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WOLFRAM NOTEBOOK
Population Dynamics with Two Competing Species
Population Dynamics with Two Competing Species
Consider two types of fish in a pond that do not prey on one another but compete for the available food. The governing equations are:
x
t
α
1
β
1
γ
1
Δ
1
y
t
α
2
β
2
γ
2
Δ
2
Restrict the discussion to the first quadrant, since and are species populations.
x
y
There are up to four steady states in the first quadrant:
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1
S
2
α
2
β
2
Δ
2
S
3
α
1
β
2
Δ
1
S
4
α
1
β
2
α
2
γ
1
β
1
β
2
γ
1
γ
2
α
2
β
1
α
1
γ
2
β
1
β
2
γ
1
γ
2
Δ
1
Δ
2
S
1
S
2
S
3
S
4
α
1
β
2
α
2
γ
1
β
1
β
2
γ
1
γ
2
α
2
β
1
α
1
γ
2
β
1
β
2
γ
1
γ
2
Vary the growth rate coefficients , , the self-inhibition parameters , , and the interaction parameters , . This Demonstration gives the corresponding stream plot.
α
1
α
2
β
1
β
2
γ
1
γ
2
The steady states and nullclines (x,y)=0 and (x,y)=0 (blue and brown) are shown in the plot.
Δ
1
Δ
2
From the snapshots, is an unstable node, and are either saddle points or stable nodes, and is either a saddle point or a stable node.
S
1
S
2
S
3
S
4
The linearized analysis (see the stability tab) confirm these conclusions. Indeed, for , both eigenvalues are positive. For , , and , there are either two negative eigenvalues or two real eigenvalues of opposite sign.
S
1
S
2
S
3
S
4
Coexistence (i.e., steady-state ) is possible only if the self-inhibition term dominates the interaction term (,>,).
S
4
β
1
β
2
γ
1
γ
2