FW364 Ecological Problem Solving

FW364 Ecological Problem Solving

Class 24: CompetitionNovember 27, 2013

Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2

From the chemostat experiment:

Daphnia wins!Consumer with the lowest R* always wins

Rotifers will take early lead, but Daphnia will win at lower resource levels

Daphnia have a R* = 20 μg/LRotifers have a R* = 40 μg/L

More TODAY

Recap from Last Class

RD*

..... .. .. .. ..

..

.

...... .. .. .. ..

..

Biom

ass (

μg/L

)

..... .. .. .. ..

..

..... ... ... .. .

..

....

.Day 1

. ...

. . ...Day 12

... . ..

.. . .

.Day 21

0

20

40

60

80

100

120

0 2 4 6 8 10 12 14 16 18 20 22

Daphnia

Algae

Days

Rotifer

Rotifers do best at high resources

Daphnia win due to lower R*

But when R drops below rotifer R*(due to Daphnia consumption)

rotifers decline

RR*

Chemostat R* Experiment – Both Consumers

Competitive Exclusion Summary

To sum up

Given these assumptions:• a stable environment• competitors that are not equivalent (different R*)• a single resource• unlimited time

Then: The species with the lowest minimum resource requirement (R*) will eventually exclude all other competitors

Let’s look at some of the other assumptions we have made more closely

Additional assumptions (from predator-prey models):

1. The consumer populations cannot exist if there are no resources2. In the absence of both consumers, the resources grow exponentially3. Consumers encounter prey randomly (“well-mixed” environment)4. Consumers are insatiable (Type I functional response)5. No age / stage structure6. Consumers do not interact with each other except through consumption (i.e., exploitative competition)


Resource: dR/dt = brR - drR – a1RP1 – a2RP2

Competition Equation Assumptions

Additional assumptions (from predator-prey models):

1. The consumer populations cannot exist if there are no resources2. In the absence of both consumers, the resources grow exponentially3. Consumers encounter prey randomly (“well-mixed” environment)4. Consumers are insatiable (Type I functional response)5. No age / stage structure6. Consumers do not interact with each other except through consumption (i.e., exploitative competition)


Resource: dR/dt = brR - drR – a1RP1 – a2RP2

Competition Equation Assumptions

Assumption 4: Consumers are insatiable

i.e., consumers eat the same proportion of the resource population (a) no matter how many resources (R) there are Type I functional response

R

aR

low

0 many

high Type I functional response (linear)

Type II functional response

Satiation

To relax assumption, we can make the consumer feeding rate (aR)a saturating function of the resource abundance Type II functional response

Adding Consumer Satiation

Let’s define an equation for Type II response


Where: fmax is the maximum feeding rateh is the half-saturation constantR is resource abundance

fmax RR + h

f =

First, we need a new symbol for feeding rate: Feeding rate: f

For a Type I functional response (linear):

f = aR

For a Type II functional response (saturating):

Let’s look at a figure…



fmax RR + h

f =

0 5 10 15 20 25 300

1

2

3

4

5

Resource abundance (R)

Feed

ing

rate

(f) fmax = 5

Consumer feeding rate approaches fmax at high

resource abundance



fmax RR + h

f =

0 5 10 15 20 25 300

1

2

3

4

5


Feed

ing

rate

(f) fmax = 5

h is the value of R when the feeding rate is half of the maximum value i.e., h is value of R when f/fmax = 0.5

Challenge Question:

What is h for this figure?



fmax RR + h

f =

0 5 10 15 20 25 300

1

2

3

4

5


Feed

ing

rate

(f) fmax = 5


fmax = 5 and half of 5 is 2.5

So, h is value of R when f is 2.5 h = 2

2



fmax RR + h

f =

0 5 10 15 20 25 300

1

2

3

4

5


Feed

ing

rate

(f) fmax = 5


A Type II functional response can apply to any type of consumer:Carnivores, herbivores, parasites, and plants

Though plants do not eat (attack) resources, their growth still increases with resource abundance to some threshold rate

(i.e., until saturated with resources)

Let’s put the Type II response into our consumer growth equation (dP/dt)

Type II Functional Response - Equation

dP/dt = acRP – dpPType I functional response:

fmax RR + h

f =

dP/dt = caRP – dpPRe-arrange to get aR adjacent:

