Interspecific competition 

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> restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
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restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
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restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
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restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
restart; -1; with(LinearAlgebra); -1; with(plots); -1; with(VectorCalculus); -1; `:=`(LV, proc (K, r, s, u, v) option operator; description
 

The model for interspecific competition is 

> IC(K1, r, s, K2, u, v); 1
 

proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(r, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vecto...
proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(r, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vecto...		 (1)
 

and the equilibria for this model are 

> equi2(IC(K1, r, s, K2, u, v)); 1
 

 

 

 

 

 

`*`(The, `*`(equilibria, `*`(are)))
[[P[1] = 0, P[2] = 0], [P[1] = K1, P[2] = 0], [P[1] = 0, P[2] = K2], [P[1] = `+`(`-`(`/`(`*`(`+`(K1, `-`(`*`(s, `*`(K2))))), `*`(`+`(`-`(1), `*`(s, `*`(v))))))), P[2] = `/`(`*`(`+`(`-`(K2), `*`(v, `*`...
`*`(The, `*`(eigenvalues, `*`(are)))
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
Vector[column](%id = 138486548), Vector[column](%id = 138980500), Vector[column](%id = 139223892), Vector[column](%id = 139231616)
`*`(The, `*`(absolute, `*`(values, `*`(of, `*`(the, `*`(eigenvalues, `*`(are)))))))
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Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)
Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)
Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)
Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)
Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)
Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)
Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)
Vector[column](%id = 135574748), Vector[column](%id = 135558912), Vector[column](%id = 136279208), Vector[column](%id = 137776728)		 (2)
 

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. 

> `:=`(F, IC(2, .5, 1.9, 1, .5, .6)); 1
 

proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...
proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...		 (3)
 

> equi2(F); 1
 

 

 

 

 

 

`*`(The, `*`(equilibria, `*`(are)))
[[P[1] = 0., P[2] = 0.], [P[1] = 2., P[2] = 0.], [P[1] = 0., P[2] = 1.], [P[1] = -.7142857143, P[2] = 1.428571429]]
`*`(The, `*`(eigenvalues, `*`(are)))
Vector[column](%id = 140847276), Vector[column](%id = 140911408), Vector[column](%id = 140962956), Vector[column](%id = 136824012)
Vector[column](%id = 140847276), Vector[column](%id = 140911408), Vector[column](%id = 140962956), Vector[column](%id = 136824012)
`*`(The, `*`(absolute, `*`(values, `*`(of, `*`(the, `*`(eigenvalues, `*`(are)))))))
Vector[column](%id = 141090032), Vector[column](%id = 141090104), Vector[column](%id = 141090176), Vector[column](%id = 141090248) (4)
 

> 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....
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....
 

Plot_2d
 

> 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 .. ...
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 .. ...
 

Plot_2d
 

> display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-1, j)), j = 1 .. 10)}, {seq(`<,>`(VectorCalculus:-`+`(.4, VectorCalculus:-`*`(.1, j)), 1.1), j = 1 .. 20)})), dir...
display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-1, j)), j = 1 .. 10)}, {seq(`<,>`(VectorCalculus:-`+`(.4, VectorCalculus:-`*`(.1, j)), 1.1), j = 1 .. 20)})), dir...
display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-1, j)), j = 1 .. 10)}, {seq(`<,>`(VectorCalculus:-`+`(.4, VectorCalculus:-`*`(.1, j)), 1.1), j = 1 .. 20)})), dir...
 

Plot_2d
 

> timeplot(F, `<,>`(.1, .5), 100); 1
 

Plot_2d
 

When s>K1/K2=2 and v<K2/K1=1/2, species 2 will dominate. 

> `:=`(F, IC(2, .5, 2.1, 1, .5, .4)); 1
 

proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...
proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...		 (5)
 

> equi2(F); 1
 

 

 

 

 

 

`*`(The, `*`(equilibria, `*`(are)))
[[P[1] = 0., P[2] = 0.], [P[1] = 2., P[2] = 0.], [P[1] = 0., P[2] = 1.], [P[1] = -.6250000000, P[2] = 1.250000000]]
`*`(The, `*`(eigenvalues, `*`(are)))
Vector[column](%id = 143115228), Vector[column](%id = 144943524), Vector[column](%id = 141299084), Vector[column](%id = 143834716)
Vector[column](%id = 143115228), Vector[column](%id = 144943524), Vector[column](%id = 141299084), Vector[column](%id = 143834716)
`*`(The, `*`(absolute, `*`(values, `*`(of, `*`(the, `*`(eigenvalues, `*`(are)))))))
Vector[column](%id = 145031328), Vector[column](%id = 143711356), Vector[column](%id = 143947100), Vector[column](%id = 144060116) (6)
 

> 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....
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....
 

Plot_2d
 

> 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 .. ...
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 .. ...
 

Plot_2d
 

> display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(VectorCalculus:-`*`(.1, i), .1), i = 1 .. 15)}, {seq(`<,>`(2.1, VectorCalculus:-`*`(.1, j)), j = 1 .. 10)})), dirfield(F, 2.1, 1.1), implicitp...
display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(VectorCalculus:-`*`(.1, i), .1), i = 1 .. 15)}, {seq(`<,>`(2.1, VectorCalculus:-`*`(.1, j)), j = 1 .. 10)})), dirfield(F, 2.1, 1.1), implicitp...
display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(VectorCalculus:-`*`(.1, i), .1), i = 1 .. 15)}, {seq(`<,>`(2.1, VectorCalculus:-`*`(.1, j)), j = 1 .. 10)})), dirfield(F, 2.1, 1.1), implicitp...
 

Plot_2d
 

> timeplot(F, `<,>`(.4, .1), 75); 1
 

Plot_2d
 

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.  

