Lyapunov-Schmidt and Centre Manifold ReductionMethods for Nonlocal PDEs Modelling Animal

Aggregations

P-L. BuonoFaculty of Science

University of Ontario Institute of TechnologyOshawa, ONT L1H 7K4, Canada

and

R. EftimieDivision of MathematicsUniversity of Dundee

DD1 4HN, UK

September 11, 2015

Abstract

The goal of this paper is to establish the applicability of the Lyapunov-Schmidtreduction and the Centre Manifold Theorem for a class of hyperbolic partial differ-ential equation models with nonlocal interaction terms describing the aggregationdynamics of animals/cells in a one-dimensional domain with periodic boundaryconditions. We show the Fredholm property for the linear operator obtained at asteady-state and from this establish the validity of Lyapunov-Schmidt reduction forsteady-state bifurcations, Hopf bifurcations and mode interactions of steady-stateand Hopf. Next, we show that the hypotheses of the Centre Manifold Theorem ofVanderbauwhede and Iooss [67] hold for any type of local bifurcation near steady-state solutions with SO(2) and O(2) symmetry. To put our results in context,we review applications of hyperbolic partial differential equation models in physicsand in biology. Moreover, we also survey recent results on Fredholm properties andCentre Manifold reduction for hyperbolic partial differential equations and equationswith nonlocal terms.

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1 Introduction

Collective self-organised behaviour is a phenomenon observed in a variety of organisms.Familiar examples include schooling fish, flocking birds, swarming insects, aggregatingbacteria, etc. The way such aggregations are formed, maintained and the transitionsbetween different patterns is a fascinating subject which has been increasingly studiedin the last twenty five years. The elegance and beauty of motion in aggregations areremarked in early writings, going back to antiquity, in Pliny the Elder’s book The NaturalHistory [63], where collective movement in flocks of birds are described. Apart fromthe appeal this problem has for curiosity-driven research, understanding of collective self-organised motion also has applications to environmental and societal problems such as theformation and motion of swarms of locusts [65], which affect rural communities in severallocations worldwide, as well as applications to the small scale phenomenon of cell-cellinteractions in developmental biology or cancer research.

Mathematical modelling has been an important aspect of the study of collective mo-tion and aggregation using both particle-based and density-based models. Those modelshave been used to probe the possible biological mechanisms leading to the formation andpersistence of aggregations, and also as a means to investigate transient aggregations.Several modelling approaches assume local interactions with conspecifics [9]. However,in many cases it is preferable for the models to assume that animals/cells can interactwith conspecifics positioned further away [58, 57, 64, 66, 29]. For instance, in migratoryflocks of birds, radar-tracking observations have shown that individuals 200-300 metersapart can fly at the same speed and in the same direction [49]. Nonlocal interactions arealso seen in developmental biology where collective cell movement results from cell-celladhesion forces with an interaction range proportional to cell size [1].

In this study, we focus on density-based models and consider a class of 1D hyperbolicfirst-order partial differential equations with nonlocal terms:

∂u+

∂t+

∂g+[u+, u−]u+

∂x= f+[u+, u−], (1a)

∂u−

∂t+

∂g−[u+, u−]u−

∂x= f−[u+, u−]. (1b)

Such equations could be used, for instance, to model the dynamics of animal aggregationsin 1D (i.e., on domains much longer than wide) [18], and in this case u± describe thedensities of left-moving (−) and right-moving (+) animals, g± are the (possibly nonlocal)density-dependent speeds, and f± incorporate (possibly nonlocal) turning behaviour andpopulations dynamics. Moreover, such equations are known to exhibit a large variety ofspatio-temporal patterns, ranging from stationary aggregations that can be time-variantor time-invariant, to different types of moving aggregations; see the patterns in [17, 6, 7,16]. Many of these patterns have complex dynamical features, which are still not fullyunderstood in terms of invariant sets of phase space.

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One way to determine the phase space origin of many of the patterns exhibited bythese nonlocal hyperbolic models is to determine whether they emerge from a sequence ofbifurcations starting with a homogeneous steady-state solution, as the main parametersof the system are varied. However, in order to study the unfolding of bifurcations near asteady-state solution, it is necessary to find out if Lyapunov-Schmidt and Centre Manifoldreduction methods can be applied to this class of nonlocal hyperbolic equations.

Our main focus in this paper is to show the applicability of Lyapunov-Schmidt (LS)and Centre-Manifold (CM) reduction methods near steady-state bifurcation points ofequations such as (1). Reduction methods are the first step to investigate the bifurcationand formation of patterns in mathematical models. While these methods have beencommonly applied to ODEs and parabolic PDEs, their use to hyperbolic PDEs is stillscarce. Moreover, the few analytical results existent in the literature for these hyperbolicPDEs are mainly applied to models describing phenomena in physics [56, 24, 8, 11].

Previous studies of nonlocal hyperbolic models for animal aggregations investigatedlocal bifurcations near codimension-two Steady-state/Hopf [7] and Hopf/Hopf [6] bifurca-tion points with O(2)-symmetry using weakly-nonlinear analysis (WNA) techniques, alsoknown as the method of multiple scales. The formal equivalence of the reduced equationsnear bifurcation, obtained with WNA and either LS and CM reduction has not beenstudied in a systematic way for these nonlocal models. Nevertheless, comparison betweenthe results of WNA and CM reductions has been performed only for some specific casesof nonlocal hyperbolic models of type (1); see [7]. Note that such comparisons have beenperformed quite often for local fluid models [24, 8]. Moreover, the equivalence between LSand CM reductions has been investigated by Chossat and Golubitsky [11] in the contextof Hopf bifurcation with symmetry.

In order to establish the validity of LS and CM reductions for nonlocal hyperbolicmodels, we first need to understand the linear operators associated with these models.The investigation of properties of linear operators coming from local hyperbolic first-orderpartial differential equations has attracted the attention of several authors [36, 47, 45]. Wereview some of these contributions in Section 3.2. Moreover, we present some details ofthe results for Fredholm operator inspired by integro-differential equations [21] and fromfunctional differential equations (FDEs), e.g. differential equations on lattices, FDEs ofmixed-type [55, 38]. We also review some Centre Manifold reduction results obtained forhyperbolic first-order partial differential equations and for general PDE systems, as wellas mentioning recent results from FDE theory. Then, in the context of nonlocal models(1), we show that for the Lyapunov-Schmidt reduction the linear operator at a steady-state solution is a Fredholm operator of index zero. Moreover, for the Centre Manifoldreduction, we verify that the nonlinear hyperbolic system (1) satisfies the conditions ofan infinite-dimensional version of the Centre Manifold Theorem of Vanderbauwhede andIooss [67] (see also Haragus and Iooss [32]). Because the nonlocal hyperbolic models (1)for animal aggregations and movement are symmetric with respect to a group isomorphicto O(2), the reduction methods also respect the symmetry group so that the equations

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obtained in the reduced space have the required symmetry properties. In this paper,we do not perform explicit computations for particular cases of bifurcations arising inthe context of nonlocal aggregation models such as (1). However, our Centre ManifoldTheorem puts on a rigorous footing the use of Weakly-Nonlinear Analysis computationsdone in [6, 7].

The content of the paper is organized as follows. We start in Section 2 with a shortliterature review of applications of hyperbolic PDEs to physics and biology. In particular,in Section 2.3, we focus on a hyperbolic first-order partial differential equations modelwith nonlocal terms for animal aggregation. Then, in Section 3 we discuss some generalresults on the Lyapunov-Schmidt and Central Manifold reductions and state our mainresults. We return to our nonlocal hyperbolic model in Section 4, where we show themain properties of the linear operator of the animal aggregation model, prove the mainresults concerning the Fredholm property and the use of Lyapunov-Schmidt reduction,and the applicability of the Centre Manifold Theorem. We conclude with Section 5,where we discuss some interesting future research directions.

2 1D hyperbolic models

Before discussing the various reduction methods, we first review briefly some 1D hyper-bolic mathematical models derived to describe phenomena in physics and biology. Thisallows us to emphasise the importance of these models, and the lack of analytical studiesto investigate the patterns exhibited by them. Moreover, by presenting some physics mod-els, it allows us to review the existent analytical results developed for these models. Sincegeneralisations of 1D hyperbolic models to 2D are more realistic but also more complex,their analytic investigation is more difficult. For this reason, we ignore them in this study.

