Eigenvalue equal to 1
If one of the eigenvalues crosses the unit circle at 1, a saddle-node bifurcation may arise in which a pair of steady states, one stable and one saddle, is created. Another possibility is that a pitchfork bifurcation arises in which two additional steady states are created. The only possibility for an eigenvalue to be equal to one is that one of the solutions, ξ
j, of the quadratic polynomial in (6.44) is equal to 1, say ξ
2=1. Then it follows from (6.43) that
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ξ1+1=1-µ + |
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+ |
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and
ξ1=(1-µ) |
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+ |
|
|
.
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Eliminating ξ
1 from these equations leads to the condition
R=v.
However, since 0 ≤
v ≤ 1 <
R, this condition can never be satisfied. Hence eigenvalues equal to 1 can never occur.
Eigenvalue equal to -1
If one of the eigenvalues crosses the unit circle at -1, a period doubling or flip bifurcation may arise in which a 2-cycle is created. Under the assumption that ξ
2=-1, equations (6.43) lead to the relations
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ξ1-1=1-µ + |
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+ |
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and
-ξ1=(1-µ) |
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+ |
|
|
.
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Eliminating ξ
1 leads to the condition
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λ (µ-2) (R+v) = 4 a γ σ2 |
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.
|
Since all parameters in the model are strictly positive and because -2<µ-2<-1, the left hand side of this condition is strictly negative. However the right hand side of the condition is strictly positive, so that this condition can never be satisfied. Hence eigenvalues equal to -1 and therefore period doubling bifurcations of the steady state can never occur.
Two complex conjugate eigenvalues of modulus 1
If a pair of complex conjugate eigenvalues crosses the unit circle in the complex plane, a Hopf or Neimark-Sacker bifurcation may arise in which an invariant circle with periodic or quasi-periodic dynamics is created. The roots ξ
1, ξ
2 of the characteristic equation are
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