> `:=`(F, IC(2, .5, 2.5, 1, .5, .6)); 1
 

proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...
proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...		 (7)
 

> equi2(F); 1
 

 

 

 

 

 

`*`(The, `*`(equilibria, `*`(are)))
[[P[1] = 0., P[2] = 0.], [P[1] = 2., P[2] = 0.], [P[1] = 0., P[2] = 1.], [P[1] = 1., P[2] = .4000000000]]
`*`(The, `*`(eigenvalues, `*`(are)))
Vector[column](%id = 138188772), Vector[column](%id = 138534836), Vector[column](%id = 138659052), Vector[column](%id = 138837616)
Vector[column](%id = 138188772), Vector[column](%id = 138534836), Vector[column](%id = 138659052), Vector[column](%id = 138837616)
`*`(The, `*`(absolute, `*`(values, `*`(of, `*`(the, `*`(eigenvalues, `*`(are)))))))
Vector[column](%id = 138866008), Vector[column](%id = 138864512), Vector[column](%id = 138858376), Vector[column](%id = 138852108) (8)
 

> 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....
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....
 

Plot_2d
 

> 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 .. ...
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 .. ...
 

Plot_2d
 

> 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...
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...
 

Plot_2d
 

> display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(1.5, VectorCalculus:-`+`(.6, Vect...
display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(1.5, VectorCalculus:-`+`(.6, Vect...
display(seq(phasedia(F, P0, 100), P0 = `union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(1.5, VectorCalculus:-`+`(.6, Vect...
 

Plot_2d
 

> display(seq(phasedia(F, P0, 50), P0 = `union`(`union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(1.5, VectorCalculus:-`+`(....
display(seq(phasedia(F, P0, 50), P0 = `union`(`union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(1.5, VectorCalculus:-`+`(....
 

Plot_2d
 

> timeplot(F, `<,>`(.1, 0.35e-1), 75); 1
 

Plot_2d
 

> timeplot(F, `<,>`(.1, 0.4e-1), 20); 1
 

Plot_2d
 

> timeplot(F, `<,>`(.1, 0.45e-1), 75); 1
 

Plot_2d
 

Case 3. The two species will coexist. 

When s<K1/K2=2 and v<K2/K1=1/2, the two species will coexist. 

> `:=`(F, IC(2, .5, 1.5, 1, .5, .4)); 1
 

proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...
proc (P) options operator, arrow; VectorCalculus:-`<,>`(VectorCalculus:-`+`(P[1], VectorCalculus:-`*`(VectorCalculus:-`*`(.5, P[1]), VectorCalculus:-`+`(1, VectorCalculus:-`-`(VectorCalculus:-`*`(Vect...		 (9)
 

> equi2(F); 1
 

 

 

 

 

 

`*`(The, `*`(equilibria, `*`(are)))
[[P[1] = 0., P[2] = 0.], [P[1] = 2., P[2] = 0.], [P[1] = 0., P[2] = 1.], [P[1] = 1.250000000, P[2] = .5000000000]]
`*`(The, `*`(eigenvalues, `*`(are)))
Vector[column](%id = 143919808), Vector[column](%id = 145964304), Vector[column](%id = 139251140), Vector[column](%id = 141077552)
Vector[column](%id = 143919808), Vector[column](%id = 145964304), Vector[column](%id = 139251140), Vector[column](%id = 141077552)
`*`(The, `*`(absolute, `*`(values, `*`(of, `*`(the, `*`(eigenvalues, `*`(are)))))))
Vector[column](%id = 141005672), Vector[column](%id = 141005736), Vector[column](%id = 141720132), Vector[column](%id = 143657576) (10)
 

> 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....
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....
 

Plot_2d
 

> 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 .. ...
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 .. ...
 

Plot_2d
 

> display(seq(phasedia(F, P0, 50), P0 = `union`({seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-1, j)), j = 0 .. 19)}, {seq(`<,>`(VectorCalculus:-`+`(.2, VectorCalculus:-`*`(.1, i)), 0.5e-1), i = 0 .. 9)})), di...
display(seq(phasedia(F, P0, 50), P0 = `union`({seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-1, j)), j = 0 .. 19)}, {seq(`<,>`(VectorCalculus:-`+`(.2, VectorCalculus:-`*`(.1, i)), 0.5e-1), i = 0 .. 9)})), di...
display(seq(phasedia(F, P0, 50), P0 = `union`({seq(`<,>`(.1, VectorCalculus:-`*`(0.5e-1, j)), j = 0 .. 19)}, {seq(`<,>`(VectorCalculus:-`+`(.2, VectorCalculus:-`*`(.1, i)), 0.5e-1), i = 0 .. 9)})), di...
 

Plot_2d
 

> display(seq(phasedia(F, P0, 20), P0 = `union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(VectorCalculus:-`*`(.1, i), 1.1), ...
display(seq(phasedia(F, P0, 20), P0 = `union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(VectorCalculus:-`*`(.1, i), 1.1), ...
display(seq(phasedia(F, P0, 20), P0 = `union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(VectorCalculus:-`*`(.1, i), 1.1), ...
 

Plot_2d
 

> display(seq(phasedia(F, P0, 50), P0 = `union`(`union`(`union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(VectorCalculus:-`*...
display(seq(phasedia(F, P0, 50), P0 = `union`(`union`(`union`({seq(`<,>`(2.1, VectorCalculus:-`+`(.5, VectorCalculus:-`*`(0.5e-1, j))), j = VectorCalculus:-`-`(4) .. 5)}, {seq(`<,>`(VectorCalculus:-`*...
 

Plot_2d
 

> timeplot(F, `<,>`(.1, .4), 50); 1
 

Plot_2d
 

> timeplot(F, `<,>`(.2, .1), 50); 1
 

Plot_2d
 

>