# Planar pulsating traveling wave solutions of non-cooperative Fisher-KPP systems in space-time periodic media

Léo Girardin, Grégoire Nadin

► To cite this version:

Léo Girardin, Grégoire Nadin. Planar pulsating traveling wave solutions of non-cooperative Fisher-KPP systems in space-time periodic media. 2025. hal-05131918

HAL Id: hal-05131918

https://hal.science/hal-05131918v1

Preprint submitted on 26 Jun 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
# PLANAR PULSATING TRAVELING WAVE SOLUTIONS OF NON-COOPERATIVE FISHER–KPP SYSTEMS IN SPACE-TIME PERIODIC MEDIA

LÉO GIRARDIN AND GRÉGOIRE NADIN

ABSTRACT. Non-cooperative Fisher–KPP systems with space-time periodic coefficients are motivated for instance by models for structured populations evolving in periodic environments. This paper is concerned with entire solutions describing the invasion of open space by a persistent population at constant speed. These solutions are important in the understanding of long-time behaviors for the Cauchy problem. Adapting methods developed for scalar equations satisfying the comparison principle as well as methods developed for systems with homogeneous coefficients, we prove, in each spatial direction, the existence of a critical speed such that: there exists no almost planar generalized transition waves with a smaller speed; if the direction is rational, each rational speed not smaller than the critical speed is the speed of a weak planar pulsating traveling wave; if the coefficients are homogeneous in space, each speed not smaller than the critical speed is the speed of a strong planar pulsating traveling wave.

## 1. INTRODUCTION

1.1. General framework. This paper is concerned with reaction–diffusion systems of the form

$$ (\text{KPP}) \qquad \operatorname{diag}(\mathcal{P})\mathbf{u} = \mathbf{L}\mathbf{u} - (\mathbf{B}\mathbf{u}) \circ \mathbf{u}, $$

where: $\mathbf{u} : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^N$ is a vector-valued function of size $N \in \mathbb{N}^*$, with a time variable $t \in \mathbb{R}$ and a space variable $x \in \mathbb{R}^n$, $n \in \mathbb{N}^* = \mathbb{N} \setminus \{0\}$ being the spatial dimension; each operator of the family $\mathcal{P} = (\mathcal{P}_i)_{i \in [N]}$, with $[N] = \mathbb{N} \cap [1, N]$, has the form

$$ \mathcal{P}_i : u \mapsto \partial_t u - \nabla \cdot (A_i \nabla u) + q_i \cdot \nabla u, $$

with $A_i : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^{n \times n}$ and $q_i : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n$ periodic functions of $(t, x)$, respectively square matrix-valued and vector-valued; $\mathbf{L}, \mathbf{B} : \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^{N \times N}$ are square matrix-valued periodic functions of $(t, x)$; and $\circ$ denotes the Hadamard product between two vectors in $\mathbb{R}^N$.

The standing assumptions on $\mathcal{P}, \mathbf{L}$ and $\mathbf{B}$ are the following.

<footnote>(L. G.) CNRS, INSTITUT CAMILLE JORDAN, UNIVERSITÉ CLAUDE BERNARD LYON-1, 43 BOULEVARD DU 11 NOVEMBRE 1918, 69622 VILLEURBANNE CEDEX, FRANCE</footnote>

<footnote>(G. N.) CNRS, INSTITUT DENIS POISSON, UNIVERSITÉ D'ORLÉANS, RUE DE CHARTRES, 45067 ORLÉANS CEDEX 2, FRANCE</footnote>

<footnote>E-mail addresses: leo.girardin@math.cnrs.fr, gregoire.nadin@cnrs.fr.</footnote>

<footnote n="2010">2010 Mathematics Subject Classification. 35K40, 35K57, 92D25.</footnote>

<footnote>Key words and phrases. KPP nonlinearities, space-time periodicity, reaction–diffusion system, traveling waves.</footnote>
(A1) The family $(A_i)_{i \in [N]}$ is uniformly elliptic:

$$0 < \min_{i \in [N]} \min_{y \in \mathbb{S}^{n-1}} \min_{(t,x) \in \mathbb{R} \times \mathbb{R}^n} (y \cdot A_i(t, x)y).$$

(A2) The matrix $\underline{\mathbf{L}} \in \mathbb{R}^{N \times N}$, whose entries are

$$\underline{l}_{i,j} = \min_{(t,x) \in \mathbb{R} \times \mathbb{R}^n} \underline{l}_{i,j}(t,x) \quad \text{for all } (i,j) \in [N]^2,$$

is essentially nonnegative: its off-diagonal entries are nonnegative.

(A3) The matrix $\overline{\mathbf{L}} \in \mathbb{R}^{N \times N}$, whose entries are

$$\overline{l}_{i,j} = \max_{(t,x) \in \mathbb{R} \times \mathbb{R}^n} \overline{l}_{i,j}(t,x) \quad \text{for all } (i,j) \in [N]^2,$$

is irreducible: it does not have a stable subspace of the form span($\mathbf{e}_{i_1}, \dots, \mathbf{e}_{i_k}$), where $k \in [N-1]$, $i_1, \dots, i_k \in [N]$ and $\mathbf{e}_i = (\delta_{ij})_{j \in [N]}$. By convention, $[0] = \emptyset$ and $1 \times 1$ matrices are irreducible, even if zero.

(A4) The matrix $\underline{\mathbf{B}} \in \mathbb{R}^{N \times N}$, whose entries are

$$\underline{b}_{i,j} = \min_{(t,x) \in \mathbb{R} \times \mathbb{R}^n} \underline{b}_{i,j}(t,x) \quad \text{for all } (i,j) \in [N]^2,$$

is positive: its entries are positive.

(A5) There exists $\delta \in (0, 1)$ such that $\mathbf{L}, \mathbf{B} \in C_{\text{per}}^{\delta/2, \delta}(\mathbb{R} \times \mathbb{R}^n, \mathbb{R}^{N \times N})$ and, for each $i \in [N]$, $A_i \in C_{\text{per}}^{\delta/2, 1+\delta}(\mathbb{R} \times \mathbb{R}^n, \mathbb{R}^{n \times n})$ and $q_i \in C_{\text{per}}^{\delta/2, \delta}(\mathbb{R} \times \mathbb{R}^n, \mathbb{R}^n)$. Moreover, $A_i = A_i^T$ for each $i \in [N]$.

The precise definition of the functional spaces appearing in (A5) will be clarified below, if not clear already. As usual in such a smooth and generic framework, the symmetry of the diffusion matrices in (A5) is actually given for free. No symmetry assumption is made on $\mathbf{L}$ and the irreducibility of $\overline{\mathbf{L}}$ in (A3) is equivalent to the irreducibility of the space-time average of $\mathbf{L}$.

In this paper, we are interested in solutions defined in $\mathbb{R} \times \mathbb{R}^n$, referred to as entire solutions. More precisely, we will seek for classical entire solutions modeling invasions of open space at constant speed in a given direction.

## 1.2. Organization of the paper.
The remainder of Section 1 is devoted to a detailed introduction and to the statement of the main results. Section 2 contains technical preliminaries. Sections 3, 4 and 5 contain the proofs.

## 1.3. State of the art.
The notion of planar traveling waves has first been investigated in the pioneering papers [14, 28] in the framework of homogeneous scalar Fisher-KPP equations ($N = 1$ and the coefficients are independent of space and time). Such solutions are entire solutions that can have the form $u(t, x) = p(x \cdot e + ct)$, with $p > 0$, $p(-\infty) = 0$ and $p(+\infty) = 1$ (where 1 is the unique positive stationary solution of the scalar equation). Here, $-e \in \mathbb{S}^{n-1}$ is referred to as the direction of propagation, $c$ as the speed of the traveling wave, and $p$ as its profile. Such solutions are important to describe the attractor of the Cauchy problem depending on the form of the initial condition; in particular, the one with minimal speed attracts, in a sense, the solutions of the Cauchy problem associated with compactly supported initial values.

When the coefficients of the scalar Fisher-KPP equation depend periodically on space and remain homogeneous in time, or vice versa, a new notion is needed in order to encapsulate this periodicity. The notion of pulsating traveling waves has been introduced in [39] for the space periodic case and later generalized to the time
periodic case. Namely, these are solutions of the form $u(t, x) = p(x \cdot e + ct, x)$ (space periodic case) or $u(t, x) = p(x \cdot e + ct, t)$ (time periodic case), where $p > 0$, $p(z, \cdot)$ is periodic for all $z \in \mathbb{R}$, $p(-\infty, \cdot) = 0$ and $p(+\infty, \cdot)$ is the unique uniformly positive entire solution. The existence of such solutions for scalar Fisher-KPP equations has been first derived in [4, 7, 27] (space periodic case) and in [16] (time periodic case).

When the coefficients are periodic in space and time both, scalar Fisher-KPP pulsating traveling waves now take the form $u(t, x) = p(x \cdot e + ct, t, x)$, where $(t, x) \mapsto p(z, t, x)$ is periodic for all $z$, $z \mapsto p(z, t, x)$ is increasing for all $(t, x)$, $p(-\infty, \cdot, \cdot) = 0$ and $p(+\infty, \cdot, \cdot)$ is the unique uniformly positive entire solution [34, 35]. The construction of such solutions requires some care since they are somehow the trace of a function $p(z, t, x)$ over the plane $z = x \cdot e + ct$.

The usual tool when studying entire solutions in scalar reaction-diffusion models is the *comparison principle*, namely the property that ordered initial conditions yield perpetually ordered solutions. It is well known that the comparison principle does not necessarily hold for systems. Systems that satisfy the comparison principle are referred to as *monotone* or *cooperative*; consistently, systems that do not satisfy the comparison principle are referred to as *non-monotone* or *non-cooperative*. A weakly coupled system of size $N$ is monotone if it satisfies the so-called *Kamke condition*: for each $i \in [N]$, the $i$-th reaction term is nondecreasing with respect to each $u_j$, $j \in [N] \setminus \{i\}$ [10, 40].

For monotone KPP-type systems (e.g., (KPP) but with $\mathbf{B} = \mathbf{I}$), arguments relying upon the comparison principle have proved the existence of traveling waves when the coefficients are homogeneous [29, 40] or of pulsating traveling waves when the coefficients are periodic [12, 13, 42].

However, due to the assumption $(A_4)$, (KPP) is monotone if and only if $N=1$. When $N \ge 2$, it is known that $N$, $\mathbf{L}$ and $\mathbf{B}$ can be chosen in such a way that a comparison principle holds true in a neighborhood of 0 or away from it [11, Proposition 2.5]. But such choices are not generic and never lead to a global comparison principle.

For non-cooperative systems, a variational approach is sometimes possible when the reaction term has a gradient structure and $\mathrm{diag}(\mathcal{P}) = \partial_t - \Delta$ [32, 33, 38]. The system (KPP) does not satisfy these assumptions in general. Even with $\mathrm{diag}(\mathcal{P}) = \partial_t - \Delta$ and $\mathbf{L}$ symmetric, the term $-\mathbf{B}\mathbf{u} \circ \mathbf{u}$ does not have a gradient structure.

New arguments are therefore needed in order to derive the existence of (pulsating) traveling waves for (KPP) when $N > 1$. Such existence results have been derived before in the following special cases:

* $N=2, n=1$, no advection and space-time homogeneous coefficients [24, 31];

* $n=1$, no advection, space-time homogeneous coefficients and pointwise essentially positive $\mathbf{L}$ [41];

* $n=1$, no advection and space-time homogeneous coefficients [19];

* $N=2, n=1, A_1 = A_2 = 1$, no advection, space periodic coefficients and pointwise essentially positive $\mathbf{L}$ [1];

* $N=2, n=1$, space periodic coefficients and pointwise essentially positive $\mathbf{L}$ [23].

In this paper, we present a unifying approach for non-cooperative KPP systems of arbitrary size $N \in \mathbb{N}^*$, in arbitrary spatial dimension $n \in \mathbb{N}^*$, with space-time
periodic coefficients, with advection and with minimal positivity requirements on L. This was raised as an open question in a previous work by the first author [21]. To the best of our knowledge, the present paper is the very first study on invasions of open space modeled by entire solutions of reaction-diffusion non-cooperative systems with space-time periodic coefficients.

1.4. Motivations. Systems of the form (KPP) arise naturally in population dynamics modeling when the population under consideration has to be divided into discrete classes. Extensive references and discussions on these models can be found in [19, 22]. Here we only suggest briefly three examples of application.

Alfaro-Griette [1] proved the existence of a pulsating traveling wave for the following system:

$$ \begin{cases} \partial_t u = \partial_{xx} u + u(r_u(x) - \gamma_u(x)(u+v)) + \mu(x)v - \mu(x)u, \\ \partial_t v = \partial_{xx} v + v(r_v(x) - \gamma_v(x)(u+v)) + \mu(x)u - \mu(x)v. \end{cases} $$

This system was conceived as a model in evolutionary epidemiology for spatio-temporal dynamics, in a one-dimensional spatially periodic medium, of a pathogen population with two phenotypes u and v. Each phenotype $i \in \{u, v\}$ has a growth rate $r_i = r_i(x)$, is subjected to Lotka-Volterra competition exerted by the total population $u + v$ at a rate $\gamma_i = \gamma_i(x)$, and mutates into the other phenotype at rate $\mu = \mu(x)$. All coefficients are uniformly positive. The one-dimensional spatial periodicity was presented as a first approach of spatial heterogeneities, which are known to influence greatly host-parasite systems.