2.1 Hyperbolic models in physics: laser models

To understand the complex dynamics of distributed feedback multi-section semiconductorlasers, [56] have investigated the following hyperbolic system describing the forward andbackward propagating complex amplitudes of the light (u1, u2), coupled to an equationfor the carrier density (v):

∂u

∂t=

(−∂u1

∂x,∂u2

∂x

)+G(x, u(x, t), v(x, t)), (2a)

∂v

∂t= I(x, t) +H(x, u(x, t), v(x, t)) +

m∑k=1

bkχSk(x)( 1

xk − xk−1

∫Sk

v(y, t)dy − v(x, t)),

(2b)

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The model describes the longitudinal dynamics of edge emitting lasers [56], and thus the1D domain [0, L] = Um

k=1Sk, which is formed of m sub-sectional intervals Sk := (xk−1, xk),k = 1, ...,m. The nonlinear operators G : (0, L)×C2×R → C2 and H : (0, L)×C2×R →R2 are differentiable with respect to (u, v), and measurable and bounded with respect tox ∈ [0, L]. Because these two operators have very complex descriptions, we will not showthem here. However, for more examples of such nonlinear operators, see [62, 56]. Model(2) was completed with reflective boundary conditions for u = (u1, u2)

u1(0, t) = r0u2(0, t) + α(t), u2(L, t) = rLu1(L, t),

and initial conditions for both u and v:

u(x, 0) = u0(x), v(x, 0) = v0(x).

This class of models have been shown to exhibit very rich dynamics: from bifurcationsto self-pulsations, hysteresis, excitability, frequency synchronisation [56]. While thereare many studies that focus on the numerical description of these behaviours, only afew studies focus on their analytical investigation; see [62, 61, 56]. The complexity ofthese equations makes it difficult to show, for example, the existence and persistence ofsmooth invariant manifolds for general cases required for bifurcation results. Fortunately,a Centre Manifold Theorem and Fredholm properties for equations similar to (2) havebeen established in [44, 51, 56]. We will return to these results in Section 3.2. Formore details on modelling of multi-section lasers, see the short introduction with manyreferences given in Lichtner [51].

2.2 Hyperbolic models in biology: predator-prey models, chemo-taxis models, aggregation models, age-dependent models

The last 20 years have seen an increase in the use of hyperbolic models to describe vari-ous biological phenomena: from self-organised biological aggregations (i.e., aggregationsin the absence of a leader or external stimuli; [17]), to chemotactic aggregations (i.e., ag-gregations in the presence of a chemotactic signal produced by the members of the group;[35, 37, 23]), predator-prey dynamics [3, 2] or age-structured models [39, 42, 69].

One of the simplest models of type (1) with constant speed and constant turning rateswas introduced and discussed extensively in [37, 34, 36]. The general form of this modelfor particles/animals aggregations, which includes a turning behaviour (λ±) as well as abirth/death processes (h±(u+, u−)), is given by

∂u+

∂t+ γ

∂u+

∂x= f+[u+, u−] = −λ+u+ + λ−u− +

1

2h+(u+, u−), (3a)

∂u−

∂t− γ

∂u−

∂x= f−[u+, u−] = λ+u+ − λ−u− +

1

2h−(u+, u−). (3b)

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Since the change in particles/animals movement directions is not always constant, butusually depends on (local or nonlocal) interactions with other particles, Eftimie et al. [18]considered a model of type (1) with constant speed g±[u+, u−] = γ (constant), nonlocaldensity-dependent turning rates f+[u+, u−] = −λ+[u+, u−]u++λ−[u+, u−]u−, f− = −f+,and no birth/death dynamics (h± = 0). This model was introduced to describe theformation and movement of self-organised biological aggregations in response to nonlocalsocial interactions among group members. Because of the complex spatial and spatio-temporal dynamics exhibited by this model we will review it in more detail in Section2.3.

The chemotactic movement of animal/cell aggregations can also be described by (1),which is now coupled with an equation for the dynamics of the chemical c(x, t) [35]:

∂c(x, t)

∂t= p(c, u+, u−) +D

∂2c(x, t)

∂x2, (4)

where p(c, u+, u−) describes the production/degradation of this chemical, and D is itsdiffusion rate. The chemical can influence the speed of animals/cells (i.e., g+[u+, u−, c] in(1)), their turning behaviour and even the birth-death dynamics of the population (i.e.,f±[u+, u−, c] in (1)).

The 1D hyperbolic predator-prey models do not usually consider turning behaviour,i.e., λ± = 0 (however, the 2D kinetic models can incorporate changes in movement di-rection in response to prey/predator behaviour; see [22]). In this case, the functionsf±[u+, u−] incorporate only the predator-prey dynamics between the two populations.Usually, this dynamics is described by Lotka-Volterra-type terms [13], but other termssuch a Holling-type functional responses can also be used [3]. Moreover, the interactionsbetween the prey and predator populations can affect the speed of either prey or predator[13], as the animals speed up to avoid or to catch up with the other population. Notehere that not all 1D predator-prey models are of the type (1). For example, Barbera etal. [3] derived a hyperbolic model where the hyperbolic equations for the two populationsare coupled with transport equations for the dissipative fluxes.

A final type of hyperbolic model that we would like to mention briefly describes age-structured populations. The hyperbolic age-structured models (of the McKendrick-vonFoerster type) have the general form [42]

∂u(a, t)

∂t+

∂u(a, t)

∂a= −λ(a)u(a, t), (5)

with u(t, a) representing the density of the population of age a at time t, and λ(a) de-scribing the mortality rate. The description of the model is completed with conditions forthe initial population u(0, a) = Q(a), a ≥ 0, and conditions for the newborn population:

u(0, t) =

∫ β

α

u(x, t)m(x)dx, (6)

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with m the maternity function.While all these models can exhibit a large variety of spatial and spatio-temporal pat-

terns ranging from stationary and moving aggregations of animals/cells (e.g., stationarypulses, travelling pulses, breathers, ripples, zigzags; see [17]) to networks of cells [23], thor-ough investigations of these patterns are still not the common approach in mathematicalbiology. For a more in-depth review of pattern formation in hyperbolic models in biology,and the analytical and numerical techniques available to investigate them, see [15, 68].Existence of reduction methods (e.g., Centre Manifold reduction) for local bifurcationsof various types of equations described in this section has been established for parabolicequations [32] and for hyperbolic age-structured models [53].

Next, we focus on a particular class of nonlocal mathematical models for self-organisedbiological aggregations, for which there are a few preliminary studies on the local bifur-cation of patterns near codimension-1 and codimension-2 bifurcation points [6, 7].

2.3 Self-organised animal aggregation models

Here, we present in more detail a class of 1D nonlocal hyperbolic models derived todescribe the formation and movement of various animal, cell and bacterial aggregationsas a result of inter-individual communication [18, 17]. The evolution of densities of right-moving (u+) and left-moving (u−) individuals, which travel with constant velocity γ andchange their movement direction from right to left (with rate λ+) and from left to right(with rate λ−) [17] is given by:

∂tu+(x, t) + ∂x(γu

+(x, t)) = −λ+[u+, u−]u+(x, t) + λ−[u+, u−]u−(x, t), (7a)

∂tu−(x, t)− ∂x(γu

−(x, t)) = λ+[u+, u−]u+(x, t)− λ−[u+, u−]u−(x, t), (7b)

u±(x, 0) = u±0 (x). (7c)

The turning rates are defined as

λ±[u+, u−] = λ1 + λ2f(y±r [u

+, u−]− y±a [u+, u−] + y±al[u

+, u−]), (8)

=(λ1 + λ2f(0)

)+ λ2

(f(y±r − y±a + y±al)− f(0)

).

