Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / B2 / d / Solution

Eliminate the juvenile variable from the stage-structured population model. Since $y_{t+1}=2z_t$,The characteristic polynomial of this linear recurrence isso $z_t=A(3/2{)}^t+B(-1{)}^t$. The initial conditions give $z_0=0$ and $z_1=(3/4)10=15/2$, hence $A+B=0$ and $(3/2)A-B=15/2$. ThereforeThe alternating term records the initial imbalance between life stages; the growing $(3/2{)}^t$ term dominates eventually.