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: 0 Resource density (R) dC/Cdt Consumer growth Mortality R*

3. The basics: A chemostat Consumer demographics; resource "non-dynamic" Inflow of resource Loss of resource and consumer

4. The basics: dynamics Consumer demographics; resource "non-dynamic" Inflow of resource Loss of resource and consumer

5. The basics: 0 Time Consumer or Resource Density Consumer Resource Set by concentration in inflow R* Introduce consumer

6. What happens to C* and R* if we increase the flow?

7. The basics: 0 Resource density (R) dC/Cdt Consumer growth Mortality R*

8. What if we have two consumers?

9. Two consumers: 0 Resource density (R) dC/Cdt Consumer 1's growth Mortality R* Consumer 2's growth

10. Synedra Asterionella From Tilman et al. 1981 (L&O)

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

12. In Competition: Synedra wins

13. What if we change the environment?

14. Synedra won at 24 o C. Who will win at 8 o C?

15. Asterionella Synedra From Tilman et al. 1981 (L&O) Who will win?

16. From Tilman et al. 1981 (L&O)

18. What about changing the mortality rate?

19. Two consumers: 0 Resource density (R) dC/Cdt Consumer 1's growth Mortality Consumer 2's growth

20. Let's extend this to >1 resource…

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

22. ZNGIs: 0 R1R1 R2R2 Zero Net Growth Isocline (ZNGI): all (R 1, R 2 ) combinations at which dC/dt=0

23. ZNGIs: R1R1 R2R2 R1R1 R2R2 R1R1 R2R2 R1R1 R2R2 EssentialSubstitutable Complementary Switching

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

25. Resource equilibrium: Supply = Consumption

26. Resource supply: "equable" vs. biotic (logistic) resources Equable (abiotic) R1R1 R2R2 S R1R1 R2R2 S Biotic (logistic)

27. Resource consumption?

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

29. Resource supply: R1R1 R2R2 Resource Supply Point Resource Supply Rates Consumption vectors

30. Stable equilibrium: R1R1 R2R2 Resource Supply Point

31. Two consumers…

32. Competition – 1 scenario: R1R1 R2R2 S What is the long- term outcome?

33. Competition – another scenario: R1R1 R2R2 S What is the long- term outcome?

34. What else do we need to specify?

35. Competition – 1 scenario: R1R1 R2R2 S We need to find R 1 * and R 2 *

36. Notice that C* is implicit

37. Competition – another scenario: R1R1 R2R2 S What is the long- term outcome?

38. Competition – another scenario: R1R1 R2R2 Neither species can make it Blue can make it, but not red (but not competitive exclusion) Red can make it, but not blue (but not competitive exclusion) What about this region?

39. Competition – another scenario: R1R1 R2R2 S Specify S and consumption vectors Now what?

40. Competition – another scenario: R1R1 R2R2 S

41. Let's look at invasibility…

42. Competition – invasibility? R1R1 R2R2 SWhere is the single species equilibrium for Blue?

43. Competition – invasibility? R1R1 R2R2 SWhere is the single species equilibrium for Red?

44. Competition – another scenario: R1R1 R2R2 S You should figure this one out.

45. Competition – another scenario: R1R1 R2R2 S Could get 2 species, but is this equilibrium stable?

46. Let's look at invasibility…

47. Invasibility? R1R1 R2R2 S Can red invade?

48. Invasibility: R1R1 R2R2 S Blue can invade

49. Co-existence!

50. Can we interpret the conditions for coexistence?

51. Resource limitation? R1R1 R2R2 S Which resource limits Red vs. Blue? Which resource is used primarily by Red vs. Blue? So, "intra vs. inter"?

52. Experimental test: vary ratio of resources R1R1 R2R2 S

56. Another possibility: R1R1 R2R2 Homework #4

57. Homework 4: 1)Determine "who wins" for each region in the previous slide. 2)Evaluate co-existence based on invasibility, when there is an equilibrium that potentially allows the two consumers to persist 3)For a supply point in the wedge, sketch out 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 5pm Monday