The terms λ1 + λ2f(0) and λ2

(f(y±) − f(0)

)describe the baseline random turning rate

and the bias turning rate, respectively. The function f is a positive function saturatingfor large values of its argument (to describe the biologically-realistic situation of boundedturning rates). An example of such function is f(y) = 0.5 + 0.5 tanh(y); see [18, 17,6, 7]. These turning rates are influenced by the social interactions among individuals:attraction towards far-away neighbours (y±a ), alignment with neighbours at intermediatedistances (y±al) and repulsion from individuals at very close distances (y±r ). Moreover,these social interactions depend on the perception of neighbours, which communicatevia different mechanisms involving visual, sound, tactile or chemical signals. Table 1

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shows the social interaction terms y±r,al,a corresponding to four examples of communicationmechanisms introduced in [17]. Note that in [17] the authors considered also a fifthmechanisms (denoted M1), which combined attraction/repulsion forces as described byM2 and alignment forces as described by M4. Since this mechanisms did not bring any newresults in terms of pattern formation or model symmetry, it was ignored in more recentstudies [6, 7] and thus we ignore it throughout this study too. The parameters qr,a,al arethe magnitudes of the repulsive (r), attractive (a) and alignment (al) interactions. Thekernels Kr,a,al that model long-distance social interactions are given by Gaussian functions

Kj(s) =1

2πm2j

e−(s−sj)2/(2m2

j ), with j = r, a, al, and mj = sj/8, (9)

with sj, j = r, a, al being the width of the interaction ranges.

Table 1: Nonlocal social interaction terms (y±j , j ∈ a, al, r) introduced in [17]. Constants qa, qal, qrdescribe the magnitudes of the attractive, alignment and repulsive interactions, respectively. KernelsKa,al,r(s) describe the spatial ranges for each of these social interactions. Note that u = u+ + u−.

Mechanisms Communic.models

Interaction terms: attraction (y±a ), repulsion (y±r ),alignment (y±al)

omnidirectionalperception,

M2 y±a,r = qr,a∫∞0

Ka,r(s)(u(x± s)− u(x∓ s)

)ds,

omnidirectionalemission

y±al = qal∫∞0

Kal(s)(u∓(x∓s)+u∓(x±s)−u±(x∓

s)− u±(x± s))ds.

unidirectionalperception,

M3 y±r,a = qr,a∫∞0

Kr,a(s)u(x± s)ds,

omnidirectionalemission

y±al = qal∫∞0

Kal(s) (u∓(x± s)− u±(x± s)) ds.

omnidirectionalperception,

M4 y±r,a = qr,a∫∞0

Kr,a(s) (u∓(x± s)− u±(x∓ s)) ds,

unidirectionalemission

y±al = qal∫∞0

Kal(s) (u∓(x± s)− u±(x∓ s)) ds.

unidirectionalperception,

M5 y±a,r = qr,a∫∞0

Ka,r(s)u∓(x± s)ds,

unidirectionalemission

y±al = qal∫∞0

Kal(s)u∓(x± s)ds.

The integrals in Table 1 can be re-written by defining the operator I±i,ℓ(u

+(x), u−(x), s),with ℓ = a, r, al, to describe the integrand for model Mi, i = 2, 3, 4, 5. The superscript ±

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in I± corresponds to the superscript in y±. Thus, the social interaction terms become

y±i,ℓ(u(x)) :=

∫ ∞

0

Kℓ(s)I±i,ℓ(u

+(x), u−(x), s) ds. (10)

Note that I± satisfies the following relation:

I±i,ℓ(v

+1 (x) + v+2 (x), v

−1 (x) + v−2 (x), s) = I±

i,ℓ(v+1 (x), v

−1 (x), s) + I±

i,ℓ(v+2 (x), v

−2 (x), s),

for all i = 2, 3, 4, 5 and ℓ = a, r, al.

2.3.1 Periodic boundary conditions

Because numerical simulations of system (7) are performed on a finite domain [0, L], wecomplete the description of the model by imposing boundary conditions. For a detaileddiscussion of biologically-realistic boundary conditions for hyperbolic systems, see [30, 36].Here, we consider periodic boundary conditions, which approximate the dynamics oninfinite domains:

u±(0, t) = u±(L, t). (11)

Hillen [36] showed the existence of solutions for local hyperbolic systems that satisfy pe-riodic, homogeneous Dirichlet and homogeneous Neumann boundary conditions. Sincemodel (7) is nonlocal, next we confirm that the integrals (10) are well-defined for u±

satisfying conditions (11). First define the space

L2per = u ∈ L2(R) | u(x) = u(x+ L) for all x ∈ [0, L).

We now show that for the interaction kernels K(s) as in (9) and for v ∈ L2per and

K±v(x) :=

∫ ∞

0

K(s)v(x± s) ds, (12)

we have K±v(x) ∈ L2per. To this end, we write v(x) =

∑∞n=−∞ cne

iknx, where kn = 2πn/L.Then,

K+v(x) =

∫ ∞

0

K(s)∞∑

n=−∞

cneikn(x+s) ds

=∞∑

n=−∞

cneiknx

∫ ∞

0

K(s)eikns ds

=∞∑

n=−∞

cnK(n)eiknx.

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Here, K(n) is the Fourier transform of K(s), and K(η) → 0 as |η| → ∞ exponentiallyfast (since K(s) is Gaussian). Next we know that |cn|2 < 1 if |n| > N for some N ∈ N:

∞∑n=0

|cnK(n)|2 ≤N∑

n=−N

|cn|2|K(n)|2 +−(N+1)∑n=−∞

|K(n)|2 +∞∑

n=N+1

|K(n)|2 < ∞.

Thus, K+v(x) ∈ L2per and the same holds for K−v(x).

Remark 1 Note that if we choose to work with functions in C0per with the sup-norm

||v||∞ = sup|v(x)| | x ∈ [0, L], it is a straightforward exercise to show that K± is abounded operator from C0

per to itself.

2.3.2 Reflective boundary conditions

Another type of boundary condition that is commonly used for systems of hyperbolicmodels (both in biology and physics; see, for example, [43, 47]), is the homogeneousNeumann condition. On the domain [0, L/2], this condition reads

u+(0, t) = u−(0, t), u+(L/2, t) = u−(L/2, t), t ≥ 0. (13)

These Neumann (reflective) conditions describe the case where cells/animals cannot leavethe domain and turn around at the boundary [52, 30, 36]. In regard to the equivalencebetween periodic and reflective boundary conditions for local hyperbolic systems, Lutscher[52] and Hillen [36] showed that for solutions that satisfy the mirror symmetry condition

u+(x) = u−(L− x), x ∈ [0, L], (14)

if one considers w±0 the initial data on [0, L/2] that satisfies the no-flux boundary condi-

tions (13), then it can be shown that

u±0 (x) =

w±

0 (x) for x ∈ [0, L/2],

w∓0 (L− x) for x ∈ [L/2, L],

defines initial data on [0, L] that satisfies periodic boundary conditions. Moreover, con-sidering solutions u± of a local version of (7) with periodic boundary conditions (11), onecan construct restrictions w±(x, t) = u±(x, t), for x ∈ [0, L/2], which are solutions of thesame system with no-flux boundary conditions (13).

The steady-state solutions of nonlocal system (7) described below do satisfy the mirrorsymmetry condition (14). Therefore, the results of the next sections obtained for periodic(or zero-flux) boundary conditions, can be easily generalised to zero-flux (or periodic)conditions.

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2.3.3 Symmetry of hyperbolic models for self-organised biological aggrega-tions

Consider functions u(x, t) = (u+(x, t), u−(x, t)) satisfying the boundary condition u(0, t) =u(L, t). We introduce the translation operator Tθ with θ ∈ [0, L), and the involution κacting on u(x, t) by

Tθ.u(x, t) := u(x− θ, t) and κ.(u+(x, t), u−(x, t)) := (u−(L− x, t), u+(L− x, t)). (15)

The elements Tθ generate a group isomorphic to SO(2) because of the periodic boundarycondition. One can check that Tθ κ = κ T−1

θ , and so Tθ and κ generate a groupisomorphic to O(2). Moreover, it is shown in [7] that system (7) is O(2)-equivariant forany of the models M2, M3, M4, M5 described in Table 1; that is, for any solution u(x, t)of (7), then κ.u(x, t) and Tθ.u(x, t) are also solutions of (7) for any θ ∈ [0, L).

