Ecology 8310 Population (and Community) Ecology - PowerPoint Presentation

Slide1

Ecology 8310Population (and Community) Ecology

Seguing into from populations to communities

Species interactions

Lotka-Volterra

equations

Competition

Adding in resources

Slide2

Species interactions:

Competition (- , -)

Predation (+ , -)

(

Herbivory

, Parasitism, Disease)

Mutualism (+ , +)

None (0 , 0)

Slide3

Species interactions:

1

a

11

a

14

=0

4

a

12

2

a

ij

>0

a

ij

<0

a

13

3

6

a

16

5

a

15

a

ij

gives the per capita effect of species j on species i’s per capita growth rate,

dN

i

/

N

i

dt

Slide4

Generalized Lotka-Volterra system:

Special cases:

Exponential model: all a’s=0

Logistic model: a

ii

<0; others =0

Slide5

N

2

N

1

dN

1

/N

1dt

Slope = a12

Slope = a

11

Slide6

N

2

N

1

dN

1

/N1dt

Slope = a12

Slope = a11

Slide7

Species interactions:

1

a

11

a

12

2

a

ij

>0

a

ij

<0

a

13

3

a

31

4

a

21

a

44

What can you say about the interactions between these species?

Which are interspecific competitors?

Which are predator and prey?

Which are mutualists? Which show self limitation?

a

4

3

a

34

Slide8

Competition:

1

a

12

2

a

21

a

11

a

22

Alternate terminology:

α

ij

=

a

ij/aii , the effect of interspecific competition relative to the intraspecific effect (e.g., how many of species i does it take to have the same effect as 1 individual of species j?)

Arises when two organisms use the same limited resource, and deplete its availability(intra. vs. interspecific)

1

2

R

Slide9

Competition:

1

a

12

2

a

21

a

11

a

22

Slide10

1

a

12

2

a

21

a

11

a

22

Can we use this model to understand patterns of competition among two species (e.g., coexistence and competitive exclusion)?

E.g.,

Paramecium

experiments by Gause…

Competition:

Slide11

Classic studies of resource competition by Gause (1934, 1935)

Paramecium aurelia

Paramecium bursaria

Paramecium caudatum

Slide12

Competitive exclusion:

P. aurelia

excludes

P. caudatum

Slide13

Paramecium

caudatum

Paramecium

bursaria

In contrast…

Why this disparity, and can we gain insights via our model?

Slide14

1

a

12

2

a

21

a

11

a

22

Competition:

Slide15

At equilibrium, dN/Ndt=0:

Competition:

Slide16

N

1

N

2

Phase planes:

K

1

/α12

Graph showing regions where dN/Ndt=0 (and +, -); used to infer dynamics

Species 1’s zero growth isocline…

dN

1

/N

1

dt=0

K

1

Slide17

N

1

N

2

Phase planes:

K

1

/α12

What if the system is not on the isocline. Will what N

1 do?

dN1/N1dt=0

K

1

Slide18

N

1

N

2

Phase planes:

K

2

dN

2

/N2dt=0

K

2

/

α

21

Slide19

N

1

N

2

Phase planes:

K

1

/α12

Putting it together…

dN1/N1dt=0

K1

dN

2/N2dt=0

Species 2 “wins”:

N

2

*

=K

2

, N

1

*

=0

(reverse to get Species 1 winning)

K

2

/

α

21

K

2

Slide20

N

1

N

2

Phase planes:

K

1

/α12

Your turn…. For A and B:Draw the trajectory on the phase-planeDraw the dynamics (N vs. t) for each system.

dN1/N1dt=0

K1

dN

2/N2dt=0

K

2

/α21

K2

A

B

Slide21

N

1

N

2

Phase planes:

K

1

/α12

Another possibility…

dN1/N1dt=0

K1

dN

2/N2dt=0

“It depends”: either species can win, depending on starting conditions

K

2

/

α

21

K

2

Slide22

N

1

N

2

Phase planes:

K

1

/α12

dN1/N1dt=0

K1

K

2/α21

K2

Your turn….

D

raw the dynamics (N vs. t) for the system that starts at:Point APoint B

A

B

Slide23

N

1

N

2

Phase planes:

K

1

/α12

dN1/N1dt=0

K1

K

2/α21

K2

Now do it for many starting points:

Separatrix

or manifold

Slide24

N

1

N

2

Phase planes:

K

1

/α12

A final possibility…

dN1/N1dt=0

K1

dN

2/N2dt=0

Coexistence!

K

2

/

α

21

K

2

Slide25

N

1

N

2

Phase planes:

K

1

/α12

“Invasibility”…

dN1/N1dt=0

K1

dN

2/N2dt=0

Mutual invasibility

 coexistence

!Why: because each species is self-limited below the level at which it prevents growth of the other

K2/α21

K

2

Slide26

N

1

N

2

Invasibility:

K

1

/α12

Contrast that with…

dN1/N1dt=0

K1

dN

2/N2dt=0

Neither species can invade the other’s equilibrium (hence no coexistence).

K2/α21

K2

Slide27

N

1

N

2

Coexistence:

K

1

/a12

dN1/N1dt=0

K1

dN

2/N2dt=0

K

2/a21

K2

Slide28

Coexistence:

“intra > inter”

Coexistence

requires that the strength of intraspecific competition be stronger than the strength of interspecific competition.

Resource partitioning

Two species cannot coexist on a single limiting resource

Slide29

Can we now explain Gause’s results?

Paramecium aurelia

Paramecium bursaria

Paramecium caudatum

Bacteria in water column

Yeast on bottom

Slide30

Resources:

But what about resources?

(they are “abstracted” in LV model)

Research by David Tilman

Slide31

Resources:

Followed population growthand resource (silicate) when alone:

Data = points.

Lines = predicted from model

Slide32

Resources:

What will happen when growth together: why?

Slide33

Resources:

R*: resource concentration after consumer population equilibrates (i.e., R at which Consumer shows no net growth)

Species with lowest R* wins (under idealized scenario: e.g., one limiting resource).

If two limiting resources, then coexistence if each species limited by one of the resources (intra>inter): trade-off in R*s.

Slide34

Next time:

Tilman's

R* framework