Replace aR with f: dP/dt = cfP – dpP

With Type II functional response:

Plug f into general equation:

dP/dt = cfP – dpPGeneral equation that we can put any functional response (f) into:

cfmax RPR + h

dP/dt = – dpP

Equation for consumer growth with a Type II functional response

cfmax RPR + h

dP/dt = – dpPOur functional response has changed,

so we need to a new R* equationi.e., R* for Type II response

cfmax R*P*R* + h

0 = – dpP*

R* occurs at steady-state,so set dP/dt = 0

cfmax R*P*R* + h

= dpP*

…a whole lot of algebra you do in Lab 10…

dp hc fmax - dp

R* =

Solve for R*

R* for Type II Functional Response

dp hc fmax - dp

R* =

Conclusions:

With a Type II functional response:R* depends on consumer death rate, half saturation constant,

conversion efficiency, and max feeding rate

If consumer death rate increases, R* increasesIf consumer half saturation constant increases, R* increasesIf conversion efficiency increases, R* decreasesIf max feeding rate increases, R* decreases

R* for Type II Functional Response

Saturation & Consumer Birth Rate

That was a lot about feeding rate…… need to get back to competition

To do that, need to make a crucial linkbetween consumer feeding rate and birth rate

R* is key for competition… and R* depends on dp

Competition winner is the consumer aliveat steady state … i.e., when bp = dp

Knowing birth rate of consumer is important for determining competition outcome

Let’s look at how a saturating feeding rate affects consumer birth rate

dp hc fmax - dp

R* =


cfmax RPR + h

dP/dt = – dpPType II functional response:

Minor re-arrangement:cfmax RR + h

dP/dt = P – dpP

This is all equivalent to our consumer birth ratei.e., consumers are born by feeding on prey

Consumer birth rate function should curve the same as the feeding rate,since birth rate is just feeding rate multiplied by a constant

(conversion efficiency)


0 5 10 15 20 25 30012345 fmax

high

high


Feed

ing

rate

(f)

0 5 10 15 20 25 30012345 bmax

high

high


birt

h ra

te (b

p)


0 5 10 15 20 25 30012345 fmax

high

high


Feed

ing

rate

(f)

0 5 10 15 20 25 30012345 bmax

high

high


birt

h ra

te (b

p)Consumer birth rate increases with resource abundance

to a threshold rate, bmax

(threshold birth rate is due to feeding rate hitting threshold)

h, the half-saturation constant, still applies:h is the value of R when the birth rate is half of the maximum value

Saturation & Consumer Death Rate

So that’s how consumer birth rate changes with resource density……now on to death rate

We have been making an (implicit) assumption abouthow consumer death rate changes with resource density

cfmax RPR + h

dP/dt = – dpP

We’ve been assuming that the consumer death rate is a constant (dp)

i.e., that the consumer death rate does NOT change with resource density

To plot this assumption on a figure…

0 5 10 15 20 25 30012345

high

high



deat

h ra

te (d

p)

Consumer death rate is just a straight line at any value along the y-axis

If we combine the death rate function with the birth rate curve…

Death rate


Consumer death rate is just a straight line at any value along the y-axis

If we combine the death rate function with the birth rate curve…we have a useful trick for graphically determining R* for a consumer…

0 5 10 15 20 25 30012345 Birth rate

high

high


birt

h ra

te (b

p) Death rate

deat

h ra

te (d

p)

(consumer birth rate and death rate must be plotted on the same scale!)

0 5 10 15 20 25 30012345 Birth rate

high

high


birt

h ra

te (b

p)

Graphical approach to R*

Death rate

deat

h ra

te (d

p)

Challenge question:

A special point on this figure represents steady state…Where is this point?

0 5 10 15 20 25 30012345 Birth rate

high

high


birt

h ra

te (b

p) Death rate

deat

h ra

te (d

p)

Challenge question:

A special point on this figure represents steady state…Where is this point?

Steady state when b = d


0 5 10 15 20 25 30012345 Birth rate

high

high


birt

h ra

te (b

p) Death rate

deat

h ra

te (d

p)

KEY feature of this graph:The resource abundance (i.e., value on x-axis) at the steady state

point (i.e., intersection of b and d functions) is R*!