Consider now the action of a group Γ on a vector space V . The isotropy subgroup ofthe point v ∈ V is

Γv := ρ ∈ Γ | ρ.v = v.The symmetry of solutions of (7) is encoded in the isotropy subgroup.

2.3.4 Steady-state solutions

Steady-state solutions of (7) are found by setting ∂tu± = 0 and solving the remaining

integro-differential system. As shown in [7], by adding the two equations in (7), onenotices that steady-state solutions (u+

∗ (x), u−∗ (x)) satisfy u+

∗ (x) = u−∗ (x) + C, where C is

a constant.For homogeneous steady-state solutions, let us first define the total conserved popu-

lation density

A =1

L

∫ L

0

(u+∗ (x, t) + u−

∗ (x, t)) dx.

Then, the homogeneous steady-state solutions are of the form (u+∗ (x), u

−∗ (x)) = (A/2, A/2)

and (u+∗ (x), u

−∗ (x)) = (A∗, A∗∗), where A∗ = A∗∗ and A∗ +A∗∗ = A. These solutions have

isotropy subgroups O(2) and SO(2), respectively.It is also possible to find non-homogeneous symmetric steady-state solutions with

isotropy subgroup Dn. Such solutions for n = 1 and n = 3 are observed in [7, 48]. Itis shown in [7] that if (u+

∗ (x), u−∗ (x)) has isotropy subgroup Σ ⊃ κ, then u+

∗ (x) = u−∗ (x).

Therefore, steady-state solutions with isotropy subgroupsO(2) andDn have this property,but not steady-state solutions with isotropy subgroup SO(2).

3 Lyapunov-Schmidt and Centre Manifold Reductions

Before discussing the application of the Lyapunov-Schmidt and Centre Manifold reductionmethods to model (7), we first present in Sections 3.1 and 3.2 some general results on

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these methods. Then, in Section 4 we verify that the reduction methods can be appliedto the nonlocal hyperbolic models (7).

3.1 General theory

Let X be a Banach space and L a closed linear operator on X with dense domain D(L).Consider a differential equation

d

dtu = L(u, µ) + F (u, µ) := G(u, µ), (16)

where F : D(L)×Rℓ → X is the nonlinear part of the operator which satisfies a Lipschitzcondition, and µ is a bifurcation parameter. Suppose that G(u0, µ0) = 0, and thatthe point spectrum of L(u0, µ0) has values on the imaginary axis. That is, (u0, µ0) isa bifurcation point of the µ family. Without loss of generality, we can assume that(u0, µ0) = (0, 0).

To unfold this bifurcation using the Lyapunov-Schmidt (LS) reduction, the linearoperator

T =d

dt− L(·, µ)

has to be Fredholm over a suitably chosen function space X ′. Recall that an operatorT : X ′ → X ′ is Fredholm if the range of T is closed, and ker T and coker T are finite. Theindex of T is dim ker T − dim coker T . Notice that if L(u0, µ0) has only zero eigenvalueson the imaginary axis (in its point spectrum), then the eigenfunctions do not dependon time, and the time derivative vanishes. Thus, it is sufficient to verify the Fredholmproperty for L only. Typically, X ′ is a function space of 2π-periodic functions (if (16) hasbeen suitably rescaled).

The Fredholm property of L enables a splitting X ′ = kerL+M = N +cokerL, whereM and N are respectively complementary subspaces to kerL and cokerL. Projectionoperators exist for each of these subspaces. Then, one can split G into the operatorsG1 : kerL×M×Rk → N and G2 : kerL×M×Rk → cokerL. Given coordinates (x1, x2, µ)for kerL×M×Rk, one can solve G1 = 0 using a properly chosen implicit function theorem(e.g. see Chicone [10]) and obtain x2 = ϕ(x1, µ) near (u0, µ0) = (x10, x20, µ0). Because Lhas finite index, this leads to a finite dimensional mapping G2 : kerL×M×Rk → cokerLwhich contains the information about particular types of bifurcating solutions, dependingon the choice of function space X ′. A detailed description of the LS reduction can befound in [26] and in Chossat and Lauterbach [12].

We now briefly discuss the Centre Manifold Theorem (CMT) for (equivariant) infinite-dimensional systems of Vanderbauwhede and Iooss [67]. See also [12] for the CMT in thecontext of equivariant systems. Suppose that the linear operator L at (0, 0) satisfies thefollowing assumptions:

13

(A0) The operator L : D(L) → X is bounded (in the graph norm).

(A1) For some k ≥ 2, there exists a neighborhood V ⊂ D(L)×Rℓ of (0, 0) and Y a Banachspace (Y ⊂ X) such that the nonlinear operator F is Ck(V , Y ) and F (0, 0) = 0 andDF (0, 0) = 0.

(A2) The spectrum σ of L can be decomposed as σ = σ+∪σ0∪σ− where σ+, σ− contain,respectively, all λ such that Re(λ) > 0 and Re(λ) < 0 while σ0 has all eigen-values λ with Re(λ) = 0. There exists δ > 0 such that infλ∈σ+ Re(λ) > δ andsupλ∈σ− Re(λ) < −δ. Moreover, σ0 consists of a finite number of eigenvalues withfinite algebraic multiplicity.

(A3) Let P0 be the projection onto the generalized eigenspaces of σ0 and Lh = I − P0,where the linear operator Lh is defined as L restricted to D(L)h = PhD(L). Thenfor any η ∈ [0, δ] and any f ∈ Cη(R, Yh) the linear problem

duh

dt= Lhuh + f(t)

has a unique solution uh = Khf , where Kh is a bounded operator from Cη(R, Yh)to Cη(R, D(L)h) and Cη(R,X ) is the space of exponentially growing functions withthe norm

||u(t)||Cη = supt∈R

e−η|t|||u(t)||X .

The norm of Kh is bounded by a continuous function of η ∈ [0, δ].

Then, there exists a parameter-dependent finite-dimensional manifold

M0(µ) = u0 +Ψ(u0, µ) | u0 ∈ E0,

where E0 = RanP0, and such that M0(µ) is locally invariant and contains the set ofall bounded solutions. Letting L0 be the restriction of L to E0, the reduced system ofequations on the centre manifold has the form

du0

dt= L0u0 + P0F (u0 +Ψ(u0, µ), µ) := g(u0, µ).

Moreover, if G is Γ-equivariant, then Γ acts on the vector space E0 and M0 can be chosento be Γ-invariant. Therefore, g satisfies a Γ-equivariant condition: g(γu0, µ) = γg(u0, µ).

Remark 2 The verification of assumption (A3) is often done by checking an inequalityestimate on the resolvent operator (λI − Lh)

−1. This is illustrated in several examplesin [32]. However, there are cases where the resolvent estimate does not hold, but (A3)does, see [40]. In our case, we do not attempt to prove the resolvent estimate due tothe complexity of the resolvent operator coming from the nonlocal nature of the linearoperator Lh. Instead, we use the symmetries to decompose the problem into a family offinite-dimensional systems for which (A3) is easily satisfied.

14

We now introduce the function spaces for which we show our results. Recall that forsome Ω ⊂ Rn, W k,p(Ω,R) ⊂ Lp(Ω,R) is the Banach space of functions for which the firstk weak derivatives are in Lp(Ω,R), and note that W 1,2 is a Hilbert subspace of L2. Welet Y = W 1,2([0, L],R2) and X = L2([0, L],R2) and so D(L) = (u+, u−) ∈ Y | u±(0) =u±(L). We also define

Yper = (u+, u−) ∈ W 1,2(R,R2) | u±(x) = u±(x+ L), x ∈ [0, L)

andXper = (u+, u−) ∈ L2(R,R2) | u±(x) = u±(x+ L), x ∈ [0, L).

