Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation

Abstract

The study of self-organised collective animal behaviour, such as swarms of insects or schools of fish, has become over the last decade a very active research area in mathematical biology. Parabolic and hyperbolic models have been used intensively to describe the formation and movement of various aggregative behaviours. While both types of models can exhibit aggregation-type patterns, studies on hyperbolic models suggest that these models can display a larger variety of spatial and spatio-temporal patterns compared to their parabolic counterparts. Here we use stability, symmetry and bifurcation theory to investigate this observation more rigorously, an approach not attempted before to compare and contrast aggregation patterns in models for collective animal behaviors. To this end, we consider a class of nonlocal hyperbolic models for self-organised aggregations that incorporate various inter-individual communication mechanisms, and take the formal parabolic limit to transform them into nonlocal parabolic models. We then discuss the symmetry of these nonlocal hyperbolic and parabolic models, and the types of bifurcations present or lost when taking the parabolic limit. We show that the parabolic limit leads to a homogenisation of the inter-individual communication, and to a loss of bifurcation dynamics (in particular loss of Hopf bifurcations). This explains the less rich patterns exhibited by the nonlocal parabolic models. However, for multiple interacting populations, by breaking the population interchange symmetry of the model, one can preserve the Hopf bifurcations that lead to the formation of complex spatio-temporal patterns that describe moving aggregations.

Appendices

Appendix A General results: $\mathbf{O(2)}$ scalar reaction/advection-diffusion equations

A general scalar reaction/advection-diffusion equation with nonlocal terms can be written as

$$\begin{aligned} u_t&= u_{xx} + h\big (u,K_1^{+}*u,K_1^{-}*u,\cdots ,K_m^{+}*u,K_m^{-}*u,u_x,J_1^{+}*u_x,\nonumber \\&J_1^{-}*u_x,\cdots ,J_n^{+}*u_x,J_n^{-}*u_x\big ), \end{aligned}$$

(39)

where $h$ is a $C^r$ function ($r\ge 1$) of its arguments. The convolutions $K_{j}^{\pm }*u$ and $J_{i}^{\pm }*u_{x}$, $j=1,\ldots ,m$, $i=1,\ldots ,n$, are given by

$$\begin{aligned} K_{j}^{\pm }*u=\int _{-\infty }^{\infty } K_j(s)u(x\pm s,t)\,ds \quad \text{ and }\quad J_{i}^{\pm }*u_{x}=\int _{-\infty }^{\infty } J_i(s)u_{x}(x\pm s,t)\,ds. \end{aligned}$$

Equation (39) is invariant for the $\mathbf{SO(2)}$-symmetry $\theta .u(x,t)=u(x-\theta ,t)$ (because of the translation invariance of the integrals in the nonlocal terms). The symmetry $\kappa .u(x,t)=u(L_0-x,t)$ acts on the arguments of $h$ as follows:

$$\begin{aligned}&u(x,t)\mapsto u(L_0-x,t),\quad u_x(x,t)\mapsto -u_x(L_0-x,t),\\&K_{j}^{+}*u(x,t) \mapsto K_{j}^{-}*u(L_0-x,t), \quad J_{i}^{+}*u_{x}(x,t) \mapsto -J_{i}^{-}*u_{x}(L_0-x,t). \end{aligned}$$

To simplify the calculations, let us introduce some notation. Let $p_0=u$, $s_0=u_x$, and for $j=1,\ldots ,m$ and $i=1,\ldots ,n$, set

$$\begin{aligned} (p_j,r_j) = (K_j^{+}*u,K_j^{-}*u) \quad \text{ and }\quad (s_i,t_i) = (J_i^{+}*u_{x},J_i^{-}*u_{x}), \end{aligned}$$

for which we have $\kappa (p_j,r_j)(x) = (r_j,p_j)(L_0-x)$ and $\kappa (s_i,t_i)(x) = -(t_i,s_i)(L_0-x)$. This leads to the following $\kappa$ invariant monomials. Define, for $j=1,\ldots ,m$ and $i=1,\ldots ,n$

$$\begin{aligned} M_j = p_j + r_j, \quad N_j = p_j - r_j, \quad F_i = s_i + t_i, \quad G_i = s_i-t_i, \end{aligned}$$

and

$$\begin{aligned} \chi _1&= (M_1,p_0 N_1,\ldots ,M_m,p_0 N_m),\quad \chi _2 = (G_1,p_0 F_1,\ldots ,G_n,p_0 F_n),\\ \chi _3&= (N_1^2,\ldots ,N_j N_k,\ldots ,N_m^2),\quad \chi _4 = (F_1^2,\ldots ,F_j F_k,\ldots , F_n^2),\\ \chi _5&= \sum _{i=1}^{m} N_i(F_1,\ldots ,F_n). \end{aligned}$$

From these definitions, it is straightforward to verify the characterization of $\mathbf{O(2)}$ symmetry described here.

