On CuckerSmale model with noise and delay^{†}^{†}thanks: The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (Fp7/20072013)/ Erc grant agreement No. 239870.
Abstract
A generalization of the CuckerSmale model for collective animal behaviour is investigated. The model is formulated as a system of delayed stochastic differential equations. It incorporates two additional processes which are present in animal decision making, but are often neglected in modelling: (i) stochasticity (imperfections) of individual behaviour; and (ii) delayed responses of individuals to signals in their environment. Sufficient conditions for flocking for the generalized CuckerSmale model are derived by using a suitable Lyapunov functional. As a byproduct, a new result regarding the asymptotic behaviour of delayed geometric Brownian motion is obtained. In the second part of the paper results of systematic numerical simulations are presented. They not only illustrate the analytical results, but hint at a somehow surprising behaviour of the system  namely, that an introduction of intermediate time delay may facilitate flocking.
Keywords: CuckerSmale system, flocking, asymptotic behaviour, noise, delay, geometric Brownian motion.
1 Introduction
Collective coordinated motion of autonomous selfpropelled agents with selforganization into robust patterns appears in many applications ranging from animal herding to the emergence of common languages in primitive societies [26]. Apart from its biological and evolutionary relevance, collective phenomena play a prominent role in many other scientific disciplines, such as robotics, control theory, economics and social sciences [5, 31, 23]. In this paper we study the interplay of noise and delay on collective behaviour. We investigate a modification of the well known CuckerSmale model [6, 7] with multiplicative noise and reaction delays.
We consider autonomous agents described by their phasespace coordinates , , where (resp. ) are timedependent position (resp. velocity) vectors of the th agent. The governing equations are given as the following system of delayed Itô stochastic differential equations
(1)  
(2) 
where the delayed velocity is given by and is the reaction delay. The parameters and , measure the alignment and noise strength, respectively, and , , are independent dimensional white noise vectors. In general, the communication rates are functions of the mutual distances , however, in most of our paper we will consider them as given functions of time satisfying certain assumptions. The standard CuckerSmale model [6, 7] is a special case of equations (1)–(2) for and . Our aim is to investigate equations (1)–(2) for general values of reaction delay and noise strength parameters ,
The CuckerSmale model was introduced and studied in the seminal papers [6, 7], originally as a model for language evolution. Later the interpretation as a model for flocking in animals (birds) prevailed. In general, the term flocking refers to the phenomena where autonomous agents reach a consensus based on limited environmental information and simple rules. The CuckerSmale model is a simple relaxationtype model that reveals a phase transition depending on the intensity of communication between agents. Using and in (1)–(2), we can write the standard CuckerSmale model as the following system of ordinary differential equations
(3)  
(4) 
where the dots denote the time derivatives. We note that the scaling by in (4) is significant to obtain a Vlasovtype kinetic equation in the meanfield limit , see, for example [28]. The communication rates introduced in [6, 7] and most of the subsequent papers are of the form
(5) 
If , then the model exhibits the socalled unconditional flocking, where for every initial configuration the velocities converge to the common consensus value as . On the other hand, with the flocking is conditional, i.e., the asymptotic behaviour of the system depends on the value of and on the initial configuration. This result was first proved in [6, 7] using tools from graph theory (spectral properties of graph Laplacian), and slightly later reproved in [28] by means of elementary calculus. Another proof has been provided in [14], based on bounding – by a system of dissipative differential inequalities, and, finally, the proof of [4] is based on bounding the maximal velocity.
Various modifications of the generic model – have been considered. For instance, the case of singular communication rates was studied in [14, 24]. Motsch and Tadmor [21] scaled the communication rate between the agents in terms of their relative distance, so that their model does not involve any explicit dependence on the number of agents. The dependence of the communication rate on the topological rather than metric distance between agents was introduced in [15]. The influence of additive noise in individual velocity measurements was studied in [13] and [29]. Stochastic flocking dynamics with multiplicative white noises was considered in [1]. Delays in information processing were considered in [18], however, their analysis only applies to the MotschTadmor variant of the model.
In this paper, we are interested in studying the combined influence of noise and delays on the asymptotic behaviour of the CuckerSmale system. In particular, we derive a sufficient condition in terms of noise intensities and delay length that guarantees flocking. Our analysis is based on a construction of a Lyapunov functional and an estimate of its decay rate. To prove our main results, we make an additional structural assumption about the matrix of communication rates which, loosely speaking, means that the communication between agents is strong enough.
