Saturday, December 2, 2017

Beverton–Holt: a discrete-time logistic model

The Beverton–Holt equation is a recurrence relation relating abundance of a population in generation $n$ to the abundance of the previous generation $n - 1$: \[ N_n = \frac{PN_{n-1}}{1 + PC^{-1}N_{n-1}}, \] where $N_n$ is the abundance at generation $n$, $P$ is the density-independent productivity of the population, and $C$ is the capacity of the population. This recurrence relation can be solved to explicitly relate the abundance at generation $n$ with the initial abundance $N_0$. Let $Q$ be the equilibrium abundance of the population given by the equation \[ Q = C \left (1-P^{-1} \right). \] Using this definition, the abunance at generation $n$ is given by the equation \[ N_n = \frac{QN_0}{N_0 + P^{-n}\left (Q - N_0 \right )}. \] The standard Beverton–Holt recurrence relation has an alternative form that is very similar to the solution of the recurrence relation. \begin{align} N_{n} &= \frac{P N_{n-1}}{1 + P C^{-1} N_{n-1}} \\ &= \frac{P N_{n-1}}{1 + P \left (1 - P^{-1 } \right ) Q^{-1} N_{n-1}} \\ &= \frac{P N_{n-1}}{1 + \left (P-1 \right ) Q^{-1} N_{n-1}} \\ &= \frac{P Q N_{n-1}}{Q + P N_{n-1} - N_{n-1}} \\ &= \frac{QN_{n-1}}{N_{n-1} + P^{-1} \left (Q - N_{n-1} \right )}. \end{align} With this alternative recurrence relation in hand, we can now prove the solution to the recurrence relation for $P > 0$ and $C > 0$.

Proof

We proceed by induction on $n$. For the initial generation $n = 1$, by the recurrence relation \[ N_1 = \frac{QN_0}{N_0 + P^{-1} \left (Q - N_0 \right )}. \] Suppose there exists a generation $t \ge 0$ such that \[ N_t = \frac{QN_0}{N_0 + P^{-t}\left (Q - N_0 \right )}. \] Then

\begin{align} N_{t+1} &= \frac{Q N_t}{N_t + P^{-1} \left ( Q - N_t \right )} \\ &= \frac{Q N_t}{P^{-1} Q + \left (1 - P^{-1} \right )N_t} \\ &= \frac{Q^2 N_0 \left (N_0 + P^{-t} \left ( Q - N_0 \right ) \right )^{-1}} {P^{-1} Q + \left (1 - P^{-1} \right ) \left (Q N_0 \left (N_0 + P^{-t} \left ( Q - N_0 \right ) \right )^{-1} \right )} \\ &= \frac{Q N_0}{P^{-1} \left (N_0 + P^{-t} \left (Q - N_0 \right ) \right ) + \left (1 - P^{-1} \right ) N_0} \\ &= \frac{Q N_0}{P^{-1} N_0 + P^{-(t + 1)} \left (Q - N_0 \right ) + N_0 - P^{-1} N_0} \\ &= \frac{Q N_0}{N_0 + P^{-(t + 1)} \left (Q - N_0 \right )}. \end{align}

Consider the differential equation describing logistic population growth: \[ \frac{\mathrm{d}N}{\mathrm{d}t} = (P - 1) N \left (1 - \frac{N}{Q} \right ) \] and its solution \[ N(t) = \frac{Q N(0)}{N(0) + (Q - N(0))e^{-(P - 1) t}}. \] Note that the limit as the time step approaches 0 of the Beverton–Holt model is the logistic equation.

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