In a theoretical evolutionary biology article, Ress et al. [37] analyzed the following ODE model:

$$ \begin{cases} \frac{du_1}{dt} = -b_1 u_1 - d_1 u_1 + Nb_N \pi_1 u_N - u_1 \sum_{j=1}^{N} k_{1,j} u_j, \\ \frac{du_i}{dt} = (i-1)b_{i-1} u_{i-1} - ib_i u_i - d_i u_i + Nb_N \pi_i u_N - u_i \sum_{j=1}^{N} k_{i,j} u_j \quad \text{for } 2 \le i \le N. \end{cases} $$

This is a model for the population dynamics of cell groups that grow in size and fragment, giving rise to new cell groups. The cell group densities $u_i$ contain groups of $i$ cells and are characterized by division rates $b_i > 0$ (thus the cell group grows at rate $ib_i$), death rates $d_i > 0$, fragmentation rates $\pi_i \ge 0$, and competition rates $k_{i,j} > 0$. The cell group density $u_N$ contains groups with the maximal number of cells. When a cell group reaches after a division the size $N+1$, it immediately fragments and gives rise to $\pi_1$ groups of size 1, $\pi_2$ groups of size 2, etc., so that $\sum_{i=1}^N i\pi_i = N+1$.

Gueguezo et al. [25] investigated the following model in a bounded spatial domain $\Omega \subset \mathbb{R}^n$ with homogeneous Neumann boundary conditions:

$$ \begin{cases} \partial_t u_1 = d_1 \Delta u_1 + r(t, x) u_2 - s(t, x) u_1 - (a(t, x) + b(t, x)) u_1 + c(t, x) u_2 u_1, \\ \partial_t u_2 = d_2 \Delta u_2 + s(t, x) u_1 - (e(t, x) + f(t, x)) u_2 + g(t, x) u_1 u_2. \end{cases} $$

This system models the dynamics of a stage-structured population with juveniles $u_1$ and adults $u_2$. The parameters $r, s, a, b, c, e, f, g$ are positive periodic functions of time and heterogeneous functions of space. In particular, $r$ is the birth rate of juveniles and $s$ is the maturation rate measuring the transition between the juvenile stage and the adult stage.
Our framework makes it possible to complexify these three models in various biologically meaningful ways and to consider invasion waves.

We especially point out that, in the second example, the corresponding matrix $\mathbf{L}$ is as follows:

$$ \mathbf{L} = \begin{pmatrix} -b_1 - d_1 & 0 & \dots & 0 & Nb_N \pi_1 \\ b_1 & -2b_2 - d_2 & \dots & 0 & Nb_N \pi_2 \\ 0 & 2b_2 & \dots & 0 & Nb_N \pi_3 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \dots & (N-1)b_{N-1} & Nb_N \pi_N - Nb_N - d_N \end{pmatrix} $$

It is neither symmetric nor essentially positive. It is irreducible if $\pi_1 > 0$ and reducible otherwise. If, for instance, coefficients are time periodic due to external stress and fragmentations give rise to groups of size 1 during the first half-period but not during the second half-period, then $\mathbf{L}$ is not always irreducible but its time average is irreducible indeed. This exemplifies the relevancy of the assumption $(A_3)$.

1.5. **Nondimensionalization.** Let $(T, L_1, \dots, L_n)$ be a family of $n+1$ positive numbers and denote temporarily by $\tilde{\tau}$ the composition with the diffeomorphism $(t, x) \mapsto (Tt, L_1x_1, \dots, L_nx_n)$. For any entire solution $\mathbf{u}$ of (KPP),

$$ \frac{1}{T} \partial_t \tilde{\mathbf{u}} = \operatorname{diag} \left( \operatorname{diag} \left( \frac{1}{L_\alpha} \right)_{\alpha \in [n]} \nabla \cdot \left( \tilde{A}_i \operatorname{diag} \left( \frac{1}{L_\alpha} \right)_{\alpha \in [n]} \nabla \right) \right) \tilde{\mathbf{u}} \\ - \operatorname{diag} \left( \tilde{q}_i \cdot \operatorname{diag} \left( \frac{1}{L_\alpha} \right)_{\alpha \in [n]} \nabla \right) \tilde{\mathbf{u}} + \tilde{L} \tilde{\mathbf{u}} - (\tilde{B} \tilde{\mathbf{u}}) \circ \tilde{\mathbf{u}}. $$

Therefore, up to redefining the coefficients of the system in a way that preserves $(A_1)-(A_5)$, we assume without loss of generality and from now on that the coefficients of (KPP) are 1-periodic with respect to each temporal or spatial variable.

1.6. **Entire solutions and invasions of open space.** It is natural to introduce a wave direction $-e \in \mathbb{S}^{n-1}$ (or direction for short) and a wave speed $c \in \mathbb{R}$ (or speed for short), and to consider the change of variable corresponding to the moving frame. Denoting by $\llcorner$ the composition with the diffeomorphism $(t, x) \mapsto (t, x-cte)$, an entire solution $\mathbf{u}$ of (KPP) satisfies:

$$ (\text{KPP}^{\llcorner}) \quad \partial_t \mathbf{u}^{\llcorner} - \operatorname{diag}(\nabla \cdot (A_i^{\llcorner} \nabla)) \mathbf{u}^{\llcorner} + \operatorname{diag}((q_i^{\llcorner} + ce) \cdot \nabla) \mathbf{u}^{\llcorner} = \mathbf{L}^{\llcorner} \mathbf{u}^{\llcorner} - (\mathbf{B}^{\llcorner} \mathbf{u}^{\llcorner}) \circ \mathbf{u}^{\llcorner}. $$

1.6.1. **Almost planar generalized transition waves.** A first class of entire solutions, relevant to describe the planar invasion of $\mathbf{0}$ by positive population densities, can be defined as follows, in the spirit of [5], but taking into account that the asymptotics as $x \cdot e \to +\infty$ are unclear and at least not unique (refer for instance to [18,31] for examples with two stable positive steady states).

**Definition 1.1.** A classical entire solution $\mathbf{u}$ of (KPP) is an almost planar generalized transition wave in direction $-e \in \mathbb{S}^{n-1}$ with global mean speed $c \in \mathbb{R}$ if:

(1) $\mathbf{u}$ is nonnegative;

(2) $\mathbf{u}$ is globally bounded in $\mathbb{R} \times \mathbb{R}^n$;

(3) $\mathbf{u}^{\llcorner} : (t, x) \mapsto \mathbf{u}(t, x - cte)$ satisfies the following two limits:

$$ (\text{downstream}) \quad \lim_{z \to -\infty} \sup_{(t,x) \in \mathbb{R} \times e^{\perp}} \max_{i \in [N]} u_i^{\llcorner}(t, x + ze) = 0, $$
$$ (\text{upstream}) \qquad \liminf_{z \to +\infty} \inf_{(t,x) \in \mathbb{R} \times \mathbb{R}^\perp} \min_{i \in [N]} u_i^\diamond(t, x + ze) > 0. $$

However this notion of solution is in the present context unsatisfying; the space-time periodicity of the coefficients of (KPP) should be somehow inherited by the profile $u^\diamond$ of the wave. We are therefore seeking for almost planar generalized transition waves that satisfy in addition some form of periodicity.

### 1.6.2. Planar pulsating traveling waves.
This leads to the notion of planar pulsating traveling wave solution in space-time periodic media. For consistency, it should contain as particular cases several notions already investigated in the literature and mentioned above:

(1) if $N = 1$, the notion of scalar pulsating traveling wave solution when coefficients are space-time periodic [34, 35];

(2) if $N = 2$ and $n = 1$, the notion of pulsating traveling wave solution when coefficients are space periodic and time homogeneous [1, 23];

(3) if $N \ge 2$, $n = 1$ and coefficients are space-time homogeneous, the notion of (classical) traveling wave solution [19, 24, 31, 41].

In particular, we recall that a scalar pulsating traveling wave solution is an almost planar generalized transition wave in direction $-e$ with global mean speed $c$ and such that $u(t, x) = p(x \cdot e + ct, t, x)$, with a profile $p$ which is increasing in its first variable and periodic in its last two variables.

However, in view of the space-time homogeneous case, some monotonicity loss with respect to the first variable seems inevitable as soon as $N \ge 2$ (cf. [24] where a non-monotonic profile is exhibited for $N = 2$); at best, the requirement could be that $p$ is monotonic in its first variable in a left half-line [19]. This leads to the following candidate.

**Definition 1.2.** A classical entire solution $u$ of (KPP) is a *planar pulsating traveling wave in direction $-e \in \mathbb{S}^{n-1}$ at speed $c \in \mathbb{R}$ if:

(1) it is an almost planar generalized transition wave in direction $-e$ with global mean speed $c$;

(2) it has the form $u(t, x) = p(x \cdot e + ct, t, x)$;

(3) with respect to its second and third variable, the wave profile $p$ is 1-periodic.

Interestingly, this definition could also encompass the notion of point-to-periodic traveling wave solution, useful at least for systems with $N \ge 3$, space-time homogeneous coefficients, and Hopf bifurcations [20].

Unfortunately, it turns out that the monotonicity loss in a right half-line breaks the known method of proof of existence [34, 35]. Despite substantial efforts, we did not manage to circumvent this obstacle. Hence it remains unclear whether this definition has any new value at all.

Instead, we will exploit a weaker notion, inspired in particular by the definition of *almost pulsating wave* introduced in [13] for monotone systems in one-dimensional spaces.

By virtue of the space-time periodicity of the coefficients of (KPP), if $e \in \mathbb{Q}^n$ and $c \in \mathbb{Q}$, then the coefficients of (KPP)$^\diamond$ still satisfy certain space-time periodicity properties. Indeed, there exists a family $(e'_\alpha)_{\alpha \in [n]}$ such that $e'_n = e$, $e'_\alpha \in \mathbb{Q}^n$ for all $\alpha \in [n-1]$, and $(e'_\alpha)_{\alpha \in [n-1]}$ is an orthonormal basis of $e^\perp$ [6, Lemma 4.2]. Hence there exists a large integer $M \in \mathbb{N}$ such that $Me'_\alpha \in \mathbb{Z}^n$ for any
$\alpha \in [n]$ and $cMe \in \mathbb{Z}$, and therefore, for any coefficient $f$ of (KPP$^\diamond$) and any family $(\epsilon_\alpha)_{\alpha+1 \in [n+1]} \in \{0, 1\}^{n+1}$,

$$ 
\begin{aligned}
f^\diamond \left( t + \epsilon_0 M, x + M \sum_{\alpha \in [n]} \epsilon_\alpha e'_\alpha \right) &= f \left( t + \epsilon_0 M, x - cte + \sum_{\alpha \in [n]} M \epsilon_\alpha e'_\alpha - cM \epsilon_0 e \right) \\
&= f(t, x - cte) \\
&= f^\diamond(t, x)
\end{aligned}
$$

This means that $f^\diamond$ is $M$-periodic with respect to time and with respect to each new orthogonal spatial coordinate $x'_\alpha$. From now on, we denote $T_{e,c}$ and $L_{e,\alpha}$ the respective minimal periods in the variables $t$ and $x'_\alpha$, $\alpha \in [n]$, for the whole (finite) family of coefficients (note that $L_e = (L_{e,\alpha})_{\alpha \in [n]}$ does not depend on $c$).

**Definition 1.3.** Let $e \in \mathbb{Q}^n$ and $c \in \mathbb{Q}$.

A classical entire solution $\mathbf{u}$ of (KPP) is a weak planar pulsating traveling wave in direction $-e \in \mathbb{S}^{n-1}$ at speed $c \in \mathbb{R}$ if:

(1) it is an almost planar generalized transition wave in direction $-e$ with global mean speed $c$;

(2) $\mathbf{u}^\diamond : (t, x) \mapsto \mathbf{u}(t, x - cte)$ is $T_{e,c}$-periodic with respect to $t$ and, for each $\alpha \in [n-1]$, $L_{e,\alpha}$-periodic with respect to the spatial coordinate $x'_\alpha$ of $e^\perp$.

Theorems 1.2 and 1.3 below will confirm the existence of such solutions.

**1.7. The linear part and the nonlinear part.** For brevity, we will denote from now on

$$ \mathcal{Q} = \mathrm{diag}(\mathcal{P}) - \mathbf{L} $$

the linear operator derived from the linearization of (KPP) at $\mathbf{u} = \mathbf{0}$. By virtue of the assumptions $(A_2)-(A_3)$, this linear operator is cooperative and fully coupled, and this will be a key property in the forthcoming analysis of the non-cooperative semilinear system (KPP). On the contrary, the nonlinear remainder of the reaction term $-(\mathbf{B}\mathbf{u}) \circ \mathbf{u}$ plays a secondary role, and results can be generalized with minor technical adaptations to reaction terms of the form $\mathbf{L}(t, x)\mathbf{u}-\mathbf{f}(t, x, \mathbf{u}) \circ \mathbf{u}$ satisfying the following assumption of KPP type:

$$ \forall (t, x) \in \mathbb{R} \times \mathbb{R}^n, \begin{cases} \mathbf{f}(t, x, \mathbf{0}) = \mathbf{0}, \\ \mathbf{f}(t, x, \mathbf{v}) \ge \mathbf{0} \quad \text{if } \mathbf{v} \ge \mathbf{0}. \end{cases} $$

Such a generalization can be found in a previous paper by the first author [19]. The motivation of the restriction in the present paper is twofold: on one hand, the form $\mathbf{L}\mathbf{u} - \mathbf{B}\mathbf{u} \circ \mathbf{u}$ is sufficient for the applications we have in mind; on the other hand, it minimizes the verbosity and highlights the role of the linear part.

**1.8. Notations.** Generally speaking, notations are chosen consistently with a previous paper on space-time homogeneous coefficients by the first author [19] and with a paper by I. Mazari and the first author on the principal spectral analysis of $\mathcal{Q}$ [22].

