1. Tilman’s Model of Competition For Resources Mark McGinley Associate Professor Honors College and Department of Biological Sciences Texas Tech University

2. R* model- single species At equilibrium B = D C = S B < D C < S B > D C > S R* Resource Level

3. R* model- 2 species At equilibrium Ba = Da Bb <Db C = S Ba > Da Bb >Db C > S Ba > Da Bb < Db C > S Ba < Da Bb < Db C < S R*b R*a Resource Level Species A wins because it has the lower R*

4. Tilman’sModel- competition for 2 limiting resources Zero Net Growth Isocline (ZNGI) Islocline is a line made up of all combination of points that share the same value ZNGIs are the line made up of all combinations of the level of R1 and R2 at which the population growth rate of a species is = 0 ZNGIs are characteristics of species and have to be determined empirically

5. Tilman’s model- ZNGIs R2 R2 R1 R1 Completely substitutable resources e.g., big macs and whoppers Non- substitutable resources e.g., light and nitrogen

6. Tilman’s Model- Consumption Vectors Consumption vectors show the change in resource availability caused by consumption The total amount of consumption depends on Consumption of an individual Number of individuals Consumption vectors are characteristics of species and have to be measured

7. Consumption vectors Consumption of R1 Consumption or R2 Total consumption R2 R1

8. Interpreting Consumption Vectors This species consumes more R1 than R2 R2 R1

9. Tilman’s Model- Supply Vectors The supply vectors measure the rate of supply of resources in the environment The supply vectors depend upon The current resource levels The Resource Supply Point (S)

10. Resource Supply Point (S) The resource supply point measures the total amount of resources in an environment Resources can either be found In the environment In living organisms The resource supply point is a characteristic of an environment and can be measured

11. Supply Vectors S R2 R1 Supply vectors point from the current resource level To the resource supply point

13. S

14. Consumption vectorsS R2 R1

15. 2 resources, 1 species S R2 R1 If we add some individuals into an environment where the resource level equals the resource supply point then we expect the resource level to decrease over time. The resource level should reach equilibrium somewhere along the ZNGI, But where??

16. 2 resources, 1 species E The equilibrium resource level (E) occurs where the Consumption vectors and the supply vectors are equal and opposite. R2 R1 When the resource level is at E then B = D and C = S. This is a stable equilibrium.

17. 2 resources, 2 species ZNGIs R2 Sp B Sp A R1 Species B is more limited by R1 than Species A Species B is more limited by R2 than Species A

18. 2 resources, 2 speciesConsumption Vectors A R2 B R1 Species A consumes more R1 than it does R2 Species B consumes more R2 than it does R1

19. 2 resources, 2 specieswhat is outcome of competition?Depends on S. R2 Sp B Sp A S R1 S below both isoclines, therefore neither species can survive

20. 2 resources, 2 specieswhat is outcome of competition? R2 Sp B Sp A R1 Species B can not survive and Species A can survive

21. 2 resources, 2 species What is the outcome of competition? S R2 Sp B Sp A R1 Coexistence is only potentially possible if S falls in this region. Because species A can reduce the levels of both R1 and R2 below the ZNGI for species B, species A will drive species B to extinction and thus be the winner of competition. What will the equilibrial resource level be?

22. R2 Sp B Sp A Species A wins E S R1

23. Practice Problem What is the outcome of competition and the equilibrial resource level if the position of the two species’ ZNGIs are reversed?

24. 2 resources, 2 speciesWhat is required for coexistence? In order for both species to coexist there needs to be a resource level in the environment at which the growth rate of both species is equal to zero. This can only happen if the ZNGIs intersect

25. 2 resources, 2 speciesintersecting ZNGIs R2 Sp B Sp A R1 Species A is more limited by R1 than species B Species B is more limited by R2 than Species A

26. 2 resources, 2 speciesintersecting ZNGIs The ultimate outcome of competition between two species when their ZNGIs intersect depends upon Consumption vectors Resource Supply Point

27. Add consumption vectors R2 Sp B A Sp A B R1 Species A consumes more R1 than R2 Species B consumes more R2 than R1

28. Importance of the location of S 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 Extending the consumption curves allow us to divide the graph into 6 regions. The outcome of competition depends upon in which region the resource supply point falls.

29. S falls in region 1 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 Neither species A or B can survive

30. S falls in region 2 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 Only species B can survive

31. S falls in region 6 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 Only species A can survive

32. When is coexistence even imaginable? Both species must be able to grow to have any chance of co-existence Thus, coexistence is only potentially possible if S falls in regions 3, 4, or 5

33. S in region 3 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 If S is in region 3, eventually consumption of resources will move The current resource level into region 2. When that happens species A is unable to survive so species B wins.

34. S in region 5 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 If S is in region 5, eventually consumption of resources will move The current resource level into region 6. When that happens species B is unable to survive so species A wins.

35. Coexistence is only possible if S is in region 4 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 If S is in region 4 the level of resources will be reduced To the point where the two ZNGIs cross. At that point Consumption = supply and the growth rate of both species Is equal to zero.

36. If S is in region 4, then coexistence is possible But is this a stable or unstable equilibrium? If the equilibrium is unstable then we expect that any slight changes in population sizes or resource level that moves the system away from the equilibrium will move the system to another point If the equilibrium is stable, then we expect to see that situation continuing in nature Whether the equilibrium is stable or unstable depends on the consumption vectors This is a difficult math proof, so let’s just believe it and see what we can learn

37. Stable Equilibrium if S is in region 4 3 2 4 1 R2 5 Sp B 6 A Sp A B R1 Species A consumes more R1 than R2 Species B consumes more R2 than R1

38. Unstable Equilibrium if S is in Region 4 3 2 4 1 R2 5 Sp B B 6 Sp A A R1 Species A consumes more R2 than R1 Species B consumes more R1 than R2

39. Coexistence Coexistence is possible when species compete more strongly with themselves than with their competitor That occurs when species A consumes more of the resources that most limits its growth and species B consumes more of the resource that most limits its growth

40. Test Yourself What is the outcome of competition when Species A is more limited by R1 than species B Species B is more limited by R2 than species A And Species A consumes more R2 than R1 Species B consumes more R1 than R2