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Further matrices: Leslie models — Question 156

Original QCE Vault practice · 7 marks

Q156 · Practice questionTechnology-activeComplex unfamiliar7 marks

QUESTION 156 (7 marks)

An insect population has juvenile and adult stages. The model is
(Jn+1An+1)=(020.50.5)(JnAn),(J0A0)=(200100).\begin{pmatrix}J_{n+1}\\ A_{n+1}\end{pmatrix} =\begin{pmatrix}0&2\\0.5&0.5\end{pmatrix} \begin{pmatrix}J_n\\ A_n\end{pmatrix},\qquad \begin{pmatrix}J_0\\A_0\end{pmatrix}=\begin{pmatrix}200\\100\end{pmatrix}.
The entries are expected population counts; fractional values are permitted. The stage diagram shows the transitions in one time step.
Two-stage directed diagram. Adults produce juveniles with coefficient 2; juveniles transition to adults with coefficient 0.5; adult survival has coefficient 0.5. No juvenile self-transition.
a)
Determine the stage counts after two time steps and the proportion of adults at that time.
[2 marks]
b)
Suppose the ratio An/JnA_n/J_n remains constant at a positive value rr. Derive and solve an equation for rr.
[3 marks]
c)
Determine the population growth factor at this stable composition. Give it exactly and to three decimal places.
[2 marks]
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