Wator: A Predator-Prey simulation

Wa-Tor is the name of a computer simulation of a simple predator-prey model set in a toroidal world. The simulation was originally described by Alexander K. Dewdney in an article for Scientific American [1]. It simulates the hypothetical toroidal planet Wa-Tor (from "Water Torus"), whose surface is entirely covered with water. Only two species inhabit this ocean: sharks and their prey fish. The sharks survive solely by hunting fish, while the fish feed on a never-ending supply of plankton. Sharks and prey fish behave according to a strict, fixed set of rules. Over the course of the simulation an interesting dynamic emerges, with the populations oscillating between extinction and overpopulation.

Pixel Size Prey Breed Time Energy per Prey Required Shark Spawn Energy Chart Skip

The simulation can also be downloaded as a screensaver. The screensaver is written in C++ and the graphics output is rendered in 2D using the OpenGL library. The fundamentals of programming screensavers with MFC and OpenGL are described and summarised very well by Rachel Grey in [2]. The source code of the Wa-Tor screensaver can be downloaded from GitHub and compiled with Visual Studio.

Simulation Rules

In this section we will briefly cover the simulation rules. Simulation time advances in discrete time steps. The toroidal world is simulated on a two-dimensional grid whose opposing edges are connected. When an individual leaves the simulation area on one side it re-enters on the opposing side. Fish and sharks move in every time step (if possible) and interact according to the rules described below.

Rules for prey fish

In every time step a prey fish moves randomly to one of its four neighbouring cells, provided that cell is empty. Fish age with every time step. Once a fish reaches a certain age (the "breed time"), its life ends with the creation of two offspring. The age counters of both offspring are reset to zero. One offspring takes the cell of the old fish, the second is created randomly in a free neighbouring cell. If no neighbouring cell is free, no further offspring is created. The descendants then go through the same life cycle.

The image on the left highlights the prey's movement options with arrows. Fish are not allowed to move into cells already occupied by sharks. If none of the neighbouring cells is free, the fish does not move.

Summary of rules for prey movement:

Fish move orthogonally to randomly chosen, free neighbouring cells.
Fish will never voluntarily swim into a cell occupied by a shark.
On reaching the "breed_time" offspring are spawned in a randomly chosen free neighbouring cell.

Rules for sharks

Sharks move randomly to neighbouring cells. Each move costs the shark one point of energy. If a shark's energy level drops to zero it dies. If the neighbouring cell into which the shark moves contains a fish, the fish is eaten and the shark is credited with a fixed amount of energy. To survive, a shark needs regular hunting success. As soon as a shark has accumulated enough energy it can spawn an offspring in a neighbouring cell. The offspring receives half of the parent's energy. The parent's age counter is reset on reproduction.

Summary of rules for predator (shark) movement:

Sharks move orthogonally to randomly chosen neighbouring cells that are either empty or occupied by prey fish.
If the neighbouring cell into which the predator moves contains a fish, the fish is eaten and the predator's energy increases by a fixed amount.
If sufficient energy is available, offspring are spawned in a free neighbouring cell.
With every time step the predator loses a small, fixed amount of energy.
If the predator's energy drops to zero, it dies.

Modifications to the original algorithm

A few changes to the algorithm were made to improve simulation performance. In the original algorithm sharks do not move randomly; they specifically target neighbouring cells containing fish. This movement pattern has been simplified here to speed up the simulation. The modification is minimal, so there is no visible difference compared to the original. Alexander K. Dewdney's implementation resembles a cellular automaton. In such an implementation every cell must be tested individually, even when it is empty, which reduces simulation speed. The implementation used here therefore stores shark data in a linked list. As a result, the algorithm only has to deal with individuals that actually exist.

The Lotka-Volterra equations

Population size over time.

Plotting the number of individuals of both species in a diagram yields a curve characteristic of the Lotka-Volterra equations. This behaviour is described by two differential equations.

\begin{equation} \frac{dx}{dt} = x(\alpha - \beta y), \\ \frac{dy}{dt} = -y(\gamma - \delta x), \end{equation}

where:

y is the number of predators (sharks)
x is the number of prey (prey fish)
t represents time
\(\alpha, \beta, \gamma\) and \(\delta\) are parameters describing the interaction between the two species.

Although the solution of these differential equations exhibits periodic behaviour, it cannot be expressed in terms of normal trigonometric functions. A simplified linearised solution shows, however, that the predator-population curve leads the prey curve by 90 degrees. A more detailed description of the information presented here can be found in the Wikipedia article on the Lotka-Volterra differential equations [3].

Wator as a screensaver

A Windows screensaver is an executable file with the extension "*.scr". In addition, a screensaver must implement a fixed set of command-line parameters. These parameters allow the screensaver to be configured or executed.

Command-line parameter	Screensaver action
/P:###	Starts the screensaver in preview mode. ### is the placeholder for a window handle (`HWND`) that should be used as the parent window. The screensaver runs inside this window. This option lets you display the screensaver inside any Windows window.
/C:###	Opens the configuration dialog. If the placeholder ### is not present, the return value of `GetForegroundWindow()` is used as the dialog's parent window. Otherwise the content of the placeholder "###" is interpreted as a decimal value representing a window handle to be used as the parent of the dialog.
/S	Starts the screensaver in full-screen mode. As soon as keyboard or mouse activity is detected, the screensaver terminates.

Installation and Configuration

After installation, the settings can be changed via a right-click on the Windows desktop. Installation is done by moving the EXE file into the windows/system32 folder. There is no separate installer. The configuration dialog can be found in the display options (right-click on the desktop). It allows you to set the parameters described below.

Simulation parameters

All parameter changes are reflected immediately in the preview window. The following table lists the available parameters:

Parameter	Description
Prey breed time	The number of time steps required before a predator can reproduce.
Energy per hunted prey	The amount of energy the predator gains from eating prey. With every time step the predator loses one energy point. Once the energy is depleted, the predator dies.
Energy to create offspring	The minimum energy required for a predator to reproduce. When reproducing, the energy is split between mother and child. Yes, that's right — reproduction occurs by division, because by a whim of nature there are no fathers on Wator.
Pixelsize	The size of an individual in pixels. The larger this value, the better the simulation performance.

Download

Download Wator-Screensaver

C++ Source Code for Visual Studio

To install the software, simply run the installer provided here. The screensaver runs on all Windows versions from Windows 7 onwards. The software is provided free of charge; use is at your own risk.

References

A.K. Dewdney: "Computer Recreations; Sharks and Fish wage an ecological war on the toroidal planet of Wa-Tor" Scientific American. pp. 14–22.
Rachel Grey: "Writing an OpenGL screensaver for Windows" online, 2002
Wikipedia: "Lotka–Volterra equations" The Free Encyclopedia. Wikimedia Foundation, Inc.