Suppose the search survival rate of the juvenile owls is 50%, so the (2,1)-entry in the stage-matrix A in (10) is .3 instead of .18. What does the stage-matrix model predict about this spotted owl population?

$A = \left [ \begin{matrix} 0 & 0 & .33 \\ .18 & 0 & 0 \\ 0 & .71 & .94 \end{matrix} \right ]$ (10)

Question

Suppose the search survival rate of the juvenile owls is 50%, so the (2,1)-entry in the stage-matrix A in (10) is .3 instead of .18. What does the stage-matrix model predict about this spotted owl population?

$A = \left [ \begin{matrix} 0 & 0 & .33 \\ .18 & 0 & 0 \\ 0 & .71 & .94 \end{matrix} \right ]$ (10)

[latex] A = \left [ \begin{matrix} 0 & 0 & .33 \ .18 & 0 & 0 \ 0 & .71 & .94 \end{matrix} ight ] [/latex] (10)

Accepted Answer

Now the eigenvalues of A turn out to be approximately [latex] \lambda_{1} = 1.01 , \lambda_{2} = -.03 + .26i , ext{and} \lambda_{3} = -.03 - .26i [/latex]. An eigenvector for [latex] \lambda_{1} [/latex] is approximately [latex] v_{1} = (10,3,31) [/latex]. Let [latex] v_{2} [/latex] and [latex] v_{3} [/latex] be (complex) eigenvectors for [latex] \lambda_{2} [/latex] and [latex] \lambda_{3} [/latex]. In this case, equation (11) becomes

[latex] x_{k} = c_{1}(\lambda_{1})^{k}v_{1} + c_{2}(\lambda_{2})^{k}v_{2} + c_{3}(\lambda_{3})^{k}v_{3} [/latex] (11)

[latex] x_{k} = c_{1}(1.01)^{k}v_{1} + c_{2}(-.03 + .26i)^{k}v_{2} + c_{3}(-.03 - .26i)^{k}v_{3} [/latex]

As [latex] k ightarrow \infty [/latex], the second two vectors tend to zero. So [latex] x_{k} [/latex] becomes more and more like the (real) vector [latex] c_{1}(1.01)^{k}v_{1} [/latex]. The approximations in equations (6) and (7), following Example 1, apply here. Also, it can be shown that the constant [latex] c_{1} [/latex] in the initial decomposition of [latex] x_{0} [/latex] is positive when the entries in [latex] x_{0} [/latex] are nonnegative. Thus the owl population will grow slowly, with a long-term growth rate of 1.01. The eigenvector [latex] v_{1} [/latex] describes the eventual distribution of the owls by life stages: for every 31 adults, there will be about 10 juveniles and 3 subadults.

[latex] x_{k+1} \approx \lambda_{1}x_{k} [/latex] (6)

[latex] x_{k} \approx c_{1}(\lambda_{1})^{k}v_{1} [/latex] (7)

Question 5.6.7: Suppose the search survival rate of the juvenile owls is 50%...

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