Ecology 8310Population (and Community) Ecology

Seguing into from populations to communities

• Species interactions

• Lotka-Volterra equations

• Competition

• Adding in resources

Species interactions:

Competition (- , -)

Predation (+ , -)

(Herbivory, Parasitism, Disease)

Mutualism (+ , +)

None (0 , 0)

Species interactions:

1

a11

a14=04

a122

aij>0

aij<0

a13

3

6

a16

5

a15

aij gives the per capita effect of

species j on species i’s per capita growth

rate, dNi/Nidt

Generalized Lotka-Volterra system:

.../

.../

.../

333232131333

323222121222

313212111111

NaNaNardtNdN

NaNaNardtNdN

NaNaNardtNdN

Special cases:

1. Exponential model: all a’s=0

2. Logistic model: aii<0; others =0

N2

N1

dN1/N1dt

Slope = a

12

Slo

pe

= a

11

N2

N1

dN1/N1dt

Slope = a 12

Slop

e =

a11

Species interactions: 1

a11

a122

aij>0

aij<0

a13

3

a31

4

a21

a44

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?

444343242141444

434333232131333

424323222121222

414313212111111

/

/

/

/

NaNaNaNardtNdN

NaNaNaNardtNdN

NaNaNaNardtNdN

NaNaNaNardtNdN

a43a34

Competition:

1 a122

a21

a11 a22

Alternate terminology:

αij = aij/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

Competition:

1 a122

a21

2121222

222121222

1212111

212111111

/)(

/

/)(

/

KNNKr

NaNardtNdN

KNNKr

NaNardtNdN

α

α

a11 a22

1 a122

a21

a11 a22

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:

Classic studies of resource competition by Gause (1934,

1935)

Paramecium aurelia

Paramecium bursaria

Paramecium caudatum

Competitive exclusion: P. aurelia excludes P.

caudatum

Paramecium caudatum

Paramecium bursaria

In contrast…

Why this disparity, and can we gain insights via

our model?

1 a122

a21

2121222

222121222

1212111

212111111

/)(

/

/)(

/

KNNKr

NaNardtNdN

KNNKr

NaNardtNdN

α

α

a11 a22

Competition:

212122222

121211111

/)(/

/)(/

KNNKrdtNdN

KNNKrdtNdN

αα

At equilibrium, dN/Ndt=0:

1212*2

2121*1

NKN

NKN

α

α

Competition:

N1

N2

Phase planes:

K1/α12

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

Species 1’s zero growth isocline…

2121*1 NKN α

dN1/N1dt=0

K1

N1

N2

Phase planes:

K1/α12

What if the system is not on the isocline. Will what N1 do?

dN1/N1dt=0

K1

N1

N2

Phase planes:

K2

dN2/N2dt=0

K2/α21

1212*2 NKN α

N1

N2

Phase planes:

K1/α12

Putting it together…

dN1/N1dt=0

K1

dN2/N2dt=0

Species 2 “wins”:N2

* =K2, N1* =0

(reverse to get Species 1 winning)

K2/α21

K2

N1

N2

Phase planes:

K1/α12

Your turn…. For A and B:

1) Draw the trajectory on the phase-plane

2) Draw the dynamics (N vs. t) for each system.

dN1/N1dt=0

K1

dN2/N2dt=0

K2/α21

K2

A

B

N1

N2

Phase planes:

K1/α12

Another possibility…

dN1/N1dt=0

K1

dN2/N2dt=0

“It depends”: either species can win,

depending on starting conditions

K2/α21

K2

N1

N2

Phase planes:

K1/α12

dN1/N1dt=0

K1K2/α21

K2

Your turn….

Draw the dynamics (N vs. t) for the system that starts at:

• Point A

• Point BA

B

N1

N2

Phase planes:

K1/α12

dN1/N1dt=0

K1K2/α21

K2

Now do it for many starting points:

Separatrix or manifold

N1

N2

Phase planes:

K1/α12

A final possibility…

dN1/N1dt=0

K1

dN2/N2dt=0

Coexistence!

K2/α21

K2

N1

N2

Phase planes:

K1/α12

“Invasibility”…

dN1/N1dt=0

K1

dN2/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

K2

N1

N2

Invasibility:

K1/α12

Contrast that with…

dN1/N1dt=0

K1

dN2/N2dt=0

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

K2/α21

K2

N1

N2

Coexistence:

K1/a12 dN1/N1dt=0

K1

dN2/N2dt=0

K2/a21

K2

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

Can we now explain Gause’s results?

Paramecium aurelia Paramecium bursaria

Paramecium caudatum

Bacteria in water

column

Yeast on bottom

Resources:But what about resources?

(they are “abstracted” in LV model)

Research by David Tilman

Resources:Followed population growthand resource (silicate) when

alone:

Data = points.Lines = predicted from model

Resources:What will happen when growth

together: why?

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.

Next time: Tilman's R* framework