Coexistence in competitive communities
Logistics
Reflections 1 and 2 graded in Moodle
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Third biweekly activity is on Moodle; deadline extended to next Tuesday (15 Oct)
Interspecific competition
Lotka-Volterra competition equations:
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.
How to distinguish between the possible options?
Zero net-growth isocline (ZNGI) analysis
Key question: Under what conditions does each species reach equilibrium?
dN1dt=0 and dN2dt=0
Species 1’s equilibrium analysis:
- dN1dt=0 at two extreme conditions:
- Species 1 is at its carrying capacity, and Species 2 is absent (N1=1/α11, N2=0)
- Species 2 is very abundant, and Species 1 is nearly absent (N1=0, N2=1/α12)
- dN2dt=0 at two extreme conditions:
- Species 2 is at its carrying capacity, and Species 1 is absent (N1=0, N2=1/α22)
- Species 1 is very abundant, and Species 2 is nearly absent (N1=1/α21, N2=0)
Visualizing the equilibrium points
Species 1’s equilibrium analysis:
- Species 1 is at its carrying capacity, and Species 2 is absent (N1=1/α11, N2=0)
- Species 2 is very abundant, and Species 1 is nearly absent (N1=0, N2=1/α12)
Visualizing the equilibrium points
Species 2’s equilibrium analysis:
Species 2 is at its carrying capacity, and Species 1 is absent (N1=0, N2=1/α22)
Species 1 is very abundant, and Species 2 is nearly absent (N1=1/α21, N2=0)
Jointly visualizing both species’ isoclines
Putting numbers to the variables
Consider the following interaction coefficients:
α11=0.005; α12=0.002; α22=0.007; α21=0.002
1/α11=200; 1/α12=500; 1/α22=142; 1/α21=500
Coexistence in competitive communities
Putting numbers to the variables
Consider the following interaction coefficients:
α11=0.005; α12=0.002; α22=0.01; α21=0.007
1/α11=200; 1/α12=500; 1/α22=100; α21=0.007=142
Recall that the equilibrium points for the two species are:
- Species 1: (N1=1/α11, N2=0) and (N1=0, N2=1/α12)
- Species 2: (N1=0, N2=1/α22) and (N1=1/α21, N2=0)
Your turn
- Draw the isoclines for a pair of species whose interaction coefficients are as follows:
α11=0.008; α12=0.008; α22=0.006; α21=0.005
Your turn
- Draw the isoclines for a pair of species whose interaction coefficients are as follows:
α11=0.002; α12=0.005; α22=0.002; α21=0.007
Recall that the equilibrium points for the two species are:
- Species 1: (N1=1/α11, N2=0) and (N1=0, N2=1/α12)
- Species 2: (N1=0, N2=1/α22) and (N1=1/α21, N2=0)
Possible outcomes in Lotka-Volterra competition
- Both species can coexist
- Each species limits itself more than it limits the other
- More “intra-specific competition” than “inter-specific competition”
- Species 1 wins, while Species 2 is excluded
- Species 2 wins, while Species 1 is excluded
- “It depends”
Semester Project Discussion
In groups of 4, discuss the following:
- What medium do you hope to pursue for your semester project?
- What are some examples of work in this format that you find inspiring?
- What excites you most about this format?
- What do you see as the most challenging barrier to your completing this project?
Conclusion:
Coexistence in competitive communities
Photo by Gaurav, of Tejon Ranch in southern California
Why study species coexistence?
- Ecological systems are often incredibly diverse
- But sometimes, they are not – as in the case of species invasion rapidly eroding biodviersity
- So, the question arises: Why can certain species coexist, but others cannot?
Photo by Gaurav, of Tejon Ranch in southern California
How to study coexistence?
- There’s often lots of species coexisting, but we can take a reductionist approach
- Model the dynamics of two species at a time
- Analyze whether the species can coexist at equilibrium
- Graphical analysis using zero-net growth isoclines on Phase Spaces
dN1dt=r1N1(1−α11N1−α12N2)
dN2dt=r2N2(1−α21N1−α22N2)
- Isocline analysis depends only on the values of the four α parameters.
- What does each αij mean?
What supports species coexistence?
- Individuals competes with conspecifics more strongly than with heterospecifics
α11>α21 and α22>α12 ##
If this is true:
α11>α21 and α22>α12
Then, the following is true:
1α11<1α21 and 1α22<1α12
1α11<1α21 and 1α22<1α12
In phase-space graphs…
- If coexistence is not possible, three possible outcomes:
- Species 1 always wins
- Species 2 always wins
- Who wins depends on where the population starts
Next week
- Moving on from competitive interactions, to consumer–resource interactions
- Predator-prey
- Plant-herbivore
- Host-parasite
Semester Project Discussion
In groups of 4, discuss the following:
- What medium do you hope to pursue for your semester project?
- What are some examples of work in this format that you find inspiring?
- What excites you most about this format?
- What do you see as the most challenging barrier to your completing this project?