In the whole paper, $\mathbb{N}$ is the set of nonnegative integers, which contains 0.

Unless specified otherwise, time and space periodicities refer to, respectively, 1-periodicity with respect to $t$ and 1-periodicity with respect to $x_\alpha$ for each $\alpha \in [n]$ (or 1-periodicity with respect to $x$ for short).
Vectors in $\mathbb{R}^N$ and matrices in $\mathbb{R}^{N \times N}$ are denoted in bold font. Functional operators are denoted in calligraphic typeface (bold if they act on functions valued in $\mathbb{R}^N$). Functional spaces, e.g. $W^{1,\infty}(\mathbb{R} \times \mathbb{R}^n, \mathbb{R}^N)$, are also denoted in calligraphic typeface. A functional space $\mathcal{X}$ denoted with a subscript $\mathcal{X}_{\text{per}}$, $\mathcal{X}_{t-\text{per}}$ or $\mathcal{X}_{x-\text{per}}$ is restricted to functions that are space-time periodic, time periodic or space periodic respectively. Lebesgue and Sobolev spaces restricted to space-time, time or space periodic functions correspond naturally to integrals over the space-time, time or space periodicity cell respectively.

For clarity, Hölder spaces of functions with $k \in \mathbb{N}$ derivatives that are all Hölder-continuous with exponent $\alpha \in (0, 1)$ are denoted $C^{k+\alpha}$; when the domain is $\mathbb{R} \times \mathbb{R}^n$, it should be unambiguously understood that $C^{k+\alpha, k'+\alpha'}$ denotes the set of functions that have $k$ derivatives in time and $k'$ derivatives in space, which are all uniformly $\alpha$-Hölder-continuous in time and $\alpha'$-Hölder-continuous in space.

For any two vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^N$, $\mathbf{u} \le \mathbf{v}$ means $u_i \le v_i$ for all $i \in [N]$, $\mathbf{u} < \mathbf{v}$ means $\mathbf{u} \le \mathbf{v}$ together with $\mathbf{u} \ne \mathbf{v}$ and $\mathbf{u} \ll \mathbf{v}$ means $u_i < v_i$ for all $i \in [N]$. If $\mathbf{u} \ge \mathbf{0}$, we refer to $\mathbf{u}$ as *nonnegative*; if $\mathbf{u} > \mathbf{0}$, as *nonnegative nonzero*; if $\mathbf{u} \gg \mathbf{0}$, as *positive*. The sets of all nonnegative, nonnegative nonzero, positive vectors are respectively denoted $[0, \infty)$, $[0, \infty) \setminus \{0\}$ and $(0, \infty)$. The vector whose entries are all equal to 1 is denoted **1** and this never refers to an indicator function. Similar notations and terminologies might be used in other dimensions and for matrices. The identity matrix is denoted **I**.

Similarly, a function can be nonnegative, nonnegative nonzero, positive. For clarity, a positive function is a function with only positive values.

To avoid confusion between operations in the state space $\mathbb{R}^N$ and operations in the spatial domain $\mathbb{R}^n$, Latin indexes $i, j, k$ are assigned to vectors and matrices of size $N$ whereas Greek indexes $\alpha, \beta, \gamma$ are assigned to vectors and matrices of size $n$. We use mostly subscripts to avoid confusion with algebraic powers, but when both Latin and Greek indexes are involved, we move the Latin ones to a superscript position, e.g. $A_{\alpha,\beta}^i(t, x)$. We denote scalar products in $\mathbb{R}^N$ with the transpose operator, $\mathbf{u}^\mathrm{T}\mathbf{v} = \sum_{i=1}^N u_i v_i$, and scalar products in $\mathbb{R}^n$ with a dot, $x \cdot y = \sum_{\alpha=1}^n x_\alpha y_\alpha$.

For any vector $\mathbf{u} \in \mathbb{R}^N$, $\mathrm{diag}(\mathbf{u})$, $\mathrm{diag}(u_i)_{i \in [N]}$ or $\mathrm{diag}(u_i)$ for short refer to the diagonal matrix in $\mathbb{R}^{N \times N}$ whose $i$-th diagonal entry is $u_i$. These notations can also be used if $\mathbf{u}$ is a function valued in $\mathbb{R}^N$.

Finite dimensional Euclidean norms are denoted $|\cdot|$ whereas the notation $\|\cdot\|$ is reserved for norms in functional spaces.

The notation $\circ$ is reserved in the paper for the Hadamard product (component-wise product of vectors or matrices) and never refers to the composition of functions. For any vector $\mathbf{v} \in \mathbb{R}^N$ and $p \in \mathbb{R}$, $\mathbf{v}^{\circ p}$ denotes the vector $(v_i^p)_{i \in [N]}$.

1.9. Main result. Before stating the main result, we need to introduce the periodic principal eigenvalue of $\mathcal{Q}$:

$$ (1) \quad \lambda_{1, \mathrm{per}} = \inf \left\{ \lambda \in \mathbb{R} \mid \exists \mathbf{u} \in W^{1, \infty} \cap C_{t-\mathrm{per}}^{1, 2}(\mathbb{R} \times \mathbb{R}^n, (0, \infty)) \mathcal{Q} \mathbf{u} \le \lambda \mathbf{u} \right\}, $$

and to recall results from our work on persistence, extinction and spreading properties of the solutions of the Cauchy problem associated with (KPP) [21].

By virtue of [21, Theorem 1.1], if $\lambda_{1, \mathrm{per}} \ge 0$, all solutions of the Cauchy problem go extinct uniformly in space and, in particular, there exists no pulsating traveling
wave solution. On the contrary, by [21, Theorem 1.3, Theorem 1.4], if $\lambda_{1,\text{per}} < 0$, solutions of the Cauchy problem whose initial values have a sufficiently slow exponential decay persist locally uniformly and spread in space, in the following sense:

$$ (2) \qquad \liminf_{t \to +\infty} \inf_{i \in [N]} \inf_{|x| \le R} u_i(t, x) > 0 \quad \text{for all } R > 0. $$

These solutions are natural candidates for long-time convergence (in some sense) to pulsating traveling waves.

To state our main result, we also need to define a modified operator parameterized by a pair $(\mu, e) \in [0, +\infty) \times \mathbb{S}^{n-1}$:

$$ (3) \qquad \mathcal{Q}_{\mu e} = \mathcal{Q} - \operatorname{diag}(2\mu A_i e \cdot \nabla) - \operatorname{diag}(\mu^2 e \cdot A_i e + \mu \nabla \cdot (A_i e) - \mu q_i \cdot e) $$

and its periodic principal eigenvalue $\lambda_{1,\mu e} = \lambda_{1,\text{per}}(\mathcal{Q}_{\mu e})$. Defining $e_{\mu e} : x \mapsto e^{\mu e \cdot x}$, it can be verified that the periodic principal eigenfunction $\mathbf{u}_{\mu e}$ of $\mathcal{Q}_{\mu e}$, normalized by $\max_{i \in [N]} \max_{(t,x) \in \mathbb{R} \times \mathbb{R}^n} u_{\mu e, i}(t, x) = 1$, satisfies:

$$ (4) \qquad \mathcal{Q}(e_{\mu e} \mathbf{u}_{\mu e}) = \lambda_{1,\mu e} e_{\mu e} \mathbf{u}_{\mu e}. $$

We also define

$$ (5) \qquad c_e^* = \min_{\mu > 0} \frac{-\lambda_{1,\mu e}}{\mu}. $$

The fact that the function $\mu \mapsto \frac{-\lambda_{1,\mu e}}{\mu}$ admits a unique global minimum is standard in KPP-type problems and will be recalled below in Lemma 2.2. Note that this minimum could be negative.

Our main results are the following ones.

**Theorem 1.1.** Assume $\lambda_{1,\text{per}} < 0$.

Then, for any $e \in \mathbb{S}^{n-1}$ and $c \in (-\infty, c_e^*)$, there exists no almost planar generalized transition wave solution in direction $-e$ with global mean speed $c$.

**Theorem 1.2.** Assume $\lambda_{1,\text{per}} < 0$.

Then, for any $e \in \mathbb{S}^{n-1} \cap \mathbb{Q}^n$ and $c \in (c_e^*, +\infty) \cap \mathbb{Q}$, there exists a weak planar pulsating traveling wave solution in direction $-e$ with speed $c$.

**Theorem 1.3.** Assume $\lambda_{1,\text{per}} < 0$.

Then, for any $e \in \mathbb{S}^{n-1} \cap \mathbb{Q}^n$ such that $c_e^* \in \mathbb{Q}$, there exists a weak planar pulsating traveling wave solution in direction $-e$ with speed $c_e^*$.

In the special case where coefficients are spatially homogeneous, the same method of proof of existence works with no rationality assumption and gives the existence of almost planar generalized transition waves with profiles that are 1-periodic with respect to $t$ and homogeneous with respect to the $n-1$ coordinates $x'_\alpha$ of $e^\perp$. Usually, when coefficients are independent of space and periodic in time, such solutions form the definition of strong planar pulsating traveling waves [2, 16]. This corresponds indeed to a particular case of Definition 1.2.

**Corollary 1.4.** Assume $\lambda_{1,\text{per}} < 0$.

Assume moreover that the coefficients of (KPP) are homogeneous with respect to space.

Then, for any $e \in \mathbb{S}^{n-1}$ and $c \in \mathbb{R}$, there exists a (strong) planar pulsating traveling wave solution in direction $-e$ with speed $c$ if and only if $c \ge c_e^*$.
1.10. **Comments.** Despite the density of $\mathbb{Q}$ in $\mathbb{R}$, the notion of weak planar pulsating traveling wave solution presented in Definition 1.3 is not sufficiently rigid to prove anything more than the existence of almost planar generalized transition waves when $e \notin \mathbb{Q}^n$ or $c \notin \mathbb{Q}$. Nonetheless, the following conjecture is natural.

**Conjecture 1.5.** For all $e \in \mathbb{S}^{n-1}$ and $c \ge c_e^*$, there exists a strong planar pulsating traveling wave solution in direction $-e$ with speed $c$, in the sense of Definition 1.2.

Conjecture 1.5 is known to be true in particular cases mentioned above (cf. Section 1.6.2 and Corollary 1.4). Nevertheless, at the highest degree of generality, its proof remains entirely open. It is indeed not clear that the weak planar pulsating traveling waves provided by Theorems 1.2 and 1.3 are actually strong planar pulsating traveling waves in the sense of Definition 1.2.

In [21], where the Cauchy problem associated with (KPP) was studied, the construction of pulsating traveling waves was mentioned as an open problem. Our results bring partial answers. The proof of Conjecture 1.5 would complete the answer for planar waves.

The fact that $c_e^*$ might be of any sign is in particular due to the presence of arbitrary advection terms $(q_i)_{i \in [N]}$. For instance, recalling that the one-dimensional scalar KPP equation $\partial_t u = \partial_{xx}u + u(1-u)$ admits 2 as bidirectional minimal wave speed, it follows that the shifted equation $\partial_t u = \partial_{xx}u + q\partial_x u + u(1-u)$, with any $q \in \mathbb{R}$, admits $2 \pm q$ as leftward and rightward minimal wave speeds.

Circumventing the absence of existence-comparison principle for the semilinear non-cooperative system will be achieved with a fixed point argument similar to the one in [19], itself inspired by [8]. In this sense, this technical circumvention is not a novelty of this paper, even though it is the first time to the best of our knowledge it is applied in a non-cooperative space-time periodic framework.

It follows from [21, Corollary 2.2] that all almost planar generalized transition waves are globally bounded from above by the same constant vector $K1$, with a constant $K > 0$ that only depends on bounds on $L$ and $B$. From this uniform boundedness, the uniform positivity in the upstream asymptotic in Definition 1.1 will follow from an argument involving the Harnack inequality, as in [19].

Apart from its uniform positivity and boundedness, the behavior of the profiles $u^{\triangle}(t, x + ze)$ as $z \to +\infty$ is unknown when $N > 1$. We recall that, contrarily to the scalar case, there might be multiple uniformly positive entire solutions of (KPP) susceptible of being the limit as $z \to +\infty$. In fact it is not even clear that the profiles converge as $z \to +\infty$. This observation is neither new nor specific to the space-time periodic setting [18, 19].

The behavior of the profiles $u^{\triangle}(t, x + ze)$ as $z \to -\infty$ is less elusive. As in the scalar case [34] or in the case of systems with homogeneous coefficients [17], exponential super- and sub-estimates follow from the construction. However these apply *a priori* only to the constructed profiles, not to all existing profiles. The exponential approximation of all profiles requires much more work, cf. [18] in the case of homogeneous coefficients or [26] in the case of scalar equations with space periodic coefficients. This is outside the scope of this paper.

We emphasize that the minimal wave speed in direction $-e$ is denoted $c_e^*$ and not $c_{-e}^*$. This change in notation compared to our previous work [21] is done for notational convenience and for consistency with the work on the scalar case [34]. It should not be a cause for confusion.
## 2. PRELIMINARIES

In this section, we establish or recall basic results that will be used repeatedly in the main proofs.

### 2.1. Harnack inequality for linear cooperative systems.
For self-containment and ease of reading, we recall here the Harnack inequality that was proved in [22]. It is a refinement of Földes-Poláčik's Harnack inequality [15, Theorem 3.9] for parabolic cooperative systems and Arapostathis-Ghosh-Marcus's Harnack inequality [3, Theorem 2.2] for elliptic cooperative systems.