For time-periodic solutions, we introduce

X2π = u ∈ L2([0, L]× R,R2) | u(x, t+ 2π) = u(x, t),

Y2π = X2π ∩W 1,2([0, L]× R,R2)

and D(T ) = u = (u+, u−) ∈ Y2π | u±(0, t) = u±(L, t). Note that Hilbert spaces are cho-sen in order to exploit the orthogonal projection properties when showing that assumption(A3) of the CMT is satisfied. If one chooses to perform the analysis using Banach spacesY = C1([0, L],R2) and X = C0([0, L],R2) along with their periodic counterparts, thenassumption (A3) of the CM theorem of [67] becomes much more cumbersome to satisfyas we need to consider explicitly the resolvent operator (λI − L)−1 in order to defineprojections using the Dunford integral formula [19].

We are now ready to state the main results of this paper. Our first main result is thefollowing.

Proposition 3 (Fredholm operators) Let u∗(x) be a steady-state solution of (7) (forany of the models M2,...,M5 described in Table 1) and let L be the linearized operator atu∗(x). Then, L : D(L) → X2π and

T =d

dt− L(·, µ) : D(T ) → X2π

are Fredholm operators of index zero.

A consequence of Proposition 3 is that the Lyapunov-Schmidt procedure can be performedon the operator L in the context of zero eigenvalues. If L has purely imaginary eigenvalues(after rescaling) ±i,±ik1, . . . ,±ikm where k1, . . . , km ∈ Z or a mixture of zero eigenval-ues and purely imaginary eigenvalues as above, then the Lyapunov-Schmidt reduction isperformed on T with a function space of 2π-periodic functions. The case of nonresonantpurely imaginary eigenvalues is not easily handled using the Lyapunov-Schmidt reductionbecause the choice of Banach space of periodic functions cannot be chosen to simultane-ously obtain all solutions with the two frequencies. Therefore, steady-state, Hopf (includ-ing resonances) and steady-state/Hopf (including resonances) bifurcation problems can

15

be unfolded with the reduced equations obtained via Lyapunov-Schmidt reduction. More-over, if u∗(x) has isotropy subgroup Σ ⊂ O(2), then the reduced equation is Σ-equivariantfor steady-state bifurcations and is Σ×S1-equivariant for Hopf and steady-state/Hopf bi-furcation problems, see [26].

The second result of this paper concerns the spectral properties at the linearizationnear a steady-state.

Proposition 4 (Spectral Properties) Let u∗(x) be a steady-state solution of (7) (forany of the models M2,...,M5) and let L be the linearized operator at u∗(x). Then, thespectrum of L is made up of isolated eigenvalues with finite multiplicity and with noaccumulation point in C. In particular, L can only have a finite number of eigenvalueswith finite multiplicity on the imaginary axis.

Therefore, property (A2) of the CMT of [67] is satisfied. This leads to the following result.

Proposition 5 (Centre Manifold Theorem) Let u∗(x) be a steady-state solution of (7)(for any of the models M2,...,M5) with isotropy subgroup SO(2) or O(2) and suppose thatL has a finite number of eigenvalues on the imaginary axis. Then, assumptions (A0),(A1), (A2) and (A3) of the Centre Manifold Theorem of [67] are satisfied by L and F .

The proof of these results is found in Section 4. In the next section, we describe recentresults on Fredholm properties for linear operators originating from integro-differentialequations and hyperbolic partial differential equations with local and nonlocal linearterms. We do the same for recent results about the applicability of Centre Manifoldreduction in similar contexts.

3.2 Recent results for hyperbolic and functional differential equa-tions

We summarise some recent results for functional differential equations and hyperbolicsystems that use the theory for Lyapunov-Schmidt (LS) and Central Manifold (CM)reductions.

In regard to the LS reductions, it is well-known that for ODEs and parabolic PDEs,the linear operator L is Fredholm [26]. In fact if L is a strongly continuous linear operator,then it is automatically Fredholm [19, 4]. This includes the case of functional differentialequations (retarded and neutral) [31] and evolution semigroups [14]. If equation (16) hasadditional properties such as Hamiltonian structure, symmetry (including reversibility),those are preserved by the LS reduction [26, 25].

There are also several results available for linear operators L that are not stronglycontinuous. Mallet-Paret [55] establishes that for mixed-type functional differential equa-

16

tions

x = L(ξ)xξ =N∑j=1

Aj(ξ)x(ξ + rj),

with L(ξ) asymptotically hyperbolic, the linear operator (ΛLx)(ξ) = x′(ξ) − L(ξ)xξ isFredholm. Moreover, in the case where the linear operator L(ξ) admits constant coefficientasymptotic operators L± as ξ → ±, the index depends only on L± and a formula is givenby the spectral flow, which counts the net number of eigenvalue crossings in the familyL(ξ) from L− to L+.

This work stimulated several other advances, especially in linking the Fredholm prop-erty to exponential dichotomy of the linear operator [33, 50]. Hupkes and Verduyn-Lunel [38] provided a direct generalization of [55] to nonhyperbolic autonomous linearoperators for mixed-type equations and they use their result to show a centre manifoldtheorem for mixed-type functional differential equations. A more recent development isfound in Faye and Scheel [21]. Here, the authors studied the following class of mixed-typeequations with nonlocal terms, which is commonly used in neural models [20]:

d

dξU(ξ) =

∫RK(ξ − ξ′; ξ)U(ξ′) dξ′ +

∑j∈J

Aj(ξ)U(ξ − ξj) +H(ξ),

where U(ξ), H(ξ) ∈ Cn and K(ζ; ξ), Aj(ξ) are n×n complex matrices and J is countablewith the shifts satisfying ξ1 = 0, ξk = ξk for j = k ∈ J . As in [55], the authors showedthat under some assumptions on the asymptotic operators defined for ξ → ±∞, theoperator defined by the right-hand side is Fredholm and the index can also be computedvia its spectral flow.

In a different direction, Kmit and Recke [44, 46, 43] established the Fredholm prop-erty for linear operators associated with a class of first-order local hyperbolic systems ofequations with reflective and Dirichlet boundary conditions. For example, in [44], theylooked at the system of equations

∂tu+ γ∂xu+ a(x)u+ b(x)v = f(x, t), (17a)

∂tv − γ∂xv + c(x)u+ d(x)v = g(x, t), (17b)

where x ∈ [0, 1], f, g are 2π-periodic with respect to t, and u, v are 2π-periodic and satisfythe reflection boundary conditions

u(0, t) = r0v(0, t), v(1, t) = r1u(1, t).

By letting W γ = H0,γ×H0,γ and V γ(r0, r1) = (u, v) ∈ W γ | (∂tu+∂xu, ∂v−∂xv) ∈ W γ,they showed that the linear operator on the left-hand side of (17) is Fredholm of indexzero from V γ to W γ, for a, d ∈ L∞(0, 1) and b, c ∈ BV (0, 1). This Fredholm resultwas generalized in [45] to a n-dimensional system analogous to system (26), along with

17

corresponding boundary conditions where the coefficient functions satisfy weak conditions.The same system was investigated in [46] but this time with C1 coefficients satisfying someoptimal non-resonance conditions. There, the authors showed that the linear operator isalso Fredholm of index zero, but this time from Cn → Cn, where Cn is the space ofcontinuous mappings u : [0, 1] × R → Rn with a sup-norm. The result in [46] was thenused in [47] to show a Hopf bifurcation theorem for semilinear hyperbolic systems

ω∂tuj + aj(x, λ)∂xuj + bj(x, λ, u) = 0, x ∈ (0, 1), j = 1, . . . , n,

with smooth coefficients aj, bj, where aj were satisfying some extra nondegeneracy condi-tions. Moreover, the authors assumed that the solutions satisfied uj(x, t+ 2π) = uj(x, t)for x ∈ [0, 1], j = 1, . . . , n, and the reflection boundary conditions were chosen to be

uj(0, t) =∑n

k=m+1 rjkuk(0, t), j = 1, . . . ,m

uj(1, t) =∑m

k=1 rjkuk(1, t), j = m+ 1, . . . , n.

