Rotational Symmetry and Rotating Waves in Planar Integro-Difference Equations

Mathematical models often possess symmetries, either because of actual symmetries of the situation being modelled, or as approximations. It is well known that these symmetries often impose restrictions on the solutions to these models. In this paper, we investigate the role of rotational symmetry in certain integro-difference equations and study the existence of rotating wave solutions to these equations. We perform explicit computations in the case where the integration kernel is a Gaussian distribution, which often occurs in applications.


Introduction
In the theory of dynamical systems, it is by now well known that symmetries play an important role in shaping the dynamics and bifurcations of the system (Golubitsky and Schaeffer 1985;Golubitsky et al. 1988). In this paper, we will be interested in investigating the consequences of rotational symmetries in discrete-time dynamical systems given by iteration of an integro-difference equation of the form where ⊆ R m , u and v are functions which go from into R N , F : ×R N −→ R N is some (generally nonlinear) function and κ : × −→ R is such that κ(x, y) dy = 1, ∀x ∈ .
(1.2) Systems such as (1.2) have been of interest to mathematicians and scientists for many years, especially in the general area of modelling of ecological systems and gene dispersal. See for example (Bourgeois et al. 2018(Bourgeois et al. , 2020Lui 1982a, b;Wang et al. 2002;Weinberger 1982;Weinberger et al. 2002). In these applications, the vector u represents the distribution in space of a generation of certain co-inhabiting populations and v represents the subsequent distribution in space of the next generation. Subsequently, one is interested in the discrete-time dynamical system generated through iteration of the map N in (1.1), i.e. given an initial distribution u 0 (x), compute u n (x) (for integer n ≥ 1) through the recursion u n = N [u n−1 ]. It is then quite natural to want to characterize the limiting behaviour of orbits of this integro-difference system as n → ∞.
In applications, the term κ(x, y) models dispersal in space of the populations from one generation to the next. In contrast to partial differential equations where diffusion is local, in (1.1) it is considered as a non-local (throughout space) phenomenon and κ(x, y) represents the density of the probability that an individual initially at point y moves to the point x after one generation. In addition to this spatial dispersal, the population densities are also affected by local interactions such as reproduction of a single invasive species (Bourgeois et al. 2018;Coutinho et al. 2012;Hsu and Zhao 2008;Li et al. 2009;Lutscher 2019;Wang et al. 2002;Weinberger 1978), predatorprey and cooperative interactions (Lin et al. 2020;Neubert et al. 1995;Weinberger et al. 2002;Wu 2021), and mating of heterozygotes (Carillo et al. 2009;Lin 1995;Weinberger 1978Weinberger , 1982, and are represented by the term F(x, ·) in (1.1). Recently, some studies have also incorporated the effects of climate change on these models (Lewis et al. 2018). Therefore, the range of interest and applicability of models of the form (1.1) is quite extensive.
The scope of this paper is not to focus on any one particular application, but rather to perform a mathematical analysis of the role played by any rotational symmetries which may be present in (1.1). To this end, we will assume that = R 2 , that the image of F is in R 2 , and the kernel κ and the function F are homogeneous in space such that we can write F(x, u) = f (u) and κ(x, y) = k(|x − y|), where |z| is the Euclidean norm of z ∈ R 2 . It is well known that the fact that the Euclidean norm is invariant under translations implies that (1.1) has an important symmetry: Proposition 1.1 Let u 0 be a given function from R 2 into R 2 such that u 1 (x) = R 2 k(|x − y|) f (u 0 (y)) dy exists. Let p ∈ R 2 be given, and define v 0 (x) = u 0 (x − p). Then, exists and is such that v 1 (x) = u 1 (x − p).
This translation symmetry is fundamental to proving the existence of travelling waves, i.e. orbits {u n+1 } = {N [u n ]} of the discrete-time dynamical system (1.1), such that u n+1 (x) = u n (x − p) for a given fixed p ∈ R 2 , as was outlined in the pioneering papers of Weinberger (1978Weinberger ( , 1982. Over the years, a large part of the literature on integro-difference systems such as (1.1) has been dedicated to such questions as proving existence/uniqueness of travelling waves, their stability, and critical wave speeds below which travelling waves do not exist [see for example (Lutscher 2019) for an excellent presentation and extensive literature review of the field].
However, surprisingly very little attention has been paid to the fact that the Euclidean norm is also invariant under rigid rotations and that this fact could lead to solutions compatible with this symmetry. It is this aspect that we wish to explore in this paper.
We define SO(2) to be the group of all rigid rotations on the plane, then SO(2) acts on R 2 according to the standard action where R θ is the rotation matrix R θ = cos θ − sin θ sin θ cos θ .
The natural phase-space for the discrete-time dynamical system (1.1) is a space of functions from R 2 into R 2 , for example C b (R 2 , R 2 ) the space of bounded and continuous functions with supremum norm. We distinguish two natural actions of SO(2) on such a space : where G(θ ) can be either the identity matrix, or the rotation matrix R θ . The kernel k(|x − y|) is compatible with either representations of this symmetry (since both representations rotate the space variable). We will assume that the local reaction dynamics given by the function F(x, u) = f (u) are compatible with the form G(θ ) = R θ of this symmetry in the sense that i.e. f is SO(2)-equivariant in the sense of Golubitsky et al. (1988). In this case, it also (trivially) respects the equivariance symmetry in the case where G(θ ) is the identity matrix. This leads to: Proposition 1.2 Suppose f satisfies the condition (1.4) and let u 0 be a given function from R 2 into R 2 such that exists. Let θ ∈ SO(2) be given, and Proof This is a simple computation: where we have used the rotational invariance of the norm and the fact that the determinant of a rotation matrix is equal to one.
