## Abstract

Starting from recent experimental observations of starlings and jackdaws, we propose a minimal agent-based mathematical model for bird flocks based on a system of second-order delayed stochastic differential equations with discontinuous (both in space and time) right-hand side. The model is specifically designed to reproduce self-organized spontaneous sudden changes of direction, not caused by external stimuli like predator’s attacks. The main novelty of the model is that every bird is a potential turn initiator, thus leadership is formed in a group of indistinguishable agents. We investigate some theoretical properties of the model and we show the numerical results. Biological insights are also discussed.

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## Acknowledgements

The authors want to warmly thank Andrea Cavagna, Irene Giardina, and Giovanna Nappo for the useful discussions we had during the preparation of the manuscript.

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### Contributions

EC conceived the idea, developed the model, proved the theoretical results, wrote the numerical code, wrote the manuscript, and secured funding for the research. MM developed the model, proved the theoretical results, wrote the numerical code, carried out the numerical experiments, and wrote the manuscript. MP proved the theoretical results. LB developed the model.

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This work was carried out within the research project “SMARTOUR: Intelligent Platform for Tourism” (No. SCN_00166) funded by the Italian Ministry of University and Research with the Regional Development Fund of European Union (PON Research and Competitiveness 2007–2013).

The authors also acknowledge the Italian Ministry of Instruction, University and Research for supporting this research with funds coming from the project entitled Innovative numerical methods for evolutionary partial differential equations and applications (PRIN Project 2017, No. 2017KKJP4X).

EC is member of the INdAM Research group GNCS and MM is member of the INdAM Research group GNAMPA.

## A Appendix

### A Appendix

Here we present the proof of Proposition 1.

### Proof

Let us consider the intervals

where \(k \ge 1\) integer, and \(h=T/k\). We define the following approximate solution

The functions \(\{y_k\}_k\) are uniformly bounded and equicontinuous. In fact, from A3) we get that there exists \(\varepsilon _0>0\) such that

Moreover, let \(\nu >0\) and \(t_1 \ne t_2 \) such that \(\left| t_2-t_1 \right| < \nu \). For any \(\varepsilon >0\) it holds

since \(\varphi (t) \equiv \int _{\delta }^{t} m(\tau ) d\tau \) is uniformly continuous. Since (22) and (23) hold true, Ascoli-Arzelá theorem ensures the existence of a uniformly convergent subsequence. In the following, we still denote with \(\{y_k\}_k\) that subsequence, and with *y* its limit. By (23), for \(h<\nu \), we get

hence \(y_k(\tau -h)\) converges to \(y(\tau )\). In the same way, \(y_k(\tau -h-\delta )\) tends to \(y(\tau )\). Caratheodory conditions A1) and A3) allow to pass to the limit under the integral in (21). Hence, the limit function *y*(*t*) satisfies the integral equation

which means that *y*(*t*) is a solution to (8).

We now prove the uniqueness of the solution to (8). Let \(y_1, y_2 : (0, \delta +T) \rightarrow \Omega \) be solutions to problem (8), and consider \(z(t)= y_1(t)-y_2(t) \). By A4) we get that for every \(t \in (0,\delta +T)\),

It follows that

We conclude that \(\displaystyle \sup _{0\le s \le t} \left| z(s)\right| \equiv 0\) for any \(t \in (\delta , \delta +T)\), hence \(y_1 \equiv y_2\). \(\square \)

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Cristiani, E., Menci, M., Papi, M. *et al.* An all-leader agent-based model for turning and flocking birds.
*J. Math. Biol.* **83**, 45 (2021). https://doi.org/10.1007/s00285-021-01675-2

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DOI: https://doi.org/10.1007/s00285-021-01675-2

### Keywords

- starlings
- turning
- agent-based models
- delay differential equations
- leaders
- switching leaders
- self-organization