A particle system approach to aggregation phenomena

TL;DR

This paper introduces a particle system model for aggregation phenomena, demonstrating that the empirical density converges to a PDE-ODE system solution, bridging individual interactions with macroscopic behavior.

Contribution

It presents a novel particle system approach that rigorously links individual-based models to PDE-ODE descriptions of aggregation.

Findings

Empirical density converges to PDE-ODE solution.

Provides mathematical proof of convergence.

Connects microscopic interactions with macroscopic aggregation patterns.

Abstract

Inspired by a PDE-ODE system of aggregation developed in the biomathematical literature, an interacting particle system representing aggregation at the level of individuals is investigated. It is proved that the empirical density of the individual converges to solution of the PDE-ODE system.

Equations

$$d{X}_t^{i,N}={\int }_{{\mathbb{R}}^d}\frac{y-X_t^{i,N}}{|y-X_t^{i,N}|}g(|y-X_t^{i,N}|,u_t^N(y),m_t^N(y))dydt+\sqrt{2}d{B}_t^i$$

$$\frac{\partial m_t^N\left(x\right)}{\partial t}=-\lambda u_t^N(x)(m_t^N\left(x\right){)}^{\zeta },x\in {\mathbb{R}}^d$$

$$\frac{\partial u_t}{\partial t}=\Delta u_t-\text{div}(u_{t}b(u_t,m_t))$$

$$\frac{\partial m_t}{\partial t}=-\lambda u_t{m}_t^{\zeta }$$

$$b(u,m)(x):={\int }_{{\mathbb{R}}^d}\frac{y-x}{|y-x|}g(|y-x|,u(y),m\left(y\right))dy.$$

$$S_t^N\left(dx\right):=\frac{1}{N}\sum _{i=1}^N{\delta }_{X_t^{i,N}}\left(dx\right).$$

$$W_N(x):=N^{\beta }W(N^{\beta /d}x)$$

$$u_t^N(x):=\left(W_N\ast S_t^N\right)\left(x\right)=\sum _{i=1}^N{W}_N(x-X^{i,N}).$$

$$b:L^2({\mathbb{R}}^d)\times L^2({\mathbb{R}}^d)\to L^{\infty }({\mathbb{R}}^d)$$

$$|g(r,u,m)|+|\nabla g(r,u,m)|\le C\cdot \exp (-r)$$

$$|b(u,m)(x)|\le {\int }_{{\mathbb{R}}^d}\left|g(|y-x|,u(y),m(y))\right|dy\le C{\int }_{{\mathbb{R}}^d}{e}^{-|x-y|}dy:=C^{\prime }<\infty .$$

$$|b(u,m)(x)-b(u^{\prime },m^{\prime })(x)|\le C\cdot {\int }_{{\mathbb{R}}^d}{e}^{-|x-y|}\left(\left|u\left(y\right)-u^{\prime }\left(y\right)\right|+\left|m\left(y\right)-m^{\prime }\left(y\right)\right|\right)dy$$

$$\begin{gathered} |{\nabla }_x\cdot b(u,m)(x)|\le \left|{\int }_{{\mathbb{R}}^d}{\nabla }_x\cdot \left(\frac{y-x}{|y-x|}g(|y-x|,u(y),m(y))\right)dy\right|\le \\ |g(0,u(x),m(x))|+\left|{\int }_{{\mathbb{R}}^d}\frac{y-x}{|y-x|}\cdot {\nabla }_x\left(g(|y-x|,u(y),m(y))\right)dy\right|\le \\ {C}_1+{\int }_{{\mathbb{R}}^d}|{\partial }_{r}g(|y-x|,u(y),m(y))|dy\le C_1+C_2. \end{gathered}$$

$$C\left(L^2\right):=C([0,T],L^2({\mathbb{R}}^d))$$

$$C_{+}\left(L^2\right):=\left\{u\in C\left(L^2\right):u_t\ge 0\text{ for all }t\in \left[0,T\right]\right\}$$

$$C_{0,M}\left(L^2\right)=\left\{m\in C\left(L^2\right):0\le m_t\le M\text{ for all }t\in \left[0,T\right]\right\}$$

$$m_t^N(x)=F_{\zeta }\left(m_0(x),{\int }_0^t{u}_s^N(x)ds\right)$$

$$F_{\zeta }\left(a,b\right)=a\cdot {\tilde{F}}_{\zeta }\left(a,b\right)$$

$${\tilde{F}}_{\zeta }\left(a,b\right)=\{\begin{array}{l}\exp \left(-\lambda b\right)\text{if }\zeta =1\\ \frac{1}{{\left[a^{\zeta -1}(\zeta -1)\lambda b+1\right]}^{\frac{1}{\zeta -1}}}\text{if }\zeta \ge 2\text{.}\end{array}$$

$$X_t^{i,N}=X_0^i+{\int }_0^{t}b(u_s^N,F_{\zeta }\left(m_0(\cdot ),{\int }_0^s{u}_r^N(\cdot )dr\right))(X_s^{i,N})ds+\sqrt{2}{B}_t^i\text{i=1,…,N}$$

$$\begin{gathered} J^i:E\to \mathbb{R}{J}^i(Y):=X_0^i+{\int }_0^{t}b(u_s^N,F_{\zeta }\left(m_0(\cdot ),{\int }_0^s{u}_r^N(\cdot )dr\right))(Y)ds+\sqrt{2}{B}_t^i \\ i=1,\dots ,N \end{gathered}$$

$${\left|\left|J^i(Y)-J^i(Y^{\prime })\right|\right|}_E\le C{T}^{\prime }{\left|\left|Y-Y^{\prime }\right|\right|}_E$$

$$u_t(x)=e^{t\Delta }{u}_0+{\int }_0^t\nabla \cdot e^{(t-s)\Delta }(u_{s}b(u_s,m_s))ds$$

$$m_t(x)=m_0(x)-{\int }_0^t\lambda u_s(x)m_s^{\zeta }(x)ds.$$

$$C\left(L_{loc}^2\right):=C([0,T],L_{loc}^2({\mathbb{R}}^d))$$

$$d_{L_{loc}^2}\left(f,g\right)=\sum _{n=1}^{\infty }{2}^{-n}\left({\Vert f-g\Vert }_{L^2\left(B\left(0,n\right)\right)}\wedge 1\right).$$

$${\left|\left|f\right|\right|}_{W^{\alpha ,2}({\mathbb{R}}^d)}={\left|\left|f\right|\right|}_{L^2({\mathbb{R}}^d)}+[f{]}_{\alpha ,2,{\mathbb{R}}^d}$$