In the language of dynamical systems, we formulate this property as follows. Suppose X is an appropriate space of functions from R 2 into R 2 , for example exists and is an element of X . We may view N in (1.5) as a map N : X −→ X . Considering the action (1.3) of SO(2) on X , Proposition 1.2 is equivalent to the equivariance equality θ • N = N • θ, ∀θ ∈ SO(2). (1.6) If we then consider the dynamical system on X defined by iterations of the map N in (1.5), Eq. (1.6) implies that the group SO(2) maps orbits of this dynamical system into other orbits of the dynamical system. The study of group equivariant dynamical systems has enjoyed a rich development over the past decades, see for example (Aston and Mir 2009;Charette and LeBlanc 2014;Chossat and Golubitsky 1988;Cicogna 2021;Golubitsky et al. 1997Golubitsky et al. , 2000Golubitsky et al. , 1988Lamb and Melbourne 1999). An interesting class of orbits which may exist for SO(2)-equivariant dynamical systems are so-called rotating waves. In the context of the discrete dynamical system on X defined above, a rotating wave corresponds to u * ∈ X such that there exists θ * ∈ SO(2) with N [u * ] = θ * · u * , which using (1.6) leads to i.e. the time orbit of u * is contained in its SO(2) orbit. Note that because there are two possible representations in (1.3), there are two distinct types of rotating wave solutions depending on the form of the matrix G(θ ). In the case where G(θ ) is the identity matrix, we will use the terminology untwisted rotating wave, whereas in the case where G(θ ) = R θ we will use the terminology twisted rotating wave. The existence of rotating waves for (1.5) (untwisted or twisted) is not necessarily guaranteed by the SO(2) symmetry; however, we may establish a sufficient condition in the form of a fixed-point equation: ( 1.7) The function u * is called the wave profile of the rotating wave. It should be noted that because of the translation symmetry of our integral operator, we are free to set the centre of rotation of the rotating wave at the origin of physical space, as we have done in (1.7). As mentioned earlier, although there has been considerable attention paid to the existence of travelling waves for (1.1), there are relatively few studies of its rotating wave solutions. This is somewhat surprising since the existence and dynamics of rotating waves in partial differential reaction-diffusion systems (mostly spiral waves) has been studied at great length over the past decades (Barkley 1992(Barkley , 1994Barkley and Kevrekidis 1994;Boily et al. 2007;Golubitsky et al. 2000;Sandstede et al. 1997Sandstede et al. , 1999Scheel 1998;Winfree 1974). Many models of cardiac electrophysiology are in the form of reaction diffusion PDEs, and rotating spiral solutions represent pathologies such as arrhythmias (Davidenko et al. 1992;Keener and Sneyd 1998;Mesin 2012;Pertsov et al. 1993;Roth 1998Roth , 2001. We do, however, note that for the related problem of integro-differential equations, there have been some numerical and analytical studies of spiral waves (Laing 2005;Troy and Shusterman 2007).
The study of the existence of travelling wave solutions of (1.5) typically relies heavily on properties of the kernel k, but also on the internal dynamics of the discrete- (1.8) In particular, equilibrium points (and their local asymptotic stability) of this system often correspond to limiting properties (as x → ±∞) of the wave profile u * . Of course, it is possible for systems such as (1.8) to have limiting states more complicated than equilibrium points. For example, there may exist a period-2 cycle (or more generally a periodp cycle) in the dynamics of (1.8). This question was analysed in some detail (Bourgeois et al. 2018(Bourgeois et al. , 2020 by studying the second-iterate map N 2 in (1.5) and looking for travelling waves of the resulting operator connecting two fixed points of the second iterate f 2 in (1.8). See also (Fang and Pan 2022;Hsu and Zhao 2008). In the left panel, the equilibrium is stable, whereas in the right panel, the equilibrium is unstable and there is a stable closed invariant curve Therefore, it is expected that limiting states of (1.8) will also play a role in describing rotating waves, i.e. solutions of (1.7). The equivariance condition (1.4) imposes strong restrictions on the functional form of the mapping f in (1.8). As is shown in Golubitsky et al. (1988), f must have the form where A 1 and A 2 are real-valued functions. Rather than studying the problem (1.5) in the full generality (1.9), we will limit ourselves to the following representative normal form of (1.9) (which, as we will see, already leads to an analysis and computations that are quite involved but manageable): cos ω − sin ω sin ω cos ω where β is a real parameter and ω ∈ (0, π) for which it is possible to completely describe the dynamics analytically. We note that despite its apparent simplicity, this mapping is in fact a special case of the generic cubic truncated normal form for the Neimark-Sacker bifurcation from an equilibrium point with eigenvalues of its linearization crossing the unit circle (when β = 0) at e ±iω (assuming the non-resonance conditions e i ω = 1 for = 1, 2, 3, 4) (Kuznetsov 2004). Adopting polar coordinates (u 1 , u 2 ) = (η cos ψ , η sin ψ), (1.10) reduces to for which the dynamics of (1.10) become clearer. The origin (u 1 , u 2 ) = 0 is a fixed point for all β real, locally stable when β < 0 and locally unstable when β > 0. For β > 0, there is a locally asymptotically stable closed invariant circle of radius η = √ β on which the dynamics of (1.10) reduces to a rigid rotation around the origin through angle ω. See Fig. 1 for a summary of this discussion.
Taking into account all these considerations, we will therefore focus our attention in this paper to the fixed-point problem (1.7) which reduces to The remainder of the paper is organized as follows. In Sect. 2, we will use polar coordinates for (1.12) and seek out certain rotating waves (which we call pinwheel solutions) in which the wave profile u * (x) has a discrete rotational symmetry. The radial part of the wave profile for these pinwheel solutions is then shown to satisfy a fixed-point problem (involving a single integral) for a positive scalar function, and most of the effort in the rest of the paper will be proving that this fixed-point problem admits a solution under certain hypotheses. The first of these hypotheses are presented in Sect. 3, and in Sect. 4 we show that these hypotheses hold in the case of a Gaussian kernel. In Sect. 5, we establish an appropriate space of functions on which to study the fixed-point problem for the radial part of the wave profile. For technical purposes, we need to introduce another hypothesis (involving lower bounds) and we do so in Sect. 6, along with a proof that this additional hypothesis holds in the case of a Gaussian kernel. The main results on existence of pinwheel solutions are proven in Sect. 7, using a set-up which takes advantage of Schauder's fixed-point theorem. The question of asymptotic stability of these rotating pinwheel solutions is addressed in Sect. 8 in the case of a Gaussian kernel. Numerical simulations supporting our results are presented in Sect. 9, and some concluding remarks are presented in Sect. 10. A few of the more technical proofs are relegated to an appendix.