The statement of the Hopf theorem in [47] also depended on the coefficients rjk.In regard to the Centre Manifold reduction, Renardy [59] proved a version of the

centre manifold theorem for quasilinear hyperbolic equations, and then applied it to aBenard problem describing viscoelastic fluid. On the other hand, Lichtner et al. [56]proved a version of the centre manifold theorem for the class of local semilinear hyperbolicsystems (2) describing laser dynamics. Here, the authors showed that the spectrum ofthe infinitesimal generator of the operator consists only of eigenvalues of finite algebraicmultiplicity. Then, using a spectral gap property, they constructed an exponentiallyattracting invariant manifold on which can be defined the flow of the reduced system

∂uc

∂t= Uc(v)uc + ϵGc(t, v, uc, γ(t, uc, v, ϵ)),

∂v

∂t= ϵF (t, v, B(v)uc + C(v)γ(t, uc, v, ϵ) + g(t, v)),

with u = B(v)uc +C(v)us (and B and C smooth bases), B(v) the spectral projection forthe critical eigenvalues, and us = γ(uc, v, t, ϵ) the Ck smooth graph representation of theinvariant manifold.

Moreover, Hillen [36] investigated the existence of solutions for a local, linear ver-sion of the hyperbolic system (7) (i.e., λ±[u+, u−] = λ± =const.). He showed that thelinear operator L with Neumann or periodic boundary conditions generates a stronglycontinuous semigroup on Lp[0, L]×Lp[0, L]. Moreover, he calculated the spectrum of thelinear operator for different types of boundary conditions. In a separate study, Hillen [34]showed the existence of an invariant manifold for the class of local hyperbolic reactionrandom-walk systems (3).

Finally, we mention the work of Magal and Ruan for hyperbolic semilinear equationswith non-dense domains, which model age-structured populations [53, 54]. In reformu-lating the equation in operator form, the nonlocal boundary condition describing the

18

fertility of the population enters the nonlinear terms of the functional equation. Theauthors adapted the approach of [67] to the context of integrated semigroups to prove theexistence of centre manifolds near steady-state solutions. The main difficulty to overcomewith non-dense domains was to determine a spectral decomposition of the whole functionspace X, while for densely defined domains of the linear operator, only the decompositionof the domain is necessary.

We can conclude from here that for hyperbolic systems, these reductions have beenapplied mainly to local systems. Next, we will focus on applying such results to nonlocalfirst-order hyperbolic models.

4 Application of LS and CM reductions to nonlocal

hyperbolic systems for biological aggregations

Due to the nonlocal nature of the linear operator associated with system (7), the resultsof the previous section do not apply directly to the nonlocal hyperbolic model (7). Inthis section, we discuss the particularities of the Lyapunov-Schmidt and Centre Manifoldreductions for this model. First, we focus on the linear operator associated with system(7), and investigate its compactness. Then, we prove the Fredholm property for thisnonlocal operator. Finally, we discuss the spectrum of L and the application of theCentral Manifold Theorem to model (7), therefore providing proofs of Propositions 3, 4and 5.

4.1 The linear operator

In this section, we extract the linear operator of (7) at an equilbrium solution u∗(x) =(u+

∗ (x), u−∗ (x)). We rewrite equations (7) as

±∂tu± = −γ∂xu

± − λ+(x)u+ + λ−(x)u− (19)

Consider a small perturbations u±(x) = u±∗ (x) + u±

1 (x) near u∗(x), where |u±1 (x)| << 1,

which we substitute in (7). Taking a Taylor expansion of λ± = λ1 + λ2f(y±r − y±a + y±al),

we obtain after truncating beyond order one

λ± ≈ (λ1 + λ2f(0)) + λ2f′(0)(y±r − y±a + y±al).

Let L1 = λ1+λ2f(0), R1 = λ2f′(0). Performing a Taylor expansion of (19) near (u+

∗ , u−∗ ),

and keeping only the linear terms we obtain

−λ+u+ + λ−u− = −L1(u+1 (x)− u−

1 (x))−R1u+∗ (x)K+

i (u1(x))−R1u+1 (x)K+

i (u∗(x))

+R1u−∗ (x)K−

i (u1(x)) +R1u−1 (x)K−

i (u∗(x)).

19

Here, we define K±i (u(x)) := y±i,a[u(x)]− y±i,r[u(x)] + y±i,al[u(x)]. Then, the linear system is

given by

±∂tu±1 = −γ∂xu

±1 − L1(u

+1 − u−

1 )−R1u+∗ K+

i (u1)

−R1u+1 K+

i (u∗) +R1u−∗ K−

i (u1) +R1u−1 K−

i (u∗).

The linear operator on the right-hand side is denoted by L(v, µ), where µ is a vector ofparameters - the main ones being qa, qr and qal. We write L = Lu + Lc, where

Lu(v+, v−)T =

(−γ∂xv

+

γ∂xv−

)+

(−L1v

+

−L1v−

)

and

Lc(v+, v−, µ)T =

(L1v

−

L1v+

)−R1

(K+

i (u∗)v+ −K−

i (u∗)v− + u+

∗ K+i (v)− u−

∗ K−i (v)

−K+i (u∗)v

+ +K−i (u∗)v

− − u+∗ K+

i (v) + u−∗ K−

i (v)

).

4.2 Compactness of Lc

We now show that Lc is a compact operator. It is sufficient to show that K±u is compactas an operator from L2

per to itself. We proceed in a standard way (see [60] for instance) bydefining a family of operators with finite range (and therefore compact) which convergesin the L2 norm to the operators K±, and thus the limit is also compact. To this end, weuse a windowed orthonormal basis of L2(R) over intervals of length L defined by

gn,j(x) =1√Leiknxχ[jL,(j+1)L)(x), (20)

where kn = 2πni/L, χ is the characteristic function of the interval, and n, j ∈ Z. Then,

K(s) =∑n,j∈Z

αn,jgn,j(s), where∑n,j∈Z

|αn,j|2 < ∞. (21)

Consider the approximations of K(s) on (0,∞) given by

KN,M(s) =M∑j=0

N∑n=−N

αn,jgn,j(s)

and define the operators K±N,M : L2

per → L2per by

K±N,Mu :=

∫ ∞

0

KN,M(s)u(x± s) ds.

20

Letting u(x) =∑∞

ℓ=−∞ cℓeikℓx ∈ L2

per and substituting (20) into K+N,Mu gives us

K+N,Mu =

∫ ∞

0

KN,M(s)u(x+ s) ds

=

∫ ∞

0

M∑j=0

N∑n=−N

αn,jgn,j(s)∞∑

ℓ=−∞

cℓeikℓxeikℓs ds

=M∑j=0

∫ (j+1)L

jL

N∑n=−N

αn,j1√Leikns

∞∑ℓ=−∞

cℓeikℓxeikℓs ds

=M∑j=0

N∑n=−N

∞∑ℓ=−∞

αn,jcℓ1√Leikℓx

∫ (j+1)L

jL

eiknseikℓs ds

=M∑j=0

N∑n=−N

αn,jc−n

√Leik−nx,

where the last equality comes from the integral being nonzero (and equal to L) if andonly if n = −ℓ. Thus, KN,M has finite range and so is compact. Consider now

||K − KN,M ||2 =

= sup||u||2=1

∫ L

0

|(K −KN,M)(u)|2 dx

= sup||u||2=1

∫ L

0

|(K(s)−KN,M(s))u(x+ s) ds|2 dx

≤ sup||u||2=1

∫ L

0

(∫ ∞

0

|(K(s)−KN,M(s))u(x+ s)| ds)2

dx

= sup||u||2=1

1

L

∫ L

0

(∞∑

j=M+1

∫ (j+1)L

jL

χ[jL,(j+1)L)(s)

∣∣∣∣∣∞∑

n=N+1

αn,jeikns + αn,je

−ikns

∣∣∣∣∣ |u(x+ s)| ds

)2

dx

= sup||u||2=1

1

L

∫ L

0

∞∑j=M+1

(∫ L

0

χ[jL,(j+1)L)(s)

∣∣∣∣∣∞∑

n=N+1

αn,jeikns + αn,je

−ikns

∣∣∣∣∣ |u(x+ s)| ds

)2

dx

21

and using the Cauchy-Schwarz inequality we obtain

||K − KN,M ||2 ≤

≤ sup||u||2=1

1

L

∫ L

0

∞∑j=M+1

∫ L

0

χ[jL,(j+1)L)(s)∞∑

n=N+1

∣∣αn,jeikns + αn,je

−ikns∣∣2 ds ||u||22 dx

≤ 1

L

∫ L

0

∞∑j=M+1

∫ L

0

∞∑n=N+1

2|Re(αn,jeiknx)|2 ds dx

≤ 1

L

∫ L

0

∫ L

0

∞∑j=M+1

∞∑n=N+1

2|αn,j|2 ds dx

= 2L∞∑

j=M+1

∞∑n=N+1

|αn,j|2 → 0

as N,M → ∞ because the series of coefficients in (21) is finite. Therefore, KN,M convergesto K in the L2-norm, which implies that K is also a compact operator. The purely matrixportion of Lc is compact because it has finite range and the sum of compact operators iscompact. We conclude that Lc is a compact operator.