$$[f{]}_{\alpha ,2,{\mathbb{R}}^d}={\int }_{{\mathbb{R}}^d}{\int }_{{\mathbb{R}}^d}\frac{|f(x)-f(y){|}^2}{|x-y{|}^{2\alpha +d}}dxdy.$$

$${\left|\left|u_0^N\right|\right|}_{W^{\alpha ,2}}={\Vert u_0^N\Vert }_{L^2\left({\mathbb{R}}^d\right)}+{\Vert {\left(-\Delta \right)}^{\alpha }{u}_0^N\Vert }_{L^2\left({\mathbb{R}}^d\right)}.$$

$${\left(-\Delta \right)}^{s}f\left(x\right)=C\left(s,d\right){\int }_{{\mathbb{R}}^d}\frac{f\left(x+y\right)+f\left(x-y\right)-2f\left(x\right)}{{\left|y\right|}^{d+2s}}dy,x\in {\mathbb{R}}^d$$

$$\left(e^{tA}f\right)\left(x\right):={\int }_{{\mathbb{R}}^d}\frac{1}{(2\pi {\sigma }^{2}t{)}^{d/2}}{e}^{-\frac{|x-y{|}^2}{2{\sigma }^{2}t}}f(y)dy$$

Full text

1 Introduction

The mathematical literature applied to Biology and Social Science is rich of models devoted to the description of aggregation. Motivations come from several problems like embryo development, tissue homeostasis, tumor growth, animal swarming and flocking. The literature presents heterogeneous mathematical tools: discrete and continuous individual based model, ordinary and partial differential equations (resp. ODE and PDE) and mixture of the previous ones. Also because of this heterogeneity, an interesting issue is to justify the PDE models through the investigation of scaling limits of models based on interaction between individuals. Following this general program, in this work we propose an individual based model and prove convergence, when the number of individuals goes to infinity, to a class of PDE-ODE systems which includes the so called Armstrong-Painter-Sherratt model proposed in [1], [18], evolution of a previous model of [20], including in particular a form of delay by coupling the system with an ODE.

We assume that individuals interact between each other by looking at the density field produced by the others: think for instance to the motion of animals in a swarm or a flock; presumably each animal moves driven by a general overview of the others, not computing several pairwise interactions. Let $N$ be the number of individuals and $X_{t}^{i,N}$, $i=1,...,N$, be their positions. We model this particle-density interaction by the following equations:

[TABLE]

[TABLE]

for $i=1,...,N$ and $t\in\left[0,T\right]$, where $u_{t}^{N}(y)$ is a density associated to the population of particles, defined below, $m_{t}^{N}\left(x\right)$ is the field which allows a dependence on the past or may be used to model external effects like those of the Extracellular Matrix, $\zeta$ is typically equal to 1 or 2, and $B_{t}^{i}$ are independent Brownian motions $B_{t}^{i}$ accounting for a random component of the motion. Each particle $X_{t}^{i,N}$ interacts with each location $y$; the direction of the force is given by the unitary vector $\frac{y-X_{t}^{i,N}}{|y-X_{t}^{i,N}|}$ which spans the line between the particles; the strength of the interaction is $g(|y-X_{t}^{i,N}|,u_{t}^{N}(y))$, namely it is modulated by the distance $|y-X_{t}^{i,N}|$, by the density $u_{t}^{N}(y)$ and by the external field $m_{t}^{N}(y)$. At positions $y$ where $g>0$, particle $X_{t}^{i,N}$ moves towards $y$, namely have a tendency to aggregate. Using different functions $g$ we may describe different kinds of attraction;a wide discussion is presented in the last section of the paper. A technical issue concerns the definition of the density $u_{t}^{N}(x)$, see the discussion below. Under suitable assumptions, our main theorem is the convergence of the previous particle model to the PDE-ODE system

[TABLE]

on $[0,T]\times\mathbb{R}^{d}$, where

[TABLE]

Let us finally discuss the concept of density $u_{t}^{N}(x)$. Given the particles $X_{t}^{i,N}$, one first associates to them the classical concept of empirical measure:

[TABLE]

Its direct use, however, in the previous modelling would oblige us to choose functions $g$ depending on measures, instead of functions, which are less easy to formulate in examples. And, more importantly, we could not speak of $u_{t}^{N}\left(y\right)$, the density at position $y$. In numerics it is common to overcome this difficulty by the so called kernel smoothing, which consists in mollifying the measure by convolution with a smooth kernel. We adopt this procedure. We choose a smooth, compactly supported, probability density $W$ (the kernel) and rescale it with $N$ in a suitable way. A general form of rescaling is

[TABLE]

for some $\beta\in\left(0,1\right)$, as suggested by K. Oelschlager [17]. The density $u_{t}^{N}(x)$ is thus given by

[TABLE]

Thanks to the semigroup approach that we implement in the estimates on the particle system, we are able to consider any choice of $\beta\in\left(0,1\right)$. This is not a trivial task, since other approaches require more restrictions on $\beta$, see [17], [16].

The paper is structured as follows: in Section 2 we give some notations, formulate the main result and prove some preliminary facts; in Section 3 we prove tightness of the density $u_{t}^{N}(x)$ in suitable spaces; in Section 4 we show the passage to the limit and complete the proof of the main result; finally in Section 5 we discuss several examples of interaction function $g$ and show by numerical simulations that the previous model may catch different kinds of aggregation pattern.

