## 5 Dynamic Prey-Predator Spatial Model via African Buffalo Optimization
### Season Definition
#### 5.3.2 Migration

Table 13: Simulation performed using agents—Case E.

of the prey were between 80 and 505, and of the predators between 1243 and 1364 giving a discrete range of 425 prey and 121 predators. Therefore, we deduce that the model without using agents allowed a direct variation of the population of prey and predators using a modification of the values of the inputs parameters \(lp1\) and \(lp2\) of the ABO metaheuristics. These parameters did not generate a visible effect when performing an autonomous regulation of the population with agents. Figures 8 and 9 describe the behavior of Case E, showing the lattice in six stages of the simulation (Seasons 1, 10, 20, 30, 40 and 50). When considering behavior without using the agent (see Figure 8), was found that prey kept in the central sector of the lattice, and predators on both sides of the prey. However, when considering the behavior



Figures 10 and 11 describe a simulation from the beginning to the end. The six life cycles of a season were considered. Figure 10 describes a simulation without using agents. Prey and predator populations remained almost static during the simulation. This is easily observable when viewing a straight line in the center of the figure. In contrast, Figure 11 depicts a simulation using agents. This simulation started with a high population of prey and with a low population of predators. In the figures, it can be observed that, during the simulation, the population of prey decreased, and the population of predators increased. The equilibrium was reached in the final stages of the simulation. Figure 9: Population behavior using agents. ## 7 Conclusions

In this research, a dynamic prey-predator spatial model was solved using cellular automata. For the movement of the cellular automata, the movement equation of the African buffalo optimization metaheuristics was applied. The results indicate that the prey and predators maintained a population without a considerable variation during the execution of the simulation. In addition, it has been proposed to use a multi-agent system for the resolution of the prey-predator model. In the results obtained, it managed to adjust and maintain a balance of prey and predators, allowing autonomous regulation of the population of prey and predators. The architecture composed of two agents could give real-time descriptions by adjusting the reproduction capacity in real time. As future work, it is proposed to use the equations of movements of various metaheuristics of the literature as well as perform simulations on geo-referential maps. Figure 11: Simulation from beginning to end using agents—Case E.

Figure 10: Simulation from the beginning to the end without using agents—Case E.

**Author Contributions:** Conceptualization, B.A. and F.A. ; Data curation, B.A., F.A. and F.Y. ; Formal analysis, B.A. and F.A. ; Investigation, B.A. ; Methodology, B.A. ; Project administration, B.A. ; Resources, B.A. and F.A. ; Software, B.A., F.A. and F.Y. ; Supervision, B.A. ; Validation, B.A., F.A. and F.Y. ; Visualization, F.A. ; Writing--original draft, B.A. ; and Writing--review and editing, B.A. **Funding:** This research received no external funding. **Acknowledgments:** [PERSON] is supported by Ph.D (h.c) [PERSON], Chile. Also, we thank the anonymous reviewers for their constructive comments. **Conflicts of Interest:** The authors declare no conflict of interest. ## Abbreviations

The following abbreviations are used in this manuscript: \begin{tabular}{l l} ABO & African Buffalo Optimization \\ MAS & Multi-Agent System \\ GIS & Geographic Information System \\ CA & Cellular Automaton \\ \end{tabular} 

## References

* (1) [PERSON] _Nature-Inspired Metaheuristic Algorithms_; Luniver Press: Bristol, UK, 2010. * (2) [PERSON] Firefly algorithms for multimodal optimization. In Proceedings of the International Symposium on Stochastic Algorithms, Sapporo, Japan, 26-28 October 2009; Springer: Berlin, Germany, 2009; pp. 169-178. * (3) [PERSON]; [PERSON] Cuckoo search via Levy flights.