1. FW364 Ecological Problem Solving
Class 24: Competition
November 27, 2013

2. Recap from Last Class More TODAY dP1/dt = a1c1RP1 – d1P1
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
From the chemostat experiment:
More TODAY
Rotifers have a R* = 40 μg/L
Daphnia have a R* = 20 μg/L
Daphnia wins!
Consumer with the lowest R* always wins
Rotifers will take early lead, but Daphnia will win at lower resource levels

3. . . . . . . Chemostat R* Experiment – Both Consumers RR* RD*
Rotifers do best at high resources
But when R drops below rotifer R*
(due to Daphnia consumption)
rotifers decline .
Day 1 .
Day 12 .
Day 21
 Daphnia win due to lower R*
Rotifer
Daphnia
Biomass (μg/L)
RR*
Algae
RD*
Days

4. 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

5. Competition Equation Assumptions
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
Resource:
dR/dt = brR - drR – a1RP1 – a2RP2
Additional assumptions (from predator-prey models):
The consumer populations cannot exist if there are no resources
In the absence of both consumers, the resources grow exponentially
Consumers encounter prey randomly (“well-mixed” environment)
Consumers are insatiable (Type I functional response)
No age / stage structure
Consumers do not interact with each other except through consumption
(i.e., exploitative competition)

6. Competition Equation Assumptions
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
Resource:
dR/dt = brR - drR – a1RP1 – a2RP2
Additional assumptions (from predator-prey models):
The consumer populations cannot exist if there are no resources
In the absence of both consumers, the resources grow exponentially
Consumers encounter prey randomly (“well-mixed” environment)
Consumers are insatiable (Type I functional response)
No age / stage structure
Consumers do not interact with each other except through consumption
(i.e., exploitative competition)

7. Adding Consumer Satiation 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
To relax assumption, we can make the consumer feeding rate (aR)
a saturating function of the resource abundance
 Type II functional response R aR
low
many
high
Type I functional response (linear)
Type II functional response
Satiation
Let’s define an equation for Type II response

8. Adding Consumer Satiation
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):
fmax R
R + h
f =
Where: fmax is the maximum feeding rate
h is the half-saturation constant
R is resource abundance
Let’s look at a figure…

9. Adding Consumer Satiation
fmax R
R + h
f =
Where: fmax is the maximum feeding rate
h is the half-saturation constant
R is resource abundance
fmax = 5
Consumer feeding rate approaches fmax at high resource abundance

10. Adding Consumer Satiation What is h for this figure?
fmax R
R + h
f =
Where: fmax is the maximum feeding rate
h is the half-saturation constant
R is resource abundance
fmax = 5
Challenge Question:
What is h for this figure?
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

11. Adding Consumer Satiation So, h is value of R when f is 2.5
fmax R
R + h
f =
Where: fmax is the maximum feeding rate
h is the half-saturation constant
R is resource abundance
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
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

12. Adding Consumer Satiation
fmax R
R + h
f =
Where: fmax is the maximum feeding rate
h is the half-saturation constant
R is resource abundance
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)
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

13. Type II Functional Response - Equation
Type I functional response:
dP/dt = acRP – dpP
Re-arrange to get aR adjacent:
dP/dt = caRP – dpP
Replace aR with f:
dP/dt = cfP – dpP
General equation that we can put any functional response (f) into:
dP/dt = cfP – dpP
fmax R
R + h
f =
With Type II functional response:
cfmax RP
R + h
dP/dt =
– dpP
Plug f into general equation:
 Equation for consumer growth with a Type II functional response

14. R* for Type II Functional Response
Our functional response has changed,
so we need to a new R* equation
i.e., R* for Type II response
cfmax RP
R + h
dP/dt =
– dpP
R* occurs at steady-state,
so set dP/dt = 0
cfmax R*P*
R* + h
0 =
– dpP*
Solve for R*
cfmax R*P*
R* + h
= dpP*
dp h
c fmax - dp
R* =
…a whole lot of algebra you do in Lab 10…

15. R* for Type II Functional Response
dp h
c 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* increases
If consumer half saturation constant increases, R* increases
If conversion efficiency increases, R* decreases
If max feeding rate increases, R* decreases