2 Notations and basic results

2.1 The particle system

For every positive integer $N$, we consider a particle system described by equations (1) coupled with the random field $m_{t}^{N}\left(x\right)$ satisfying (2) for some integer $\zeta\geq 1$, with initial conditions $X_{0}^{i,N}=X_{0}^{i}$, $i=1,...,N$, where $B_{t}^{i}$, $i\in\mathbb{N}$, is a sequence of independent Brownian motions on a filtered probability space $\left(\Omega,\mathcal{F},\mathcal{F}_{t},P\right)$; $X_{0}^{i}$, $i\in\mathbb{N}$, is a sequence of $\mathcal{F}_{0}$-measurable independent random variables with values in $\mathbb{R}^{d}$, identically distributed with density $u_{0}$; the random function $u_{t}^{N}$ is given by $u_{t}^{N}(x):=\left(W_{N}\ast S_{t}^{N}\right)\left(x\right)$ where $S_{t}^{N}=\frac{1}{N}\sum_{i=1}^{N}\delta_{X_{t}^{i,N}}$ and $W_{N}(x):=N^{\beta}W(N^{\beta/d}x)$ for some $\beta\in\left(0,1\right)$; the random fields $m_{t}^{N}$ have initial conditions $m_{0}^{N}\left(x\right)=m_{0}\left(x\right)$ where $m_{0}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ is a measurable function with $0\leq m_{0}\leq M$, and the functional

[TABLE]

is given by (4) where $g:\mathbb{R}^{+}\times\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow\mathbb{R}$, $g=g(r,u,m)$, is differentiable, bounded with bounded derivatives, and satisfies

[TABLE]

for some constant $C>0$ (where $\nabla g$ denotes the gradient in all variables). It follows that, for every pair of measurable functions $u(x),m(x)$, the aggregation force is bounded:

[TABLE]

We also have

[TABLE]

and regarding the derivative, due to the condition on the gradient of $g$:

[TABLE]

Under these assuptions, existence and uniqueness of a solution, for finite $N$, of system (1)-(2) can be proved by classical methods. Let us explain some details. Let us denote by $C\left(L^{2}\right)$, $C_{+}\left(L^{2}\right)$, $C_{0,M}\left(L^{2}\right)$, the spaces

[TABLE]

[TABLE]

[TABLE]

We say that a random field $m_{t}^{N}\left(x\right)$, $t\in\left[0,T\right]$, $x\in\mathbb{R}^{d}$ defined on $\left(\Omega,\mathcal{F},\mathcal{F}_{t},P\right)$, is adapted of class $C_{0,M}\left(L^{2}\right)$ if $P$-a.s. the functions $\left(t,x\right)\mapsto m_{t}^{N}\left(x\right)$ belong to $C_{0,M}\left(L^{2}\right)$ and for every $t\in\left[0,T\right]$ the function $\left(x,\omega\right)\mapsto m_{t}^{N}\left(x,\omega\right)$ is $\mathcal{B}\left(\mathbb{R}^{d}\right)\times\mathcal{F}_{t}$-measurable. We say that $\left(X^{1,N},...,X^{N,N},m^{N}\right)$ is a strong solution of system (1)-(2) if $X_{t}^{1,N},...,X_{t}^{N,N}$ are continuous adapted processes on $\left(\Omega,\mathcal{F},\mathcal{F}_{t},P\right)$, $m_{t}^{N}\left(x\right)$ is adapted of class $C_{0,M}\left(L^{2}\right)$, all defined on $\left(\Omega,\mathcal{F},\mathcal{F}_{t},P\right)$, and identities (1)-(2) hold, with the equations understood integrated in time. We say that pathwise uniqueness hold if two such solutions are indistinguishable processes.

Proposition 2.1**.**

Given any positive integer $N$ and any function $m_{0}\in L^{2}(\mathbb{R}^{d})$ such that $0\leq m_{0}\leq M$, there exists a strong solution of system (1)-(2) and pathwise uniqueness hold.

Proof.

The proof is classical, we explain only the idea. Given an integer $\zeta\geq 1$, $u^{N}$ and a.e. $x\in\mathbb{R}^{d}$, the solution of equation (2) is global, unique and explicit:

[TABLE]

[TABLE]

[TABLE]

The function $\widetilde{F}_{\zeta}:\left[0,M\right]\times[0,\infty)\rightarrow\mathbb{R}$ is bounded and the function $F_{\zeta}:\left[0,M\right]\times[0,\infty)\rightarrow\mathbb{R}$ is Lipschitz continuous, with at most linear growth in $a$, uniformly in $b$. Then one may consider the system of integral equations

[TABLE]

as a closed system, with only the variables $X_{t}^{1,N},...,X_{t}^{N,N}$. It is a path-dependent equation: the past appears in the drift; but this does not change the way contraction principle applies. One can check that strong existence and pathwise uniqueness for the original system for the variables $\left(X^{1,N},...,X^{N,N},m^{N}\right)$ is equivalent to strong existence and pathwise uniqueness for this reduced path-dependent system in the variables $\left(X^{1,N},...,X^{N,N}\right)$ only; property $m^{N}\in C_{0,M}\left(L^{2}\right)$ is deduced from the explicit formula. Let us say how to prove existence and uniqueness for (9). Thanks to the property (8) the drift of equation (9) is globally Lipschitz continuous. We define the family of maps $J^{i}$ as

[TABLE]

where $E=L^{2}_{\mathcal{F}}(\Omega,C([0,T^{\prime}],\mathbb{R}^{d}))$, with $T^{\prime}<T$. Then with classical computation we get that $J^{i}$ is a contraction on the space E:

[TABLE]

choosing $CT^{\prime}<1$. Hence local existence and uniqueness of strong solutions is proved. Iterating this argument one can get the global existence result, because the amplitude of the interval of iteration depends only on $CT^{\prime}$, namely it is fixed for each iteration. ∎

Remark 2.1*.*

Existence and uniqueness of solution of the system (1)-(2) could be obtained following another approach. With less effort could be possible to obtain just weak existence and uniqueness in law for the system (1)-(2): the method of creating weak solution to SDEs is transformation of drift via Girsanov theorem, see [12]. Being the drift $b$ bounded, see condition (6), hypotheses of Proposition 3.6 and Proposition 3.10 of [12] are verified and existence and uniqueness of the system is obtained. Then $X^{i,N}_{t}$ is solution of (1). Thus also $m^{N}_{t}$ exists, is unique and explicit. This kind of existence would be enough for the purpose of the paper, but we still to decide to emphasize in Proposition 2.1 that a stronger result is attainable.

2.2 Main results

After the indentity of Lemma 2.6 below for the empirical measure is proved, it is natural to conjecture that the limit of the pair $\left(u_{t}^{N}\left(x\right),m_{t}^{N}\left(x\right)\right)$ solves the system (3) with initial condition $\left(u_{0},m_{0}\right)$, where $u_{0}$ is the density of the r.v.’s $X_{0}^{i}$ and $m_{0}$ is the limit of $m_{0}^{N}$. We interpret the first equation of this system in the so called mild form and the second one in integral form. Concerning the initial conditions, we make a choice of simplicity. We assume that $u_{0}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ (the initial distribution of individuals) is a probability density of class $C^{1}$ with compact support, see Lemma 2.9. About $m_{0}:\mathbb{R}^{d}\rightarrow\mathbb{R}$, we assume it is of class $L^{2}(\mathbb{R}^{d})$ and $0\leq m_{0}\leq M$.

