1. FW364 Ecological Problem Solving
Class 23: Competition
November 25, 2013

2. Outline for Today Shifting focus from predator-prey interactions
to two species competition
Objectives for Today:
Derive equations for two-species resource competition
Introduce R* rule to determine competition winner
Objectives for Next Three Classes:
Explore R* rule in more detail
Include Type II functional response for consumers
Examine graphical approaches for determining competition winner
Discuss limits to competitive exclusion
Discuss practical application of resource competition models
No textbook chapters!

3. Competition Types – The Basics
Two types of competition:
NEW focus
Intraspecific = within species
Interspecific = between species
Previous focus
Scramble & contest density dependence

4. Competition Types – The Basics
Two types of interspecific competition (also saw for intraspecific):
Exploitative = indirect competition
Competition through a
common resource  scramble
Interference = direct competition
Aggressive / physical encounters
for resources  contest
Plant competition
for nutrients
Plant chemical warfare
e.g., manzanita

5. Exploitative Interspecific Competition
We will only model exploitative interspecific competition
a.k.a. resource competition
Conceptual framework
Consumer 1
Consumer 2
Resource
Applies to: Examples:
Carnivores-animal prey Herons and cranes competing for fish in swamps
Herbivores-plant Zebras and wildebeest consuming grasses
Parasites-host Sea lamprey and copepods parasitizing lake trout
Plants-resource Ferns and grass competing for nutrients

6. Exploitative Interspecific Competition
Key features: R
Competition is an extension of predator-prey concepts we just studied!
 Two coupled predator-prey interactions that share same prey (resource)
Interaction between consumers mediated through resource
No direct interaction (this is an assumption!)
Only one consumer can persist at steady state (another assumption)
Competitive exclusion principle
i.e., when two species compete over a common resource, only one species (the superior competitor) can persist in the long-term

7. Exploitative Interspecific Competition
Key features: R
To reiterate:
With the competitive exclusion principle,
we are assuming there is always a winner in the long run
i.e., one consumer will out-compete (exclude) the other
The winner of the competition is the consumer that is still alive at steady state, whereas the loser is the consumer that has gone extinct
This doesn’t have to do with victims/prey, but does get to the idea of two predators enter, one predator leaves. 

8. Exploitative Interspecific Competition
Goals R
Determine the competition winner at steady state (algebra)
Describe dynamics and forecast time to extinction of inferior competitor (Stella – Lab 10)
Important to understand the difference between the two approaches
and how they complement each other
#1 tells us WHO will win
#2 tells us WHEN the loser goes extinct,
and HOW populations change through time
Let’s build equations!

9. Competition Equations
Like before, we’ll build simple models that capture the essence
of two-species resource competition
(we’ll be making many assumptions)
Start with our basic (coupled) predator-prey equations:
dV/dt = bvV - dvV - aVP
dP/dt = acVP - dpP
Victim, V:
Predator, P:
Adapt more general notation of consumer-resource
substitute R for V, but keep P
Note that I will use “consumer” and “predator” interchangeably
Resource, R:
dR/dt = brR - drR - aRP
Predator, P:
dP/dt = acRP - dpP

10. Competition Equations
Resource, R:
dR/dt = brR - drR - aRP
Predator, P:
dP/dt = acRP - dpP
Key: Need equations for each consumer!
Predator 1, P1:
dP1/dt = a1c1RP1 – d1P1
Predator 2, P2:
dP2/dt = a2c2RP2 – d2P2
Each consumer can have own a, c, and d (note subscripts specific to consumers)
 This will be important later for determining competition winner
Now need to include consumption by both consumers in resource equation:
Resource, R:
dR/dt = brR - drR – a1RP1 – a2RP2
brR Number of resources born per time (same as before)
drR Number of non-predatory resource deaths per time (same as before)
a1RP1 Number of resources eaten by Consumer 1 population per time
a2RP2 Number of resources eaten by Consumer 2 population per time

11. Competition Equations
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
Resource:
dR/dt = brR - drR – a1RP1 – a2RP2
These are our competition equations!
Some key features:
Equations are adaptable to competition between plants for resources
with slight modification to the resource equation
We can predict the outcome of competition from the consumer equations!
(do not need the resource equation)