16. 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 link
between consumer feeding rate and birth rate
R* is key for competition… and R* depends on dp
Competition winner is the consumer alive
at steady state … i.e., when bp = dp
dp h
c fmax - dp
R* =
 Knowing birth rate of consumer is important for determining competition outcome
Let’s look at how a saturating feeding rate affects consumer birth rate

17. Saturation & Consumer Birth Rate
cfmax RP
R + h
dP/dt =
– dpP
Type II functional response:
cfmax R
R + h
dP/dt =
P – dpP
Minor re-arrangement:
This is all equivalent to our consumer birth rate
i.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)

18. Saturation & Consumer Birth Rate Resource abundance (R)
high
fmax
Feeding rate (f)
Resource abundance (R)
high
bmax
high
Resource abundance (R)
birth rate (bp)

19. Saturation & Consumer Birth Rate
high
fmax
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
Feeding rate (f)
Resource abundance (R)
high
high
bmax
birth rate (bp)
Resource abundance (R)
high

20. 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 about
how consumer death rate changes with resource density
cfmax RP
R + 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…

21. Saturation & Consumer Death Rate
Consumer death rate is just a straight line at any value along the y-axis
high
Death rate
death rate (dp)
Resource abundance (R)
high
If we combine the death rate function with the birth rate curve…

22. Saturation & Consumer Death Rate
Consumer death rate is just a straight line at any value along the y-axis
high
Birth rate
Death rate
birth rate (bp)
death rate (dp)
Resource abundance (R)
high
If we combine the death rate function with the birth rate curve…
we have a useful trick for graphically determining R* for a consumer…
(consumer birth rate and death rate must be plotted on the same scale!)

23. Graphical approach to R*
high
Birth rate
Death rate
birth rate (bp)
death rate (dp)
Resource abundance (R)
high
Challenge question:
A special point on this figure represents steady state…
Where is this point?

24. Graphical approach to R*
high
Birth rate
Death rate
birth rate (bp)
death rate (dp)
Steady state when b = d
Resource abundance (R)
high
Challenge question:
A special point on this figure represents steady state…
Where is this point?

25. Graphical approach to R*
high
Birth rate
Death rate
birth rate (bp)
death rate (dp)
Steady state when b = d R*
Resource abundance (R)
high
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*!

26. Graphical approach to R* Resource abundance (R)
high
Birth rate
Death rate
birth rate (bp)
death rate (dp)
Steady state when b = d R*
Resource abundance (R)
high
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*

27. Graphical Approach to R* First, one more question for single consumer:
Birth rate
high
Resource abundance (R)
birth rate (bp)
Death rate
death rate (dp)
Quick Challenge Question:
What happens if the death rate is higher than the birth rate?

28. Graphical Approach to R* First, one more question for single consumer:
Birth rate
high
Resource abundance (R)
birth rate (bp)
Death rate
death rate (dp)
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

29. 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 h
Case 2: Different birth rates and death rates, same h
Case 3: Different birth rates, same death rates, different h
Case 4: Different birth rates, same death rate, different h w/ twist
For each case, we’ll determine competition winner

30. Case 1: Effect of different birth rates
d1d2
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
Who wins?

31. Case 1: Effect of different birth rates
d1d2
R1*
R2*
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
Consumer 1 wins: R1* < R2*
 Higher birth rate makes better competitor

32. Case 2A: Effect of different death ratesb1 d1 b2 d2
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
Who wins?

33. Case 2A: Effect of different death ratesb1 d1
R1* b2 d2
R2*
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
Consumer 2 wins: R2* < R1*
 Lower death rate makes better competitor

34. 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
Who wins?

35. 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
Consumer 1 wins: R1* < R2*
 Lower death rate makes better competitor

36. Case 3: Effect of different h At very high resource density
b1b2 b1
At very high resource density b2
d1d2
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)
Who wins?

37. Case 3: Effect of different h
b1b2 b1
At very high resource density b2
d1d2
R1*
R2*
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)
Consumer 1 wins: R1* < R2*
 Lower h makes better competitor

38. Graphical R* & Competition Summary
What makes a better competitor (i.e., lower R*)?
Higher birth rate
Lower death rate
Lower h
dp h
c fmax - dp
R* =
We reached the same conclusions looking at R* equation
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* decreases
If max feeding rate increases, R* decreases
 Birth rate is just conversion efficiency * feeding rate,
so higher birth rate makes better competitor

39. 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