Consider a simplified version of our herring model from class Jan $30$th$.$ This model

follows the population size N$\_($t$)$ in each of two age classes, recruits R$\_($t$)$ and adults A$\_($t$).$

Each age class has a survivorship s$\_($x$).$ As before, there is density$-$dependent recruitment

following Beverton$-$Holt dynamics. This leads to the model:

R$\_($t$+1)=($fA$\_($t$))/(1+$hA$\_($t$))$

A$\_($t$+1)=$s$\_($R$)$R$\_($t$)+$s$\_($A$)$A$\_($t$)$

$($a$)(20$ points$)$ Find all equilibria $($in terms of the equilibrium population size in each

stage class$).$ Remember each equilibrium will be a paired set of values for the

number of recruits $($R$)$ and the number of adults $($A$).$ When necessary, indicate

the criteria for biological existence in terms of s$\_($A$)^(1).$

$($b$)(15$ points$)$ Calculate the Jacobian matrix $([($delR$\_($t$+1))/($delR$\_($t$)),($delR$\_($t$+1))/($delA$\_($t$))],[($delA$\_($t$+1))/($delR$\_($t$)),($delA$\_($t$+1))/($delA$\_($t$))])$ for the age$-$structured

model with density dependence in terms of R and A$.$ PLease see attached image for a$)$ and b$).$