23 Discrete-time Lotka-Volterra Predator-Prey Model in R

23.1 Introduction

This exercise is the discrete-time counterpart to the continuous-time predator-prey practical, and corresponds to the student chapter Discrete-time Lotka-Volterra Predator-Prey Model in R in the main book. Students program the discrete-time model in R, adjust the time step (\(\Delta t\)), and see how the implied time lag destabilises the system into expanding spirals and eventual extinction — in direct contrast to the neutrally stable continuous-time cycles.

The update equations are: \[V_{t+1} = V_t + \Delta t\,(R V_t - a C_t V_t)\] \[C_{t+1} = C_t + \Delta t\,(a f V_t C_t - q C_t)\] With \(\Delta t = 1\) this is the year-by-year model fitted in the Excel class, in which predators respond to prey abundance with a one-year lag.

23.2 Key Concepts

Discretisation and time lag: \(\Delta t = 1\) embeds a one-step lag; the larger the step, the stronger the destabilising lag.
Expanding spirals: in the phase plane the discrete model spirals outward, unlike the closed orbits of the continuous model.
Convergence to continuous behaviour: as \(\Delta t \to 0\) the discrete model approaches the continuous-time dynamics.

23.3 Learning Outcomes

By the end of this exercise, students will be able to: - Implement a discrete-time predator-prey model with a for loop in R. - Explain how the time step introduces a lag that affects stability. - Compare discrete-time and continuous-time dynamics directly.

23.4 Activity Overview

Suggested Timings: - 5 minutes: Recap the discrete update equations and the meaning of \(\Delta t\). - 10 minutes: Build the model function and run the default simulation. - 15–20 minutes: Vary \(\Delta t\) (0.1, 1, 2) and compare phase-plane behaviour; relate to the continuous-time results. - 5 minutes: Wrap-up on why discretisation matters.

23.5 Instructions for Facilitating

Build the discrete_predator_prey_model() function together, mapping each line to a term in the update equations.
Run with the defaults (\(\Delta t = 1\)) and produce the time-series and phase-plane plots; point out the outward spiral.
Have students change \(\Delta t\) to 0.1 and then to 2, re-running each time, and describe the change in stability.
Put the discrete phase plane next to the continuous-time one from the previous exercise for a side-by-side comparison.

23.6 Questions & Model Answers

Decrease \(\Delta t\) to 0.1 and compare with \(\Delta t = 1\).
- With \(\Delta t = 0.1\) the spirals expand much more slowly and the trajectory stays close to a near-closed orbit — it approaches the neutrally stable continuous-time behaviour. Smaller steps mean a weaker lag.
Increase \(\Delta t\) to 2 — does extinction happen faster?
- Yes. A larger step strengthens the lag and the overshoot, so the spiral expands more rapidly and the population reaches extinction (a zero crossing) in fewer steps.
Why does the discrete model differ from the continuous one?
- The finite time step means each population responds to the previous step’s abundances rather than instantaneously. This built-in lag amplifies oscillations; the continuous model, with an infinitesimal step, has no such lag and produces sustained neutral cycles.

23.7 Teaching Tips

Same parameters, different world: run the discrete and continuous models with comparable parameters so the destabilising effect of discretisation is unmistakable.
Watch for negatives: a large \(\Delta t\) can push a population below zero — a good hook for discussing where the model stops being biologically meaningful.
Connect to Excel: remind students the \(\Delta t = 1\) case is exactly the year-by-year model they built in the spreadsheet.

23.8 Common Pitfalls

Expecting closed orbits: students who just did the continuous model may be surprised by the spiral; make the contrast explicit.
Interpreting negative populations literally: these are numerical artefacts of a large step, not biology.
Loop indexing errors: initial values go in V[1]/C[1], and the loop fills V[t+1]/C[t+1] — off-by-one mistakes are common.