Polar Coordinates Representation and Pinwheel Solutions
It will be useful to adopt polar coordinates for both variables x = x 1 + i x 2 = re iϕ and y = y 1 + iy 2 = ρe is so that if we write u = u 1 + iu 2 , the integral operator N in then the operator (2.1) admits as invariant subspaces each of the Fourier modes, as follows: We will exploit Proposition 2.1 and search for solutions u * of the fixed-point problem (1.12) of the form where m ∈ {1, 2, 3, . . .} is the degree of rotational symmetry. If P(r ) ≥ 0 is realvalued, we call the solution a pinwheel solution (see Fig. 2). The perhaps better-known class of rotating wave solution, the spiral wave (see Fig. 2), which requires a complexvalued P(r ), will not be addressed in this paper.
Remark 2.2 1. For rotating wave profiles of the form (2.3), the difference between twisted and untwisted rotating waves is easy to describe. We have which implies that for the untwisted action (δ = 0), the wave profile u * is rotated through the origin by an angle mα, whereas for the twisted action the angle is (m − 1)α. 2. We want to address an issue of potential relevance to the applications in spatial ecology we mentioned in the introduction. It is clear that rotating wave solutions of the form (2.3) oscillate in space and time between positive and negative values. As such, one may believe that these solutions have limited relevance to describing populations, which obviously cannot take on negative values. However, we note that system (2.1) need not represent absolute population, but could represent a deviation from a spatially homogeneous and nonzero baseline population. For example, consider a biological or ecological system modelled by an integro- and suppose that there exists a positive number P 0 such that F(P 0 ) = P 0 , i.e. P 0 is a fixed-point of the local dynamics. Then, the spatially homogeneous and constant state p(x) = P 0 is a fixed point for (2.4). If we are interested in variation of the population around this baseline P 0 , we may write p(x) Negative values for u (provided they are small in comparison with P 0 ) would still represent positive values for p in (2.4). Setting and substituting (2.3) in (1.12) using the polar representation (2.1), we get after some simplification where δ = 0 leads to untwisted rotating waves and δ = 1 leads to twisted rotating waves. Using the 2π -periodicity of the integrand and settings = s − ϕ − α, (2.6) becomes (upon dropping the tildes) The term k(|r − ρe is |) which appears in (2.7) is equal to We note that for any integrable function h, we have 2π 0 h(cos(s)) sin ms ds = 0. Therefore, is a real-valued function, and we note that A m (r , ρ) = A m (ρ, r ).