Proposition 5.1

Equation (39) is $\mathbf{O(2)}$-equivariant if

$$\begin{aligned} h(p_0,p_1,r_1,\ldots ,p_m,r_m,s_0,s_1,t_1,\ldots ,s_n,t_n) = g(p_0,s_0^2,\chi _1,\chi _2,\chi _3,\chi _4,\chi _5), \end{aligned}$$

(40)

where $g$ is a $C^r$ function.

Expression (40) is used below to obtain the eigenvalues of the linearised operator at a $\mathbf{O(2)}$-symmetric homogeneous equilibrium. We can now prove the $\mathbf{O(2)}$-equivariance of equation (8).

Proof of Proposition 2.1

In this case, we have

$$\begin{aligned} (p_0,p_1,r_1,s_0,s_1,t_1)=(u,K^{+}*u,K^{-}*u,u_x,K^{+}*u_x,K^{-}*u_x). \end{aligned}$$

The $\mathbf{O(2)}$ invariant polynomials in (40) can be obtained by taking the Taylor series of $f$. We present also a direct proof. For models M2 and M4, the function $h$ in (8) has the form

$$\begin{aligned} s_0 (f(-(p_1-r_1)) - f(p_1-r_1)) + s_0 (-f'(-(p_1-r_1))-f'(p_1-r_1))(s_1-t_1). \end{aligned}$$

The first term is such that $(s_0,p_1,r_1)\mapsto (-s_0,r_1,p_1)$ becomes $- s_0 (f(-(r_1-p_1))-f(r_1-p_1))= s_0 (f(-(p_1-r_1))-f(p_1-r_1) )$. The second term is also invariant because

$$\begin{aligned} (p_0,p_1,r_1,s_0,s_1,t_1)\mapsto (p_0,r_1,p_1,-s_0,-t_1,-s_1) \end{aligned}$$

implies

$$\begin{aligned}&p_0 (-f'(-(p_1-r_1))-f'(p_1-r_1))(s_1-t_1) \mapsto p_0 (-f'(p_1-r_1)\\&\quad -f'(-(p_1-r_1)))(s_1-t_1). \end{aligned}$$

This confirms the $\mathbf{O(2)}$-equivariance of (8). For models M3 and M5, we have

$$\begin{aligned} h = s_0 (f(r_1)-f(p_1)) + p_0 (f(t_1)-f(s_1)), \end{aligned}$$

and interchanging $(s_0,p_1,r_1)\mapsto (-s_0,r_1,p_1)$ leaves the first term invariant. Then, $(p_0,s_1,t_1)\mapsto (p_0,-t_1,-s_1)$ yields

$$\begin{aligned} f(-s_1)-f(-t_1)&= \displaystyle \frac{1}{2}(\tanh (-s_1)-\tanh (-t_1)) \\&= \displaystyle \frac{1}{2}(\tanh (t_1)-\tanh (s_1)) = f(t_1)-f(s_1) \end{aligned}$$

and the $\mathbf{O(2)}$-equivariance holds. $\square$

We now describe the spectrum of the linearization of equation (39). The function $u=u^{*}$, where $u^{*}$ is a constant, is a homogeneous $\mathbf{O(2)}$-symmetric equilibrium for (39) if $h$ vanishes at the point

$$\begin{aligned} U^{*} = (p_0^{*},p_1^{*},r_1^{*},\ldots ,p_m^{*},r_m^{*},s_0,s_1^{*},t_1^{*},\ldots ,s_n^{*},t_n^{*}), \end{aligned}$$

where $p_0^{*}=u^{*}$, $p_j^{*}=p_j|_{u=u^{*}}$,$r_j^{*}=r_j|_{u=u^{*}}$ for $j=1,\ldots ,m,$ and $s_0^{*}=s_1^{*}=t_{1}^{*}=\cdots =s_n^{*}=t_n^{*}=0$. The linearization of the operator that appears in the right-hand-side of (39), calculated at $u=u^{*}$, has the form

$$\begin{aligned} \xi _{xx} + \nabla h(U^{*})V(\xi ), \end{aligned}$$

where

$$\begin{aligned} V(\xi )&= (\xi ,K_1^{+}*\xi ,K_1^{-}*\xi ,\cdots ,K_m^{+}*\xi ,K_m^{-}*\xi ,\xi _x,J_1^{+}*\xi _x,\\&\quad J_1^{-}*\xi _x,\cdots ,J_n^{+}*\xi _x,J_n^{-}*\xi _x)^{T}. \end{aligned}$$

