Pattern formation in the doubly-nonlocal Fisher-KPP equation

We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a local diffusion term and a nonlocal reaction term. If the diffusion is approximated by a nonlocal kernel, we show that our results are consistent and reduce to the local ones in the local singular diffusion limit. Furthermore, we prove that there are parameter regimes, where no bifurcations can occur for the doubly-nonlocal Fisher-KPP equation. The results demonstrate that intricate different parameter regimes are possible. In summary, our results provide a very detailed classification of the multi-parameter dependence of the stationary solutions for the doubly-nonlocal Fisher-KPP equation.


Introduction
The aim of this paper is to study existence and stability of stationary solutions to the doubly-nonlocal Fisher-KPP equation. Namely, we consider bounded non-negative solutions u = u(x) on the real line R to the following equation where κ + , κ − and m are (strictly) positive real numbers, a + and a − are probability densities, and the convolution terms are defined as follows The evolution equation corresponding to (1.1) first appeared, for the case κ + a + = κ − a − , m = 0, in [30,29]. For the case κ + a + = κ − a − , m > 0 we refer to [15] and for different kernels to [11], where the so-called Bolker-Pacala model of spatial ecology was considered. The equation (1.1) was rigorously derived from the Bolker-Pacala model in [23] for integrable u and in [18] for bounded u. The long-time behavior was studied in [19,20,21,22]. In [2], the term κ + (a * u − u) was (formally) approximated by the Laplace operator using the Taylor expansion of the convolution term; see Section 6 below for more detail. In this approximation limit, one obtains the Fisher-KPP equation with a non-local reaction, (1.2) where d := κ + 2 R y 2 a + (y) dy, θ is given in (1.3), and (1.2) also often referred to as the nonlocal Fisher-KPP equation. Observe that there are two constant solutions to (1.1) and (1.2), namely, It was pointed out in [9] that under additional assumptions the nonlocal Fisher-KPP equation (1.2) admits a steady state bifurcation of u ≡ θ leading to existence of spatially periodic solutions. Later, more detailed analysis was carried out for a more general reaction in [8]. Numerical analysis of bifurcations and traveling waves to (1.2) was considered in [1,2,13,14,24]. Analytical results for stationary solutions and traveling waves to (1.2) can be found in [4,3,10,17,26]. We also remark that there is a variant of the Fisher-KPP equation with a nonlocal operator replacing the Laplacian and with a local reaction [12,25].
In this paper we demonstrate that under additional assumptions there exists a steady-state bifurcation of u ≡ θ for the doubly-nonlocal Fisher-KPP equation (1.1). This bifurcation leads to existence of periodic solutions to (1.1) and is connected with results to (1.2) (specifically [17]) as we show in Section 6 via the singular local diffusion limit. Up to our knowledge, in contrast to (1.2), the only results on bifurcations in (1.1) were done heuristically in recent publications [5,6]. Thus we present the first rigorous statements of this sort.
We stress that the problem of existence of stationary solutions to the equation (1.1) depends on relations between parameters of the equation (1.1). In particular, if κ + < m, then u ≡ 0 is the only non-negative bounded solution to (1.1), which follows from the Duhamel formula (θ < 0 in this case). If κ + > m, κ + a + (x) ≥ (κ + − m)a − (x), for x ∈ R, and a + , a − are symmetric, then the constant solutions given by (1.3) are the only non-negative bounded solutions to (1.1) (see Proposition 7.1 below). If a + , a − are non-symmetric, it is possible that there exist a traveling wave with a speed 0, namely, there can exist decreasing u : R → [0, θ] which satisfies (1.1) and such that u(+∞) = 0, u(−∞) = θ (see [19]). Therefore, we have to carefully investigate, under which conditions bifurcations are possible. In Section 2 we formulate assumptions sufficient for a steady-state bifurcation of u ≡ θ. First, we introduce a small parameter ε in (1.1) substituting κ + , κ − by κ + ε = (1 + ε)κ + and κ − ε = 1 + ε κ + κ + −m κ − correspondingly, which turns out to be a more suitable compact notation to state our results. Studying the problem for symmetric a ± in the space of square-integrable periodic function on the real line, we show that the spectrum of the linearization of the left-hand side of (1.1) at u ≡ θ equals to the following set, where a ± is the Fourier transform of a ± defined below in (2.6). Next, we require, that for small ε < 0 the spectrum belongs to the negative half-plane {z ∈ C | Rez < 0}, it touches the imaginary axis {z ∈ C | Re(z) = 0} for ε = 0, and it intersects the positive half-plane {z ∈ C | Re(z) > 0} for small ε > 0. Thus, we have the following picture: These assumptions impose constraints on the parameters, yet they imply existence of periodic solutions to (1.1), that we prove in Section 3 applying the Lyapunov-Schmidt reduction method. Namely, we demonstrate that for any sufficiently small ε > 0 and δ (probably negative), there exists a periodic solution to (1.1) with a period 2π Section 4 is devoted to the study of stability of the solutions in a space of square-integrable periodic functions. We show, that the solutions are (locally) asymptotically stable with respect to perturbations with the same period and phase. Section 5 provides several examples of the probability densities a + , a − , which satisfy assumptions of the previous sections. Since our article is related to the results for (1.2), specifically to [17], we demonstrate this detailed connection in a certain limiting case in Section 6. Section 7 presents some results on non-existence of solutions to (1.1), which shows that one really has to distinguish fundamentally different behaviour already for the stationary solutions of doubly-nonlocal equation.
