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Tutorial 1: Building recursion equations

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Let's get some more practice building recursion equations.

Review

In Lecture 1 we asked, how does immigration affect population size?

We then built a model with a single variable, \(n(t)\), denoting population size at time \(t\).

In discrete time the parameters were the average number of immigrants per time step (\(M\)), the average number of offspring per individual per time step (\(B\)), and the fraction of individuals that die each time step (\(D\)).

Assuming migration, then birth, then death each time step, we drew the following life-cycle diagram:

graph LR;
    A((n)) --migration--> B((n'));
    B --birth--> C((n''));
    C --death--> A;

We then built an equation for the population size in the next generation, \(n(t+1)\), based on the life-cycle diagram above, by constructing an equation for each event

\[n' = n(t) + M\]
\[n'' = n' + Bn'\]
\[n(t+1) = n'' - Dn''\]

We then substituted \(n''\) and then \(n'\) into the equation for \(n(t+1)\) to write \(n(t+1)\) in terms of \(n(t)\)

\[ \begin{aligned} n(t+1) &= n'' − Dn'' \\ &= (n' + Bn') − D(n' + Bn') \\ &= n'(1 + B − D − DB) \\ &= (n(t) + M)(1 + B − D − DB) \\ \end{aligned} \]

This recursion equation correctly takes into account the order of the life cycle (migration, birth, death) and the point at which the census is taken (immediately after death).

Problem

Show that the six different life-cycle orders give four distinct recursion equations.