Definition 2.2**.**

By mild solution of system (3) we mean a pair $\left(u,m\right)$ belonging to $C_{+}\left(L^{2}\right)\times C_{0,M}\left(L^{2}\right)$ such that

[TABLE]

Where $e^{tA}$ denote the heat semigroup, more precisely defined in Section 2.3. Notice that the $L^{2}(\mathbb{R}^{d})$-norm of $u_{s}b(u_{s},m_{s})$ is bounded, since $b$ is bounded and $u\in C\left(L^{2}\right)$. Hence $\nabla\cdot e^{(t-s)\Delta}(u_{s}b(u_{s},m_{s}))$ is integrable by property (3.4) below. Convergence of the particles system is proved only locally in space, hence we introduce the space

[TABLE]

where the topology on $L_{loc}^{2}(\mathbb{R}^{d})$ is given by the metric

[TABLE]

Theorem 2.3**.**

System (3) has one and only one mild solution $\left(u,m\right)$ in $C_{+}\left(L^{2}\right)\times C_{0,M}\left(L^{2}\right)$; and the pair $\left(u^{N},m^{N}\right)$ converges to $\left(u,m\right)$ in $C\left(L_{loc}^{2}\right)\times C\left(L_{loc}^{2}\right)$, in probability.

2.3 Some useful properties of Analytic Semigroup

We denote with $W^{\alpha,2}(\mathbb{R}^{d})$ the fractional sobolev space, which is a Banach space with the norm

[TABLE]

where

[TABLE]

Or equivantely

[TABLE]

where the fractional laplacian can be characterized by the following lemma.

Lemma 2.4**.**

Let $s\in\left(0,1\right)$. Then there exists a constant $C\left(s,d\right)$ such that

[TABLE]

for every compact support twice differentiable function $f$.

Notice that boundedness of $f$ guarantees integrability at infinity, while twice differentiability implies that the numerator is, for small $\left|y\right|$, infinitesimal of order two, which compensates the singularity of the denominator. Another very useful property on the fractional laplacian is that it is a local operator, namely it preserves compact support of functions.

Let us recall some well known properties of analytical semigroups. The family of operators

[TABLE]

for $t\geq 0$, defines an analytic semigroup (the heat semigroup) on the space $W^{\alpha,2}(\mathbb{R}^{d})$, for every $\alpha\geq 0$. The infinitesimal generator in $L^{2}(\mathbb{R}^{d})$ is the operator $A:D(A)\subset L^{2}(\mathbb{R}^{d})\longrightarrow L^{2}(\mathbb{R}^{d})$, $D(A)=W^{2,2}(\mathbb{R}^{d})$, given by $Af=\frac{\sigma^{2}}{2}\Delta f$. It is possible to define fractional power of the operator $(I-A)^{\delta}$ for $\delta\in\mathbb{R}$ and a well known fact is the equivalence of norms:

[TABLE]

Another property, often used in the sequel, is that for every $\delta,T>0$

there is a constant $C_{\delta,T}$ such that for $t\in(0,T]$

[TABLE]

Finally, we remark that the operator $\nabla(I-A)^{-1/2}$ is bounded in $L^{2}$

[TABLE]

where, here and below, we continue to write simply $L^{2}$ also when the functions are vector valued, as in the case of $\nabla(I-A)^{-1/2}f$.

It will be useful to know the following result on the improvement of regularity.

Lemma 2.5**.**

If $u\in L^{p}\left(0,T;L^{2}(\mathbb{R}^{d})\right)$ for some $p>2$ and satisfies

[TABLE]

for some bounded measurable function $b$, then $u\in C([0,T],L^{2}(\mathbb{R}^{d}))$.

Proof.

The product $ub$ is in $L^{p}\left(0,T;L^{2}(\mathbb{R}^{d})\right)$. Using the bound

[TABLE]

we deduce that $t\mapsto\int_{0}^{t}\nabla\cdot e^{(t-s)A}(u_{s}b_{s})ds$ is of class $C([0,T],L^{2}(\mathbb{R}^{d}))$; the same is true for $t\mapsto e^{tA}u_{0}$ because $u\in L^{2}(\mathbb{R}^{d})$ as a byproduct of our assumptions. Hence $u\in C([0,T],L^{2}(\mathbb{R}^{d}))$. ∎

2.4 Preliminary results

Lemma 2.6**.**

For every $\varphi\in C^{2}([0,T]\times\mathbb{R}^{d})$, $S_{t}^{N}$ satisfies the following identity:

[TABLE]

where

[TABLE]

In particular, choosing $\varphi\left(\cdot\right)=\varphi_{x}\left(\cdot\right)=W_{N}(\cdot-x)$, for $x\in\mathbb{R}^{d}$, we get

[TABLE]

where

[TABLE]

Proof.

The proof follows by Itô formula and Gauss Green formula. ∎

Concerning the family of mollifiers, we have the following useful properties, whose proof is an elementary computation, see for instance [9].

Lemma 2.7**.**

Recall that $W_{N}(x)=N^{\beta}W(N^{\beta/d}x)$. Then

[TABLE]

[TABLE]

with $\gamma^{\ast}=\frac{\beta}{d}(2\gamma+d)$.

We shall use also the following tightness result.

Lemma 2.8**.**

Let $X_{1}$ and $X_{2}$ be two metric spaces with their Borel $\sigma$-fields $\mathcal{B}_{1},\mathcal{B}_{1}$ and let $\varphi:X_{1}\rightarrow X_{2}$ be a continuous function. Let $\mathcal{G}_{1}$ be a family of probability measures on $\left(X_{1},\mathcal{B}_{1}\right)$. Denote by $\mathcal{G}_{2}$ the family of probability measures on $\left(X_{2},\mathcal{B}_{2}\right)$ obtained as image laws of the measures in $\mathcal{G}_{1}$ under the map $\varphi$. If $\mathcal{G}_{1}$ is tight, then $\mathcal{G}_{2}$ is tight.

Proof.