Remark 2.3
Note that the case m = 0 is not of interest since it corresponds to a radially symmetric steady state, i.e. not a rotating wave. Also, it is clear from (2.8) and the parity of the cosine function that A −m (r , ρ) = A m (r , ρ), for all m ≥ 1. Therefore, we restrict our analysis to m ≥ 1.
Since the left-hand side of (2.7) is real, we are immediately led to the compatibility condition for the rotational frequency α of the rotating wave (2.9) and have reduced (2.1) to the following fixed-point problem for the function P: Since we have assumed that ω ∈ (0, π), condition (2.9) implies that there are no twisted solutions to (2.1) of the form (2.3) with m = 1. Untwisted solutions with m = 1 are possible provided α = 2 π − ω.
To the extent possible, we will keep the analysis general, but in one important case of integration kernel k in (2.10), i.e. the Gaussian kernel (2.11) explicit computations are possible and we will elaborate on these computations throughout the sequel.
The analysis required to prove the existence of solutions to (2.10) will depend on an appropriate choice of function space and on properties of the function A m . We will further explore these issues in the next sections.

Hypotheses on the Integration Kernel
In this section, we will discuss some general assumptions on the integration kernel k for the fixed-point problems (2.1) and (2.10), and then in the next section we will analyse the specific case of the Gaussian kernel (2.11).
We first note that the function A m (r , ρ) defined in (2.8) is such that when r = 0 or ρ = 0, we get (3.1) We will suppose
The function M(r ) defined in (3.5) will serve an important purpose in our analysis. We note that item (i) in the above hypothesis and the condition A m ≥ 0 (item (iii) above) guarantee that M(0) = 0, M(r ) ≥ 0 for r > 0, and so M is positive, bounded and strictly increasing (item (v) in the above hypothesis) on [0, ∞).

Gaussian Kernel
In the case where the integration kernel k is as in (2.11), we will show that all elements of Hypothesis 3.1 are satisfied (clearly item (i) is satisfied, so we will focus on the other items). We compute that where for ν ≥ 0, I ν (t) is the modified Bessel function of order ν (see Watson (1995)), which in terms of the ordinary Bessel function of order ν, J ν , is given by For any integer ν ≥ 0, the function I ν (t) is strictly increasing, I 0 (0) = 1, I ν (0) = 0 for ν > 0, and we have that for large positive real t and for ν ≥ 0 (Watson 1995) so that asymptotically the function A m (r , ρ) behaves like which resembles a "travelling" Gaussian (in (r , ρ)-space) with decaying amplitude (see left panel of Fig. 3). The function M(r ) in (3.5) can also be computed analytically, using formula (3) on page 394 of Watson (1995) and relationships between confluent hypergeometric functions and Bessel functions (e.g. pages 100-105 of Watson (1995) and Chapter 13 of Olver et al. (2010)) (4.3) See the right panel in Fig. 3, and note in particular that inequality (3.7) is manifested in that figure. We note that a straightforward computation using recurrence relations involving modified Bessel functions and their derivatives (for example, see page 79 of Watson (1995)): where we have used the inequality (Soni 1965) (4.5) Moreover, M (0) = 0 unless m = 1 in which case we have M (0) = √ 2π 4σ > 0. So item (v) in the hypothesis is satisfied.
We thus want to show that Using the recurrence relation (see page 79 of Watson (1995)), (4.7) can be rewritten as For t > 0, the positive root of the quadratic tw 2 +2mw−t is w + = −m + √ m 2 + t 2 t . In Theorem 1.1 of Laforgia and Natalini (2010), it is shown that from which it follows that Q(t; m) > 0 for all t > 0, and thus in (4.6) we have Q(r , ρ; m, σ ) > 0 for all r > 0, ρ > 0. So the kernel ρ A m (r , ρ) for the fixed point problem (2.10) is of total positivity class T P 2 , i.e. item (iii) is satisfied. Next, we check Hypothesis 3.1 (ii). In this case, we compute using formula (3) on page 394 of Watson (1995)  Note that for m = 1, this formula can be simplified to using Bessel function identities. For m > 1, the leading term in the Taylor expansion of I m 2 (r 2 /4σ 2 ) is ∼ r m , so that for m > 1, Also, for r > 0 we have So K (r ) is positive, increasing (strictly increasing for m > 1), and using the asymptotic formula (4.2) we get i.e. K (r ) is bounded, see Fig. 4 for an example. For Hypothesis 3.1 (iv), we leave the somewhat lengthy computation to the appendix.