We denote by $\sigma > 0$ the smallest positive entry of $\bar{\mathbf{L}}$ (cf. (A$_3$)) and by $K \ge 1$ the smallest positive number such that

$$ 
\begin{aligned}
K^{-1} &\le \min_{i \in [N]} \min_{y \in \mathbb{S}^{n-1}} \min_{(t,x) \in [0,1]^{1+n}} (y \cdot A_i(t,x)y), \\
\max_{i \in [N]} \max_{y \in \mathbb{S}^{n-1}} \max_{(t,x) \in [0,1]^{1+n}} (y \cdot A_i(t,x)y) &\le K, \\
\max_{i \in [N]} \max_{\alpha \in [n]} \max_{(t,x) \in [0,1]^{1+n}} |q_{i,\alpha}(t,x)| &\le K, \\
\max_{i,j \in [N]} \sup_{(t,x) \in [0,1]^{1+n}} |l_{i,j}(t,x)| &\le K.
\end{aligned}
$$

The existence of $K$ is given by (A$_1$) and (A$_5$).

**Proposition 2.1** (Proposition 2.4 in [22]). Let $\theta \ge \max(T, L_1, \dots, L_n)$ and $\mathbf{f} \in \mathcal{L}^\infty \cap \mathcal{C}^{\delta/2, \delta}(\mathbb{R} \times \mathbb{R}^n, \mathbb{R}^N)$, where $\delta \in (0, 1)$ is as in (A$_5$). Let $F > 0$ such that

$$ \max_{i \in [N]} \sup_{(t,x) \in \mathbb{R} \times \mathbb{R}^n} |f_i(t,x)| \le F. $$

There exists a constant $\bar{\kappa}_{\theta,F} > 0$, determined only by $n, N, \sigma, K$ and the parameters $\theta$ and $F$ such that, if $\mathbf{u} \in \mathcal{C}([-2\theta, 6\theta] \times [-\frac{3\theta}{2}, \frac{3\theta}{2}]^n, [0, \infty))$ is a solution of $\mathbf{Q}\mathbf{u} = \text{diag}(\mathbf{f})\mathbf{u}$, then

$$ \min_{i \in [N]} \min_{(t,x) \in [5\theta, 6\theta] \times [-\frac{\theta}{2}, \frac{\theta}{2}]^n} u_i(t,x) \ge \bar{\kappa}_{\theta,F} \max_{i \in [N]} \max_{(t,x) \in [0,2\theta] \times [-\frac{\theta}{2}, \frac{\theta}{2}]^n} u_i(t,x). $$

### 2.2. Existence of the minimal wave speed.

**Lemma 2.2** (Lemma 3.5 in [21]). Let $e \in \mathbb{S}^{n-1}$. Assume $\lambda_{1,\text{per}} < 0$.

The infimum $c_e^* \in \mathbb{R}$ of the image of the function $\mu \in [0, +\infty) \mapsto \frac{-\lambda_{1,\mu e}}{\mu}$ is a minimum attained at a unique $\mu_e^* > 0$.

Moreover, for any $c > c_e^*$, there exists $\mu_e^\wedge, \mu_e^\vee > 0$ such that $\mu_e^\wedge < \mu_e^* < \mu_e^\vee$ and $\frac{-\lambda_{1,\mu_e^\wedge e}}{\mu_e^\wedge} = \frac{-\lambda_{1,\mu_e^\vee e}}{\mu_e^\vee} = c$.

## 3. PROOF OF THEOREM 1.1

*Proof.* Assume there exists $e \in \mathbb{S}^{n-1}$, $c < c_e^*$ and an almost planar generalized transition wave $\mathbf{u}$ in direction $-e$ with global mean speed $c$. Note that the limit downstream implies

$$ (6) \qquad \mathbf{u}\left(t, -\frac{c+c_e^*}{2}te\right) = \mathbf{u}^\triangle\left(t, -\frac{c_e^*-c}{2}te\right) \to \mathbf{0} \quad \text{as } t \to +\infty. $$

By applying Gaussian estimates and global bounds, we are in a position to compare, in present time, certain powers of the components of $\mathbf{u}$. This is stated in [21, Corollary 2.7] and inspired by [9]. Thanks to this nonlinear comparison and
using the fact that $\mathbf{u}$ is an entire solution defined for all times $t \in \mathbb{R}$, it is possible
to prove that, for a certain $D > 0$ and $p \in (0, 1)$,

$$
\mathbf{L}\mathbf{u} - \mathbf{B}\mathbf{u} \circ \mathbf{u} \geq \mathbf{L}\mathbf{u} - D\mathbf{u}^{\circ(1+p)} \quad \text{in } \mathbb{R} \times \mathbb{R}^n.
$$

This is a key comparison between the non-cooperative reaction term and a semilin-
ear cooperative reaction term. Hence, by virtue of the comparison principle applied
to the cooperative operator $\mathbf{v} \mapsto \mathbf{Q}\mathbf{v} + D\mathbf{v}^{\circ(1+p)}$, for any initial condition $\mathbf{u}_0$ satisf-
fying, for all $x \in \mathbb{R}^n$, $\mathbf{u}_0(x) \le \mathbf{u}(0,x)$, the solution $\mathbf{u} : [0, +\infty) \times \mathbb{R}^n \to \mathbb{R}^N$ of the
Cauchy problem

$$
\begin{cases}
\mathbf{Q}\underline{\mathbf{u}} = -D\underline{\mathbf{u}}^{\circ(1+p)} & \text{in } (0, +\infty) \times \mathbb{R}^n, \\
\underline{\mathbf{u}}(0, \cdot) = \underline{\mathbf{u}}_0 & \text{in } \mathbb{R}^n,
\end{cases}
$$

satisfies $\mathbf{u} \ge \mathbf{u}$ in $[0, +\infty) \times \mathbb{R}^n$. Hence, from (6),

$$
(7) \qquad \underline{\mathbf{u}} \left( t, -\frac{c+c_e^*}{2}te \right) \to \mathbf{0} \quad \text{as } t \to +\infty.
$$

Now we specify $\underline{\mathbf{u}}_0 : x \mapsto \varepsilon H(x \cdot e)\mathbf{1}$, where $H$ is the Heavyside function equal to 0 in $(-\infty, 0)$ and to $+1$ in $(0, +\infty)$, and where $\varepsilon$ is chosen appropriately small so that $\underline{\mathbf{u}}_0(\cdot) \le \mathbf{u}(0, \cdot)$ holds true (this is indeed possible because $\mathbf{u}$ is pointwise positive and, as $x \cdot e \to +\infty$, it satisfies upstream).

Since $\mathbf{v} \mapsto \mathbf{Q}\mathbf{v} + D\mathbf{v}^{\circ(1+p)}$ is a cooperative operator with space-time periodic coefficients, we can apply to $\mathbf{u}$ a known spreading result for front-like initial data [12, Theorem 2.1] and deduce that $\mathbf{u}$ spreads at speed $c_e^*$, and in particular

$$
\liminf_{t \to +\infty} \min_{i \in [N]} \underline{u}_i \left( t, -\frac{c+c_i^*}{2} te \right) > 0.
$$

But this contradicts directly (7).

□

4. PROOF OF THEOREM 1.2

In this section, we assume $\lambda_{1,\text{per}} < 0$ and we prove the existence of pulsating traveling waves for all $e \in S^{n-1} \cap \mathbb{Q}^n$ and $c \in (c_e^*, +\infty) \cap \mathbb{Q}$.

The wave direction $e$ and the wave speed $c$ are fixed once and for all. Recall
the notations $T_{e,c}$ and $L_e$ for the minimal periods of the coefficients of (KPP$^<$).
For brevity, we omit thereafter the subscripts $e$ and $c$ in all notations introduced
earlier.

Let $P \in \mathbb{R}^{n \times n}$ be the orthogonal matrix associated with the change of variables
replacing $x$ into $x'$, i.e.

$$
(8) \qquad \begin{pmatrix} x_1 \\ \vdots \\ x_{n-1} \\ x_n \end{pmatrix} = P \begin{pmatrix} x'_1 \\ \vdots \\ x'_{n-1} \\ x'_n \end{pmatrix}.
$$

Remarkning that $\nabla_{x'} = P^T \nabla_x = P^{-1} \nabla_x$, we set

$$
\begin{align*}
A'_i : (t, x') &\mapsto P^T A_i^\dagger(t, P x') P, \\
q'_i : (t, x') &\mapsto P^T (q_i^\dagger + c e)(t, P x'), \\
\mathbf{L}' : (t, x') &\mapsto \mathbf{L}^\dagger(t, P x'),
\end{align*}
$$

$$
(9) \quad \mathcal{R} = \partial_t - \operatorname{diag}(\nabla_{x'} \cdot (A_i' \nabla_{x'})) + \operatorname{diag}(q_i' \cdot \nabla_{x'}) - \mathbf{L}'.
$$
By construction, if $\mathbf{u}'$ is a solution of $\mathcal{R}\mathbf{u}' = -\mathbf{B}'\mathbf{u}' \circ \mathbf{u}'$ (where, consistently, $\mathbf{B}'(t, x') = \mathbf{B}^{\dagger}(t, P x')$), then $\mathbf{u} : (t, x) \mapsto \mathbf{u}'(t, P^T(x + cte))$ is a solution of $\mathcal{Q}\mathbf{u} = -\mathbf{B}\mathbf{u} \circ \mathbf{u}$.

In the following construction, we only work with the variables $(t, x')$, never with the variables $(t, x)$. In order to simplify notations, we sometimes use the generic notation $x' = (y, z) \in \mathbb{R}^{n-1} \times \mathbb{R}$, i.e. $x'_\alpha = y_\alpha$ if $\alpha \in [n-1]$ and $x'_n = z$. The spatial periodicity cell $(0, L) = (0, L_1) \times (0, L_2) \times \cdots \times (0, L_n)$ is then denoted $(0, L_y) \times (0, L_z)$, with $L_y \in \mathbb{R}^{n-1}$ and $L_z \in \mathbb{R}$. The notation $\nabla$ should be unambiguously understood as $\nabla_{x'} = \nabla_{(y,z)}$ in this construction. Also, in this coordinate system, $e$ becomes $e' = P^T e = (0, 0, \dots, 0, 1)^T$, $e \cdot \nabla_x$ becomes $e' \cdot \nabla = \partial_{x'_n} = \partial_z$ and $\nabla_x \cdot (A_i^\dagger e)$ becomes $\nabla \cdot (A_i^\dagger e')$.

The family $(A'_i)_{i \in [N]}$ of diffusivity matrices in the new coordinate system remains uniformly elliptic, i.e. it still satisfies $(\mathbf{A}_1)$. For any $\mu \ge 0$, the operator $\mathcal{Q}_\mu$ defined in (3) becomes

$$ \mathcal{R}_\mu = \mathcal{R} - \operatorname{diag}(2\mu A'_i e' \cdot \nabla) - \operatorname{diag}(\mu^2 e' \cdot A'_i e' + \mu \nabla \cdot (A'_i e') - \mu q'_i \cdot e'). $$

By construction, the coefficients of this operator are space-time periodic (with periods $L$ and $T$). Also, by construction, $\mathbf{u}'_\mu : (t, x') \mapsto \mathbf{u}_\mu(t, P x' - cte)$ is a space-time periodic principal eigenfunction of $\mathcal{R}_\mu$ (with periods $L$ and $T$). Applying the Krein-Rutman theorem, we deduce by uniqueness that all space-time periodic principal eigenfunctions of $\mathcal{R}_\mu$ with the same periods are positively proportional to $\mathbf{u}'_\mu$.

Finally, $(t, x) \mapsto e^{\mu e \cdot x} \mathbf{u}_\mu(t, x)$ becomes $(t, y, z) \mapsto e^{\mu c t} e^{\mu z} \mathbf{u}'_\mu(t, y, z)$, and the identity (4) becomes:

$$ (10) \qquad \mathcal{R}(e_{\mu e'} \mathbf{u}'_\mu) = (\lambda_{1,\mu} + c\mu)e_{\mu e'} \mathbf{u}'_\mu. $$

For clarity, we divide the proof into parts.

**4.1. Systems in truncated cylinders.** In this subsection we fix $a > 0$. We define the truncated cylinder $\Sigma_a = (0, T) \times (0, L_y) \times (-a, a)$. We are going to construct a solution $\mathbf{u}_a$ of the semilinear system $\mathcal{R}\mathbf{u} = -\mathbf{B}'\mathbf{u} \circ \mathbf{u}$ in $\Sigma_a$, with periodic boundary conditions in $(t, y)$ and appropriate inhomogeneous Dirichlet boundary conditions in $z$.

Although the semilinear non-cooperative system does not satisfy the comparison principle, the construction, inspired by the traveling wave construction in homogeneous media [19], relies upon super- and sub-solutions, making it possible to apply a fixed point argument.

Recall that, by virtue of the space-time periodicity and of the Harnack inequality of Proposition 2.1 (see also [22, Remark 2.1]) applied, *mutatis mutandi*, to the operator $\mathcal{R}_\mu$, there exists $\kappa_\mu > 0$ such that $\mathbf{u}'_\mu \ge \kappa_\mu \mathbf{1}$ globally in $\mathbb{R} \times \mathbb{R}^n$. Moreover, it follows from the proof of the Harnack inequality that the Harnack constant can be chosen in such a way that $\mu \in \mathbb{R} \mapsto \kappa_\mu$ is continuous.