12. Competition Equations dR/dt = brR - drR – a1RP1 – a2RP2
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
Resource:
dR/dt = brR - drR – a1RP1 – a2RP2
Assumptions:
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)

13. Steady State Winner dP1/dt = a1c1RP1 – d1P1 dP2/dt = a2c2RP2 – d2P2
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
Resource:
dR/dt = brR - drR – a1RP1 – a2RP2
Most important question for competition:
Who will win in the long-term?
i.e., which competitor will be alive once the system has reached steady state?
Key Point:
The competitor that wins in the long-term MAY NOT be the
competitor that does best initially
Different factors are involved in short-term vs. long term competitive ability

14. Steady State Winner dP1/dt = a1c1RP1 – d1P1 dP2/dt = a2c2RP2 – d2P2
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
Resource:
dR/dt = brR - drR – a1RP1 – a2RP2
Short vs. long-term competitive ability
We all know an example of this:
One consumer may have advantage
of speed in the short-term,
but endurance is what matters in
the long-run for competition
Let’s look at a figure
The tortoise and the hare fable

15. Which consumer is the winner?
Steady State Winner
Resource
Consumer 1
Consumer 2
Biomass (g/L)
Which consumer is the winner?
 Consumer 2

16. Two consumers introduced
Steady State Winner
Resource
Consumer 1
Consumer 2
Biomass (g/L)
Two consumers introduced
Consumer 1 dominates at first when resources are abundant
Consumer 2 wins in long-run when resources are low
Steady state reached

17. Steady State Winner Resource Consumer 1 Consumer 2
Biomass (g/L)
Reminder: Steady state is when there is no change
in abundance through time
i.e., dP1/dt = 0, dP2/dt = 0, and dR/dt = 0
Steady state reached

18. Use our competition equations to derive
Steady State Winner
Resource
Consumer 1
Consumer 2
Next step:
Use our competition equations to derive
general equations that tell us which consumer will survive and which consumer will go extinct
at steady state
 R* rule
Biomass (g/L)
Reminder: Steady state is when there is no change
in abundance through time
i.e., dP1/dt = 0, dP2/dt = 0, and dR/dt = 0
Steady state reached

19. R* Rule R* Rule determines competitive dominance at steady state
R* is the resource level at equilibrium
The R* Rule:
The resource abundance at equilibrium (R*) is that level of the resource at which the consumer is just maintained in the system
i.e., R* is the lowest resource level at which a consumer can be sustained
If the resource level were to decrease, the consumer would go extinct
Each consumer has it’s own R*
Competing consumers will (almost always) have different R*
 R* will determine which consumer wins

20. R* Rule R* Rule determines competitive dominance at steady state
R* is the resource level at equilibrium
The R* Rule:
The resource abundance at equilibrium (R*) is that level of the resource at which the consumer is just maintained in the system
i.e., R* is the lowest resource level at which a consumer can be sustained
If the resource level were to decrease, the consumer would go extinct
R* for a
rotifer consumer
Let’s look at an experimental example

21. . . Chemostat R* Experiment – Consumer 1 R*
Day 1 .
Day 12
R* is the lowest level of algae that maintains rotifers in the system
= steady state abundance (biomass) of algae
For rotifers, R* = 40 μg/L
Rotifer
Biomass (μg/L)
Algae R*
Days

22. . . Chemostat R* Experiment – Consumer 1 R* Challenge Question:
Day 1 .
Day 12
Challenge Question:
What would happen if we increased the amount of algae being delivered?
Think about last class!
Rotifer
Biomass (μg/L)
Algae R*
Days

23. . . Chemostat R* Experiment – Consumer 1 R* Challenge Question:
Day 1 .
Day 12
Challenge Question:
What would happen if we increased the amount of algae being delivered?
Algae level stays SAME
Rotifer level would increase
Rotifer
Biomass (μg/L)
Algae R*
Days

