Flocking Behavior of Cucker–Smale Model with Processing Delay

,e dynamics of a delay multiparticle swarm, which contains symmetric and asymmetric pairwise influence functions, are analyzed. Two different sufficient conditions to achieve conditional flocking are obtained. One does not have a clear relationship with this delay, and the other proposes a range of processing delays that affect the emergence of a flock. It is also pointed out that if the interparticle communication function has tail dissipation, unconditional flocking can be guaranteed. Compared with the previous results, the range of the communication rate β that allows a flock to emerge has been expanded from 1/4 to 1/2.


Introduction
ere are many survival-oriented clusters in nature, such as ant colonies that coordinate food transportation, birds that increase the success rate of foraging, and fish that unite against danger, and so on. e research on the colony of biological groups should be traced back to Reynolds' simulation experiments on birds in [1]; further some scholars have proposed many motion models to mathematically characterize them. Among them, a second-order model proposed by Cucker and Smale in [2,3] to explain selforganizing behavior in complex adaptive systems has been favored by researchers and continuously improved. For example, Motsch and Tadmor in [4] modified the symmetry of the influence intensity between particles to be asymmetric to explore the aggregation behavior of nonuniformly distributed particle swarms. Some scholars have carried out the impact of the time delay on flocking or consensus of the system in [5][6][7] and the references therein.
Liu and Wu in [5] proposed a model with processing delay, which is described as _ x i (t) � v i (t), i � 1, 2, . . . , N, where x i , v i ∈ R d and d is a positive integer, α > 0 indicates the intensity of the influence between particles, and τ > 0 represents the time lag, which includes the response time of the particle i and the communication time between particles i and j. e communication function can be defined as where . e initial conditions are In this study, we further consider the flocking conditions of the delayed model proposed in [5]. e significant contributions of our results are reflected in the following three aspects. (1) Compared with eorem 3.1 in [5], the unconditional flocking condition ∞ ψ 2 (r)dr � ∞ is improved to ∞ ψ(r)dr � ∞, that is, the communication rate β is expanded from 1/4 to 1/2. (2) Note that with τ � 0 in (1), the communication rate β of unconditional flocking in [4] has also been expanded from 1/4 to 1/2. (3) It is clearly pointed out that processing delay can affect the occurrence of aggregation behavior, which is specifically manifested in the controllable range of the delay in flocking conditions. e following two variables (D x and D v ) are used to analyze the evolution of the aggregation behavior of systems (3) and (4), for t ≥ −τ: us, for both I CS (r) and I MT (r), it follows from a few simple calculations that a uniform result can be directly verified about the estimation of influence function I(r) as We still adopt the definition of time-asymptotic flocking proposed in [4].

Main Results
is section proposes two different sufficient conditions for systems (3) and (4) with I CS or I MT to achieve the conditional flocking in eorem 1 and eorem 2. We have also established certain conditions for the completion of unconditional flocking in eorem 3.

Conditional Flocking.
To establish the flocking solution of systems (3) and (4), the following important auxiliary lemmas are introduced first.
Lemma 1 (see [8]). Let x(t) be the solution of the linear functional differential equation, be a solution to systems (3) and (4); then, we have where D x and D v are defined in (5).
Proof. Making use of system (3) yields Using inequality (6) and the normalization assumptions for communication functions, that is, and this proof is completed.

