors:• Intra-specific competition

• food• Space

• contagious disease• waste production• Interspecific competition• Other species interactions!

• Density-independent factors:• disturbance, environmental conditions

• 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?• Start with logistic growth

= r * N (1 – )= r * N (1 – )N K

dN dt

= r * N ( - )= r * N ( - )N K

dN dt

K K

= r * N ( )= r * N ( )dN dt

K-N K

Use this equation for 2 different species

Species Interactions• Population 1 N1

• Population 2 N2

• But the growth of one population should have an effect the size of the other population

= r= r11 * N * N11 ( ) ( )dN

1 dt

K1-N1 K1

= r= r22 * N * N22 ( ) ( )dN

2 dt

K2-N2 K2

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 sizethe 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

Population 2 N2

= r= r11 * N * N11 dN

1 dt

K1 - N1 - a12 N2 K1

= r= r22 * N * N22 dN

2 dt

K2 - N2 - a21 N1 K2

Competition If two species are competing, the growth of one

population should reduce the size of the other

Population 1 N1

Population 2 N2

= r= r11 * N * N11 dN

1 dt

K1 - N1 - a12 N2 K1

= r= r22 * N * N22 dN

2 dt

K2 - N2 - a21 N1 K2

Because this is a negative term, K is reduced

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

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

B

SBBBB

B

K

NNKNr

dt

dN 4

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

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

s

Bssss

S

K

NNKNr

dt

dN 25.

COMPETITION

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 growthrate of species 1, measured relative to the effect of species 1.

Predator-prey

Predator-Prey Relationships

• Prey defenses: avoid conflict!• coevolution

• as predator evolves, prey evolves to evade it

• warning coloration and mimicry• Camouflage

Red = Fox’s Carrying Capacity

It takes 10 rabbits to support 1 fox

aFR =.10

⎟⎟⎠

⎞⎜⎜⎝

⎛ +−=

F

RFFFF

F

K

NNKNr

dt

dN 10.

Predator-PreyPredator-Prey

Yellow = Rabbits Carrying Capacity

It takes 10 rabbits to support 1 fox

aRF = 10

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

R

FRRRR

R

K

NNKNr

dt

dN 10

Predator-PreyPredator-Prey

•Bottom-up vs. Top-Down control•Predators can promote diversity by keeping competition in check

Predator-Prey Relationships

Predatory-Prey If it is a predator-prey relationship, then the two

populations have opposite effects on one another

Prey (N1)

Predator (N2)

= r= r11 * N * N11 dN

1 dt

K1 - N1 - a12 N2 K1

= r= r22 * N * N22 dN

2 dt

K2 - N2 + a21 N1 K2

Because this is a negative term, K is reduced

Because this is a positive term, K is increased

Mutualism

Both species benefit

Mutualism If it is a mutually beneficial relationship, then the two

populations increase each other’s size

Population 1 N1

ti

Population 2 N2

= r= r11 * N * N11 dN

1 dt

K1 - N1 + a12 N2 K1

= r= r22 * N * N22 dN

2 dt

K2 - N2 + a21 N1 K2

Because this is a positive term, K is increased

Because this is a positive term, K is increased

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

Population 2 N2

= r= r11 * N * N11 dN

1 dt

K1 - N1 + a12 N2 K1

= r= r22 * N * N22 dN

2 dt

K2 - N2

K2

Because this is a positive term, K is increased

Because there is no a21 term, K is unchanged

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)

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

1

2121111

1 ?

K

NaNKNr

dt

dN⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

2

1212222

2 ?

K

NaNKNr

dt

dN

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. Carrying capacity is an arbitrary, set value

3. No age structure

1. Separate Births and Deaths

= Births - Deaths

Births = b*N

Deaths = d*N

dN dt

Births and deaths may be density dependent

1. Separate Births and Deaths

= Births - Deaths

Births = b*N

Deaths = d*N

dN dt

Births rate may be density dependentDeath rate may be dominated by predator effects

Example:Births = b*N(1- N )

KDeaths = db+a21N2

2. Refine Carrying Capacity

If the population is a herbivore, K may depend on the population of plants

= r= rHH * N * NHH (1 – ) (1 – )dNH

dtNH NP

Kherbivore= Nplant

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 structureTest model to see what it is most sensitive toDo 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

Eagles

Foxes

Mice

Plants 1st trophic level

2nd trophic level

3rd trophic level

4th trophic level

autotroph/ primary producer

herbivore/ primary

consumer

predator/ secondary consumer

predator/ tertiary consumer

How would we Model the Fox Population?

Why not include the effect of the plant population?

What if foxes had a competitor?

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+−=

F

EFEMFMFFFF

F

K

NaNaNKNr

dt

dN

Trophic Cascade

Eagles

Foxes

Mice

Plants 1st trophic level

2nd trophic level

3rd trophic level

4th trophic level

if eagles go extinct, what could happen to…

foxes?

mice?

plants?

Trophic Cascade

Eagles

Foxes

Mice

Plants 1st trophic level

2nd trophic level

3rd trophic level

4th trophic levelIf a new

predator on mice is introduced, what could happen to…

mice?

plants?

foxes?

eagles?

Trophic Cascade

Eagles

Foxes

Mice

Plants 1st trophic level

2nd trophic level

3rd trophic level

4th trophic levelIf drought

caused a dip in plant production, what would happen to…

mice?

foxes?

eagles?

Simplified Temperate Forest Food Web

What happens to when it’s a WEB instead of a CHAIN?

Oak seedling

Deer

Wolf

Fox

Rabbit

Grasses Herbs

Caterpillars

Shrews

Eagle

In long term, balance is restored

Food Web doesn’t account for Keystone Species

Kelp provides otter habitat

Sea urchins eat kelp

Otters eat sea urchins

Summary

Modeling Species Interactions Competition Predator-prey

Mutualism Commensalism

Community Dynamics Food Webs

Keystone Species