**Lemma 4.1.** Let

$$ (11) \qquad \bar{\mathbf{u}} : (t, y, z) \mapsto e^{\mu^\wedge z} \mathbf{u}'_{\mu^\wedge} (t, y, z). $$

Then it satisfies:

$$ (12) \qquad \mathcal{R}\bar{\mathbf{u}} \ge 0 \quad \text{in } \Sigma_a. $$

*Proof.* The function $\bar{\mathbf{u}}$ actually satisfies in $\Sigma_a$:

$$ \mathcal{R}\bar{\mathbf{u}} = (\lambda_{1,\mu^\wedge} + c\mu^\wedge)\bar{\mathbf{u}} = 0. $$
<figure><img src="image_1.png" /></figure>

**Lemma 4.2.** Let

$$ (13) \qquad \gamma = \frac{1}{2} \min \{\mu^{\wedge}, \mu^{\vee} - \mu^{\wedge}\}, $$

$$ (14) \qquad M = \max \left\{ \frac{1}{\kappa_{\mu^{\wedge}+\gamma}}, \frac{N \max_{(i,j) \in [N]^2} \max_{(t,x) \in [0,1]^{1+n}} b_{i,j}(t,x)}{((\lambda_{1,(\mu^{\wedge}+\gamma)e} + c(\mu^{\wedge} + \gamma)) \kappa_{\mu^{\wedge}+\gamma})} \right\}, $$

and

$$ (15) \qquad \underline{\mathbf{u}} : (t, y, z) \mapsto e^{\mu^{\wedge} z} \mathbf{u}'_{\mu^{\wedge}}(t, y, z) - M e^{(\mu^{\wedge}+\gamma)z} \mathbf{u}'_{\mu^{\wedge}+\gamma}(t, y, z). $$

Then it satisfies:

$$ (16) \qquad \underline{\mathbf{u}} \le \mathbf{0} \quad \text{in } \Sigma_a \cap \{z \ge 0\}, $$

$$ (17) \qquad (\mathcal{R} + \operatorname{diag}(\mathbf{B}'\bar{\mathbf{u}})) \underline{\mathbf{u}} \le \mathbf{0} \quad \text{in } \Sigma_a. $$

*Proof.* We fix an arbitrary $(t, x') = (t, y, z) \in \Sigma_a$ and we omit arguments in $(t, y)$ for readability.

On one hand, with the same calculations as in the proof of Lemma 4.1,

$$ \mathcal{R} \underline{\mathbf{u}} = -M (\lambda_{1,\mu^{\wedge}+\gamma} + c(\mu^{\wedge} + \gamma)) e^{(\mu^{\wedge}+\gamma)z} \mathbf{u}'_{\mu^{\wedge}+\gamma}. $$

By definition of $\gamma$, $\mu^{\wedge} + \gamma \in (\mu^{\wedge}, \mu^{\vee})$, whence, by virtue of Lemma 2.2,

$$ \lambda_{1,\mu^{\wedge}+\gamma} + c(\mu^{\wedge} + \gamma) > 0. $$

Using $\mathbf{u}'_{\mu^{\wedge}+\gamma} \ge \kappa_{\mu^{\wedge}+\gamma} \mathbf{1}$ and then the definition of $M$,

$$ \begin{aligned} -\mathcal{R}\underline{\mathbf{u}} &= M(\lambda_{1,\mu^{\wedge}+\gamma} + c(\mu^{\wedge} + \gamma))e^{(\mu^{\wedge}+\gamma)z}\mathbf{u}'_{\mu^{\wedge}+\gamma} \\ &\ge M(\lambda_{1,\mu^{\wedge}+\gamma} + c(\mu^{\wedge} + \gamma))e^{(\mu^{\wedge}+\gamma)z}\kappa_{\mu^{\wedge}+\gamma}\mathbf{1} \\ &\ge e^{(\mu^{\wedge}+\gamma)z}N \max_{(i,j) \in [N]^2} \max_{[0,1]^{1+n}} b_{i,j}\mathbf{1}. \end{aligned} $$

On the other hand, by definition of $\underline{\mathbf{u}}$ and using $\mathbf{u}'_{\mu^{\wedge}} \le \mathbf{1}$ and $\mathbf{u}'_{\mu^{\wedge}+\gamma} \ge \kappa_{\mu^{\wedge}+\gamma}\mathbf{1}$,

$$ \underline{\mathbf{u}} \le e^{\mu^{\wedge} z} (1 - M e^{\gamma z} \kappa_{\mu^{\wedge}+\gamma}) \mathbf{1} \le e^{\mu^{\wedge} z} (1 - M e^{\gamma z} \kappa_{\mu^{\wedge}+\gamma})^{+} \mathbf{1}, $$

whence, by nonnegativity of $\mathbf{B}'\bar{\mathbf{u}}$, definition of $\bar{\mathbf{u}}$ and using again $\mathbf{u}_{\mu^{\wedge}} \le \mathbf{1}$,

$$ (\mathbf{B}'\bar{\mathbf{u}}) \circ \underline{\mathbf{u}} \le e^{2\mu^{\wedge}z} (1 - M e^{\gamma z} \kappa_{\mu^{\wedge}+\gamma})^{+} N \max_{(i,j) \in [N]^2} \max_{[0,1]^{1+n}} b_{i,j} \mathbf{1}. $$

Note that $(1 - M e^{\gamma z} \kappa_{\mu^{\wedge}+\gamma})^{+} > 0$ if and only if $z < z_0 = -\frac{1}{\gamma} \ln (M \kappa_{\mu^{\wedge}+\gamma})$. By definition of $M$, $z_0 \le 0$. By definition of $\gamma$, $\mu^{\wedge} > \gamma$. Hence we deduce the inequality $e^{(\mu^{\wedge}+\gamma)z} \ge e^{2\mu^{\wedge}z} (1 - M e^{\gamma z} \kappa_{\mu^{\wedge}+\gamma})^{+}$.

Consequently $-\mathcal{R}\underline{\mathbf{u}} \ge (\mathbf{B}'\bar{\mathbf{u}}) \circ \underline{\mathbf{u}}$ in $\Sigma_a$ and the proof is ended.

<figure><img src="image_2.png" /></figure>

The function $\mathbf{0}$ is obviously another sub-solution for the linear cooperative operator $\mathcal{R}+\operatorname{diag}(\mathbf{B}'\bar{\mathbf{u}})$. Therefore the component-by-component maximum of $\underline{\mathbf{u}}$ and $\mathbf{0}$, denoted hereafter $\underline{\mathbf{u}} \vee \mathbf{0}$, still provides a sub-solution, that can be used in a generalized version of the existence-comparison principle for linear cooperative operators. This is stated in the following corollary.
**Corollary 4.3.** With the notations of Lemma 4.2,

$$ (18) \qquad (\mathcal{R} + \operatorname{diag}(\mathbf{B}'\bar{\mathbf{u}})) (\mathbf{u} \vee \mathbf{0}) \leq \mathbf{0} \quad \text{in } \Sigma_a $$

where partial derivatives are understood in distributional sense where necessary. The function $\mathbf{u} \vee \mathbf{0}$ is globally Lipschitz-continuous.

We are now in a position to construct a bounded nonnegative entire solution of $\mathcal{R}\mathbf{u} = -\mathbf{B}'\mathbf{u} \circ \mathbf{u}$ in $\Sigma$ with appropriate boundary conditions.

**Proposition 4.4.** Let $a^* > 0$ such that

$$ (19) \qquad (\mathbf{u} \vee \mathbf{0})(t, y, -a^*) = \mathbf{u}(t, y, -a^*) \gg \mathbf{0} \quad \text{for all } (t, y) \in [0, T] \times [0, L_y]. $$

Then, for all $a \ge a^*$, there exists a solution $\mathbf{u}$ of

$$ (20) \qquad \begin{cases} \mathcal{R}\mathbf{u} = -(\mathbf{B}'\mathbf{u}) \circ \mathbf{u} & \text{in } \Sigma_a, \\ \mathbf{u}(t, y, \pm a) = (\mathbf{u} \vee \mathbf{0})(t, y, \pm a) & \text{for all } (t, y) \in [0, T] \times [0, L_y], \\ (t, y) \mapsto [z \mapsto \mathbf{u}(t, y, z)] & \text{is periodic.} \end{cases} $$

Furthermore, it satisfies

$$ (21) \qquad \mathbf{u} \vee \mathbf{0} \le \mathbf{u} \le \bar{\mathbf{u}} \quad \text{in } \Sigma_a. $$

*Proof.* The existence of $a^*$, and the fact that any value $a \ge a^*$ actually satisfies the same condition (19), can be reformulated as the boundedness from below of the set

$$ \{z \in \mathbb{R} \mid \exists (t, y) \in [0, T] \times [0, L_y] \ \exists i \in [N] \ \underline{u}_i(t, y, z) < 0\}. $$

This bound is a direct consequence of the definition of $\mathbf{u}$.

The core of the proof will use the following two differential inequalities, valid for any Hölder-continuous function $\mathbf{r} \in [\mathbf{0}, \bar{\mathbf{u}}]$:

$$ (22) \qquad (\mathcal{R} + \operatorname{diag}(\mathbf{B}'\mathbf{r})) (\mathbf{u} \vee \mathbf{0}) \leq \mathbf{0} \quad \text{in } \Sigma_a, $$

$$ (23) \qquad (\mathcal{R} + \operatorname{diag}(\mathbf{B}'\mathbf{r})) \bar{\mathbf{u}} \geq \mathbf{0} \quad \text{in } \Sigma_a. $$

These are consequences of obvious monotonicity properties of $\mathbf{v} \mapsto \mathbf{B}'\mathbf{v}$ implied by (A$_4$) and of Corollary 4.3 and Lemma 4.1.

To begin the proof, we fix temporarily a Hölder-continuous function $\mathbf{r} \in [\mathbf{0}, \bar{\mathbf{u}}]$ and we study the well-posedness of:

$$ (24) \qquad \begin{cases} (\mathcal{R} + \operatorname{diag}(\mathbf{B}'\mathbf{r}))\mathbf{u} = \mathbf{0} & \text{in } \Sigma_a, \\ \mathbf{p}(t, y, \pm a) = (\mathbf{u} \vee \mathbf{0})(t, y, \pm a) & \text{for all } (t, y) \in [0, T] \times [0, L_y], \\ (t, y) \mapsto [z \mapsto \mathbf{p}(t, y, z)] & \text{is periodic.} \end{cases} $$

Since it is more convenient to manipulate homogeneous Dirichlet boundary conditions, let us transform the system into an equivalent one. By definition of $\bar{\mathbf{u}}$ and $\mathbf{u}$, there exists $f \in C^\infty([-a, a], [0, +\infty))$ such that $f(-a) = 1$, $f(a) = 0$ and, for all $(t, y, z) \in \bar{\Sigma}_a$,

$$ (\mathbf{u} \vee \mathbf{0})(t, y, z) \le f(z)\mathbf{u}(t, y, -a) \le \frac{1}{2}((\mathbf{u} \vee \mathbf{0})(t, y, z) + \bar{\mathbf{u}}(t, y, z)). $$

By setting $\mathbf{f} = f\mathbf{u}(-a)$ and changing $\mathbf{u}$ into $\tilde{\mathbf{u}} = \mathbf{u} - \mathbf{f}$, we find that $\mathbf{u}$ solves (24) if and only if $\tilde{\mathbf{u}}$ solves:

$$ (25) \qquad \begin{cases} (\mathcal{R} + \operatorname{diag}(\mathbf{B}'\mathbf{r}))\tilde{\mathbf{u}} = -(\mathcal{R} + \operatorname{diag}(\mathbf{B}'\mathbf{r}))\mathbf{f} & \text{in } \Sigma_a, \\ \tilde{\mathbf{u}}(t, y, \pm a) = \mathbf{0} & \text{for all } (t, y) \in [0, T] \times [0, L_y], \\ (t, y) \mapsto [z \mapsto \tilde{\mathbf{u}}(t, y, z)] & \text{is periodic.} \end{cases} $$
In order to verify that this new system is well-posed, we have to verify the invertibility of the associated linear operator. Since the principal eigenvalue of a monotone linear operator is the leftmost point of its spectrum in $\mathbb{C}$, it is sufficient to prove that this principal eigenvalue is positive. Denote $\lambda_{1,\text{per-Dir}}(\mathcal{R}+\text{diag}(\mathbf{B}'\mathbf{r}))$ the principal eigenvalue of $\mathcal{R}+\text{diag}(\mathbf{B}'\mathbf{r})$ with time periodicity and periodic-Dirichlet boundary conditions in space. Assume by contradiction

$$ \lambda_{1,\text{per-Dir}}(\mathcal{R} + \text{diag}(\mathbf{B}'\mathbf{r})) \leq 0. $$

On one hand, let $\varphi$ be its associated periodic-Dirichlet principal eigenfunction. It satisfies

$$ (\mathcal{R} + \text{diag}(\mathbf{B}'\mathbf{r}))\varphi = \lambda_{1,\text{per-Dir}}(\mathcal{R} + \text{diag}(\mathbf{B}'\mathbf{r}))\varphi \leq 0 \quad \text{in } \Sigma_a. $$

On the other hand, the function $\bar{u} \gg 0$ satisfies (23). Therefore, for a sufficiently small $\kappa > 0$, the following inequalities hold true in $\Sigma_a$:

$$ 0 \leq \kappa\varphi \leq \bar{u} \quad \text{and} \quad (\mathcal{R} + \text{diag}(\mathbf{B}'\mathbf{r}))\bar{u} - \kappa\varphi \geq 0. $$

For the supremum $\kappa^*$ of such $\kappa$, there exists $(t^*, y^*, z^*) \in \bar{\Sigma}_a$ and $i^* \in [N]$ such that $\bar{u}_{i^*}(t^*, y^*, z^*) - \kappa^*\varphi_{i^*}(t^*, y^*, z^*) = 0$. In view of the boundary conditions, it occurs in fact at $(t^*, y^*, z^*) \in \Sigma_a$. Then the strong comparison principle for linear cooperative operators and the continuity show that $\kappa^*\varphi = \bar{u}$ in $\bar{\Sigma}_a$, a contradiction with the boundary conditions. Hence $\lambda_{1,\text{per-Dir}}(\mathcal{R} + \text{diag}(\mathbf{B}'\mathbf{r})) > 0$.