4.3 Fredholm property and the Lyapunov-Schmidt reduction

It is shown in [41] (Chap IV, Theorem 5.26) that if E,F are Banach spaces and T : E → Fis a closed Fredholm operator and A : E → F is a compact operator, then T + A is alsoFredholm with the index of T and T + A being equal. We use this result to show theFredholm property for L and for the operator

T =d

dt− L

where T : D(T ) → X and D(T ) is a subspace of a space of 2π time periodic solutions.As mentioned above, for steady-state bifurcations, it is sufficient to show that L is Fred-

holm. For studying Hopf bifurcations (including resonance cases) and steady-state/Hopfmode-interactions, we need to show that T is a Fredholm operator.

L operator: We use the splitting L = Lu + Lc above. We show that Lu is an iso-morphism. Our goal is to solve Luv = h with v ∈ D(L) and h = (h1(s), h2(s))

⊤ ∈ X.Let v(x) = (v+(x), v−(x))⊤, M = γ−1L1 (

−1 00 1 ) and h(s) = γ−1(−h1(s), h2(s))

⊤. Theequation is rewritten as a differential equation system v′(x) = Mv(x) + h(s) and hassolution

v(x) = eMxC + eMx

∫ x

0

e−Msh(s) ds.

22

Applying the boundary condition C = v(0) = v(L) implies

v(0) = eMLv(0) + eML

∫ L

0

e−Msh(s) ds.

Because γ−1L1L = 0, then I − eML is invertible and the system has a unique solutionwith

v(0) = (I − eML)−1eML

∫ L

0

e−Msh(s) ds.

Thus, Lu : D(L) → X is an isomorphism, and so it is a Fredholm operator of index 0.Because Lc is a compact operator, we conclude that L is also a Fredholm operator ofindex zero.

T operator: We proceed in a similar way as for L and use the splitting

T = Tu − Lc :=d

dt− Lu − Lc.

We show that Tu is an isomorphism when defined with appropriate function spaces of2π-time-periodic functions (since we are interested in time-periodic solutions emergingfrom Hopf bifurcations). For 2π/ω-periodic solutions a time-rescaling gives a one-to-onecorrespondence with the 2π periodic solutions, see [26]. Let h(x, t) = (h1(x, t), h2(x, t)) ∈X2π and consider the equation Tuv = h where v ∈ D(T ). This equation is transformedinto the decoupled transport system

∂tv+ + γ∂xv

+ = −L1v+ + h1(x, t) (22a)

∂tv− − γ∂xv

− = −L1v− + h2(x, t) (22b)

The existence and uniqueness of solutions of (22) with periodic boundary conditionsfollows the proof in [36]. Thus Tu is an isomorphism and therefore it is Fredholm of indexzero. Again, the compact perturbation preserves the Fredholm property.

4.4 Centre Manifold Theorem

We now show that we can apply the CMT as stated in [32]. We begin by investigatingthe spectrum of L in order to show that assumptions (A2) and (A3) of Section 3.1 aresatisfied. Assumptions (A0) and (A1) are straightforward to verify and are discussedbelow.

The operator L : D(L) ⊂ Y → X is a compact perturbation of the closed differentialoperator γ(−∂x, ∂x)

T : D(L) ⊂ Y → X. Therefore, they have the same essential spectrum(Kato [41], Chap. IV, Thm 5.35). The differential operator γ(−∂x, ∂x) (with periodic

23

boundary conditions) has compact resolvent and its spectrum is a point spectrum givenby

2kγπ

Li | k ∈ Z

,

see Hillen [36]. Therefore, L has empty essential spectrum and so the resolvent set ofL is non-empty. It is straightforward to conclude that L also has compact resolvent.This is done by noticing that for two invertible operators U, V one can verify that U−1 −V −1 = U−1(V − U)V −1. Choose λ in the resolvent set of −γ(∂x, ∂x) and letting V =λI − γ(−∂x, ∂x), U = λI − L, then

U−1 = V −1(I − (L − γ(−∂x, ∂x))V−1)−1

is compact because V −1, L− γ(−∂x, ∂x) are compact, the product of a compact operatorand a bounded operator is compact [41] (Chap III, Theorem 4.8), and λ is chosen so that(I−(L−γ(−∂x, ∂x))V

−1)−1 exists and is bounded. The spectrum of L is a point spectrumconsisting of isolated eigenvalues with finite multiplicity and with no accumulation points.Thus, assumption (A2) is automatically satisfied.

We now turn to assumption (A3). We consider only steady-state solutions u∗(x) whichhave the isotropy subgroups Σ = SO(2) and Σ = O(2). For a steady-state solution u∗(x)with isotropy subroup Σ, the tangent space to X at u∗(x) (which is isomorphic to X as aHilbert space) is Σ-invariant. Since Σ ⊂ O(2) is a compact group acting on the separableHilbert space X, a consequence of the Peter-Weyl theorem [5] implies that the action ofΣ leads to a decomposition of the space as a direct sum of finite-dimensional irreduciblerepresentations; that is,

X =∞⊕k=1

Uk

where Uk, k = 1, 2, . . . are irreducible representations of Σ. The isotypic decomposition ofX with respect to the Σ action is obtained by grouping Σ-isomorphic representations intoso-called isotypic components Uℓ, with ℓ ∈ I, where I is the indexing set for isomorphismclasses of irreducible representations of Σ; see [28, 27] for details. The cases Σ = SO(2)and Σ = O(2) have a countably infinite number of non-isomorphic irreducible represen-tations and are studied together. The case Dn has a finite number of isomorphism classeswhich leads to a decomposition of the tangent space into infinite-dimensional isotypicblocks. We do not consider this case in this paper.

SO(2) and O(2) symmetric steady-states: Let ej, j = 1, 2, be the standard basisvectors of C2. The subspaces

V jn =

zeje

iknx + c.c. | z ∈ C

24

are irreducible with respect to the SO(2) action (15) and V jn , V

jm are not isomorphic if

n = m. Thus, we have the decomposition

X =∞⊕n=1

V 1n ⊕ V 2

n

where for all n ∈ N, Xn := V 1n ⊕ V 2

n are the isotypic components. In the case of O(2),the decomposition derived in [7] has the following form:

X =∞⊕n=1

Xn, (23)

where the subspaces Xn are defined as follows. Let kn = 2πn/L. For all n ≥ 1,

Xn =aeiknx + c.c. | a = (a+, a−)⊤ ∈ C2

⊂ X,

are isomorphic to C2, O(2)-invariant, and can be decomposed into isomorphic irreduciblerepresentations. Let f1 = (1, 1)⊤ and f2 = (1,−1)⊤, then

X1n = (v0eiknx + v0e

−iknx)f1 | v0 ∈ C and X2n = (v1eiknx + v1e

−iknx)f2 | v1 ∈ C,(24)

are real two-dimensional O(2)-irreducible representations (written in complex notation).The basis is given by eiknxf1, eiknxf2, e−iknxf1, e

−iknxf2. Then, Xn = X1n ⊕X2

n, and thesubspaces Xj are called isotypic components of the O(2) action on X. The decomposi-tion (23) is called the isotypic decomposition of X.