Given $\epsilon>0$, let $K_{1}^{\epsilon}\subset X_{1}$ be a compact set such that $\mu\left(K_{1}^{\epsilon}\right)>1-\epsilon$ for every $\mu\in\mathcal{G}_{1}$. Set $K_{2}^{\epsilon}=\varphi\left(K_{1}^{\epsilon}\right)$; it is a compact set of $X_{2}$ and for every $\nu\in\mathcal{G}_{2}$, called $\mu$ a measure in $\mathcal{G}_{1}$ such that $\nu$ is the image of $\mu$ under $\varphi$, we have

[TABLE]

This proves tightness of $\mathcal{G}_{2}$. ∎

Regarding the initial condition, we state a result, that will be usefull in the proof of tightness

Lemma 2.9**.**

Assume that $X^{i}_{0}$ , $i=1,...,N,$ are independent identically distributed r.v with common probability density $u_{0}\in C^{2}(\mathbb{R}^{d})$, then on $u^{N}_{0}$, defined as $u^{N}_{0}(x)=(W^{N}\ast u_{0})(x)$, we get the following uniform bounds for $p>1$:

[TABLE]

where $C$ is a constant depending on $p$ and $\alpha$.

Proof.

By the definition of the norm in the fractional Sobolev space, we need to estimate uniformly in N:

[TABLE]

We recall that $u_{0}$ is compactly supported and moreover fractional laplacian is a local operator, namely it preserves compactness properties of functions. Then, for $p\geq 2$,

[TABLE]

[TABLE]

where $\mathcal{B}_{1}$, $\mathcal{B}_{2}$ are respectively compact supports of $u^{N}_{0}$ and $\left(-\Delta\right)^{\alpha}u_{0}^{N}$ Assuming that

[TABLE]

[TABLE]

we can write the estimates for (13) in the following terms:

[TABLE]

Then we need to estimate

[TABLE]

Being $Y^{i}\geq 0$ on the first summand we have:

[TABLE]

where $Y$ has the same law of $Y^{i}$. Notice that the equality

[TABLE]

follows easily from the fact that $Y^{i}$ are iid. Because also $\tilde{Y}^{i}\geq 0$, the same result holds for the second term. Then

[TABLE]

Let us estimate the first term (the same result will hold for the second term). We recall some basics inequalities $\log\left(1+x\right)\leq x$, $e^{x}-1\leq xe^{x}$ for $x\geq 0$. Then

[TABLE]

We have to estimate:

[TABLE]

Recalling the definition of $Y^{i}$,

[TABLE]

being $W$ bounded, we get that $\frac{Y(x)}{N}$ is bounded. Now we just need to estimate $\mathbb{E}\left[Y(x)\right]$.

[TABLE]

the last term is bounded because $u_{0}$ is it. Let us analyze the second term, which is a bit more delicate. By the definition of $\tilde{Y}^{i}$,

[TABLE]

Choosing an $\alpha$ small enough the term $\frac{\widetilde{Y}\left(x\right)}{N}$ is bounded. At the end we need to prove a uniform estimate on $\mathbb{E}\left[\tilde{Y}(x)\right]$.

[TABLE]

Being $\left(-\Delta\right)^{\alpha}u_{0}$ compactly supported and continuous also $W_{N}\ast\left(\left(-\Delta\right)^{\alpha}u_{0}\right)$ is uniformly bounded. In summary,

[TABLE]

∎

3 Tightness

3.1 Compactness of function spaces

We use Corollary 9 of J. Simon [22], using as far as possible the notations of that paper, for easiness of reference. Given a ball $B_{R}:=B\left(0,R\right)$ in $\mathbb{R}^{d}$, taken $\alpha>\epsilon>0$, consider the spaces

[TABLE]

We have

[TABLE]

with compact dense embeddings. Moreover, we have the interpolation inequality (see Theorem 6.4.5 in [2])

[TABLE]

for all $f\in X$, with

[TABLE]

These are preliminary assumptions of Corollary 9 of [22]. The Corollary tells us that the embedding of

[TABLE]

is relatively compact in $C\left(\left[0,T\right];B\right)$, if $s_{1}r_{1}>1$ and $r_{0}$ is so large that $s_{\theta}>\frac{1}{r_{\theta}}$ where (always following the notations of [22]) $s_{\theta}=\theta s_{1}$, $\frac{1}{r_{\theta}}=\frac{1-\theta}{r_{0}}+\frac{\theta}{r_{1}}$. Below we shall choose for instance $s_{1}=\frac{1}{3}$ (any number smaller than $\frac{1}{2}$) and $r_{1}=4$, so $s_{1}r_{1}>1$ is fulfilled. Then we need

[TABLE]

The logical sequence of our choices is: given $\beta\in\left(0,1\right)$ (think to $\beta$ close to 1, which is the most difficult choice), we shall choose $\alpha>0$ so small to satisfy a condition related to $\beta$ which appears in the proof of Lemma 3.1 below (when $\beta$ is close to 1, we have to choose $\alpha$ small). Given this small $\alpha$, we choose $\epsilon\in\left(0,\alpha\right)$ and then $\theta=\frac{\epsilon}{2+\alpha}$ is determined, typically very small. Now, we choose $r_{0}$ so large that $\frac{\theta}{3}>\frac{1-\theta}{r_{0}}+\frac{\theta}{4}$. Summarising we choose $(\alpha,s_{1},r_{1},r_{0},\epsilon)$, in the following way:

[TABLE]

The final step consists in taking $\mathbb{R}^{d}$ instead of $B_{R}$. We denote by $W_{loc}^{\alpha,2}\left(\mathbb{R}^{d}\right)$ the space of functions $f\in\cap_{R>0}W^{\alpha,2}\left(B_{R}\right)$ and we endow this space with the metric

[TABLE]

Under the same conditions on the indexes, we have now that

[TABLE]

is compactly embedded into $C\left(\left[0,T\right];W_{loc}^{\alpha-\epsilon,2}\left(\mathbb{R}^{d}\right)\right)$.

3.2 Main estimate on the empirical density $u^{N}$

Before looking into details for the derivation of main estimates for the empirical density, we state the mild formulation for $u_{t}^{N}$, see Lemma 2.6 for the identity for $u_{t}^{N}$:

[TABLE]

Lemma 3.1**.**

Given $\beta\in\left(0,1\right)$, there exists $\alpha>0$ small enough such that the following holds: for every $p>1$ there is a constant $C_{p}>0$ such that

[TABLE]

independently of $N$.

Proof.