By (22), the function $\underline{u} \vee 0 - \underline{f} \leq 0$ is a sub-solution of (25). By (23), the function $\bar{u} - \underline{f} \geq 0$ is a super-solution of (25). By monotonicity of the operator, the unique classical solution $\tilde{u}$ of (25) satisfies $\underline{u} \vee 0 - \underline{f} \leq \tilde{u} \leq \bar{u} - \underline{f}$ in $\Sigma_a$.

Therefore, the equivalent problem (24) is well-posed, and its solution, denoted $\mathbf{u}_r$, satisfies $\underline{u} \vee 0 \leq \mathbf{u}_r \leq \bar{u}$ in $\Sigma_a$.

Subsequently, the mapping $\Phi: \mathbf{r} \mapsto \mathbf{u}_r$ defined on the following convex set:

$$ \mathcal{F} = \{ \mathbf{r} \in C_{\text{per}}^{\delta/2, \delta} (\mathbb{R} \times \mathbb{R}^{n-1}, \mathcal{C}^{\delta}([-a, a], \mathbb{R}^N)) \mid \underline{\mathbf{u}} \vee 0 \leq \mathbf{r} \leq \bar{\mathbf{u}} \} $$

is well-defined and satisfies $\Phi(\mathcal{F}) \subset \mathcal{F}$. Furthermore, by classical parabolic estimates and compact embeddings of Hölder spaces, $\Phi$ is a compact mapping. Therefore, by virtue of the Schauder fixed point theorem, there exists a solution $\mathbf{u}$ of (20), that satisfies (21). $\square$

4.2. **System in the non-truncated cylinder.** In this subsection, we pass to the limit $a \to +\infty$ and obtain an entire solution of $\mathcal{R}\mathbf{u} = -\mathbf{B}'\mathbf{u} \circ \mathbf{u}$. Recall that $\Sigma_a = (0, T) \times (0, L_y) \times (-a, a)$.

To do so, we first need a pointwise bound uniform with respect to $a \geq a^*$, where $a^*$ is defined in Proposition 4.4. To emphasize its dependency on $a$, we denote $\mathbf{u}_a$ the solution of (20) constructed in the previous subsection.

**Lemma 4.5.** There exists $K > 0$, dependent only on bounds on $\mathbf{L}$ and $\mathbf{B}$, such that, up to increasing $a^*$ in a way that only depends on $K$ and $\mu^>$,

$$ (26) \qquad 0 \leq \mathbf{u}_a \leq K1 \quad \text{in } \bar{\Sigma}_a \quad \text{for all } a \geq a^*. $$

*Proof.* The proof relies upon the exact same idea as the one in [21, Proposition 2.1] and is actually an *a priori* estimate on solutions of (20). For the sake of completeness and to show precisely how boundary conditions are accounted for, we recall the proof.
By assumptions (A5) and (A4), there exist constants $r, K > 0$ such that, for any $\mathbf{u} \ge 0$,

$$L'\mathbf{u} - B'\mathbf{u} \circ \mathbf{u} \le r (\mathbf{1}^{\mathrm{T}}\mathbf{u})(K\mathbf{1} - \mathbf{u}).$$

By definition of $\mathbf{u}$, it satisfies $\mathbf{u}(t, y, z) \le e^{\mu^\top z} \mathbf{1}$ globally in $\mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$. Therefore, up to increasing $a^*$ in an obvious way, for any $a \ge a^*$, $\mathbf{u}(t, y, -a) \le K\mathbf{1}$ in $[0, T] \times [0, L_y]$.

To ease the reading we denote, for each $i \in [N]$,

$$\tilde{\mathcal{P}}_i = (\mathcal{R} + \mathbf{L}')_i = \partial_t - \nabla \cdot (A_i' \nabla) + q_i' \cdot \nabla$$

Solutions $\mathbf{u}$ of (20) satisfy, for each $i \in [N]$,

$$\tilde{\mathcal{P}}_i u_i \le r(K - u_i) \sum_{j=1}^{N} u_j.$$

Whenever $u_i \ge K$,

$$\tilde{\mathcal{P}}_i u_i \le r u_i (K - u_i).$$

In particular, $v_i = \max(u_i, K)$ is a sub-solution of the periodic-Dirichlet semilinear scalar problem:

$$(27) \qquad \begin{cases} \tilde{\mathcal{P}}_i u = r u(K-u) & \text{in } \Sigma_a, \\ u(t,y, \pm a) = K & \text{for any } (t,y) \in [0,T] \times [0,L_y], \\ (t,y) \mapsto [z \mapsto u(t,y,z)] & \text{is periodic.} \end{cases}$$

Let $\tilde{K} = \max_{\Sigma_a} v_i$ and assume by contradiction $\tilde{K} > K$. The constant $\tilde{K}$ is a super-solution of (27) and, by definition, it satisfies $\tilde{K} \ge v_i$. Also by definition, it satisfies $\tilde{K} = v_i(t^*, y^*, z^*)$ for certain $(t^*, y^*, z^*) \in \Sigma_a$. In view of the boundary conditions, $z^* \in (-a, a)$. By virtue of the scalar comparison principle [36], $v_i = \tilde{K}$ globally in $\Sigma_a$. But this contradicts the boundary conditions satisfied by $v_i$. Hence $\tilde{K} = K$. This ends the proof. $\square$

Now, for all $a \ge a^*$, we extend $\mathbf{u}_a$ in $\mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$ by setting $\mathbf{u}_a = \mathbf{0}$ outside $\mathbb{R} \times \mathbb{R}^{n-1} \times [-a, a]$. This extension is continuous and periodic with respect to $(t, y)$. It also satisfies the estimate (26).

Thanks to this estimate, classical parabolic estimates [30] and a diagonal extraction process, we can extract a sequence $(a_k, \mathbf{u}_{a_k})_{k \in \mathbb{N}}$ such that $a_k \to +\infty$ and $\mathbf{u}_{a_k}$ converges in $C_{\mathrm{per}}^{1,2}(\mathbb{R} \times \mathbb{R}^{n-1}, C_{\mathrm{loc}}^2(\mathbb{R}, \mathbb{R}^N))$ to some limit $\mathbf{u}'$ satisfying $\mathbf{u} \vee \mathbf{0} \le \mathbf{u}' \le \bar{\mathbf{u}} \wedge (K\mathbf{1})$ globally, where $\bar{\mathbf{u}} \wedge (K\mathbf{1})$ denotes the component-by-component minimum of $\bar{\mathbf{u}}$ and $K\mathbf{1}$. For future reference, this is stated in the following corollary.

**Corollary 4.6.** There exists a solution $\mathbf{u}' \in C_{\mathrm{per}}^{1,2}(\mathbb{R} \times \mathbb{R}^{n-1}, C^2(\mathbb{R}, \mathbb{R}^N))$ of

$$(28) \qquad \mathcal{R}\mathbf{u}' = -(\mathbf{B}'\mathbf{u}') \circ \mathbf{u}' \quad \text{in } \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}.$$

Furthermore, it satisfies

$$(29) \qquad \mathbf{u} \vee \mathbf{0} \le \mathbf{u}' \le \bar{\mathbf{u}} \wedge (K\mathbf{1}) \quad \text{in } \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}.$$
4.3. Upstream positivity. In this subsection, we prove the asymptotic uniform positivity of the profile, (upstream). This will end the proof of Theorem 1.2.

**Proposition 4.7.** *Necessarily,*

$$ \liminf_{z \to +\infty} (t, y) \in [0, T] \times [0, L_y] \min_{i \in [N]} u'_i(t, y, z) > 0. $$

*Proof.* Assume by contradiction

$$ (30) \qquad \liminf_{z \to +\infty} \min_{(t,y) \in [0,T] \times [0,L_y]} \min_{i \in [N]} u'_i(t, y, z) = 0. $$

Let $(i_k, t_k, y_k)_{k \in \mathbb{N}} \in ([N] \times [0, T] \times [0, L_y])^{\mathbb{N}}$ and $(z_k)_{k \in \mathbb{N}}$ such that $z_k \to +\infty$ and $u'_{i_k}(t_k, y_k, z_k) \to 0$ as $k \to +\infty$. Up to extraction, $(i_k, t_k, y_k) \to (i_\infty, t_\infty, y_\infty) \in [N] \times [0, T] \times [0, L_y]$. Moreover, let $Z_k \in [0, L_z]$ such that, for each $k \in \mathbb{N}$, $\zeta_k = z_k - Z_k \in L_z \mathbb{Z}$. Up to another extraction, $(\zeta_k)_{k \in \mathbb{N}}$ is increasing and $Z_k \to Z_\infty \in [0, L_z]$. It follows by continuity that

$$ u'_{i_\infty}(t_\infty, y_\infty, \zeta_k + Z_\infty) \to 0 \quad \text{as} \quad k \to +\infty. $$

Applying the Harnack inequality of Proposition 2.1 to $(t, y, z) \mapsto \mathbf{u}'(t + t_\infty, y + y_\infty, z + \zeta_k + Z_\infty)$ and to the translated operator, denoted for simplicity $\mathcal{R}(t + t_\infty, y + y_\infty, z + Z_\infty) + \text{diag}(\mathbf{B}'\mathbf{u}')(t + t_\infty, y + y_\infty, z + Z_\infty)$, we easily deduce that

$$ \max_{i \in [N]} u'_{i}(t, y, z + \zeta_k) \to 0 $$

uniformly with respect to $(t, y, z) \in [0, T] \times [0, L_y] \times [0, L_z]$.

In general this shows that limit points as $z \to +\infty$ are either **0** or in $(0, K1)$. By (30), here it is **0**. Thanks to this convergence, we are now going to replace the semilinear system by a linearized one.

For any $i \in [N]$ and any $k \in \mathbb{N}$, the function

$$ \mathbf{v}_k : (t, y, z) \mapsto \frac{1}{u'_{i}(t_\infty, y_\infty, \zeta_k + Z_\infty)} \mathbf{u}'(t + t_\infty, y + y_\infty, z + \zeta_k + Z_\infty) $$

satisfies, for all $(t, y, z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$ and $k \in \mathbb{N}$,

$$ \mathcal{R}(t+t_\infty, y+y_\infty, z+z_\infty)\mathbf{v}_k(t, y, z) = -(\mathbf{B}'\mathbf{u}')(t+t_\infty, y+y_\infty, z+\zeta_k+Z_\infty) \circ \mathbf{v}_k(t, y, z), $$ or else

$$ \mathcal{R}(t, y, z)\mathbf{v}_k(t - t_\infty, y - y_\infty, z - Z_\infty) = -(\mathbf{B}'\mathbf{u}')(t, y, z) \circ \mathbf{v}_k(t - t_\infty, y - y_\infty, z - Z_\infty), $$ as well as $v_{k,i}(0, 0, 0) = 1$. By classical parabolic estimates [30] and a diagonal extraction process, the sequence $(\mathbf{v}_k(\cdot - t_\infty, \cdot - y_\infty, z - Z_\infty))_{k \in \mathbb{N}}$ converges, up to extraction, locally uniformly, to a nonnegative nonzero solution $\mathbf{v} \in C_{\text{per}}^{1,2}(\mathbb{R} \times \mathbb{R}^{n-1}, \mathcal{C}^2(\mathbb{R}, \mathbb{R}^N))$ of

$$ \mathcal{R}\mathbf{v} = 0. $$

In other words, $\mathbf{v}$ is a generalized principal eigenfunction for the generalized principal eigenvalue 0 of the operator $\mathcal{R}$ [22]. By the Harnack inequality of Proposition 2.1, it is positive.

Now we are going to adapt the proof of [22, Proposition 3.9] to prove an exponential estimate, that will lead to a contradiction.
Define the translation $\tau: z \in \mathbb{R} \mapsto z + L_z$ and denote $\mathbf{v}^{\tau}: (t, y, z) \mapsto \mathbf{v}(t, y, \tau(z))$ and $\mathbf{w} = (v_i^{\tau}/v_i)_{i \in [N]}$. By virtue of the Harnack inequality of Proposition 2.1 and periodicity of the coefficients of $\mathcal{R}$, $\mathbf{w}$ is globally bounded. Let

$$ \bar{\mu} = \frac{1}{L_z} \ln \left( \max_{i \in [N]} \sup_{(t,y,z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}} w_i(t, y, z) \right). $$

Recalling that $\mathbf{v}$ and consequently $\mathbf{w}$ are periodic with respect to all variables except maybe $z$, there exists $\bar{i} \in [N]$ and a new sequence still denoted $(t_k, y_k, z_k)_{k \in \mathbb{N}} \in ([0, T] \times [0, L_y] \times \mathbb{R})^\mathbb{N}$ such that $w_{\bar{i}}(t_k, y_k, z_k) \to e^{\bar{\mu}L_z}$ as $k \to +\infty$. Moreover, there exists another sequence still denoted $(Z_k)_{k \in \mathbb{N}} \in [0, L_z]^\mathbb{N}$ such that, for all $k \in \mathbb{N}$, $z_k - Z_k \in L_z\mathbb{Z}$. Up to extraction, we assume that $(t_k, y_k, Z_k) \to (t_\infty, y_\infty, Z_\infty) \in [0, T] \times [0, L_y] \times [0, L_z]$.

