I am asked to onsider a population of annual plants with the following life cycle: Seeds are produced at the end of summer, after which all adult plants die. A proportion of the seeds survives winter and some of these seeds then germinate in the following spring. Some of the seeds that don’t germinate survive another winter and may germinate in the spring following this second winter. No seeds germinate after more than two winters.
From this, I am given the following model: $$N_{t+2} = \alpha\sigma\gamma N_{t+1} + \beta(1 − \alpha)\sigma^2\gamma N_t $$.
I interpreted it physically as γ representing the number of seeds produced by one plan per year, σ as the survival rate per winter, α as the germination rate in the first spring, and β as the germination rate in the second spring.
Where I am having trouble is constructing a Leslie Matrix. I am asked to consider the population at the flowering stage when there are $P_n$ plants and $S_n$ one year old seeds, and write this model as a Leslie Matrix.
How should I approach this?
I was able to find an example where, counting fresh seeds and one year old seeds, the matrix would be: $$\textbf{L} = \begin{pmatrix} \sigma\alpha\gamma & \sigma\beta\gamma \\ \sigma(1-\alpha) & 0 \end{pmatrix}$$
But obviously, this isn't the Leslie matrix that's being asked of me. Any suggestions?
At flowering stage, we have only two age classes: plants $P_n$ and 1-year-old seeds $S_n$. The plants don't survive, but they have produced $\sigma\gamma P_n$ new seeds which survived. A fraction $\alpha$ germinates in the first spring, while a fraction $(1-\alpha)$ doesn't. A fraction $\beta$ of the surviving seeds $\sigma S_n$ has germinated (remaining seeds die later on): $$ P_{n+1} = \alpha\sigma\gamma P_n + \beta\sigma S_n ,\qquad S_{n+1} = (1-\alpha)\sigma\gamma P_n \, . $$ Introducing the vector ${\bf u}_n = (P_n, S_n)$, we can construct the Leslie matrix $$ {\bf L} = \begin{pmatrix} \alpha\sigma\gamma & \beta\sigma\\ (1-\alpha)\sigma\gamma & 0 \end{pmatrix} $$ such that ${\bf u}_{n+1} = {\bf L} {\bf u}_n$. We have \begin{aligned} P_{n+2} &= \alpha\sigma\gamma P_{n+1} + \beta\sigma S_{n+1} \\ &= \alpha\sigma\gamma P_{n+1} + \beta (1-\alpha)\sigma^2\gamma P_n \end{aligned} which is the proposed updating formula. This is a linear recursion similar to the Fibonacci sequence.