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)
Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2
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)
Predator 1: dP1/dt = a1c1RP1 – d1P1 Predator 2: dP2/dt = a2c2RP2 – d2P2
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
Adding Consumer Satiation
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…
Adding Consumer Satiation
Where: fmax is the maximum feeding rateh is the half-saturation constantR is resource abundance
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
Adding Consumer Satiation
Where: fmax is the maximum feeding rateh is the half-saturation constantR is resource abundance
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
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?
Adding Consumer Satiation
Where: fmax is the maximum feeding rateh is the half-saturation constantR is resource abundance
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
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
fmax = 5 and half of 5 is 2.5
So, h is value of R when f is 2.5 h = 2
2
Adding Consumer Satiation
Where: fmax is the maximum feeding rateh is the half-saturation constantR is resource abundance
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
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
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* =
Saturation & Consumer Birth Rate
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)
Saturation & Consumer Birth Rate
0 5 10 15 20 25 30012345 fmax
high
high
Resource abundance (R)
Feed
ing
rate
(f)
0 5 10 15 20 25 30012345 bmax
high
high
Resource abundance (R)
birt
h ra
te (b
p)
Saturation & Consumer Birth Rate
0 5 10 15 20 25 30012345 fmax
high
high
Resource abundance (R)
Feed
ing
rate
(f)
0 5 10 15 20 25 30012345 bmax
high
high
Resource abundance (R)
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
Resource abundance (R)
Saturation & Consumer Death Rate
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
Saturation & Consumer 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
Resource abundance (R)
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
Resource abundance (R)
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
Resource abundance (R)
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
Graphical approach to R*
0 5 10 15 20 25 30012345 Birth rate
high
high
Resource abundance (R)
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*
Steady state when b = d
Graphical approach to R*
0 5 10 15 20 25 30012345 Birth rate
high
high
Resource abundance (R)
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*
Steady state when b = d
Graphical approach to 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
Resource abundance (R)
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
Resource abundance (R)
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
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
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
Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2)Consumer 2 has a higher death rate than Consumer 1 (d2 > d1)Monod curves never cross
d1
Who wins?
d2
Resource level (R)
Birt
h an
d de
ath
rate b1
b2
Case 2B: Effect of different death rates
Consumer 1 has a higher birth rate than Consumer 2 at all R levels (b1 > b2)Consumer 2 has a higher death rate than Consumer 1 (d2 > d1)Monod curves never cross
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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