Now, define, for all $k \in \mathbb{N}$,

$$ \hat{\mathbf{v}}_k: (t, y, z) \mapsto \frac{1}{v_{\bar{i}}(t_k, y_k, z_k)} \mathbf{v}(t + t_k, y + y_k, z + z_k), $$

$$ \hat{\mathbf{v}}_k^{\tau}: (t, y, z) \mapsto \hat{\mathbf{v}}_k(t, y, \tau(z)). $$

Once more by virtue of the Harnack inequality and the periodicity of the coefficients of $\mathcal{R}$, $(\hat{\mathbf{v}}_k)_{k \in \mathbb{N}}$ is globally bounded. By periodicity of the coefficients of $\mathcal{R}$, it satisfies:

$$ \mathcal{R}(t + t_k, y + y_k, z + Z_k)\hat{\mathbf{v}}_k(t, y, z) = \mathbf{0} \quad \text{for all } (t, y, z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}, \quad k \in \mathbb{N}. $$

Therefore, by classical regularity estimates [30], $(\hat{\mathbf{v}}_k)_{k \in \mathbb{N}}$ converges up to a diagonal extraction to $\hat{\mathbf{v}}_\infty \in C_{\text{per}}^{1,2}(\mathbb{R} \times \mathbb{R}^{n-1}, C^2(\mathbb{R}, \mathbb{R}^N))$ which satisfies:

$$ \mathcal{R}(t + t_\infty, y + y_\infty, z + Z_\infty)\hat{\mathbf{v}}_\infty(t, y, z) = \mathbf{0} \quad \text{for all } (t, y, z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}, $$

or else:

$$ \mathcal{R}(t, y, z)\hat{\mathbf{v}}_\infty(t - t_\infty, y - y_\infty, z - Z_\infty) = \mathbf{0} \quad \text{for all } (t, y, z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}. $$

Moreover, $\hat{v}_{\bar{i},\infty}(0,0,0) = 1$, whence $\hat{\mathbf{v}}_\infty$ is nonzero. By the strong maximum principle, it is in fact positive.

We can now define $\mathbf{f}_\infty = e^{\bar{\mu}L_z}\hat{\mathbf{v}}_\infty - \hat{\mathbf{v}}_\infty^\tau$. We deduce by definition of $\bar{\mu}$ that $\mathbf{f}_\infty \ge \mathbf{0}$ with $f_{\bar{i},\infty}(0,0,0) = 0$. Moreover, $\mathbf{f}_\infty$ satisfies the same equation as $\hat{\mathbf{v}}_\infty$. Therefore, by virtue of the strong maximum principle, $\mathbf{f}_\infty$ is the zero function. This exactly means that $e^{\bar{\mu}L_z}\hat{\mathbf{v}}_\infty = \hat{\mathbf{v}}_\infty^\tau$.

It is now clear that $\mathbf{v}^1: (t, y, z) \mapsto \hat{\mathbf{v}}_\infty(t-t_\infty, y-y_\infty, z-Z_\infty)$ is positive, periodic with respect to $t$ and the $n-1$ first spatial variables, is a solution of $\mathcal{R}\mathbf{v}^1 = \mathbf{0}$, and that the function $(t, y, z) \mapsto e^{-\bar{\mu}z}\mathbf{v}^1(t, y, z)$ is $L_z$-periodic with respect to $z$.

This implies that $0 = \lambda_{1,\text{per}}(\mathcal{R}_{\bar{\mu}})$. Applying (10), it follows that $0 = \lambda_{1,\bar{\mu}e} + c\bar{\mu}$. The function $\nu \in \mathbb{R} \mapsto \lambda_{1,\nu e} + c\nu$ is strictly concave and coercive (cf. [21, Lemma 3.5]), and thus $\mu^\wedge$ and $\mu^\vee$ given by Lemma 2.2 are its only two zeros. Hence $\bar{\mu} \in \{\mu^\wedge, \mu^\vee\}$.

Repeating the same process with $\bar{\mu}$ replaced by

$$ \underline{\mu} = \frac{1}{L_z} \ln \left( \min_{i \in [N]} \inf_{(t,y,z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}} w_i(t, y, z) \right), $$

we end up similarly with $\underline{\mu} \in \{\mu^\wedge, \mu^\vee\}$.

Hence

$$ e^{\mu^\wedge L_z} \mathbf{1} \le \mathbf{w} \le e^{\mu^\vee L_z} \mathbf{1} \quad \text{in } \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}. $$
By definition of $\mathbf{w}$ and $\mathbf{v}^\tau$, for all $(t, y, z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$,

$$e^{\mu^{\wedge} L_z} \mathbf{v}(t, y, z) \leq \mathbf{v}(t, y, z + L_z) \leq e^{\mu^{\vee} L_z} \mathbf{v}(t, y, z).$$

By definition of $\mathbf{v}$ and by its positivity and global boundedness, there exists $k_0 \in \mathbb{N}$ such that, for all $(t, y, z) \in [0, T] \times [0, L_y] \times [0, L_z]$ and all $k \in \mathbb{N}$,

$$e^{\frac{\mu^{\wedge}}{2} L_z} \mathbf{u}'(t, y, z + \zeta_{k+k_0}) \leq \mathbf{u}'(t, y, z + \zeta_{k+k_0} + L_z).$$

Recall that, by construction, for all $k \in \mathbb{N}$, $\zeta_k \in L_z \mathbb{Z}$ and $\zeta_k < \zeta_{k+1}$. By iteration, for all $(t, y, z) \in [0, T] \times [0, L_y] \times [0, L_z]$ and all $k \in \mathbb{N}$,

$$e^{(\zeta_{k+k_0+1} - \zeta_{k+k_0}) \frac{\mu^{\wedge}}{2} L_z} \mathbf{u}'(t, y, z + \zeta_{k+k_0}) \leq \mathbf{u}'(t, y, z + \zeta_{k+k_0+1}).$$

Subsequently, after another iteration, for all $(t, y, z) \in [0, T] \times [0, L_y] \times [0, L_z]$ and all $k \in \mathbb{N}$,

$$(31) \qquad e^{(\zeta_{k+k_0} - \zeta_{k_0}) \frac{\mu^{\wedge}}{2} L_z} \mathbf{u}'(t, y, z + \zeta_{k_0}) \leq \mathbf{u}'(t, y, z + \zeta_{k+k_0}).$$

However the exponential estimate (31), with $\mu^{\wedge} > 0$ (cf. Lemma 2.2), obviously contradicts the convergence $\mathbf{u}'(t, y, z + \zeta_k) \to \mathbf{0}$ as $k \to +\infty$. Thus (30) is contradicted and, consequently, (upstream) is established. $\square$

## 5. PROOF OF THEOREM 1.3

In this section, we assume again $\lambda_{1,\text{per}} < 0$, we fix $e \in \mathbb{Q}^n \cap \mathbb{S}^{n-1}$, we assume $c_e^* \in \mathbb{Q}$, and we prove the existence of a pulsating traveling wave. Again, we omit the subscripts $e$ and $c_e^*$ in all forthcoming notations and we use the notations $T$ and $L$ for the time and space periods of the coefficients of the rotated operator $\mathcal{R}$.

The proof is quite similar to that of Theorem 1.2. In fact, except minor obvious adaptations, the only substantial difference is in the construction of $\bar{\mathbf{u}}$ and $\mathbf{u}$ before proving Proposition 4.4. For the sake of brevity, below, we only detail these two new constructions in the truncated cylinder $\Sigma_a = (0, T) \times (0, L_y) \times (-a, a)$.

The constructions will use repeatedly the mapping

$$\Theta : \mu \in (0, +\infty) \mapsto [(t, y, z) \mapsto e^{\mu z} \mathbf{u}'_{\mu}(t, y, z)].$$

By the Krein–Rutman theorem, the principal eigenvalue $\lambda_{1,\mu} = \lambda_{1,\text{per}}(\mathcal{Q}_{\mu})$ is always simple, and by the Kato–Rellich theorem, it depends smoothly on $\mu$. It follows that $\Theta$ is of class $\mathcal{C}^1$. Below, we denote its derivative $\dot{\Theta}$.

We also need two new notations to circumvent possible oscillations of the functions at hand. If $\mathbf{v}$ is a function of the variable $z \in \mathbb{R}$ of class $\mathcal{C}^1$ such that, for each $i \in [N]$, there exists $z_i \in \mathbb{R}$ such that $v_i(z) > 0$ in $(-\infty, z_i)$, $v_i(z_i) = 0$ and $v_i'(z_i) < 0$, then $\mathbf{v} \vee_{-\infty} \mathbf{0}$ denotes the continuous, piecewise $\mathcal{C}^1$, function whose $i$-th component equals $v_i$ in $(-\infty, z_i)$ and $0$ in $[z_i, +\infty)$. Consistently, $\mathbf{v} \wedge_{-\infty} \mathbf{1}$ denotes the function $\mathbf{1} - (\mathbf{1} - \mathbf{v}) \vee_{-\infty} \mathbf{0}$.

**Lemma 5.1.** Let $M_1 > 0$, $M_2 > 0$ and

$$(32) \qquad \bar{\mathbf{u}} = M_2 ((-M_1 \dot{\Theta}(\mu^*)) \wedge_{-\infty} \mathbf{1}).$$

If $M_1$ is large enough, then $\bar{\mathbf{u}} \geq \mathbf{0}$ in $\mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$.

Furthermore, if $M_2$ is large enough, then $\bar{\mathbf{u}}$ satisfies:

$$(33) \qquad \mathcal{R}\bar{\mathbf{u}} \geq -\operatorname{diag}(b'_{i,i})\bar{\mathbf{u}}^{\circ 2} \quad \text{in } \Sigma_a$$

where partial derivatives are understood in distributional sense where necessary.
*Proof.* On one hand, differentiating the identity (10) with respect to $\mu$, we find directly

$$ (34) \qquad \mathcal{R} \dot{\Theta}(\mu) = \left( \frac{\partial \lambda_{1,\mu}}{\partial \mu} + c^* \right) \Theta(\mu) + (\lambda_{1,\mu} + c^* \mu) \dot{\Theta}(\mu). $$

Note that the derivative of $\mu \in (0, +\infty) \mapsto \frac{-\lambda_{1,\mu}}{\mu}$ is $\mu \in (0, +\infty) \mapsto \frac{-1}{\mu} \frac{\partial \lambda_{1,\mu}}{\partial \mu} + \frac{\lambda_{1,\mu}}{\mu^2}$. In particular, evaluating the derivative at $\mu = \mu^* > 0$ where the global strict minimum $\frac{\lambda_{1,\mu^*}}{\mu^*} = c^*$ is achieved, we deduce:

$$ (35) \qquad c^* = - \left( \frac{\partial \lambda_{1,\mu}}{\partial \mu} \right)_{|\mu=\mu^*}. $$

Therefore, evaluating (34) at $\mu^*$ and using the linearity,

$$ \mathcal{R}(-M_1 \dot{\Theta}(\mu^*)) = 0 \geq -\operatorname{diag}(b'_{i,i}) (-M_1 \dot{\Theta}(\mu^*))^{\circ 2}. $$

On the other hand, if $M_2$ is large enough, then $-\mathbf{L}'\mathbf{1} + M_2 \operatorname{diag}(b'_{i,i})\mathbf{1} \geq \mathbf{0}$, whence $\mathcal{R}(M_2\mathbf{1}) \geq -\operatorname{diag}(b'_{i,i})(M_2\mathbf{1})^{\circ 2}$.

As a matter of fact, for any $(t, y, z) \in \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$

$$ (36) \qquad \dot{\Theta}(\mu^*)(t, y, z) = e^{\mu^* z} \left( z \mathbf{u}_{\mu^*}'(t, y, z) + \left( \frac{\partial \mathbf{u}_{\mu}'}{\partial \mu} \right)_{|\mu=\mu^*} (t, y, z) \right), $$

so that $-M_1 \dot{\Theta}(\mu^*)$ is positive in a neighborhood of $z = -\infty$. Increasing appropriately $M_1$, it follows that $(-M_1 \dot{\Theta}(\mu^*)) \wedge_{-\infty} \mathbf{1}$ is well-defined and positive in $\mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$.

Therefore $M_2((-M_1 \dot{\Theta}(\mu^*)) \wedge_{-\infty} \mathbf{1})$ is also positive, and finally it satisfies indeed (33) in distributional sense. $\square$

Below, the two constants $M_1$ and $M_2$ are fixed accordingly.

**Lemma 5.2.** Let $M_3 > 0$, $\gamma = \frac{\mu^*}{2}$ and

$$ (37) \qquad \mathbf{u} = M_1 M_2 ((-\dot{\Theta}(\mu^*) - M_3 \Theta(\mu^*) + \Theta(\mu^* + \gamma)) \vee_{-\infty} \mathbf{0}). $$

If $M_3$ is large enough, then $\mathbf{u}$ satisfies:

$$ (38) \qquad \mathbf{u} \leq \bar{\mathbf{u}} \quad \text{in } \mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}, $$

$$ (39) \qquad \mathcal{R}\mathbf{u} \leq -\mathbf{u} \circ (\mathbf{B}'\bar{\mathbf{u}}) \quad \text{in } \Sigma_a $$

where partial derivatives are understood in distributional sense where necessary.