Theorem 5.2

Let $u=u^{*}$ be a $\mathbf{O(2)}$-symmetric homogeneous equilibrium of (39). Then, the spectrum of the linearised operator $\mathcal{L}$ can only have real eigenvalues. In particular, there cannot be $\mathbf{O(2)}$-symmetric Hopf bifurcations.

Proof

Consider the subspaces

$$\begin{aligned} X_{\ell } = \{ a e^{i k_\ell x} + \text {c.c} \mid a\in {\mathbb {C}}, k_\ell = 2\pi \ell /L_{0}\}. \end{aligned}$$

Then, $\mathcal{L}_{\ell } = \mathcal{L}|_{X_{\ell }}: X_{\ell } \rightarrow X_{\ell }$ has the form

$$\begin{aligned} \mathcal{L}_{\ell }(a e^{i k_{\ell } x})= -k_\ell ^2 a e^{ik_\ell x} +\nabla h(U^{*}) V(a e^{ik_\ell x}) \end{aligned}$$

with

$$\begin{aligned} V(ae^{i k_\ell x})&= (1,K_1^{+}(k_\ell ),K_1^{-}(k_\ell ),\ldots ,K_{m}^{+}(k_\ell ),i k_\ell ,ik_\ell J_1^{+}(k_\ell ),ik_\ell J_1^{-}(k_\ell ),\ldots ,\\&\quad ik_\ell J_n^{+}(k_\ell ),ik_\ell J_n^{-}(k_\ell ))^{T})a e^{ik_\ell x} \end{aligned}$$

and

$$\begin{aligned} K_{j}^{\pm }(k_\ell ) = \int _{0}^{\infty } K_{j}(s) e^{\pm i k_\ell s}\,ds, \quad J_{i}^{\pm }(k_\ell ) = \int _{0}^{\infty } J_{i}(s) e^{\pm i k_\ell s}\,ds. \end{aligned}$$

(41)

Note that $\overline{K_i^{-}(k_\ell )}=K_i^{+}(k_\ell )$ and $\overline{J_i^{-}(k_\ell )}=J_i^{+}(k_\ell )$. We derive a more explicit form of $\nabla h(U^{*})$ using the expression in Proposition A. Evaluating the partial derivatives of $h$ and $g$ at $U^{*}$, we obtain for $i=1,\ldots ,m$ and $j=1,\ldots ,n$

$$\begin{aligned} h_{p_0} = g_{p_0}, \quad h_{p_i} = g_{M_i},\quad h_{r_i} = g_{M_i},\\ h_{s_0} = 0,\quad h_{s_j} = g_{G_j}, \quad h_{t_j} = -g_{G_j}. \end{aligned}$$

Therefore,

$$\begin{aligned}&\mathcal{L}_{\ell }(a e^{i k_\ell x}) \\&\quad = (-k_\ell ^2 a e^{i k_{\ell }x}+ (g_{p_0},g_{M_1},g_{M_1},\ldots ,g_{M_m},g_{M_m},0,g_{G_1},-g_{G_1},\ldots ,g_{G_n},-g_{G_n})V(ae^{i k_{\ell }x})\\&\quad = \left( -k_\ell ^2 + g_{p_0} + \displaystyle \sum _{i=1}^{m} g_{M_i} (K_i^{+}(k_\ell )+K_i^{-}(k_n)) + i k_\ell \displaystyle \sum _{j=1}^{n} g_{G_j} (J_j^{+}(k_\ell )-J_j^{-}(k_\ell ))\right) \,a e^{i k_\ell x}. \end{aligned}$$

Using (41), we see from here that the coefficient of $a e^{i k_\ell x}$ must be real. Hence all eigenvalues of $\mathcal{L}$ are real. $\square$

Appendix B Basic notions of equivariant bifurcation theory

For the convenience of readers, we have collected basic definitions of bifurcation theory of dynamical systems with symmetry (a.k.a equivariant bifurcation theory).