The paper is organized as follows: In Section 2 we show how our model – is derived from the CuckerSmale model – and define what is meant by flocking. Moreover, we consider a simplified version of the model to provide an intuitive understanding of what qualitative properties may be expected. In Section 3 we derive a sufficient condition for flocking in terms of the parameters , and , based on a micromacro decomposition and construction of a Lyapunov functional. Moreover, as a byproduct of our analysis, we provide a new result about the asymptotic behaviour of delayed geometric Brownian motion. Section 4 is devoted to a systematic numerical study of the model. First, we focus on simulation of delayed geometric Brownian motion, in particular, we study the dependence of its asymptotic behaviour on the delay and noise levels. Then, we perform the same study for system (1)–(2). This leads to the interesting observation that, for weak coupling and small noise levels, an introduction of intermediate delays may facilitate flocking. A systematic study of this effect concludes the paper.
2 The stochastic CuckerSmale model with delay
In order to make the generic model (3)–(4) more realistic, we amend it with two additional features. First, we note that measurements in the real world are subject to errors and imprecisions that are typically modeled in terms of white noise. In particular, we assume that the state (velocity) of agent measured by agent is given by the expression
(6) 
where represents the imprecision of ’s measurement device, and are independent identically distributed dimensional Brownian motions with zero mean and the covariance relations
with iff and otherwise, and similarly for .
Note that the multiplicative structure of the noise term ensures that . Substituting given by (6) for in (4) and defining , we obtain the following system of stochastic differential equations (SDEs) for velocities
with a positive coupling strength.
The second amendment is the introduction of delays, motivated by the fact that agents react to information received from their surroundings with some time lag. However, we assume that information propagates instantaneously, so the delay does not depend on the physical distance between agents. For simplicity, we assume the reaction lag to be the same for all agents, so that at time they react to information perceived at time for a fixed .
Convention 1
Throughout the paper, we denote by the quantity evaluated at time , i.e., , and by the same quantity evaluated at time , i.e., . We will also write resp. for the vectors of locations resp. velocities of the agents.
In general, the communication rates may be functions of the mutual distances . However, our analysis is based on a certain structural assumption about the communication matrix and the particular form of the dependence on the mutual distances is irrelevant. Therefore, we consider the rates as given adapted stochastic processes, so that decouples from . Moreover, we assume that are uniformly bounded,
(7) 
Thus, we finally arrive at the stochastic system of delayed differential equations that we will study,
(8) 
which is supplemented with the deterministic constant initial datum ,
(9) 
Let us note that we interpret the noise term in in terms of the Itô calculus [22, 19].
Theorem 1
The stochastic delay differential system with initial condition admits a unique global solution on which is an adapted process with for all , i.e., a martingale.
Proof: The proof follows directly from Theorem 3.1 and the subsequent remark on p. 157 of [19]. Indeed, (8) is of the form
for suitable functions and . In particular, the righthand side is independent of the present state , so that the solution can be constructed by the method of steps. The second order moment is bounded on because of the linear growth of the righthand side of (8) in .
Definition 1
We say that the system exhibits asymptotic flocking if the solution for any initial condition satisfies
where denotes the expected value of a stochastic process.
2.1 Simplified case with
To get an intuition of what qualitative properties we may expect from the solutions of , we consider the case when the communication rate is constant, i.e., ; in other words, we set in (5). We also assume that is equal to the same constant for all , i.e. , and, moreover, that for some and all . Then, defining , we obtain
Since, by assumption, for , we have for all . Consequently, decouples into copies of the delayed SDE
(10) 
where we denoted for any . We are not aware of any results concerning the asymptotic behaviour of . The method developed in [3] suggests that
where is the fundamental solution of the delayed ODE
(11) 
i.e., formally, solves (11) subject to the initial condition for . The fundamental solution can be constructed by the method of steps [25], however, evaluation of its norm is an open problem. From this point of view, the analysis carried out in Section 3 provides new and valuable information about the asymptotics of (10), see Section 3.4. Let us note that setting in the above criterion recovers the wellknown result about geometric Brownian motion [22]: the mean square fluctuation tends to zero if and only if .
Finally, for the convenience of the reader, we give an overview of the qualitative behaviour of solutions to (11) with , subject to a constant nonzero initial datum (see, e.g., Chapter 2 of [25]):