Step 1 (preliminary estimates). We shall use the equivalence between norms (10):

[TABLE]

Then, up to a constant, denoting with $f_{s}^{N}(x)=\mathrm{div}(W_{N}\ast\left(b(u_{s}^{N},m_{s}^{N})S_{s}^{N}\right))(x)$

[TABLE]

On the first term, using (11) we prove the following estimate

[TABLE]

The last expected value is bounded by the assumption that $u_{0}$ is $C^{1}$ compact support: in this case one can show convergence of the empirical means of the i.i.d. r.v.’s $X_{0}^{i}$ , that imply a uniform in $N$ bound on $\mathbb{E}\left[\left\|u_{0}^{N}\right\|_{W^{\alpha,2}(\mathbb{R}^{d})}^{p}\right]$, for every $p$ (see [9] for similar results).

On the third term, we use the following fact. For every $p>1$ there is a constant $C_{p}>0$ such that, if $\Phi_{t}^{1},...,\Phi_{t}^{N}$ are adapted square integrable processes with values in a Hilbert space $H$,

[TABLE]

Therefore

[TABLE]

Moreover, the gradient commutes with the heat semigroup and the fractional powers of the Laplacian. Hence, using (11) and (12), the integrand can be estimated as follows

[TABLE]

From Lemma 2.7 we get

[TABLE]

and thus we can estimate the martingale term in the following way:

[TABLE]

Choosing $\alpha$ so small that $(\alpha+\epsilon)^{\ast}\leq 1$, i.e. $\beta\leq\frac{d}{2(\alpha+\epsilon)+d}<1$, we get a uniform bound on the martingale term.

Finally, thanks to the boundness on $b$ we get the following estimate

[TABLE]

hence

[TABLE]

Step 2 (estimate in $L^{2}(\mathbb{R}^{d})$). Consider the case $\alpha=0$ in the previous computations. We have proved, with the notation $H=L^{2}(\mathbb{R}^{d})$, that

[TABLE]

Thus

[TABLE]

We have

[TABLE]

using properties on the analytical semigroup (11), (12) and the last bound of Step 1. Therefore

[TABLE]

A generalised form of Gronwall lemma implies

[TABLE]

where the constant $C$ depends on $p$ but not on $N$.

Step 3 (estimate in $W^{\alpha,2}(\mathbb{R}^{d})$). Similarly to the beginning of Step 2, we have

[TABLE]

where now $\widetilde{H}=W^{\alpha,2}(\mathbb{R}^{d})$; and recalling some properties of the analytical semigroup, see (10),(11), (12) similarly we get,

[TABLE]

But from Step 2 we know that $\left\|u_{s}^{N}\right\|_{L^{p}\left(\Omega;H\right)}$ is uniformly bounded, hence for $\alpha<1$ we deduce the claim of the Lemma. ∎

3.3 Tightness of $\left(u_{t}^{N},m_{t}^{N}\right)$

Recall from Section 3.1 that the space there denoted by $\mathcal{W}$ is compactly embedded into $C\left(\left[0,T\right];W_{loc}^{\alpha-\epsilon,2}\left(\mathbb{R}^{d}\right)\right)$, when $s_{1}=\frac{1}{3}$ and $r_{1}=4$ and when, having chosen $\alpha$ small enough related to the original choice of $\beta$ in order that the result of Lemma 3.1 is true, we take $r_{0}$ large enough.

In order to prove tightness of the family of laws of $u_{t}^{N}$ in $\mathcal{W}$ we have to prove that $u_{t}^{N}$ is bounded in probability in $L^{r_{0}}\left(0,T;W^{\alpha,2}\left(\mathbb{R}^{d}\right)\right)$ and in $W^{s_{1},r_{1}}\left(0,T;W^{-2,2}\left(\mathbb{R}^{d}\right)\right)$. For the first claim it is sufficient to prove that

[TABLE]

and this is true by Lemma 3.1, because

[TABLE]

The second claim is proved in the next Proposition.

Proposition 3.2**.**

The family $\{u_{t}^{N}\}_{N}$ is bounded in probability in $W^{s_{1},r_{1}}\left(0,T;W^{-2,2}\left(\mathbb{R}^{d}\right)\right)$.

Proof.

Let us recall that a norm on $W^{s_{1},r_{1}}\left(0,T;W^{-2,2}\left(\mathbb{R}^{d}\right)\right)$ is given by the sum

[TABLE]

The property

[TABLE]

is a consequence of Lemma 3.1, because $\left\|u_{t}^{N}\right\|_{W^{-2,2}}$ is a weaker norm than $\left\|u_{t}^{N}\right\|_{W^{\alpha,2}}$. We have to prove

[TABLE]

Thus, for $t>s$, we have to estimate

[TABLE]

From the equation satisfied by $u_{t}^{N}$, proved in lemma 2.6 and Hölder inequality, we have

[TABLE]

Being $A$ a bounded operator from $L^{2}$ to $W^{-2,2}$, we have

[TABLE]

thanks to the estimate of Lemma 3.1. Notice that the spaces $L^{2}$ and $W^{1,2}$ are continuously embedded in $W^{-2,2}$, namely there exists a constant $C>0$ such that $||f||_{W^{-2,2}}\leq C||f||_{L^{2}}$ and $||f||_{W^{-2,2}}$ $\leq C||f||_{W^{-1,2}}$. We shall use this in the next computations. We have

[TABLE]

where the last inequality is similar to one proved in Lemma 3.1. Hence

[TABLE]

as above. Therefore, until now, we have proved

[TABLE]

Estimating the martingale as in Lemma 3.1, we have

[TABLE]

by Lemma 2.7, hence (being $\beta<1$)

[TABLE]

Summarising,

[TABLE]

It follows that

[TABLE]

which is finite if $s_{1}r_{1}-\frac{r_{1}}{2}<0$; with our choice $s_{1}=\frac{1}{3}$ and $r_{1}=4$, this is true. ∎

Corollary 3.3**.**

The family $\{u_{t}^{N}\}_{N}$ is bounded in probability in $\mathcal{W}$, and therefore the family of laws of $\{u_{t}^{N}\}_{N}$ is tight in $C\left(\left[0,T\right];W_{loc}^{\alpha-\epsilon,2}\left(\mathbb{R}^{d}\right)\right)$. In particular, it is tight in $C\left(L_{loc}^{2}\right)$. If $Q_{u}$ is any limit measure of this family and $u$ is a r.v. with law $Q_{u}$, we also have the property

[TABLE]

namely $Q_{u}$ is supported on $L^{p}\left(0,T;L^{2}(\mathbb{R}^{d})\right)$ for some $p>2$. Therefore, if we prove that $Q_{u}$ is supported on mild solutions, by Lemma 2.5 we deduce that $Q_{u}$ is supported on $C\left(L^{2}\right)$.