*Proof.* Evaluating at $z = 0$ and $(t, y) \in \mathbb{R} \times \mathbb{R}^{n-1}$,

$$ \underline{\mathbf{u}}(t, y, 0) = M_1 M_2 \left( \left( - \left( \frac{\partial \mathbf{u}'_{\mu}}{\partial \mu} \right)_{|\mu=\mu^*} (t, y, 0) - M_3 \mathbf{u}_{\mu^*}'(t, y, 0) + \mathbf{u}_{\mu^*+\gamma}'(t, y, 0) \right) \vee_{-\infty} \mathbf{0} \right), $$

whence there exists $M_3^0$ such that, provided $M_3 \geq M_3^0$, $\underline{\mathbf{u}}(t, y, 0) = \mathbf{0}$. Subsequently, by definition of $\vee_{-\infty}$, $M_3 \geq M_3^0$ implies $\underline{\mathbf{u}} = \mathbf{0}$ in $\mathbb{R} \times \mathbb{R}^{n-1} \times [0, +\infty)$.

Also, by positivity of $\gamma$ and up to increasing again $M_3$, $-M_3 \dot{\Theta}(\mu^*) + \Theta(\mu^* + \gamma) \leq \mathbf{0}$ in $\mathbb{R} \times \mathbb{R}^{n-1} \times (-\infty, 0]$. Hence $\underline{\mathbf{u}} \leq -M_1 M_2 \dot{\Theta}(\mu^*)$ in $\mathbb{R} \times \mathbb{R}^{n-1} \times (-\infty, 0]$.

Up to increasing once more $M_3$, $\underline{\mathbf{u}} \leq M_2 \mathbf{1}$ in $\mathbb{R} \times \mathbb{R}^{n-1} \times \mathbb{R}$, whence (38) is proved.
It remains to verify (39) provided $M_3$ is sufficiently large. In fact, it is sufficient
to verify that $M_3$ can be chosen so large that, in $\mathbb{R} \times \mathbb{R}^{n-1} \times (-\infty, 0]$,

$$
(40) \quad (\mathcal{R} + \operatorname{diag}(\mathbf{B}'\bar{\mathbf{u}})) (-\dot{\Theta}(\mu^*) - M_3\Theta(\mu^*) + \Theta(\mu^* + \gamma)) \le 0.
$$

By (35) and the Taylor theorem,

$$
(41) \quad \lambda_{1,\mu^*+\gamma} = -c^*\mu^* - c^*\gamma + \int_{\mu^*}^{\mu^*+\gamma} (\mu^* + \gamma - \mu) \frac{\partial^2 \lambda_{1,\mu}}{\partial \mu^2} d\mu.
$$

By virtue of the concavity of $\mu \mapsto \lambda_{1,\mu}$ (cf. [22, Theorem 1.3]), the integral remainder is nonpositive. The concavity being in fact strict (cf. [22, Theorem 1.3]), it is zero if and only if $\gamma = 0$. Since $\gamma = \mu^*/2$, this is not the case. In other words, there exists a strictly concave function $g : \mathbb{R} \to \mathbb{R}$ such that $g(0) = g'(0) = 0$, $g(\gamma) < 0$ and $\lambda_{1,\mu^*+\gamma} \le -c^*\mu^* - c^*\gamma + g(\gamma)$.

Hence we deduce from (10) that

$$
(42) \qquad \mathcal{R}\Theta(\mu^* + \gamma) \leq g(\gamma)\Theta(\mu^* + \gamma).
$$

This inequality does not depend on $M_1$, $M_2$ or $M_3$.

By linearity, $\mathcal{R}\dot{\Theta}(\mu^*) = 0$, $\mathcal{R}\Theta(\mu^*) = 0$ and (42), (40) reduces to:

$$
(43) \quad g(\gamma)\Theta(\mu^* + \gamma) + (\mathbf{B}'\bar{\mathbf{u}}) \circ (-\dot{\Theta}(\mu^*) - M_3\Theta(\mu^*) + \Theta(\mu^* + \gamma)) \le 0.
$$

On one hand, recall the definition of $\bar{u}$, (32), as well as the expansion (36).
Considering the asymptotics as $z \to -\infty$, we find that, omitting the arguments in
$(t, y)$,

$$
(\mathbf{B}'\bar{\mathbf{u}}) \circ (-\dot{\Theta}(\mu^*) - M_3\mathbf{\Theta}(\mu^*) + \mathbf{\Theta}(\mu^* + \gamma)) (z) \underset{z \to -\infty}{\sim} z^2 e^{2\mu^*z} \mathbf{B}'\mathbf{u}_{\mu^*}'(z) \circ \mathbf{u}_{\mu^*}'(z).
$$

Therefore, as $z \to -\infty$, this term is negligible in (43) provided $\mu^* + \gamma < 2\mu^*$.
Recalling again $\gamma = \mu^*/2$, this is true indeed.

On the other hand, away from $z = -\infty$, it suffices to increase again $M_3$ to ensure
the negativity of the left-hand side in (43).

This ends the proof.

<figure><img src="image_3.png" /></figure>

Remark 5.1. Contrarily to the construction in the super-critical case $c > c^*$, here in the critical case the super-solution accounts for a certain amount of nonlinearity (the quadratic term in (33)). This is required in order to circumvent the sign changes of $-\dot{\Theta}(\mu^*)$. Yet this nonlinearity has no coupling effect on the components of $\mathbf{u}$, and therefore it preserves the maximum principle for cooperative systems. Thanks to this key fact, the rest of the proof is not altered significantly. In the proof of Proposition 4.4, it suffices to replace the cooperative linear operator $\mathcal{R} + \operatorname{diag}(\mathbf{B}'\mathbf{r})$ by the cooperative semilinear operator $\mathbf{u} \mapsto \mathcal{R}\mathbf{u} + \operatorname{diag}(b'_{i,i}) \mathbf{u}^{\circ 2} + \operatorname{diag}((\mathbf{B}' - \operatorname{diag}(b'_{i,i})) \mathbf{r}) \mathbf{u}$.

6. ACKNOWLEDGEMENTS

L. G. acknowledges support from the ANR via the project Indyana under grant agreement ANR-21-CE40-0008. L. G. and G. N. acknowledge support from the ANR via the project Reach under grant agreement ANR-23-CE40-0023-01.
REFERENCES

[1] Matthieu Alfaro and Quentin Griette. Pulsating fronts for Fisher-KPP systems with mutations as models in evolutionary epidemiology. *Nonlinear Anal. Real World Appl.*, 42:255-289, 2018.

[2] Nicholas D. Alikakos, Peter W. Bates, and Xinfu Chen. Periodic traveling waves and locating oscillating patterns in multidimensional domains. *Trans. Amer. Math. Soc.*, 351(7):2777-2805, 1999.

[3] Ari Arapostathis, Mrinal K. Ghosh, and Steven I. Marcus. Harnack's inequality for cooperative weakly coupled elliptic systems. *Comm. Partial Differential Equations*, 24(9-10):1555-1571, 1999.

[4] Henri Berestycki and François Hamel. Front propagation in periodic excitable media. *Comm. Pure Appl. Math.*, 55(8):949-1032, 2002.

[5] Henri Berestycki and François Hamel. Generalized transition waves and their properties. *Comm. Pure Appl. Math.*, 65(5):592-648, 2012.

[6] Henri Berestycki, François Hamel, and Grégoire Nadin. Asymptotic spreading in heterogeneous diffusive excitable media. *J. Funct. Anal.*, 255(9):2146-2189, 2008.

[7] Henri Berestycki, François Hamel, and Lionel Roques. Analysis of the periodically fragmented environment model. II. Biological invasions and pulsating travelling fronts. *J. Math. Pures Appl. (9)*, 84(8):1101-1146, 2005.

[8] Henri Berestycki, Grégoire Nadin, Benoît Perthame, and Lenya Ryzhik. The non-local Fisher-KPP equation: travelling waves and steady states. *Nonlinearity*, 22(12):2813, 2009.

[9] Emeric Bouin, Christopher Henderson, and Lenya Ryzhik. The Bramson logarithmic delay in the cane toads equations. *Quart. Appl. Math.*, 75(4):599-634, 2017.

[10] Robert Stephen Cantrell and Chris Cosner. *Spatial ecology via reaction-diffusion equations*. Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003.

[11] Robert Stephen Cantrell, Chris Cosner, and Xiao Yu. Dynamics of populations with individual variation in dispersal on bounded domains. *Journal of Biological Dynamics*, 12(1):288-317, 2018.

[12] Li-Jun Du, Wan-Tong Li, and Wenxian Shen. Propagation phenomena for time-space periodic monotone semiflows and applications to cooperative systems in multi-dimensional media. *J. Funct. Anal.*, 282(9):Paper No. 109415, 59, 2022.

[13] Jian Fang, Xiao Yu, and Xiao-Qiang Zhao. Traveling waves and spreading speeds for time-space periodic monotone systems. *ArXiv e-prints*, apr 2015.

[14] Ronald Aylmer Fisher. The wave of advance of advantageous genes. *Annals of eugenics*, 7(4):355-369, 1937.

[15] Juraj Földes and Peter Poláčik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. *Discrete Contin. Dyn. Syst.*, 25(1):133-157, 2009.

[16] Guillaume Fréjacques. *Ondes progressives pour des équations de réaction-diffusion avec des coefficients périodiques en temps*. PhD thesis, Université Aix-Marseille, 2005.

[17] Léo Girardin. Competition in periodic media: I – Existence of pulsating fronts. *Discrete and Continuous Dynamical Systems - Series B*, 22(4):1341-1360, 2017.

[18] Léo Girardin. Non-cooperative Fisher-KPP systems: Asymptotic behavior of traveling waves. *Mathematical Models and Methods in Applied Sciences*, 28(06):1067-1104, 2018.

[19] Léo Girardin. Non-cooperative Fisher-KPP systems: traveling waves and long-time behavior. *Nonlinearity*, 31(1):108, 2018.

[20] Léo Girardin. Two components is too simple: an example of oscillatory Fisher-KPP system with three components. *Proceedings of the Royal Society of Edinburgh: Section A Mathematics*, pages 1-24, 2019.

[21] Léo Girardin. Persistence, extinction and spreading properties of non-cooperative Fisher-KPP systems in space-time periodic media. *SIAM J. Math. Anal.*, accepted for publication in 2024.

[22] Léo Girardin and Idriss Mazari. Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems. *Memoirs of the French Mathematical Society*, accepted for publication in 2024.
[23] Quentin Griette and Hiroshi Matano. Propagation dynamics of solutions to spatially periodic reaction-diffusion systems with hybrid nonlinearity. *arXiv e-prints*, page arXiv:2108.10862, August 2021.

[24] Quentin Griette and Gaël Raoul. Existence and qualitative properties of travelling waves for an epidemiological model with mutations. *J. Differential Equations*, 260(10):7115-7151, 2016.

[25] H. M. Gueguezo, T. J. Doumatè, and R. B. Salako. Global dynamics of a two-stage structured diffusive population model in time-periodic and spatially heterogeneous environments. *Stud. Appl. Math.*, 153(4):Paper No. e12750, 39, 2024.

[26] François Hamel. Qualitative properties of monostable pulsating fronts: exponential decay and monotonicity. *J. Math. Pures Appl. (9)*, 89(4):355-399, 2008.

[27] W. Hudson and B. Zinner. Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media. In *Boundary value problems for functional-differential equations*, pages 187-199. World Sci. Publ., River Edge, NJ, 1995.

[28] Andrei N. Kolmogorov, I. G. Petrovsky, and N. S. Piskunov. Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique. *Bulletin Université d'État à Moscou*, 1:1-25, 1937.

[29] Bingtuan Li, Hans F. Weinberger, and Mark A. Lewis. Spreading speeds as slowest wave speeds for cooperative systems. *Math. Biosci.*, 196(1):82-98, 2005.

[30] Gary M. Lieberman. *Second order parabolic differential equations*. World Scientific Publishing Co., Inc., River Edge, NJ, 1996.

[31] Aled Morris, Luca Börger, and Elaine C. M. Crooks. Individual variability in dispersal and invasion speed. *Mathematics*, 7(9), 2019.

[32] C. B. Muratov. A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. *Discrete Contin. Dyn. Syst. Ser. B*, 4(4):867-892, 2004.

[33] C. B. Muratov and M. Novaga. Front propagation in infinite cylinders. I. A variational approach. *Commun. Math. Sci.*, 6(4):799-826, 2008.

[34] Grégoire Nadin. Traveling fronts in space-time periodic media. *J. Math. Pures Appl. (9)*, 92(3):232-262, 2009.

[35] James Nolen, Matthew Rudd, and Jack Xin. Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds. *Dyn. Partial Differ. Equ.*, 2(1):1-24, 2005.

[36] Murray H. Protter and Hans F. Weinberger. *Maximum principles in differential equations*. Springer-Verlag, New York, 1984. Corrected reprint of the 1967 original.

[37] Vanessa Ress, Arne Traulsen, and Yuriy Pichugin. Eco-evolutionary dynamics of clonal multicellular life cycles. *eLife*, 11:e78822, sep 2022.

[38] Emmanuel Risler. Global convergence toward traveling fronts in nonlinear parabolic systems with a gradient structure. *Ann. Inst. H. Poincaré Anal. Non Linéaire*, 25(2):381-424, 2008.

[39] Nanako Shigesada, Kohkichi Kawasaki, and Ei Teramoto. Traveling periodic waves in heterogeneous environments. *Theoret. Population Biol.*, 30(1):143-160, 1986.

[40] Aizik I. Volpert, Vitaly A. Volpert, and Vladimir A. Volpert. *Traveling wave solutions of parabolic systems*, volume 140 of *Translations of Mathematical Monographs*. American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda.

[41] Haiyan Wang. Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems. *J. Nonlinear Sci.*, 21(5):747-783, 2011.

[42] Hans F. Weinberger. On spreading speeds and traveling waves for growth and migration models in a periodic habitat. *J. Math. Biol.*, 45(6):511-548, 2002.