Functional Set-up
Proving the existence of solutions to the fixed-point equation where g(P, β) = P(1 + β − P 2 ) requires an appropriate choice of function space for the function P : [0, ∞) → R. Throughout, we will assume that β is small enough and in particular is less than 1/2. We note the following obvious property

Then, the functionP(t) defined byP(t) = (g • P)(t) also satisfies all properties (i)-(iii) above.
Proof This is a simple consequence of the fact that g(P, β) is continuous, increasing and greater than P on [0, denotes the Banach space of bounded and continuous functions from [0, ∞) into R endowed with supremum norm ∞ , we will consider the following closed subspace of X X √ β = {P ∈ X : P(0) = 0, P is increasing, P ∞ ≤ β }.
(5.2) Proposition 5.1 is equivalent to saying that the space X √ β is invariant under the mapping P → g(P, β).
Theorem 5.2 If P ∈ X √ β and if we define Proof Given Proposition 5.1, it will suffice to show that if L is the linear operator defined by and if P ∈ X √ β , thenZ = L[P] belongs to X √ β . From (3.1), we obviously havẽ Z (0) = 0. The fact that ρ A m (r , ρ) satisfies Hypothesis 3.1 implies thatZ (r ) ≥ 0 for all r ≥ 0. It also follows thatZ is continuous (in fact,Z is C 2 on (0, ∞)) and where M(r ) is as in (3.5). Thus Z ∞ ≤ √ β. It thus remains to be shown thatZ is increasing. SinceZ (0) = 0 andZ is C 2 on (0, ∞), it will suffice to show that Z (r ) > 0 for all r > 0. We will exploit the fact that A m satisfies Hypothesis 3.1 (iii), which implies that the kernel ρ A m (r , ρ) is of total positivity class T P 2 . Kernels F(r , ρ) of class T P 2 satisfy an important property (called the variation diminishing property in the literature), which is as follows: suppose the function ζ(ρ) changes sign once, then the functioñ changes sign at most once. Moreover, ifζ (r ) changes sign exactly once, thenζ (r ) and ζ(ρ) must have the same arrangements of signs as r and ρ, respectively, traverse R + from left to right (Karlin 1968).
For any a such that a > 0, consider the horizontal line ρ = a. If the function P(ρ) − a vanishes, then it does so either at one isolated root, or on an interval [ρ 1 , ρ 2 ] (in the case where P is constant on the interval). Either way, there is at most one sign change (necessarily from negative to positive) as ρ traverses R + from left to right. Since ρ A m (r , ρ) is T P 2 , the same remark on sign changes holds for the functioñ (where M is as in (3.5)). We compute (see Hypothesis 3.1 (iv)) lim r →0 +Z Suppose r 0 > 0 is such thatZ (r 0 ) = 0, and defineã =Z (r 0 )/M(r 0 ) > 0. Then, the function has a root at r 0 , and W (r 0 ) = −ã M (r 0 ) < 0. Therefore, W goes from positive to negative as r passes through r 0 , which is a contradiction. We conclude thatZ can have no root on (0, ∞). Using (5.5) and the fact thatZ (0) = 0,Z (r ) ≥ 0, we conclude thatZ (r ) > 0 for all r > 0, i.e.Z is (strictly) increasing. Therefore, we may define the nonlinear operator and we will be interested in proving the existence of a non-trivial fixed point for T in X √ β .

Lower Bounds
Of course, it is clear that the function P(r ) = 0 is a fixed point for the operator T in (5.6). We are obviously interested in non-trivial fixed points. To this end, we will need to impose an additional hypothesis on the kernel function ρ A m (r , ρ): Hypothesis 6.1 There exists a β 0 > 0 such that for every β ∈ (0, β 0 ), there exists a bounded positive function φ(r ; β) from [0, ∞) into R such that 0 < φ(·; β) ∞ < √ β and such that We note that in the case of an integration kernel of the form K(r , ρ) = K (|r − ρ|), it is shown in Hsu and Zhao (2008) how to construct a such a lower bound φ(r ; β) under some general conditions, which unfortunately we cannot exploit here since the integration kernel ρ A m (r , ρ) is not of the form K (|r − ρ|). While condition (6.1) may be difficult to verify in practice, the following gives a sufficient condition which presumably is easier to check.

