Ecology 8310Population (and Community) Ecology

Competition: the R* approach

• Consumer and resource dynamics• A graphical approach

• ZNGIs• Consumption vectors• Resource renewal• Putting it together

• Tests

The basics:

0

Resource density (R)

dC

/Cd

tConsumer growth

Mortality

R*

The basics: A chemostat

Consumer demographics; resource

"non-dynamic"

Inflow of resource Loss of resource and consumer

The basics: dynamics

Consumer demographics; resource

"non-dynamic"

Inflow of resource Loss of resource and consumer

The basics:

0

Time

Con

sum

er

or

Reso

urc

e D

en

sity Consume

r

Resource

Set by concentration in

inflow

R*

Introduce consumer

What happens to C* and R* if we increase the flow?

The basics:

0

Resource density (R)

dC

/Cd

t

Consumer growth

Mortality

R*

What if we have two consumers?

Two consumers:

0

Resource density (R)

dC

/Cd

tConsumer 1's growth

Mortality

R* R*

Consumer 2's growth

Synedra Asterionella

From Tilman et al. 1981 (L&O)

Alone:Followed population growthand resource (silicate) when

alone:

Data = points.Lines = predicted from model

In Competition:

Synedra wins

What if we change the environment?

Synedra won at 24oC.

Who will win at 8oC?

Asterionella Synedra

From Tilman et al. 1981 (L&O)

Who will win?

From Tilman et al. 1981 (L&O)

From Tilman et al. 1981 (L&O)

What about changing the mortality rate?

Two consumers:

0

Resource density (R)

dC

/Cd

tConsumer 1's growth

Mortality

Consumer 2's growth

Let's extend this to >1 resource…

We could approach this mathematically, but Tilman advanced an elegant graphical approach (underlain by

explicit math)…

ZNGIs:

0

R1

R2

Zero Net Growth Isocline (ZNGI): all

(R1, R2) combinations at which dC/dt=0

ZNGIs:

R1

R2

R1

R2

R1

R2

R1

R2

Essential Substitutable

Complementary

Switching

ZNGI's tell us when the consumer is at equilibrium.

What about the resources?

Resource equilibrium:

Supply = Consumption

Resource supply: "equable" vs. biotic (logistic) resources

Equable (abiotic)

R1

R2

S

R1

R2

S

Biotic (logistic)

Resource consumption?

We'll assume: 1) essential resources; 2) fixed stoichiometry (i.e., consumption

ratio is constant)

Resource supply:

R1

R2

Resource Supply Point

Resource Supply Rates

Consumption vectors

Stable equilibrium:

R1

R2

Resource Supply Point

Two consumers…

Competition – 1 scenario:

R1

R2

SWhat is the long-term outcome?

Competition – another scenario:

R1

R2

SWhat is the long-term outcome?

What else do we need to specify?

Competition – 1 scenario:

R1

R2

SWe need to find

R1* and R2*

Notice that C* is implicit

Competition – another scenario:

R1

R2

SWhat is the long-term outcome?

Competition – another scenario:

R1

R2

Neither species

can make it

Blue can make it, but not red (but

not competitive exclusion)

Red can make it, but not blue (but

not competitive exclusion)

What about this region?

Competition – another scenario:

R1

R2

SSpecify S

and consumption vectors

Now what?

Competition – another scenario:

R1

R2

S

Let's look at invasibility…

Competition – invasibility?

R1

R2

S Where is the single species

equilibrium for Blue?

Competition – invasibility?

R1

R2

S Where is the single species

equilibrium for Red?

Competition – another scenario:

R1

R2

S

You should figure this one

out.

Competition – another scenario:

R1

R2

S

Could get 2 species, but is this equilibrium stable?

Let's look at invasibility…

Invasibility?

R1

R2

S

Can red invade?

Invasibility:

R1

R2

S

Blue can invade

Co-existence!

Can we interpret the conditions for coexistence?

Resource limitation?

R1

R2

S

Which resource limits Red vs. Blue?

Which resource is

used primarily

by Red vs. Blue?

So, "intra vs. inter"?

Experimental test: vary ratio of resources

R1

R2

S

Another possibility:

R1

R2

Homework #4

Homework 4:

1) Determine "who wins" for each region in the previous slide.

2) Evaluate co-existence based on invasibility, when there is an equilibrium that potentially allows the

two consumers to persist

3) For a supply point in the wedge, sketch out the dynamics (densities through time for the two

resources and the two consumers) if you start the system with very low numbers of each consumer

Due by 5pm Monday