The inner product on X given by

⟨v,w⟩ =∫ L

0

(v+w+ + v−w−)dx, (25)

where v = (v+, v−) and w = (w+, w−) are O(2)-invariant. Using this inner product,one can verify that the subspaces Xj, Xk are mutually orthogonal for all j = k. LetPXk

: X → Xk be the orthogonal projection associated withXk. TheO(2)-equivariance ofL implies a block diagonalization along the isotypic decomposition. That is, L(Xk) ⊂ Xk

and we write Lk := L|Xk. Therefore, L decomposes as a direct sum of finite dimensional

matrices.Let σ(L) = σ+ ∪ σ0 ∪ σ− where σ0 has a (nonzero) finite number of elements. From

assumption (A2), we define δ > 0 such that

infλ∈σ+

(Re(λ)) > δ and supλ∈σ−

< −δ.

For each k, we define the projections Pk,+ and Pk,− onto the stable and unstable subspacesof Lk. Consider the linear system

du

dt= Lhu+ f(t), (26)

25

where f ∈ Cη(R, Yh) with η ∈ [0, δ]. We use the isotypic decomposition of X and writeu ∈ Cη(R, D(L)h) as

u(t) =∞∑k=1

Φk(x)uk(t),

where Φk(x) is the matrix with basis elements of Xk as columns and uk(t) = PXku(t). This

leads to the decomposition of (26) into an infinite family of finite dimensional systems

duk

dt= Lh,kuk + fk(t), (27)

where fk = PXkf . For isotypic blocks with no eigenvalues on the imaginary axis, Lh,k =

Lk, while for isotypic blocks with spectrum intersecting σ0 and at least one of σ+ or σ−,then Lh,k = Pk,+Lk + Pk,−Lk. We denote by σk,± the hyperbolic part of the spectrum inLh,k.

The system of equations (27) has solution

uk(t) = eLh,ktu0 + eLh,kt

∫ t

0

e−Lh,ksfk(s) ds,

with the constraint ||uk(t)||Cη(R,Xk) < ∞ forcing the unique solution

u0 = −∫ ∞

0

e−Lh,ksPk,+fk(s) ds+

∫ 0

−∞e−Lh,ksPk,−fk(s) ds,

where the splitting fk(s) = Pk,+fk + Pk,−fk guarantees that the integrals are convergent.The solution can be rewritten uk = Kk,hfk where

(Kk,hfk)(t) := eLh,kt

(−∫ ∞

t

e−Lh,ksPk,+fk(s) ds+

∫ t

−∞e−Lh,ksPk,−fk(s) ds

).

Because of the finite-dimensionality, it is straightforward to check that

Kk,h ∈ L (Cη(R, Xk), Cη(R, Xk)) and ||Kk,h|| < Ck(η),

where Ck is a continuous function of η ∈ [0, δk] with δk chosen so that

infλ∈σk,+

Re(λ) > δk and supλ∈σk,−

Re(λ) < −δk.

We define

u = Khf :=∞∑k=1

Φk(x)(Kh,kfk)(t),

26

and one can verify that this provides a unique solution of equation (26). We now showthat Kh is also a bounded operator; i.e. Kh ∈ L (Cη(R, Yh), Cη(R, D(Lh)) and that thenorm of Kh is bounded by a continuous function of η ∈ [0, δ].

Let f ∈ Yh and we write f =∑∞

k=1 fk, where fk ∈ Xk. Then,

||f ||Yh=

∫ L

0

|f |2 dx+

∫ L

0

∣∣∣∣ ddxf∣∣∣∣2 dx =

∞∑k=1

(1 + k2)||fk||2Xk.

For any η ∈ [0, δ], we write ||Khf ||Cη(R,D(Lh)) =

= supt∈R

e−η|t|||Khf ||D(Lh)

= supt∈R

e−η|t|

(∫ L

0

|(Knf)(t)|2 dx+

∫ L

0

∣∣∣∣ ddx(Knf)(t)

∣∣∣∣2 dx

)

= supt∈R

e−η|t|

∫ L

0

∣∣∣∣∣∞∑k=1

Φk(x)(Kk,hfk)(t)

∣∣∣∣∣2

dx+

∫ L

0

∣∣∣∣∣ ddx(

∞∑k=1

Φk(x)(Kk,hfk)(t)

)∣∣∣∣∣2

dx

= supt∈R

e−η|t|

∫ L

0

(∞∑k=1

Φk(x)(Kk,hfk)(t)

)·

(∞∑j=1

Φj(x)(Kj,hfj)(t)

)dx

+

∫ L

0

(d

dx

(∞∑k=1

Φk(x)(Kk,hfk)(t)

))·

d

dx

(∞∑k=1

Φk(x)(Kk,hfk)(t)

) dx

where · is the Euclidean inner product on R2. Commuting with the summations and bylinearity, the inner product turns into the inner product on each Xk and the last line isequal to

supt∈R

e−η|t|

(∞∑k=1

∞∑j=1

∫ L

0

Φk(x)(Kk,hfk)(t) · Φj(x)(Kj,hfj)(t) dx

+∞∑k=1

∞∑j=1

∫ L

0

k2Φk(x)(Kk,hfk)(t) · Φj(x)(Kj,hfj)(t) dx

).

27

By orthogonality of Xk, Xj for j = k this last line becomes

supt∈R

e−η|t|

(∞∑k=1

||(Kh,kfk)(t)||2 + k2||(Kh,kfk)(t)||2)

≤ supt∈R

e−η|t|∞∑k=1

Ck(η)2((1 + k2)||fk||2Xk

).

= C(η)2||f ||Cη(R,Yh)

where the inequality holds for all η ∈ [0, δ] because δ ≤ δk for all k = 1, 2, . . ., and so wecan factor out Ck from the summation and drop the index k. Thus, assumption (A3) issatisfied.

Assumption (A0) is automatically satisfied because L is a closed operator, see [32],while assumption (A1) is satisfied because of the tanh function in the definition of (7).Therefore, all four assumptions of the CMT of [67] are satisfied and Proposition 5 isverified.

5 Discussion and generalisation of the results

In this study, we investigated Lyapunov-Schmidt and Centre Manifold reductions for aclass of nonlocal hyperbolic systems developed to model animal aggregations. We firstpresented the general theory behind these reduction methods, and the application ofthese results to functional differential equations and local hyperbolic systems (describingphysical or biological phenomena). This approach allowed us to summarise the resultsexistent in the literature, and to identify the results that are still missing. Then, we appliedthe two reduction methods to our class of nonlocal hyperbolic models. We showed thecompactness of the operator associated with the nonlocal system, and then proved thatthe operator is Fredholm - a condition necessary for Lyapunov-Schmidt reduction. Weemphasize here that the Fredholm property for hyperbolic equations, and in particular fornonlocal hyperbolic equations, has been less studied compared to the ODE or parabolicPDE models. Hence, our study fills a gap in the literature about Fredholm operators fornonlocal hyperbolic systems.

In regard to the Central Manifold reduction, we proved that the version of the CentreManifold Theorem described in [32] can be applied to the nonlocal model (7), and hencethe CM reduction in [7] is valid near steady-state solutions with isotropy subgroups SO(2)and O(2). The extension to steady-state solutions with isotropy subgroup Dn wouldrequire a different approach than the one presented here.

An interesting consequence of our results in this paper, is that they extend automati-cally to two or more population models for animal/cellular aggregations [16]. They also

28

extend to coupled equations of animal/cellular aggregation via chemotaxis, because thechemotaxis models typically contain a Laplacian operator and the theory is well-knownthere [32].

It is possible that the LS and CM reductions extend in a straightforward way also to2D nonlocal kinetic models (generalisations of the 1D hyperbolic models (7); see [22]).However, it was not the goal of our study to investigate this aspect. Such an analysis willform the object of a future study.

Another interesting question to be addressed in the future refers to a formal compar-ative study of the unfolding and dynamics obtained from Lyapunov-Schmidt reductionand Weakly-Nonlinear Analysis for arbitrary bifurcation problems.

Acknowledgements

PLB acknowledges the financial support from NSERC in the form of a Discovery Grant.RE acknowledges support from an Engineering and Physical Sciences Research Council(UK) First Grant number EP/K033689/1. PLB would like to thank Christiane Rousseaufor her support and encouragements over the years. PLB is particularly grateful to herfor proposing and championing the Mathematics of Planet Earth initiative.

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