Proof
Since v is positive it follows that φ = e − √ β v is positive and Using the fact that the integration kernel ρ A m is positive, we may write

Now,
From the hypotheses on the function D, we claim that there exists a β 0 > 0 such that To verify the claim, we note that which is increasing for small enough β > 0 since D (0) ≤ 0 (note that the claim holds even in the case where D (0) = 0 because 2β 3 2 > 0). We may then write ∀r ∈ [0, ∞), ∀β ∈ (0, β 0 ).
The hypotheses of Proposition 6.2 are easy to verify in the case where k is a Gaussian as in (2.11) and consequently A m (r , ρ) is as in (4.1). We will use formula (3) on page 394 of Watson (1995), which after simplification leads to the following formula (for given constant λ > 0) where we note that D(0) = 1 and D (0) = 0.
Remark 6.4 From Propositions 6.2 and 6.3, it follows that in the case where k is a Gaussian as in (2.11) and consequently A m (r , ρ) is as in (4.1), Hypothesis 6.1 is satisfied, using the lower bound

Main Result
We are now ready to state and prove the main result of the paper concerning the existence of rotating pinwheel solutions to (1.12) via the existence of non-trivial fixed points for the operator T in (5.6). For small β > 0, recall the definition of the metric space X √ β in (5.2), and note that from Theorem 5.2 we have T (X √ β ) ⊂ X √ β . Let φ(r ; β) be the lower bound such as in Hypothesis 6.1. Consider the Banach space is a non-empty, closed and convex subset of Z 1 r +1 , invariant under the nonlinear operator T .

Proposition 7.1 The operator T is continuous on
(7.2) where the function M is as in (3.5), the function K is as in Hypothesis 3.1 (ii) and K 0 is an upper bound for K (r ). From this and the Lipschitz continuity (with Lipschitz constant L > 0) of the function u −→ g(u; √ β] × [0, 1/2], we have that for all p 1 , p 2 ∈ X √ β,φ , and so T : Proof It is clear that for any p ∈ U we have p = T (Y p ) for some Y p ∈ X √ β,φ , and for any r ≥ 0 we have Considering Hypothesis 3.1 (vi), let M > 0 be such that Then, for all p ∈ U we may write (since p is C 1 ) Therefore, the number M √ β is a uniform global Lipschitz constant for all elements of U, and we conclude that U is equicontinuous.
Let { p n } n∈N be an arbitrary sequence of elements of U. Then, there is a subsequence { p n } which converges uniformly on compact subsets of [0, ∞) to a continuous function p : [0, ∞) −→ R. Since each p n is such that then the limit function p enjoys the same properties because these are preserved by pointwise convergence, i.e. p belongs to Thus, for any ε > 0, there exists R > 0 such that Since { p n − p} converges uniformly to 0 on [0, R], the same holds for p n − p r +1 , so there exists N ∈ N such that It follows that . We thus conclude that We can now give the main result of this paper in the form of a theorem and a corollary.

Theorem 7.3 Consider the two component integro-difference equation on the x =
where β > 0 is a small enough parameter, ω ∈ (0, π). Let m > 0 be an integer and suppose that the kernel k satisfies the conditions of Hypotheses 3.1 and 6.1 for that value of m. There exists a function U * (x) = U * (re iϕ ) = P * (r )e imϕ which is a rotating-wave solution to the system (7.3) in the sense that where α satisfies the compatibility condition (2.9). The rotating wave is untwisted if δ = 0, and twisted if δ = 1 (in which case m = 1). Furthermore, the radial-shape function P * is C 2 smooth and such that Proof This is a simple application of Schauder's fixed-point theorem, which given Proposition 7.2 guarantees the existence of a P * β ∈ X √ β,φ which satisfies all the properties (i-iii) in the statement of the theorem (because these properties are satisfied by all elements of X √ β,φ ) and such that P * β = T [P * β ], i.e.

Radial Stability in the Gaussian Case and a Uniqueness Result
In the case of the Gaussian kernel, we will prove a partial stability result for the rotating wave solution found in Corollary 7.4. That is, we will provide sufficient conditions to guarantee that the fixed point P * of the operator is locally asymptotically stable in C b ([0, ∞), R). At this point, we do not have any results concerning stability with respect to angular perturbations. The important technical tools used in our analysis are the ordering (4.5) of the Bessel functions and the fact that the lower bound φ(r ; β) found in Remark 6.4 is independent of the parameter σ in the Gaussian distribution.
We note that the linearization of the operator T in (8.1) at the fixed point P = P * is given by Since P * ∞ ≤ √ β, for small β > 0 the term 1 + β − 3P * (ρ) 2 is positive, so the norm of the operator L on the space The main result is the following: Theorem 8.1 Let β > 0 be fixed and small enough. Then, for sufficiently large σ > 0, we have L ∞ < 1 in (8.2).