24. R* Rule
That’s how we can determine R* empirically (experimentally) for a single species
Can also determine R* by building an equation
Let’s derive equation for R*
Recall for predator-prey equations that we used the
predator equation to derive equation for V*
and prey equation to derive equation for P*
Likewise, we will use consumer equation to derive equation for R*
 We’ll start with Consumer 1

25. R* Rule 0 = a1c1R*P1* – d1P1* a1c1R*P1* = d1P1* a1c1R*P1* d1P1* =
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Let’s look at the other consumer
R* occurs at steady-state,
so set dP1/dt = 0
and denote equilibrium with *
Conclusions:
The minimum resource requirement
(R*) for a consumer is determined by
the consumer death rate, attack rate,
and conversion efficiency
If death rate increases, R* increases
If attack rate increases, R* decreases
If conversion efficiency increases,
R* decreases
Note:
We determine R* for each consumer
when ALONE!
0 = a1c1R*P1* – d1P1*
Solve for R*
a1c1R*P1* = d1P1*
a1c1P1*
a1c1R*P1*
d1P1* = d1
a1c1
R* =

26. R* Rule 0 = a1c1R*P1* – d1P1* 0 = a2c2R*P2* – d2P2* a1c1R*P1* = d1P1*
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
R* occurs at steady-state,
so set dP1/dt = 0
and denote equilibrium with *
R* occurs at steady-state,
so set dP2/dt = 0
and denote equilibrium with *
0 = a1c1R*P1* – d1P1*
0 = a2c2R*P2* – d2P2*
Solve for R*
Solve for R*
a1c1R*P1* = d1P1*
a2c2R*P2* = d2P2*
a1c1P1*
a1c1R*P1*
d1P1* =
a2c2P2*
a2c2R*P2*
d2P2* = d1
a1c1
R* = d2
a2c2
R* =

27. R* Rule 0 = a1c1R*P1* – d1P1* a1c1R*P1* = d1P1* a1c1R*P1* d1P1* =
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
R* occurs at steady-state,
so set dP1/dt = 0
and denote equilibrium with *
Let’s look at a
chemostat experiment for
the second consumer
0 = a1c1R*P1* – d1P1*
Solve for R*
a1c1R*P1* = d1P1*
a1c1P1*
a1c1R*P1*
d1P1* =
Daphnia d1
a1c1
R* = d2
a2c2
R* =

28. Chemostat R* Experiment – Consumer 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Day 21 .
Day 1 .
Day 12
Daphnia
Biomass (μg/L)
Daphnia have a R* = 20 μg/L
Algae R*
Days

29. R* Rule – Competitive Exclusion
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
From the chemostat experiment:
Rotifers have a R* = 40 μg/L
Daphnia have a R* = 20 μg/L
Challenge Question:
Given each R*, which consumer wins in long-run?
What will happen if these two consumers are put together?

30. R* Rule – Competitive Exclusion
Predator 1:
dP1/dt = a1c1RP1 – d1P1
Predator 2:
dP2/dt = a2c2RP2 – d2P2
From the chemostat experiment:
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
Challenge Question:
Given each R*, which consumer wins in long-run?
What will happen if these two consumers are put together?

31. More R*
When the resource level falls below the equilibrium level for a consumer
(when R < R*),
the consumer density will decline
When each consumer is alone, the consumer will drive R down to R*,
but when a competitor is added, the second consumer can drive R < R*!
The consumer whose biological characteristics are such that its minimum resource requirement (R*) are lowest WINS competition dp ac
R* =
According to our R* equation:
The consumer with a lower death rate, higher attack rate,
and/or greater conversion efficiency will win
i.e., any characteristic that decreases R* will provide a competitive advantage

32. Chemostat R* Experiment – Both Consumers. . . .
Day 1 .
Day 12 .
Day 21
Rotifer
Daphnia
Biomass (μg/L)
Algae
RD*
Days

33. . . . . . . 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

34. 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

35. Adding Type II functional response Stella lab – Meet in Computer Lab
Looking Ahead
Next Class:
More R*
Adding Type II functional response
Lab Tomorrow
Competition modeling
Stella lab – Meet in Computer Lab