If , the solution monotonically converges to zero as , hence no oscillations occur.

If , oscillations appear, however, with asymptotically vanishing amplitude.

If , periodic solutions exist.

If , the amplitude of the oscillations diverges as .
Hence, we conclude that the (over)simplified model , corresponding to the delayed CuckerSmale system with and no noise, exhibits flocking if and only if . In the next Section we derive a sufficient condition for flocking for the general model .
3 Sufficient condition for flocking
In this section we derive a sufficient condition for flocking in according to Definition 1. Our analysis will be based on a construction of a Lyapunov functional that will imply decay of velocity fluctuations for suitable parameter values. However, we will have to adopt an additional structural assumption on the matrix of communication rates .
Before we proceed, let us shortly point out the mathematical difficulties that arise due to the introduction of delay and noise into the CuckerSmale system. The “traditional” proofs of flocking of model (3)–(4), for instance [6, 7, 28, 14], rely on the monotone decay of the kinetic energy (velocity fluctations) of the form
However, this approach fails if processing delays are introduced, since for without noise (i.e., all ), we have
One then expects the product to be nonnegative for small enough, however, it is not clear how to prove this hypothesis.
The introduction of noise leads to additional difficulties  in particular, the classical bootstrapping argument [6, 7, 14] for fluctuations in velocity fails in this case. Similarly as in [13], we circumvent this problem by adopting, in addition to the boundedness (7), a structural assumption about the matrix of communication rates. We define the Laplacian matrix by
(12) 
and note that is symmetric, diagonally dominant with nonnegative diagonal entries, thus it is positive semidefinite and has real nonnegative eigenvalues. Due to its Laplacian structure, its smallest eigenvalue is zero [6]. Let us denote its second smallest eigenvalue (the Fiedler number) . Our structural assumption is that there exists an such that
(13) 
This can be guaranteed for instance by assuming that the communication rates are uniformly bounded away from zero, , since there exists a constant such that , see Proposition 2 in [6].
Moreover, we assume that the matrix of communication rates is uniformly Lipschitz continuous in the Frobenius norm, in particular, there exists a constant such that
(14) 
where denotes the Frobenius matrix norm.
To ease the notation and without loss of generality, we will consider the onedimensional setting , i.e., and , where is the number of agents. Then, with the definition (12), we put (8) into the form
(15) 
Our main result is the following.
Theorem 2
Let be given by satisfying , and . Let the parameters and satisfy
(16) 
then there exists a critical delay , independent of , such that for every the system exhibits flocking in the sense of Definition 1.
Moreover, if the matrix of communication rates is constant, i.e. holds with , then is of the form
(17) 
Remark 1
The system with constant communication matrix can be seen as a linearization of the system – about the equilibrium for with some . Note that in this case the formula for the critical delay does not depend on the particular value of in .
3.1 Micromacro decomposition
We introduce a micromacro decomposition [28, 13] which splits into two parts: macroscopic, that describes the coarsescale dynamics, and microscopic, that describes the finescale dynamics. The macroscopic part for the solution is set to be the mean velocity ,
(18) 
The microscopic variables are then taken as the fluctuations around their mean values,
(19) 
We denote . Then we have
(20) 
Since is the eigenvector of corresponding to the zero eigenvalue, we have . Then (15) can be rewritten as follows
(21) 
The macroscopic variable satisfies the following lemma.
Lemma 1
Let be a solution of subject to the deterministic constant initial datum . Then for and for all .
Proof: The boundedness of follows directly from the definition (18) and the martingale property of provided by Theorem 1. Using (12), we have
Summing equations (15), , using (18) and , we obtain that the macroscopic dynamics is governed by the system
(22) 
After integration in time this implies
Since is a martingale, we have (see Theorem 5.8 on p. 22 of [19]). Thus we obtain .
Remark 2
Note that and are expressed in terms of the variables only and so they form a closed system, which is equivalent to .
Clearly, due to (19), we have . Consequently, it is natural to introduce the decomposition , where is given by (20). We then have for all .
Lemma 2
Let , , be the matrix defined in and assume that and hold. Then:
(a) The maximal eigenvalue of is bounded by .
(b) We have for any vector
(c) We have for any vector .
(d) For any vectors , and we have
Proof:

