Agent-based prey-predator model

The simplest ecological system can be constructed from the two interacting species, e.g., prey and predator. This kind of system is very interesting in the terms of Physics of Risk primarily because it is nonlinear [1], and due to being real life example of competition (conflict). Also there are few known simple models for the prey-predator interaction. Among them there are both macroscopic, Lotka-Volterra equations, and microscopic, agent-based, models. In this text we continue the previous discussion by considering the agent-based model.

Interestingly due to the various applications of the Lotka-Volterra equations [2, 3, 4, 5, 6], this agent-based model provides insights not only to the ecosystem, but also into the other socio-economic systems.

Formulation of the agent-based model

The main principles of the agent-based model are pretty elementary and straightforward. Also they are self-evidently identically to the actual behavior in ecosystem. Though the mechanics are slightly over-simplified.

First we start with the two types of agents - prey (white square in the applet below) and predator (black square). The agents move one the square lattice, which is under the periodic boundary condition. This means that the agent exiting trough the bottom wall of the lattice reenters lattice at the top. Exiting through the top causes reentrance through the bottom. Same goes for the left-right wall exits and reentrances.

Now at each time step let us pick random cell on the lattice. If the cell is empty then nothing happens, while if it is occupied then another cell, in the neighborhood, is random selected. Now taking into account the two randomly picked cells, one has to apply the following rules:

• If one cell is occupied by predator and another by prey, then the prey is eaten. After doing so the predator gives a birth to another predator with a certain probability. The new predator is placed in the former prey's cell.
• If both cells are occupied by the same type of agent, then nothing happens.
• If we have a prey cell and empty cell, then prey gives a birth to another prey with a certain probability.
• If we have a predator cell and empty cell, then predator dies with a certain probability.
• If after the application of the rules above still nothing has changed and movement of the agent is possible, then the agent moves from one cell to another unoccupied cell.

Below you should an interactive HTML5 applet, which can be used to study the dynamics of the agent-based model formulated in this text. Note that unlike in the previous case of deterministic model, this model possesses certain stochastic features. Namely you can observe certain fluctuations, which could "kill" the ecosystem or extend its "life". This happens both due to the nature of random movements of our agents and due to the probabilistic formulation of the model.

HTML5 applet

To initialize modeling press button "Continue". If modeling was paused, then the program will resume instead. Modeling can be stopped by pressing "Stop" button. To reset the modeling press "New" button.

References

• N. S. Goel, S. C. Maitra, E. W. Montroll. On the Volterra and Other Nonlinear Models of Intereacting Populations. Reviews of Modern Physics 43: 231-276 (1971). doi: 10.1103/RevModPhys.43.231.
• M. Ausloos. On religion and language evolutions seen through mathematical and agent based models. Proceedings of the First Interdisciplinary Chess Interactions Conference, pp. 157-182. Canada, 2009. arXiv: 1103.5382 [physics.soc-ph].
• R. M. Goodwin. A Growth Cycle. In: Socialism, Capitalism and Economic Growth, ed. C.H. Feinstein. Cambridge University Press, 1967.
• N. L. Olivera, A. N. Proto, M. Ausloos. Information Society: Modeling A Complex System With Scarce Data. Proceedings of The V Meeting on Dynamics of Social and Economic Systems 6: 443-460 (2011). arXiv: 1201.1547 [physics.soc-ph].
• S. Solomon. Generalized Lotka Volterra (GLV) Models of Stock Markets. In: Applications of Simulation to Social Sciences, ed. G. Ballot, G. Weisbuch, pp. 301-322. Hermes Science Publications, 2000. arXiv: cond-mat/9901250.
• N. K. Vitanov, M. Ausloos, G. Rotundo. Discrete model of ideological struggle accounting for migration. Advances in Complex Systems 15 (supp01), 2012: 1250049. doi: 10.1142/S021952591250049X.