%%META
%%AUTHOR Kvyatkovskyy, Andriy
%Mathematics, UQ
%%KEYWORDS leslie matrix
%%DESCRIPTION Estimate the population at given times, using simple Leslie model.
%Module parameters allow to initialise the question with certain limits / information about the population (see LeslieMatrixModule parameters).
%%CREATED 2007-08-22
%%MODULE au.edu.uq.smartass.biology.LeslieMatrixModule
%The module generates a simple question on the usage of Leslie matrix
%%PARAMETERS Constructor LeslieMatrixModule initialises the question
%* with parameters passing.
%* In case of "values" as a first parameter:
%* @params params[0] - string "values", indicates types of parameters passing,
%* params[1] - number of age groups,
%* params[2] - number of generations,
%* params[from 3 to number number_of_ages+2] - the numbers of individuals in each age group in initial population,
%* params[from number_of_ages+3 to 2number_of_ages+2] - the fertility of the corresponding age groups,
%* params[from 2number_of_ages+3 to 3number_of_ages+1] - the fraction of individuals that survives from the corresponding age group,
%
%For instance, v1(values,3,5,10,0,0,0,4,1,0.8,0.9);
%
%* or
%@params params[0] - string "limits", indicating that there are limits being passed
%* params[1] - number of age groups,
%* parmas[2-3] - min and max limits for number of generations (including initial generation) required,
%* params[from 4 to 2number_of_ages+3] - limits for numbers of individuals in each age group in initial population,
%* params[from 2number_of_ages+4 to 4number_of_ages+3] - limits for fertility of each age group,
%* params[from 4number_of_ages+4 to 6number_of_ages+1] - limits for the fraction of individuals that survives from the corresponding age group.
%
%For example,
%v1(limits,2,2,8,5,10,0,1,0,1,10,15,0.1,0.3);
%
%%CATEGORY BIOLOGY
%%META END
%Simple question on the usage of Leslie matrix.
\input{smartass.tex}
\begin{document}
%%BEGIN DEF
%definition of variables/modules used
%/**
%* Constructor LeslieMatrixModule initialises the question
%* with parameters passing.
%* In case of "values" as a first parameter:
%* @params params[0] - string "values", indicates types of parameters passing,
%* params[1] - number of age groups,
%* params[2] - number of generations,
%* params[from 3 to number number_of_ages+2] - the numbers of individuals in each age group in initial population,
%* params[from number_of_ages+3 to 2number_of_ages+2] - the fertility of the corresponding age groups,
%* params[from 2number_of_ages+3 to 3number_of_ages+1] - the fraction of individuals that survives from the corresponding age group,
% for instance, v1(values,3,5,10,0,0,0,4,1,0.8,0.9);
%* or
%@params params[0] - string "limits", indicating that there are limits being passed
%* params[1] - number of age groups,
%* parmas[2-3] - min and max limits for number of generations (including initial generation) required,
%* params[from 4 to 2number_of_ages+3] - limits for numbers of individuals in each age group in initial population,
%* params[from 2number_of_ages+4 to 4number_of_ages+3] - limits for fertility of each age group,
%* params[from 4number_of_ages+4 to 6number_of_ages+1] - limits for the fraction of individuals that survives from the corresponding age group.
%v1(limits,2,2,8,5,10,0,1,0,1,10,15,0.1,0.3);
#<
LeslieMatrixModule v1;
#>
%%END DEF
%%BEGIN QUESTION
A particular organism's population is modelled using a simple Leslie model with # life stages.
The fertility of each life stage is:\; #.\\
The survival rate from each life stage to the next is:\; #.\\
The initial population is:\; #.
Find the Leslie matrix $L$ and initial population vector $P_0$, then
estimate the population at times $t=1$ to $t=#$.
(Round your answers to 1 decimal place at each time step.)
%%END QUESTION
%%BEGIN SOLUTION
The initial population vector $P_0$ and the Leslie matrix $L$ are:
$$P_0=#\quad\mbox{and}\quad
L=#.$$\\
Then to find the population at time step $t+1$ we calculate $P_{t+1}=L\times P_t$, as follows:\\
#\\
%%END SOLUTION
%%BEGIN SHORTANSWER
The initial population vector $P_0$, the Leslie matrix $L$, and the final
population vector $P_{#}$ are:
$$P_0=#\quad\mbox{and}\quad
L=#\quad\mbox{and}\quad
P_{#}=#.$$
%%END SHORTANSWER
\end{document}