%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 #<v1.number_of_ages#> life stages. The fertility of each life stage is:\; #<v1.fecundity_string#>.\\ The survival rate from each life stage to the next is:\; #<v1.survival_string#>.\\ The initial population is:\; #<v1.initial_population_string#>. Find the Leslie matrix $L$ and initial population vector $P_0$, then estimate the population at times $t=1$ to $t=#<v1.number_of_steps#>$. (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: $$P_0=#<v1.population0#>\quad\mbox{and}\quad L=#<v1.leslie_matrix#>.$$\\ Then to find the population at time step $t+1$ we calculate $P_{t+1}=L\times P_t$, as follows:\\ #<v1.working#>\\ %%END SOLUTION %%BEGIN SHORTANSWER The initial population vector $P_0$, the Leslie matrix $L$, and the final population vector $P_{#<v1.number_of_steps#>}$ are: $$P_0=#<v1.population0#>\quad\mbox{and}\quad L=#<v1.leslie_matrix#>\quad\mbox{and}\quad P_{#<v1.number_of_steps#>}=#<v1.last_population_vector#>.$$ %%END SHORTANSWER \end{document}