Lemma 8.3
For all r ≥ 0, ρ ≥ 0, m ≥ 1 and σ > 0, we have Proof of Lemma 8.3 Again, we only need to prove the result for m = 1. Setr = r /σ andρ = ρ/σ , then upon dropping the tildes the result follows if we can show that Write r = t cos η, ρ = t sin η where t ≥ 0 and η ∈ [0, π/2]. Then, using the fact that I 1 is an increasing function, we have ρe − r 2 +ρ 2 2 I 1 (r ρ) = t sin η e − t 2 2 I 1 sin 2η t 2 2 ≤ t e − t 2 2 I 1 t 2 2 and we invoke Lemma 8.2 for the conclusion of the proof of this lemma.
Let φ(r ; β) be as in Remark 6.4. This function attains a maximum value of e − √ β √ β at r = √ 2m 2β . Since P * (r ) ≥ φ(r ; β) and P * is increasing, we conclude that where M(r ) is as in (3.5) and (4.3) (we recall that M(r ) ≤ 1 for all r ≥ 0), and is finite (and computable in closed form expression). Thus, if we conclude from (8.2) that L ∞ < 1.

Remark 8.4
We make the following observations about the proof of the previous theorem: (i) We can compute a closed form expression for G (m, β) in terms of the function We have We have illustrated in Fig. 5 the curves (as a function of β) given by the right-hand side of inequality (8.4) for m = 1 and m = 4. (ii) As is clear from the proof of the previous theorem, condition (8.4) is sufficient, but not necessary for stability of the fixed point. We have performed numerical simulations to better characterize the stability of the fixed point P * . The results will be presented in the next section. See Fig. 11 and the description in the caption. (iii) Indeed, the inequality (8.3) is far from optimal, but is valid for all elements p of the space X √ β in (5.2) which are such that p ≥ φ. In fact, we can use this observation to improve on Corollary 7.4.
where S is as in (8.3). Define the set Then, the operator T in (8.1) has a unique fixed point in So T is a contraction on Y √ β,φ and we get the conclusion from the Banach fixed-point theorem.

