1. Ecology 8310 Population (and Community) Ecology
Competition: the R* approach
Consumer and resource dynamics
A graphical approach
ZNGIs
Consumption vectors
Resource renewal
Putting it together
Tests

2. The basics: Consumer growth dN/Ndt Mortality R* Resource density (R)
Resource density (R)
dN/Ndt
Consumer growth
Mortality
Explain Monod or Michaelis-Menton function:
rmax is asymptote (max growth rate)
K is half saturation constant: R at which per capita growth rate is ½ of max.
Initial slope is r/k R*

3. Loss of resource and consumer
The basics: A chemostat
Inflow of resource
Loss of resource and consumer

4. Write out equations for dynamics of the consumer (N) and resource (R)…
F=flow rate into (out of) chemostat
V=volume of chemostat
R0=concentration of resource in inflow
Growth of consumer follows Monod function

5. Note: F/V is the mortality rate (per capita"wash out" rate)

6. The dynamics: Set by concentration in inflow Time
Time
Consumer or Resource Density
Consumer
Resource R*
Introduce consumer

7. What happens to N* and R*
if we increase the flow?

8. Washout > maximum growth
The basics:
Resource density (R)
dN/Ndt
Consumer growth
Mortality
Washout > maximum growth R*

9. What if we have two consumers?

10. Two consumers: Consumer 1's growth Consumer 2's growth dN/Ndt
Resource density (R)
dN/Ndt
Consumer 1's growth
Mortality
Consumer 2's growth R* R*

11. Synedra Asterionella
What will happen if we put one (or both) species into a chemostat and silicate is limiting?
From Tilman et al (L&O)

12. Followed population growth and resource (silicate) when alone:
Data = points. Lines = predicted from model

13. In Competition:
Synedra wins

14. What if we change the environment?

15. Synedra won at 24oC.
Who will win at 8oC?

16. Asterionella Synedra
Who will win?
From Tilman et al (L&O)

17. From Tilman et al (L&O)

18. From Tilman et al (L&O)

19. What about changing the mortality rate?

20. Two consumers: Consumer 1's growth Consumer 2's growth dN/Ndt
Resource density (R)
dN/Ndt
Consumer 1's growth
Mortality
Consumer 2's growth

21. Let's extend this to >1 resource…

22. We could approach this mathematically, but Tilman advanced an elegant graphical approach (underlain by explicit math)…

23. ZNGIs: R1 R2
Zero Net Growth Isocline (ZNGI): all (R1, R2) combinations at which dN/dt=0

24. ZNGIs: Essential Substitutable R1 R2 R1 R2 R1 R2 R1 R2 Switching
Complementary

25. ZNGI's tell us when the consumer is at equilibrium.
What about the resources?

26. Resource equilibrium:
Supply = Consumption

27. Resource supply: "equable" vs. biotic (logistic) resources
Equable (abiotic) R1 R2 S
Biotic (logistic) R1 R2 S

28. Resource consumption?

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

30. Resource supply: Resource Supply Point Resource Supply Rates R2
Consumption vectors

31. Stable equilibrium: R1 R2
Resource Supply Point

32. Two consumers…

33. What is the long-term outcome?
Competition – 1 scenario: R1 R2
What is the long-term outcome? S

34. What is the long-term outcome?
Competition – another scenario: R1 R2
What is the long-term outcome? S

35. What else do we need to specify?

36. Competition – 1 scenario:
We need to find R1* and R2* S

37. Notice that N* is implicit

38. What is the long-term outcome?
Competition – another scenario: R1 R2
What is the long-term outcome? S

39. Competition – another scenario:
Blue can make it, but not red (but not competitive exclusion)
What about this region?
Red can make it, but not blue (but not competitive exclusion)
Neither species can make it

40. Specify S and consumption vectors
Competition – another scenario:
Specify S and consumption vectors S R1 R2
Now what?
What are the possible patterns for consumption of the system as a whole? Rcall that the Consumption vectors are a product of N x consumption, so the size of each vector (red or blue) is just scaled by the size of the consumer population. So then, what possible consumer vectors could are possible for the combined system of consumers?

41. Range of possible consumptions by N1&N2
Competition – another scenario: S
Range of possible consumptions by N1&N2 R1 R2
Point out that there is no way to combine the two consumption vectors to match the supply vector. This would suggest that we cannot achieve coexistence because the resoures canNOT be at equilibrium when the consumers are – they system will change. But how?

42. Let's look at invasibility…

43. Where is the single species equilibrium for Blue?
Competition – invasibility?
Where is the single species equilibrium for Blue?
Can Red invade? S R1 R2

44. Where is the single species equilibrium for Red?
Competition – invasibility? S
Where is the single species equilibrium for Red?
Can Blue invade? R1 R2

45. You should figure this one out.
Competition – another scenario: R1 R2 S
You should figure this one out.

46. Could get 2 species, but is this equilibrium stable?
Competition – another scenario: S R1 R2
Could get 2 species, but is this equilibrium stable?

47. Let's look at invasibility…

48. Invasibility? S R1 R2
Can red invade?

49. Invasibility: S R1 R2
Can Blue invade?

50. Mutual invasibility = Co-existence!

51. Can we interpret the conditions for coexistence?

52. Resource limitation? Which resource limits Red vs. Blue? S
Which resource is used primarily by Red vs. Blue?
So, "intra vs. inter"?

54. What if we alter the flow rate of the chemostat?
Which species will win?
What if we alter the flow rate of the chemostat?
Can we draw the ZNGIs?

55. Experimental test: vary ratio of resources
PO4
Asterionella
Cyclotella
0.01-
0.20-
0.6
1.9

56. What about consumption vectors?
Tilman got those by looking at cell quotients (ratio of resources in the cells of the two species)..

57. Experimental test Cyclotella S PO4 0.20- Asterionella 0.01- 0.6 1.9
SO2
PO4 S
Asterionella
Cyclotella
0.01-
0.20-
0.6
1.9

58. Experimental test
SO2
PO4
Asterionella
Cyclotella
0.01-
0.20-
0.6
1.9

59. Experimental test S PO4 SO2
Red wins in first S (better comp for P), but as we shift the ratio of Si:P, we move into a region of coexistence and then into a region where Blue wins (because it’s a better competitor for Si).

60. Test: vary ratio of nutrients supplied to chemostat
What do you expect? What happens as we got from hi Si/P to low Si/P?

63. Homework 6
Note that we've flipped the consumption vectors

64. R1 R2
Note that we've flipped the consumption vectors

65. Determine "who wins" for each region in the previous slide.
Homework:
Determine "who wins" for each region in the previous slide.
Evaluate co-existence based on invasibility, when there is an equilibrium that potentially allows the two consumers to persist
For a supply point in the wedge, sketch out (do not simulate – just think about it) 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 next class