Community Ecology
Nature of species interactions
| Sp 1 effect on Sp 2 |
Sp 2 effect on Sp 1 |
Shorthand |
|---|---|---|
| Benefit (+) | Benefit (+) | Mutualism |
| Harm (-) | Harm (-) | Competition |
| Benefit (+) | Harm (-) | Predation Herbivory Parasitism |
| Neutral (0) | Benefit (+) | Commensalism |
| Neutral (0) | Harm (-) | Ammensalism |
Population dynamics of two competing species
dN1dT=r1N1(1−α11N1−α12N2)
dN2dT=r2N2(1−α21N1−α22N2)
α11 and α22: competitive effect of each species on itself (e.g. how strongly do ospreys compete with one another, and how strongly do eagles compete with one another?)
α12 and α21: competitive effect of each species on the other (e.g. how strongly do ospreys compete with eagles, and how strongly do eagles compete with ospreys?)
Population dynamics of two competing species
dN1dT=r1N1(1−α11N1−α12N2)
What happens if species 1 is growing alone?
dN1dT=r1N1(1−α11N1)
Growth to Species 1’s carrying capacity (1α11=K)
Similarly, if Species 2 is growing alone, it will grow to its carrying capacity 1α22=K2
But what happens if both species are present in the system?
Possible outcomes of two species competing:
- Both species can have stable coexistence
- Species 1 can win, and exclude Species 2
- Species 2 can win, and exclude Species 1
- “It depends” – whichever species comes first, wins in competition.
How to predict the outcome for any given pair of species?
- Or, given that species coexistence is something we are trying to explain – what enables coexistence?
Population dynamics of two competing species
dN1dT=r1N1(1−α11N1−α12N2)
dN2dT=r2N2(1−α21N1−α22N2)
Under what conditions do both species co-exist?
Signature of stable coexistence:
- System is at equilibrium
- Both species are present (abundance > 0)
Under what conditions do both species co-exist?
Approach: Graphical analysis of the competition model
- New type of visualization: phase space (AKA “state space”)
- New type of analysis: null-cline analysis
(AKA “zero net-growth isocline”)
Community ecology
Coexistence of competing species
Population dynamics of two competing species
dN1dT=r1N1(1−α11N1−α12N2)
dN2dT=r2N2(1−α21N1−α22N2)
Possible outcomes of two species competing:
- Both species can have stable coexistence
- Species 1 can win, and exclude Species 2
- Species 2 can win, and exclude Species 1
- “It depends” – whichever species comes first, wins in competition.
dN1dT=r1N1(1−α11N1−α12N2)
dN2dT=r2N2(1−α21N1−α22N2)
Given that we frequently see similar species coexisting, but are now in an era of biodiversity declines, an important question is: under what conditions do both species co-exist?
dN1dT=r1N1(1−α11N1−α12N2)
dN2dT=r2N2(1−α21N1−α22N2)
Signature of stable coexistence:
System is at equilibrium (dN1dt=0 and dN2dt=0)
Both species are present (abundance > 0)
When are the conditions above satisfied?
Evaluating coexistence through phase space analysis
Graph with each axis representing a state variable
Any point on the graph represents a possible state of the system
Lines on the graph show how a system changes through time (trajectory)
Null-cline analyses (AKA zero net growth isocline analysis)
At what points in the state space does the system not change? (i.e. is at equilibrium)
Analyzed one axis at at time
Key questions:
At what points does the abundance of species 1 (N1) not change?
At what points does the abundance of species 2 (N2) not change?
At what points does the abundance of species 1 (N1) not change?
dN1dT=r1N1(1−α11N1−α12N2)
Two “extreme” cases…
- When species 1 is at its carrying capacity, and species 2 is not around
- N1=1α11,N2=0
- When there are so many individuals of Species 2 that Species 1 cannot begin to grow
- When does that happen?
- N1 is low (0), and N2 is… some high number
dN1dT=r1N1(1−α11N1−α12N2)
Solve for dN1dT=0,N1=0,N2>0
At what points does the abundance of species 1 (N1) not change?
- When species 2 is absent, and species 1 is at its carrying capacity
- N2=0,N1=1α11
- When there are so many individuals of Species 2 that Species 1 cannot begin to grow
- When does that happen?
- N1=0,N2=1α12
Summary of the null-cline analysis so far:
- We set out to find cases where N1 doesn’t change, i.e. dN1/dt=0
- We identified two extreme scenarios:
- N1=1/α11,N2=0
- N1=0,N2=1/α12
- We can put these two extreme points on the state space graph.
Draw state space with Species 1 equilibrium points
Growth of species 1 is also zero for intermediate combinations between these extremes
These intermediate combinations are defined by the line connecting the two extremes.
We can solve for the equation of this line.
dN1dT=r1N1(1−α11N1−α12N2)
N∗2=1−α11N1α12
N∗2=1−α11N1α12=1α12+α11α12N1
Growth of species 1 is also zero for intermediate combinations between these extremes
dN1dT=r1N1(1−α11N1−α12N2)
N∗2=y-intercept⏞1α12−slope⏞α11α12N1
This is the equation of the null-cline for species 1 (AKA zero net-growth isocline, or ZNGI)
Draw state space with Species 1 equilibrium points, plus intermediate line
- What happens on either side of the null-cline?
Recall the key questions of null-cline analysis:
- Key questions:
At what points does the abundance of species 1 (N1) not change?- N1=1/α11,N2=0;
N1=0,N2=1/α12;
N=21α12−α11α12N1
- N1=1/α11,N2=0;
- At what points does the abundance of species 2 (N2) not change?
At what points does the abundance of species 2 (N2) not change?
- When species 1 is not around, and species 2 is at its carrying capacity
- N1=0,N2=1/α22
- When there are so many individuals of Species 1 that Species 2 cannot begin to grow
- N1 is some high number… and N2 is low (0)
- Following algebra, N1=1/α21,N2=0
- We can add these two extremes to the state space plot
Growth of species 2 is also zero for intermediate combinations between these extremes
dN2dT=r2N2(1−α21N1−α22N2)
N∗2=1α22−α21α22N1
This is the equation of the null-cline for species 2 (AKA zero net-growth isocline, or ZNGI)
Add null-cline to state space
What happens on either side of the nullcline?
Graphical analysis of the Lotka-Volterra competition model
Review:
- Our goal is to identify conditions under which both species can coexist at equilibrium
- Being ‘at equilibrium’ means dN1/dt=dN2/dt=0
- Null-cline analysis lets us find the conditions at which dN1/dt=0, and the conditions at which dN2/dt
- There are a couple of ‘extreme’ cases (e.g. one species at carrying capacity, and the other absent), and a whole bunch of in-between cases that result in dN1/dt=0 or dN2/dt=0
Test your recollection
- Consider a pair of species that interact with the following strength:
- α11=0.01, α12=0.005, α22=0.02, α21=0.001
- (10.01=100, 10.01=200, 10.02=50, 10.001=1000)
- On separate graphs, draw the isoclines for species 1 and 2.
