Interspecific competition
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The model for interspecific competition is
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(r, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vecto...](images/competition_129.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(r, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vecto...](images/competition_130.gif) |
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and the equilibria for this model are
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Well, this becomes a little confusing.
In the following we consider models with K1=2, K2=1, and r=0.5, u=0.5. The two remaining parameters, s and v, will be chosen so that in
Case 1: One species will dominate (independent of the initial population phase)
Case 2: Either one of the two species may dominate depending on the initial population phase
Case 3: The two species will coexist
Case 1. One species will dominate.
When s<K1/K2=2 and v>K2/K1=1/2, species 1 will dominate.
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_155.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_156.gif) |
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![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.9, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_165.gif)
![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.9, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_166.gif) |
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![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.9, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_168.gif)
![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.9, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_169.gif) |
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When s>K1/K2=2 and v<K2/K1=1/2, species 2 will dominate.
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_178.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_179.gif) |
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![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.1, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_188.gif)
![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.1, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_189.gif) |
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![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.1, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_191.gif)
![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.1, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_192.gif) |
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Case 2. Both species may dominate.
When s>K1/K2=2 and v>K2/K1=1/2, both species may dominate, the outcome of the competition depends on the initial population phase.
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_201.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_202.gif) |
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![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_211.gif)
![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_212.gif) |
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![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_214.gif)
![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.6, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_215.gif) |
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![display(seq(phasedia(F, P0, 50), P0 = {seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-2, j)), j = 0 .. 20)}), dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.5, P[2])) = 2...](images/competition_217.gif)
![display(seq(phasedia(F, P0, 50), P0 = {seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-2, j)), j = 0 .. 20)}), dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(2.5, P[2])) = 2...](images/competition_218.gif) |
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Case 3. The two species will coexist.
When s<K1/K2=2 and v<K2/K1=1/2, the two species will coexist.
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![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_234.gif)
![proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...](images/competition_235.gif) |
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![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_244.gif)
![display(growth(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. 1....](images/competition_245.gif) |
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![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_247.gif)
![display(dirfield(F, 2.1, 1.1), implicitplot([VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(1.5, P[2])) = 2, VectorCalculus:-`+`(VectorCalculus:-`*`(.4, P[1]), P[2]) = 1], P[1] = 0 .. 2.1, P[2] = 0 .. ...](images/competition_248.gif) |
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