Our first main assumptions are the following where C, ξ > 0 are some fixed constants. Note that the first assumption in (A1) already hints at the fact that we must impose certain growth restrictions on the linear part to obtain bifurcation results. The further assumptions are typical technical assumptions on the kernel(s) for nonlocal and doublynonlocal Fisher-KPP equations. We introduce a small parameter ε, to study structural changes of the solutions to (1.1) under small perturbations of this parameter. To simplify our notations, we will write κ + ε := (1 + ε)κ + . Let us suppose that the constant solution u ≡ θ = κ + −m κ − is independent of ε and so we have where subscripts denote new parameters. Let us also assume that m is not changed so that m ε ≡ m > 0. As a result of the parameter change, the coefficients in (1.1) are transformed as follows, If we set w := u − θ, then w satisfies the following equation We will study bifurcations of u ≡ θ in the class of periodic functions, i.e., bifurcations of w from the branch of trivial solutions a w ≡ 0. Therefore, we introduce the following (complex) Hilbert space of periodic square-integrable functions with a period p > 0, Let us ensure that w ∈ L 2 p (R) implies w(a − * w) ∈ L 2 p (R). Proposition 2.3. Let w ∈ L 2 p (R) and a ∈ L 1 (R → R + ) be such that Then The last statement of the proposition follows from the estimate which finishes the proof.
Proof. By (2.3), the proof follows from the following estimate As a next step, it is helpful to introduce the wave number k. For a ∈ L 1 (R) and f ∈ L 2 2π which is the main bifurcation problem we study near the point (v, ε) = (0, 0). In particular, instead of considering (2.2) with w ∈ ∪ k>0 L 2 2π k (R), we consider (2.4) on L 2 2π (R), passing to the space with the fixed period 2π. Remark 2.5. Since, by (A1), a ± (x) ≡ a ± (−x), we can conclude that A ε,k given by (2.4) is a bounded self-adjoint operator in L 2 2π (R). Therefore, the spectrum of A ε,k , denoted by σ(A ε,k ) ⊂ R, is a bounded set.
To observe a bifurcation at ε = 0, we assume that the spectrum of the operator A ε,k passes through the imaginary axis at ε = 0, and some k = k c > 0, namely, there exist k c , ε 0 , δ 0 > 0, such that for all It is helpful to re-interpret the last assumption more concretely in Fourier space. Consider the Fourier transform of f ∈ L 2 2π (R) defined by and the Fourier transform of a ∈ L 1 (R) given by Obviously, if a ∈ L 1 (R) and f ∈ L 2 2π (R), then a * f ∈ L 2 2π (R), and (F(a * f ))(j) = a(j)(Ff )(j), j ∈ Z.
We denote By the Plancherel formula F : L 2 2π → l 2 is a unitary operator, which implies Since, for a k (x) := 1 k a( x k ),â k (j) =â(kj), it follows we get for f ∈ L 2 2π (R), Therefore, σ(A ε,k ) is the closure of the set {α(ε, jk)|j ∈ Z}. Since α(ε, p) → −κ + ε as |p| → ∞, the condition (2.5) follows from the assumption (2.8) We want to re-formulate the last bifurcation condition more concisely in terms of α and its derivatives. First, we have to assume that We also assume that there are well-distinguished critical modes . By (A1) and (2.7), we have, In order for (2.8) to hold, by (2.9), we assume, We can understand the spectrum near the critical wave number by considering the following expansion Hence, we automatically get a transversality condition for bifurcation parameter In order to satisfy the first inequality in (2.8) (consider e.g. −ε = δ 3 < 0 in (2.10)), we assume To ensure the second inequality in (2.8) it is sufficient to suppose that Ω(ε, δ) > 0, which, by (2.11) and (A5), holds if We denote A c = A 0,kc . Now we can check that our assumptions limit the critical modes to a twodimensional space: Lemma 2.6. Let (A1), (A2) and (A3) hold, then Proof. By (A1) and (A2) one easily concludes {α(0, ±k c ) = 0} ∈ σ(A c ). Moreover, the following equalities hold a * e ±ikcx = e ±ikcxâ (±k c ), a k * e ±ix = a * e ±ikx . Hence, we obtain A c e ±ix = e ±ix α(0, ±k c ) = 0. Thus e ix , e −ix are eigenvectors for the eigenvalue λ = 0. If f ∈ ker A c , then we find So our assumption (A3) yields (f, e ijx ) = 0 for j = ±1. Thus f ∈ span{e ix , e −ix } follows. As a result, (2.12) holds, which finishes the proof.
In order to prove existence of non-constant solutions to (1.1) we also need the assumption This last assumption is not yet transparent but it will be interpreted below as a local solvability condition to obtain a real branch of non-trivial solutions; see equation (3.1).
We will follow a similar strategy as in [27] and apply the Lyapunov-Schmidt reduction method in order to give a proof of Theorem 3.1.