Proposition 3.4**.**

The family of laws of $\{m_{t}^{N}\}_{N}$ is tight in $C\left(L_{loc}^{2}\right)$. If $Q_{m}$ is any limit measure of this family and $m$ is a r.v. with law $Q_{m}$, we also have the property

[TABLE]

namely $Q_{m}$ is supported on $L^{p}\left(0,T;L^{2}(\mathbb{R}^{d})\right)$ for some $p>2$. Therefore, if we prove that $Q_{m}$ is supported on mild solutions, by Lemma 2.5 we deduce that $Q_{m}$ is supported on $C\left(L^{2}\right)$.

Proof.

Call $C_{+}\left(L_{loc}^{2}\right)$ the space of nonnegative functions of class $C\left(L_{loc}^{2}\right)$. Recall the explicit form of the solution of equation (2) given in Proposition 2.1. We want to apply Lemma 2.8 with $X_{1}=C_{+}\left(L_{loc}^{2}\right)$, $X_{2}=C_{+}\left(L_{loc}^{2}\right)$, $\mathcal{G}_{1}$ given by the family of laws of $\{u_{t}^{N}\}_{N}$, $\mathcal{G}_{2}$ given by the family of laws of $\{m_{t}^{N}\}_{N}$, and $\varphi$ given by (for $f\in C_{+}\left(L_{loc}^{2}\right)$, it is here that we use non negativity)

[TABLE]

where $F_{\zeta}\left(a,b\right)$ has been introduced in Proposition 2.1. Tightness of the family $\mathcal{G}_{1}$ is given by Proposition 3.2. To prove continuity of $\varphi$, we just notice that the map

[TABLE]

is continuous from $C\left(L_{loc}^{2}\right)$ to $C\left(L_{loc}^{2}\right)$ and then we have to compose with a bounded continuous map. ∎

4 Passage to the limit

Denote by $Q^{N}$ the law of $(u^{N},m^{N})$, on the space $C\left(L_{loc}^{2}\right)\times C\left(L_{loc}^{2}\right)$. We have proved above that the family $\left\{Q^{N}\right\}$ is tight. Hence, by Prohorov theorem, there is a subsequence $Q^{N_{k}}$ which converges weakly to some probability measure $Q$ on $C\left(L_{loc}^{2}\right)\times C\left(L_{loc}^{2}\right)$. Moreover, from Corollary 3.3, the marginal $Q_{u}$ on the first component is supported on the space $L^{p}\left(0,T;L^{2}(\mathbb{R}^{d})\right)$ for some $p>2$. We want to prove first that $Q$ is supported on the class of mild solutions of system (3). Second, we shall prove that this class has a unique element $\left(u,m\right)$; it will follow that the full sequence $\left\{Q^{N}\right\}$ converges to $\delta_{\left(u,m\right)}$ in the weak sense of measures; and that $(u^{N},m^{N})$ converges in probability to $\left(u,m\right)$, because the limit is deterministic. This will complete the proof of Theorem 2.3; verification of the properties $u_{t}\geq 0$ and $0\leq m_{t}\leq M$ for every $t\in\left[0,T\right]$ are done with the same technique used in the proof of the next proposition, through suitable continuous functionals; we omit the details. The regularity $C\left(L^{2}\right)\times C\left(L^{2}\right)$ of $\left(u,m\right)$ comes from Corollary 3.3 and Proposition 3.4. In the following proof, we will prove that $Q$ is supported on the class of mild solution of system (3). The proof of the following result is quite classical. It has been widely used in the mean field theory, see [23]. Our case is very close to the mean field framework , but it can not be considered a particular case of the known mean field theories, in particular because of the presence of $u^{N}$ and the dependence of the function $g$ on a density of particles. To prove this step, we adopt the approach of [14], see Chapter 4, although presumably it can be given along several classical lines, see [23]. Before going in to some details of the proof, we introduce a family of functionals which characterizes the solution of the system:

[TABLE]

where $\varphi\in C_{c}^{\infty}(\mathbb{R}^{d})$. On these family we prove a preliminary result, to Proposition 4.2.

Lemma 4.1**.**

Let $Q^{N_{k}}$ be the subsequence of measure of $Q^{N}$ that convergesp to $Q$ on $C\left(L_{loc}^{2}\right)\times C\left(L_{loc}^{2}\right)$, then

[TABLE]

Proof.

One has that

[TABLE]

The second term of the functional is clearly zero, because of the equation satisfied by $m_{t}^{N_{k}}$. Using the identity satisfied by $u_{t}^{N_{k}}$, we get

[TABLE]

Hence it is sufficient to prove that, for given $\delta>0$, both the following probabilities

[TABLE]

and

[TABLE]

converge to zero as $k\rightarrow\infty$. The first probability is bounded above as follows:

[TABLE]

We have

[TABLE]

having used property (8),

[TABLE]

having used the form $W_{N}(x):=N^{\beta}W(N^{\beta/d}x)$ and the property of compact support of $W$,

[TABLE]

Hence the last probability above is

[TABLE]

which goes to zero (recall the bound of Lemma 3.1).

Finally,

[TABLE]

as in the proof of Lemma 3.1 (with $\alpha=0$, $p=2$, without semigroup, using Doob’s inequality), hence it goes to zero. ∎

Proposition 4.2**.**

Let $Q$ be the limit probability measure of some subsequence $Q^{N_{k}}$. Then $Q$ is supported on the set of mild solutions of system 3.

Proof.

Firstly we observe that the functional is continuous with respect to the topology of $C\left(L_{loc}^{2}\right)\times C\left(L_{loc}^{2}\right)$. It holds because $\varphi$ is compact support with its derivatives (this is sufficient to treat the terms $\left\langle u_{t},\varphi\right\rangle$ and $\int_{0}^{t}\left\langle u_{s},\Delta\varphi\right\rangle ds$), by property (7) (this fact plus the previous ones is used to treat the term $\int_{0}^{t}\left\langle u_{s},b(u_{s},m_{s})\cdot\nabla\varphi\right\rangle ds$) and $F_{\zeta}$ is continuous, $\widetilde{F}_{\zeta}$ is bounded (these facts are used to deal with the $m$-term). Moreover $b(u,m)$ converges locally uniformly in space when $\left(u,m\right)$ converges in $C\left(L_{loc}^{2}\right)\times C\left(L_{loc}^{2}\right)$. This last point is a delicate one, so in the next lines we will prove it. Let us consider a sequence $(u^{N},m^{N})$ converging in $L^{2}_{loc}$, fixed $\epsilon>0$ and $x\in B(0,K)$, with $K>0$

[TABLE]

where thanks to hypothesis (5), $R$ can be chosen such that $I_{2}(R,N)\leq\epsilon/2$, uniformly in $x$. Regarding $I_{1}$, there exists $N_{0}$ such that

[TABLE]

for all $N>N_{0}$. Computation including time component are straightforward. So one get that $b(u^{N},m^{N})$ converges locally uniformly in space to $b(u,m)$.

