An all-leader agent-based model for turning and flocking birds

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.

A Appendix

Here we present the proof of Proposition 1.

Proof

Let us consider the intervals

$$\begin{aligned} \delta +ih \le t \le \delta + \left( i+1\right) h, \qquad \forall \ i=0,...,k-1 \end{aligned}$$

(20)

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

$$\begin{aligned} \left\{ \begin{array}{ll} y_k(t)= \phi _{0} + \displaystyle \int _{\delta }^{t} f\left( \tau , y_k(\tau -h), y_k(\tau -h-\delta ) \right) \, d\tau , &{} \quad \delta< t <\delta + T, \\ y_k(t) \equiv \phi _{0}, &{} \quad t \in [-\delta ,\delta ]. \end{array} \right. \end{aligned}$$

(21)

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

$$\begin{aligned} \left| y_k(t)- \phi _{0} \right| \le \displaystyle \int _{\delta }^{t} m(\tau ) d\tau < \varepsilon _0. \end{aligned}$$

(22)

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

$$\begin{aligned} \left| y_k(t_1)- y_k(t_2) \right| \le \displaystyle \left| \int _{t_1}^{t_2} m(\tau ) d\tau \right| = \left| \varphi (t_2)-\varphi (t_1)\right| < \varepsilon , \end{aligned}$$

(23)

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

$$\begin{aligned} \left| y_k(\tau -h)- y(\tau ) \right| \le \left| y_k(\tau -h)- y_k(\tau ) \right| + \left| y_k(\tau )- y(\tau ) \right| < \varepsilon , \end{aligned}$$

(24)

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

$$\begin{aligned} y(t)=\phi _0+ \displaystyle \int _{\delta }^{t} f\left( \tau , y(\tau -h), y(\tau -h-\delta ) \right) \, d\tau , \end{aligned}$$

(25)

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 $t \in (0,\delta +T)$,

$$\begin{aligned} \left| z(t) \right| \le \displaystyle \int _{\delta }^{t} l(\tau )\left( \left| z(\tau )\right| + \left| z(\tau -\delta )\right| \right) d\tau \le 2 \displaystyle \int _{\delta }^{t} l(\tau ) \sup _{0\le s \le \tau } \left| z(s)\right| d\tau . \end{aligned}$$

(26)

It follows that

$$\begin{aligned} \sup _{0\le s \le t} \left| z(s)\right| \le 2 \displaystyle \int _{\delta }^{t} l(\tau ) \sup _{0\le s \le \tau } \left| z(s)\right| d\tau . \end{aligned}$$

(27)

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