a solution of systems (3) and (4); then, the upper Dini derivative of D x (t) and D v (t) satisfies
where p, q ∈ Γ. One can obtain Similarly, without loss of generality, let Complexity (13) erefore, (11) is proven. To establish the flocking conditions, we define a set containing all the initial configurations of asymptotic flocking allowed, that is, where D x and D v are shown in (5) and ψ(r) � (1+ □ Theorem 1. For β > (1/2), suppose that the initial conditions (4) are selected from the set S; then systems (3) and (4) with I MT (r) or I CS (r) can complete the conditional flocking.
Proof. Inspired by the work in [5,9], we take the following Lyapunov function: us, the upper Dini derivative of E(D x , D v )(t) along (D x , D v ) with respect to t is shown below.
Furthermore, combining with Lemma 3, we have Since the initial conditions (4) are selected from the set S, it follows from (16) that Due to the fact that ψ has a divergent tail, there must be a constant D * < ∞ such that D x (t − τ) ≤ D * for t ≥ 0. Considering inequality (6) yields Using the second inequality in (11) in Lemma 3, we can further derive that Making use of Lemma 1, we can show that D v (t) ⟶ 0 as t ⟶ ∞ and systems (3) and (4) converge to a flock as shown in Definition 1. e proof is completed.
Another flocking condition closely related to processing delay is proposed in the following theorem. □ Theorem 2. For β > (1/2), suppose that the initial configurations (4) are met as follows: and the processing delay τ satisfies where R τ ≔ max θ∈[−τ,0] D v (θ) > 0 and D x , D v are defined in (5); then, systems (3) and (4) with I MT (r) or I CS (r) converge to a flock.
Proof. We only need to prove that the initial conditions which satisfy (22) all exist in the set S defined in (8). Note that which means that the initial conditions which satisfy (22) all exist in set S. Consequently, systems (3) and (4)  (1) Note that with τ � 0, eorem 1 and eorem 2 will then the flocking condition is further written as D v (0) < ∞ ψ(r)dr, thereby improving eorem 3.1 in [4]. (2) Comparing eorem 1 and eorem 2, we can get the following two points worthy of attention. First, it is clear from the set of allowed initial conditions that the former is larger than the latter. Second, the latter helps us realize that the occurrence of aggregation behavior is indeed affected by the size of τ.   (1) e fundamental reason for the unconditional flocking of systems (3) and (4) is that the following condition always holds for any initial configuration: (2) Note that with τ � 0, the unconditional flocking result in [4] is improved to ∞ ψ(r)dr � ∞, which means that the communication rate β is expanded from 1/4 to 1/2. (3) Compared with eorem 3.1 in [5], the range of communication rate β has been expanded. Specifically, we promote the results from ∞ ψ 2 (r)dr � ∞ to ∞ ψ(r)dr � ∞, that is, we extend the communication rate β from 1/4 to 1/2. us, the results in [5] have been improved.

Numerical Simulations
Some numerical simulations will be enumerated to illustrate the effect of processing delay on the aggregation behavior of systems (3) and (4). For the convenience of calculation,    (3) and (4) converge asymptotically to form a flock. e convergence rate is slower than that in Example 1 due to processing delays. e diameter of the population is larger than that in Example 1, which means that the cohesion and aggregation density between particles are not as good as the case in Example 1. Example 1. α � 15, β � 0.51, τ � 0 (see Figure 1). It can be seen from Figure 1 that without a time delay, the particle swarm can converge asymptotically to form a flock under the fixed initial configurations and the above parameters.
As shown in Figure 3, under the same initial condition with Example 2, this group cannot converge to a flock, and its fatal factor is that τ � 12 > τ 0 , that is, (22) in eorem 2 is broken.
To realize the emergence of flocking in this group, the following method can be adopted, that is, to appropriately adjust the communication rate between particles so that β < 0.5. It may be selected as β � 0.2 < 0.5 and then combined with the discussion in eorem 3; the system unconditionally converges to a flock. e simulation results are shown in Figure 4.

Conclusions
We study the emergence conditions of flocking of multiple particle swarms with processing delays, establish two   Complexity sufficient conditions for conditional flocking in eorem 1 and eorem 2, and give an unconditional flocking result in eorem 3. In particular, eorem 2 intuitively explains the fact that processing delays affect the emergence of flocking, which is reflected in the time-lag range (22) that affects the emergence of flocking. For 0 ≤ β ≤ 1/2, we note that τ � 0, which is obtained from the analysis in Remark 1 and Remark 2, and the flocking results in [4] have been improved from D v (0) < ∞ ψ 2 (r)dr to D v (0) < ∞ ψ(r)dr. It means that the communication rate β has been expanded from 1/4 to 1/2. Compared with the work on the flocking results in [5], we have also expanded β from 1/4 to 1/2.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.