Tilman’s Model of Competition For Resources

Mark McGinleyAssociate Professor

Honors College and Department of Biological SciencesTexas Tech University

R* model- single species

Resource Level

R*

B > DC > S

B < DC < S

At equilibriumB = DC = S

R* model- 2 species

Resource Level

R*a

Ba > DaBb < DbC > S

Ba < DaBb < DbC < S

At equilibriumBa = DaBb <DbC = S

R*b

Ba > DaBb > DbC > S

Species A wins because it has the lower R*

Tilman’s Model- 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

Tilman’s model- ZNGIs

R1

R2

R1

R2

Completely substitutable resourcese.g., big macs and whoppers

Non- substitutable resourcese.g., light and nitrogen

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

Consumption vectors

R1

R2

Consumption of R1Consumption or R2Total consumption

Interpreting Consumption Vectors

R1

R2

This species consumes moreR1 than R2

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)

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

Supply Vectors

R1

R2

S

Supply vectors point from the current resource levelTo the resource supply point

Tilman’s Model- 2 resources, 1 species

R1

R2

S Need to determine- ZNGI- S- Consumption vectors

2 resources, 1 species

R2

S

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??

R1

2 resources, 1 species

R1

R2

The equilibrium resource level(E) occurs where the Consumption vectors and the supply vectors are equal and opposite.

E

When the resource level is at E then B = D and C = S. This is a stable equilibrium.

2 resources, 2 speciesZNGIs

R1

R2

Sp A

Sp B

Species B is more limited by R1 than Species ASpecies B is more limited by R2 than Species A

2 resources, 2 speciesConsumption Vectors

R1

R2

A

B

Species A consumes more R1 than it does R2Species B consumes more R2 than it does R1

2 resources, 2 specieswhat is outcome of competition?

Depends on S.

R2

Sp A

Sp B

R1

S

S below both isoclines, therefore neither species can survive

2 resources, 2 specieswhat is outcome of competition?

R2

Sp A

Sp B

R1

Species B can not survive and Species A can survive

2 resources, 2 speciesWhat is the outcome of competition?

R1

R2

Sp A

Sp B

Coexistence is only potentially possible if S falls in this region.Because species A can reduce the levels of both R1 and R2 below theZNGI 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?

S

R2

Sp A

Sp B

Species A wins

R1

E

S

Practice Problem

• What is the outcome of competition and the equilibrial resource level if the position of the two species’ ZNGIs are reversed?

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

2 resources, 2 speciesintersecting ZNGIs

R2

Sp A

Sp B

R1

Species A is more limited by R1 than species BSpecies B is more limited by R2 than Species A

2 resources, 2 speciesintersecting ZNGIs

• The ultimate outcome of competition between two species when their ZNGIs intersect depends upon

– Consumption vectors

– Resource Supply Point

Add consumption vectors

R2

Sp A

Sp B

R1

A

B

Species A consumes more R1 than R2Species B consumes more R2 than R1

Importance of the location of S

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

Extending the consumption curves allow us to divide the graph into 6 regions. The outcome of competition depends upon in whichregion the resource supply point falls.

S falls in region 1

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

Neither species A or B can survive

S falls in region 2

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

Only species B can survive

S falls in region 6

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

Only species A can survive

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

S in region 3

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

If S is in region 3, eventually consumption of resources will moveThe current resource level into region 2. When that happens speciesA is unable to survive so species B wins.

S in region 5

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

If S is in region 5, eventually consumption of resources will moveThe current resource level into region 6. When that happens speciesB is unable to survive so species A wins.

Coexistence is only possible if S is in region 4

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

If S is in region 4 the level of resources will be reducedTo the point where the two ZNGIs cross. At that pointConsumption = supply and the growth rate of both speciesIs equal to zero.

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

Stable Equilibrium if S is in region 4

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

Species A consumes more R1 than R2Species B consumes more R2 than R1

Unstable Equilibrium if S is in Region 4

R2

Sp A

Sp B

R1

A

B

12 3

4

5

6

Species A consumes more R2 than R1Species B consumes more R1 than R2

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

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