The claim follows from the Gershgorin circle theorem. Indeed, since , the diagonal entries satisfy , and for all .

The smallest eigenvalue of is zero with the corresponding eigenvector . The second smallest eigenvalue (Fiedler number) is assumed to be positive by (13). Thus, is a symmetric, positive operator on the space and there exists an orthonormal basis of composed of eigenvectors of corresponding to the positive eigenvalues . Then, every vector can be decomposed as
(23) Thus, due to the above bound on the eigenvalues , we have

With the orthonormality of the basis and the positivity of the eigenvalues , we have by the CauchySchwartz inequality
and with any ,
3.2 Lyapunov functional
The proof of Theorem 2 relies on estimating the decay rate of the following Lyapunov functional for (21)–(22),
where , are positive constants depending on , and .
Lemma 3
Let the assumptions of Theorem 2 be satisfied. Then there exist positive constants , and such that for every solution of – the Lyapunov functional satisfies
(25) 
Proof: We apply the Itô formula to calculate . Note that the Itô formula holds in its usual form also for systems of delayed SDE, see page 32 in [12] and [17, 8, 20]. Therefore, we obtain
With the identity (formula (6.11) on p. 36 of [19]), we have
Consequently, summing over , using and the identity
we obtain
Consequently, we have
(26)  
Our goal is to estimate from above. First of all, we note that by the elementary property of the Itô integral (Theorem 5.8 on p. 22 of [19]),
For the first term of the righthand side in , we write
and apply Lemma 2(d) with ,
Using Lemma 2(c), we have
(27) 
Now we write for , componentwise, using ,
Thus, we have for the expectation of the square
An application of the CauchySchwartz inequality and Fubini’s theorem for the first term of the righthand side yields
For the second term we use the fundamental property of the Itô integral (Theorem 5.8 on p. 22 of [19]),
Similarly, the third term is estimated as
Thus, we get from , estimating ,
An application of Lemma 2(b) gives
To balance this term with , we use assumption and Lemma 2(c) in
Collecting all the terms in finally leads to
We set
(28) 
then the above expression simplifies to
(29) 
We want . Substituting (28) into this inequality leads to a third order polynomial inequality in . This polynomial has all positive coefficients but the zero order one, which is . If is satisfied, then choosing such that
makes negative. Consequently, there exists a such that for any ,
This completes the proof of .
It remains to study the case when is a constant matrix, i.e. in . Then simplifies to
and we have to find such that . Again, substituting (28) for and leads to
(30) 
The maximum value of the right hand side is obtained for which is positive because of the first inequality in (17). Substituting into (30), we obtain
Finally, resolving in leads to
If the above sharp inequality is satisfied, there exists an such that and, consequently, (25) holds.
3.3 Proof of Theorem 2
An integration of in time gives
An application of Lemma 2(c) gives then
so that the last integral is convergent as and, consequently, for all