The set of translations on the real line given by operators $T_{y}$ (with $y\in \mathbb {R}$), defined by $T_{y}(x)=x+y$, form a group. If one considers a periodic lattice on $\mathbb {R}$, say of length $L_0$, translations satisfy $T_{y+L_{0}} = T_{y}$. Therefore, we can only consider the translations $T_{\theta }$ for $\theta \in [0,L_0]$, which in fact correspond to the group of translations on the circle. This group is denoted by $\text{ S }^1$, and it has the same algebraic structure as the group $\mathbf{SO(2)}$ of $2\times 2$ orthogonal matrices with determinant $1$; this latter terminology is the one used predominantly. The group consisting of $\mathbf{SO(2)}$ along with a reflection symmetry $\kappa$ satisfying $T_{\theta }\circ \kappa = \kappa \circ T_{-\theta }$, has the same algebraic structure as the group $\mathbf{O(2)}$ of $2\times 2$ orthogonal matrices. (Note that the symmetry $\kappa$ depends on the context, as seen in the various models discussed in this paper.) To keep with the tradition, we use the $\mathbf{O(2)}$ notation to describe all those groups with identical algebraic structure.

Let $\Gamma$ be a group and $C$ be a function space (with finite dimensional spaces as a special case). Then, $\Gamma$ acts on elements $u\in C$, written $\gamma .u$ for $\gamma \in \Gamma$, if for any $\gamma _1,\gamma _2\in \Gamma$, then

$$\begin{aligned} \gamma _1.(\gamma _2.u) = (\gamma _1 \gamma _2).u. \end{aligned}$$

Consider now a partial differential equation for $u(x_1,\ldots ,x_d,t)\in \mathbb {R}^m$ given by

$$\begin{aligned} u_t = F(x_1,\ldots ,x_d,u_{x_1},\ldots ,u_{x_d},u_{x_1 x_1},\ldots ,u_{x_1,x_d},\ldots ) \end{aligned}$$

(42)

Let $\Gamma$ be a group and $u(x_1,\ldots ,x_d,t)$ be any solution of (42). Then, (42) is said to be $\Gamma$-symmetric or $\Gamma$ -equivariant if $\gamma .u$ is also a solution of (42), for all $\gamma \in \Gamma$. To each point $u\in C$ is attached its symmetry group $\Gamma _u :=\{\gamma \in \Gamma \mid \gamma .u=u\}$, also called the isotropy subgroup.

One benefit of using symmetry concepts is that it often enables simplifications in the computation of the spectrum of linear operators, via the following properties of group actions and equivariant differential equations.

A subspace $V\subset C$ is $\Gamma$ -invariant if $\gamma .v \in V$, for all $\gamma \in \Gamma$ and $v\in V$. The subspace $V$ is a $\Gamma$ -irreducible representation if it is $\Gamma$-invariant and does not contain any proper $\Gamma$-invariant subspace. If $\Gamma$ is a compact Lie group (such as $\mathbf{SO(2)}$ and $\mathbf{O(2)}$), then all $\Gamma$-irreducible representations are finite dimensional. Two $\Gamma$-irreducible representations $V_1,V_2$ are called $\Gamma$ -isomorphic if there exists a nonsingular linear map $A:V_1\rightarrow V_2$, which commutes with the actions of $\Gamma$ on $V_1$ and $V_2$ respectively. For compact Lie groups, $C$ has a (non-unique) direct sum decomposition into irreducible representations. The direct sum of isomorphic irreducible representations in the decomposition is called a $\Gamma$ -isotypic component and the decomposition of $C$ into a direct sum of its isotypic components is called the $\Gamma$ -isotypic decomposition.

A linear mapping $L:C\rightarrow C$ is $\Gamma$ -equivariant if $L(\gamma .u)=\gamma .L(u)$, for all $u\in C$ and $\gamma \in \Gamma$. Note that the linearization $L$ of (42) at $u^{*}$ with $\Gamma _{u^{*}} = \Gamma$ is always $\Gamma$-equivariant. An important result is that $L$ maps $\Gamma$-isotypic components $V_k$ to themselves; that is, $L:V_{k} \rightarrow V_k$ and so one can “block diagonalize” $L$ along its isotypic decomposition:

$$\begin{aligned} L = L_0 \oplus L_1 \oplus \cdots \oplus L_k \oplus \cdots \end{aligned}$$

where $L_k := L|V_k$. Therefore, the point spectrum of $L$ can be obtained as the union of the point spectra of $L_k$.