Simulations
We present the results of iterations of the nonlinear operator T defined by (8.1). The numerical method we have used is explained below and was implemented in the symbolic programming language Maple. We have used a spatial discretization of the ρ and the r axes with 10,000 intervals of length 1/400, which gives values of r and ρ between 0 and 25. For fixed values of the parameters m and σ , we compute once a 10000 × 10000 matrix M such that The initial condition P 0 (r ) is written as a 10,000-dimensional vector and the vector v 0 = (1 + β)P 0 − P 3 0 is computed. Then, for every i value between 1 and 10,000, the value P 1 (r [i]) is computed by using a trapezoidal rule to sum the points with spatial distancing 1/400. Since the true integral is an integral from 0 to ∞, obviously our implementation will give problematic results near the endpoints at ρ = 25 and at r = 25. For this reason, for purposes of viewing P j [r ], we use only the first 8000 datapoints, corresponding to r values between 0 and 20. The iteration process is halted (i.e. considered to have converged) when the maximum value of |P n+1 (r ) − P n (r )|/ √ β for r between 0 and 20 is less than 5 × 10 −5 . We report here the results of two such simulations: one for parameter values β = 0.2, σ = 0.5, m = 4, and another for parameter values β = 0.08, σ = 0.3 and m = 1. See Figs. 6 and 7 for a summary of the results (Figs. 8,9).
For both simulations we have reported on here, we have numerically computed the function i.e. we have used P 63 as an approximation of the fixed point P * , and the results are reported in Fig. 10. As we can see, the maximum value of L 63 (r ) is bounded away from one which would indicate that the linearization L of the operator T at the fixed point P * is a contraction for the chosen parameter values. Note that the value of σ for each of these simulations is considerably less than the lower bound given in (8.4) (taking into account formula (8.5)): in the first case, the right-hand side of inequality (8.4) yields a value of ∼ 15.4, whereas for the second case the value is ∼ 5.9. In Fig. 11, we have numerically computed the fixed point P * up to an accuracy of 5 × 10 −5 (by the iteration procedure described above) for values of β between 0 and 0.1 and then computed the maximum value of the linearized operator The parameter values are β = 0.08, σ = 0.3 and m = 1. The initial condition is P 0 (r ) = √ β r 3 for r ∈ [0, 3] and = √ β for r ≥ 3, and is in blue. The iterate P 63 is in solid red and the dashed horizontal redline is at height √ β = √ 0.08. The residual ratio max r ∈[0,20] |P 64 (r ) − P 63 (r )|/ √ β is less than 5 × 10 −5 . The ordering of the curves is such that the "steep initial part" of the curve y = P 7(n+1) (r ) is to the left of that of y = P 7n (r ) (Color figure online) which suggests (as expected) that the bifurcating solution is stable for all β > 0 small enough.   ). In this case, the analysis essentially reduced to proving the existence of a pair (P(r ), α) which solves the system (2.9) and (2.10): In contrast, spiral wave solutions to (2.1), for example of the ansatz form u * (re iϕ ) = P(r )e i(r +mϕ) , would need to satisfy for (P(r ), α) the system which has similarities with (10.1), but is sufficiently different (for example, the kernels oscillate and change sign) that our techniques here don't immediately apply. This is currently being investigated. We wish to make a few comments on our choice (1.10) for the local dynamics in the integro-difference equation, which has the expression (1.11) in polar coordinates. As previously noted, the dynamics of this map are a combination of expansion (or contraction) in the η-direction, coupled with a uniform (in η) rotation of angle ω around the origin. However, we note that the general form of the cubic truncation would be (after rescaling and in complex notation) where the coefficient d 2 is not zero in general and the real part of the cubic coefficient has been rescaled to 1. In polar coordinates, mapping (10.2) becomes (Kuznetsov 2004) and should be compared to (1.11). In particular, although the dynamics consist of rigid rotations around the origin, the angle of rotation is now dependent on η, and following the approach in this paper, the relevant equations to solve (instead of (10.1)) would be where C 1 = cos(ω + (m − δ)α) and S 1 = sin(ω + (m − δ)α), which reduce to (10.1) when d 2 = 0. Although clearly more complicated structurally than (10.1), we believe that our results in this paper will be critical to a full analysis of (10.3). Another important remark is that we have chosen the local dynamics to possess the necessary symmetry with respect to the twisted action (1.4), which consequently imposed the algebraic restrictions (1.9). It is clear from our analysis that these algebraic restrictions were fully exploited and immensely simplified the subsequent analysis. Such simplified computations would not have been possible for a general form of f (i.e. different than (1.9)). The functional form (1.9) is not required for equivariance in the case of the untwisted action. Therefore, we believe that a full analysis of the untwisted case would be considerably more difficult, although we are confident that our analysis here would prove beneficial. Another possible generalization to our analysis would be to allow the kernel κ(x, y) in (1.1) to be a 2 × 2 matrix of functions which would also lead to the SO(2) symmetry (1.6). While of great potential interest, this falls outside the scope of this paper and will be the subject of future investigation. The techniques used in this paper would also prove the existence of rotating pinwheel solutions to integro-differential equations of the form ∂u ∂t (x, t) = −(1 + iω)u(x, t) + R 2 k(|x − y|)(1 + β − |u(y, t)| 2 )u(y, t) dy 1 dy 2 .
Whereas we have determined the existence of rotating wave solutions in invariant spaces (corresponding to Fourier modes) and characterized their stability within these invariant spaces (in the case of a Gaussian kernel), we have not investigated stability properties with respect to perturbations which are not in the form of a pure Fourier mode, nor have we investigated existence of rotating wave solutions outside of these invariant subspaces. While these are important questions, they fall outside the scope of this paper.
We note that while we have used the origin of the physical two-dimensional space as the centre of rotation for the rotating waves, the translation equivariance of our system implies that any point can be made to correspond to the centre of rotation; therefore, there exist rotating wave solutions rotating about any point in physical space. Initial conditions would determine which is observed. g 2 (r , ρ) = ρ 2 σ 4 e − r 2 +ρ 2 2σ 2 I m+1 r ρ σ 2 g 3 (r , ρ) = mρ r σ 2 e − r 2 +ρ 2 2σ 2 I m r ρ σ 2 .
Since I m (t) ∼ t m for small t, we have that (uniformly in ρ ≥ 0) the dominant term in (A.2) for small r > 0 is the term g 3 which is of order r m−1 . Since g 3 is positive, we have where M(r ) is as in (4.3) and M (r ) in (4.4) is bounded by some positive number K 1 (m, R).
We easily compute which is bounded above for r ≥ 0 by some positive constant K 2 (m) because of the asymptotic formula (4.2). Therefore, it remains to show that and so from (A.6) we have 2σ 2 dρ.
(A.7) So our final step will be to show the boundedness (in r ∈ [R, ∞)) of both integrals in (A.7).
We note that for every t ≥ 0, we can show using simple calculus arguments that 2 e t 2 /2 ≥ 2 3/4 t 3/2 from which it follows (using t → t √ 2 σ ) that t 3/2 σ 3 e − t 2 2σ 2 ≤ 2 σ 3/2 e − t 2 4σ 2 , so for the first term in (A.7) we may write A similar argument is used to show the boundedness of the second integral in (A.7) using the relation (for t ≥ 0) Thus, we conclude that the function B(r ) in (A.6) is bounded on r ∈ [R, ∞), and thus Hypothesis 3.1 (iv) is satisfied by the Gaussian.