R*



0 5 10 15 20 25 30012345 Birth rate

high

high


birt

h ra

te (b

p) Death rate

deat

h ra

te (d

p)

Key application:If we plot the birth and death rates of two competing species on same figure, we can determine which consumer will win based on who has the lower R*

R*



Graphical Approach to R*

First, one more question for single consumer:

0 5 10 15 20 25 30-0.5

1.5

3.5

5.5

7.5

Birth rate

high

high


birt

h ra

te (b

p)

Death rate

deat

h ra

te (d

p)

Quick Challenge Question:

What happens if the death rate is higher than the birth rate?

Graphical Approach to R*

First, one more question for single consumer:

0 5 10 15 20 25 30-0.5

1.5

3.5

5.5

7.5

Birth rate

high

high


birt

h ra

te (b

p)

Death rate

deat

h ra

te (d

p)

What happens if the death rate is higher than the birth rate? Consumer goes extinct, even without the competitor

Now let’s look at resource competition

Graphical R* & Competition

Outline:

Look at four graphical cases of two-species competition(competition with Type II functional response)

Consumers will have:

Case 1: Different birth rates, same death rate and hCase 2: Different birth rates and death rates, same hCase 3: Different birth rates, same death rates, different hCase 4: Different birth rates, same death rate, different h w/ twist

For each case, we’ll determine competition winner

Case 1: Effect of different birth rates

Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2)Both consumers have same death rate, d1 = d2

Monod curves never cross

b1

b2

d1d2

Who wins?

Resource level (R)

Birt

h an

d de

ath

rate

b1

b2

d1d2

Case 1: Effect of different birth rates

Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2)Both consumers have same death rate, d1 = d2

Monod curves never cross

Resource level (R)

Birt

h an

d de

ath

rate

R2*R1*

Consumer 1 wins: R1* < R2* Higher birth rate makes better competitor

Case 2A: Effect of different death rates

Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2)Consumer 1 has a higher death rate than Consumer 2 (d1 > d2)Monod curves never cross

d1

Who wins?

Resource level (R)

Birt

h an

d de

ath

rate

d2

b1

b2

Case 2A: Effect of different death rates


Resource level (R)

Birt

h an

d de

ath

rate

R1*R2*

d1

d2

b1

b2

Consumer 2 wins: R2* < R1* Lower death rate makes better competitor

Case 2B: Effect of different death rates


d1

Who wins?

d2

Resource level (R)

Birt

h an

d de

ath

rate b1

b2

Case 2B: Effect of different death rates


d1

d2

Resource level (R)

Birt

h an

d de

ath

rate

R1* R2*

b1

b2

Consumer 1 wins: R1* < R2* Lower death rate makes better competitor

Case 3: Effect of different h

Both consumers have same maximum birth rate, b1max = b2max

Both consumers have same death rate, d1 = d2

Consumer 1 has a lower h ( Consumer 1 approaches bmax at lower R)

d1d2

Who wins?

b1b2

Resource level (R)

Birt

h an

d de

ath

rate

At very high resource density

b1

b2

Case 3: Effect of different h

Both consumers have same maximum birth rate, b1 = b2

Both consumers have same death rate, d1 = d2

Consumer 1 has a lower h ( Consumer 1 approaches bmax at lower R)

d1d2

b1b2

Resource level (R)

Birt

h an

d de

ath

rate

At very high resource density

R2*R1*

Consumer 1 wins: R1* < R2* Lower h makes better competitor

b1

b2

What makes a better competitor (i.e., lower R*)?

If consumer death rate increases, R* increases so lower dp makes better competitor

If consumer half saturation constant increases, R* increases so lower h makes better competitor

If conversion efficiency increases, R* decreasesIf max feeding rate increases, R* decreases

Birth rate is just conversion efficiency * feeding rate, so higher birth rate makes better competitor

dp hc fmax - dp

R* = We reached the same conclusions looking at R* equation

Higher birth rate Lower death rate Lower h

Graphical R* & Competition Summary

Graphical R* & Competition Summary

What makes a better competitor (i.e., lower R*)?

Higher birth rate Lower death rate Lower h

Does this perfect competitor exist in nature?

Not really… there are always trade-offs in nature

e.g., high max birth rate requires more resources, foraging exposes consumers to predation, and so high bmax associated with high death rates

High birth rate rabbit takes risks to forage

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FW364 Ecological Problem Solving