Thanks to continuity of the functional $\Psi_{\varphi}$, by Portmanteau theorem,

[TABLE]

Then for Lemma 4.1

[TABLE]

for every $\delta>0$. By a classical argument, see [14]

[TABLE]

Thus $Q$ is supported on weak solutions. In addition, by Corollary 3.3, $u$ is also of class $L^{p}\left(0,T;L^{2}(\mathbb{R}^{d})\right)$ for some $p>2$. With proper choice of $\varphi$ related to the heat kernel $\frac{1}{(4\pi t)^{d/2}}e^{-\frac{|x-y|^{2}}{4t}}$, one proves that $u$ satisfies the mild formulation; and it is straightforward to see that $m$ satisfies the differential equation. Hence we have proved that $Q$ is supported by the set of mild solutions. ∎

Proposition 4.3**.**

Assume that $u^{1},m^{1},u^{2},m^{2}$ are functions of class $C(L^{2})$, such that $(u^{1},m^{1})$ and $(u^{2},m^{2})$ are mild solutions of the system 3 corresponding to the same initial condition $(u_{0},m_{0})$, with $u_{t},m_{t}\geq 0$ for every $t\in\left[0,T\right]$. Then $(u^{1},m^{1})=(u^{2},m^{2})$.

Proof.

Each $u^{i}$, $i=1,2$, satisfies the identity

[TABLE]

where we have used the explicit formula for equation (2). This is a closed equation and we are going to prove from it that $u^{1}=u^{2}$. A fortiori we get also $m^{1}=m^{2}$, again from the explicit formula for equation (2).

Assume by contradiction that $u^{1}\neq u^{2}$. Let $t_{0}\in[0,T)$ the infimum of all $t\in\left[0,T\right]$ such that $u_{t}^{1}\neq u_{t}^{2}$. On $\left[0,t_{0}\right]$ we have $(u^{1},m^{1})=(u^{2},m^{2})$. On $\left[t_{0},T\right]$ we use the mild formula and property (3.4) to get

[TABLE]

Recall that $b$ is bounded, see (6), and that $||u_{s}^{2}||_{L^{2}}$ is bounded by assumption. Hence

[TABLE]

From property (7) and Hölder inequality we have

[TABLE]

hence

[TABLE]

Recalling the explicit formula for equation (2), we have

[TABLE]

whence

[TABLE]

Summarising, the function $v_{t}:=\left\|u_{t}^{1}-u_{t}^{2}\right\|_{L^{2}}$ satisfies

[TABLE]

Given $t_{1}\in\left[t_{0},T\right]$ we set $A\left(t_{1}\right)$ $:=\sup_{t\in\left[t_{0},t_{1}\right]}v_{t}$. Then on the interval $t\in\left[t_{0},t_{1}\right]$ we have

[TABLE]

It follows that

[TABLE]

If $t_{1}-t_{0}>0$ is small enough, we deduce $A\left(t_{1}\right)=0$, hence $u^{1}=u^{2}$ on $\left[t_{0},t_{1}\right]$, in contradiction with the definition of $t_{0}$. ∎

5 Simulations

The aim of this section is to show the flexibility of the model, namely how it may catch different kinds of aggregations. For instance, we may avoid arbitrary concentration (even with infinitesimal noise), opposite to most of the models in the literature; but we cover also the case of concentration, both in the case of single and multiple concentration points. Each numerical simulation shown below is given by the following choice of parameters: number of particles $N=100$, parameter of diffusion $\sigma^{2}=0.1$, discretization of time $dt=10^{-4}$, Kernel smoothing parameter $\beta=0.9$ and on the initial condition we made a simple choice, choosing just a realization of uniform distribution on the square $[0,2]\times[0,2]$.

5.1 Degenerate aggregation

Let us start from the most basic example, the case when each particle is attracted by the others. Recall we model interaction between individuals and density of population; hence each individual is pushed to high population density regions. A standard choice for $g$ could be the following one:

[TABLE]

Notice that with this choice, cells continue to aggregate even at high density. The population mass tends to concentrate into a single point (see figure $1$)

5.2 Moderate aggregation

Let us now include also a repulsive component in the force, to avoid collapse of the total mass. (see figure $3$). The function $g(r,u)$ we look for should have the following features (see figure $2$):

• •

given the distance $r$, $g(r,u)$ is such that the force is aggregative for small density and repulsive for huge density. This behavior is natural in certain cases for animals: each individual is attracted by its similar, but it does not where there are too many;

• •

but there is an issue when we quantify small and huge density: this quantification should depend on distance. At big distances, we expect that aggregation is more relevant, and the individual tends to avoid only really huge densities. On the contrary, at short distances, each individual is attracted only by very small densities.

The function we propose is the following one:

[TABLE]

Another example could be

[TABLE]

where the parameter $\alpha$ can be interpreted as an index of overcrowding; choosing properly $\alpha$, particles aggregate, without collapsing. The main drawback we have observed in simulations about this alternative is its strong sensibility to the choice of the parameter $\alpha$ with respect to the initial configuration. The first option we propose is more stable.

Notice that the functions $g(r,u)$ of this subsection are not product of functions of the two single variables, namely $g(r,u)=g_{1}\left(r\right)g_{2}\left(u\right)$.

5.3 Aggregation in clusters

Going back to the first model, $g(r,u)=e^{-r}\cdot\frac{u}{1+u}$, an interesting variant is when attraction happened only up to a certain distance (see figure $4$):

[TABLE]

With this choice we observe the formation of clusters of individuals. Clearly, the parameter $R$ influence on the number of clusters that are generated: for big $R$, population aggregate in a reduced number of clusters.

When $t$ goes to infinity, if the noise is infinitesimal, each cluster reduces to a point, and maybe due to noise different clusters may meet and collapse.

5.4 Moderate aggregation in clusters

We may mix-up the previous two features. The following example has a tendency to construct clusters (see figure $5$), but they remain of finite size (independently of the noise):

[TABLE]