Stability of periodic solutions
Although we have shown the existence of non-trivial stationary solutions, we also would like to check whether one actually observe these solutions as long-time limit of the evolution problem. Therefore, we are going to study stability of the periodic solutions u ε,δ obtained in Theorem 3.1 for small values of ε and δ. Since u ε,δ (· − h) solves (1.1) for any h ∈ [0, 2π) we consider v ε,δ (·) = u ε,δ ( · kc+δ ) − θ only in the following subspace of L 2 2π (R) (cf. (3.2)) where we reduce ker A c = {se ix + te −ix } to its subspace with t =s and arg(s) = 0. In other words Y is chosen such that the phase of u ε,δ is fixed, namely, there exists unique h ∈ [0, 2π), such that u ε,δ (· − h) ∈ Y (in fact h = 0). The main result of the section is the following theorem.

Corollary 4.2. Under conditions of Theorem 4.1, for any
. We start with a lemma on compactness of the convolution operator, which also explains the reason for the condition (4.2) in Theorem 4.1. Then the operator Ah = a * h is compact in L 2 k (R), for any k > 0.
By the Implicit Function Theorem applied above we can redefine ε 0 > 0 such that there is no new eigenvalue around 0 and thus in the positive half-space for all ε < ε 0 .

Examples
We still have to show that there exist kernels satisfying all our assumptions so that we can get bifurcations. We are going to provide two examples. Both examples are motivated by the goal to find simple, yet non-trivial kernels, where can check our conditions.

Example 5.1. We start with Gaussians, respectively linear combinations of Gaussians, and consider
.
In this case, the Fourier transforms of a ± have the following form for p ∈ R, a + (p) = e − lp 2 2 ; a − (p) = cos(hp)e − qp 2 2 .
One can see on Figure 5.2 that ω = ω(0, k c ) > 0. In fact, the condition is evidently not close to being violated in this case and the argument would be easy to make completely rigorous by just using interval arithmetic to validate the sign.
Example 5.2. The second example is in spirit similar to the first one, so we are a bit more brief. We consider uniform distributions: In this case, the Fourier transform of a ± has the following form, for p ∈ R, a + (p) = sin(lp) lp ; a − (p) = sin(qp +hp) − sin(hp) qp = 2 cos(hp + qp 2 ) sin( qp 2 ) qp .

Relation to the Fisher-KPP equation with a non-local reaction
In this section we establish the connection between Theorem 3.1 and and [17, Theorem 1.1] for the nonlocal Fisher-KPP equation. For the convenience of the reader we are going to formulate [17, Theorem 1.1] here again for reference. We consider the equation where µ > 0. We need to discuss the relevant hypotheses before stating the result.
To finish the proof of the statement 6.2 it is left to notice that by (6.6), (6.10) and (6.11) (c.f. (A6) σ ) The proof is fulfilled.

Remark 6.3.
It is worth to point out that typically a diffusive scaling is considered for κ = 0. We introduce κ ∈ R to get an additional 'degree of freedom' that allows us to choose κ = κ(σ) such that the spectrum of the linearization of (6.5) at u ≡ θ touches the imaginary axis for all small σ > 0 (c.f. (6.7)).

On nonexistence of stationary solutions
One might now ask, whether all the assumptions are really crucial to obtain a non-trivial bifurcating solution.
Then there exist only two non-negative bounded solutions to (1.1), namely u ≡ 0 and u ≡ θ.
In fact, one can even describe that nothing can happen "between" the two homogeneous solutions, even for other parameter ranges as the following results shows: Proposition 7.2. Let a + ∈ L 1 (R → R + ) be such that R a + (y) dy = 1, R |y|a + (y) dy < ∞.
Proof. We argue by contradiction and suppose there exists u ∈ L ∞ (R) satisfying (1.1) and 0 ≤ u(x) ≤ l for x ∈ R. Then, we must have

This implies
We distinguish two cases. Suppose first that u ∈ L 1 (R), then we get which implies u ≡ 0 as a + has mass one. For the second case let u ∈ L 1 (R). For any r > 0 we compute As a result, by (7.1), where the left-hand side is infinite because u ∈ L 1 (R). Therefore, we have obtained again a contradiction.
The results in this section show that there are also many cases, where bifurcations are impossible.