Claire Gallois

Jean-François Emile

Stefano Kim

Carole Monterymard

Marine Gilabert

Jérémie Bez

Astrid Lièvre

Laetitia Dahan

Pierre Laurent-Puig

Laurent Mineur

Astrid Lievre

Romain Coriat

Jean-Louis Legoux

Vincent Hautefeuille

Jean-Marc Phelip

Thierry Lecomte

Harry Sokol

Claude Capron

Violaine Randrian

Come Lepage

Nicolas Lomenie

Camille Kurtz

Julien Taieb

David Tougeron
email: david.tougeron@chu-poitiers.fr

Pembrolizumab with Capox Bevacizumab in patients with microsatellite stable metastatic colorectal cancer and a high immune infiltrate: The FFCD 1703-POCHI trial

Keywords: Clinical trial, Tumor-infiltrating lymphocytes, Colorectal cancer, Checkpoint blockade

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Background

About 85% of non-metastatic colorectal cancers (CRC) are related to chromosomal instability and have a proficient DNA mismatch repair system (pMMR). Other CRCs, i.e. 15%, are microsatellite unstable (MSI) with deficient DNA mismatch repair system (dMMR).

They are characterised by high tumor mutational burden (TMB) and the generation of many neo-antigens, which result in a high anti-tumour immune response and a high peri-and/or intra-tumour lymphocyte infiltration (TIL) [START_REF] Tougeron | Tumor-infiltrating lymphocytes in colorectal cancers with microsatellite instability are correlated with the number and spectrum of frameshift mutations[END_REF]. We recently showed, with a prospectively validated immune score called TuLIS (Tumor Lymphocyte Infiltration Score), that 22% of localised MSS/pMMR CRCs are also highly infiltrated by CD3+ lymphocytes [START_REF] Emile | Prospective validation of a lymphocyte infiltration prognostic test in stage III colon cancer patients treated with adjuvant FOLFOX[END_REF]. This score has also been used to measure lymphocyte infiltration in hepatic metastases, particularly in patients treated with CAPOX/FOLFOX [START_REF] Tanis | Prognostic impact of immune response in resectable colorectal liver metastases treated by surgery alone or surgery with perioperative FOLFOX in the randomised EORTC study 40983[END_REF].

Pembrolizumab, an anti-PD1 (programmed death-1) monoclonal antibody, is an immune checkpoint inhibitor (ICI) of PD1/PD-L1 axis, recently approved in many cancers.

Anti-PD1 antibodies have recently been reported as being very effective in patients with MSI/dMMR metastatic CRC (mCRC) [START_REF] Le | PD-1 Blockade in Tumors with Mismatch-Repair Deficiency[END_REF][START_REF] André | Pembrolizumab in Microsatellite-Instability-High Advanced Colorectal Cancer[END_REF]. The recently published phase III KEYNOTE-177 trial, showing a significant improvement of progression-free survival (PFS) in patients with MSI/dMMR tumors treated with pembrolizumab compared to those treated with chemotherapy as first-line treatment [START_REF] André | Pembrolizumab in Microsatellite-Instability-High Advanced Colorectal Cancer[END_REF], raised pembrolizumab as the new standard of care in this setting. By contrast, MSS/pMMR mCRC has not benefited from monotherapy with anti-PD1 antibodies [START_REF] Le | PD-1 Blockade in Tumors with Mismatch-Repair Deficiency[END_REF], and the results of early phase trials combining atezolizumab (anti-PD-L1) with fluoropyrimidine +/-oxaliplatin and bevacizumab in these patients have been disappointing [START_REF] Grothey | Fluoropyrimidine (FP) + bevacizumab (BEV) + atezolizumab vs FP/BEV in BRAFwt metastatic colorectal cancer (mCRC): Findings from Cohort 2 of MODUL -a multicentre, randomized trial of biomarker-driven maintenance treatment following first-line induction therapy[END_REF][START_REF] Mettu | BACCI: A phase II randomized, double-blind, multicenter, placebo-controlled study of capecitabine (C) bevacizumab (B) plus atezolizumab (A) or placebo (P) in refractory metastatic colorectal cancer (mCRC): An ACCRU network study[END_REF][START_REF] Wallin | Abstract 2651: Clinical activity and immune correlates from a phase Ib study evaluating atezolizumab (anti-PDL1) in combination with FOLFOX and bevacizumab (anti-VEGF) in metastatic colorectal carcinoma[END_REF]. However, 7% of MSS/pMMR CRCs are hypermutated [START_REF] Zaidi | Landscape of somatic single nucleotide variants and indels in colorectal cancer and impact on survival[END_REF], and it remains possible that the 20% of MSS/pMMR mCRCs presenting tumors highly infiltrated by CD3+ may be sensitive to ICI.

Finally, immunogenic cell death induced by oxaliplatin, the regulation of myeloid derived suppressor cells (MDSC) reported with fluoropyrimidine and the modulation of regulatory T cells and PD1 expression on T lymphocytes reported with bevacizumab, may boost ICI efficacy [START_REF] Vincent | 5-Fluorouracil selectively kills tumor-associated myeloid-derived suppressor cells resulting in enhanced T cell-dependent antitumor immunity[END_REF][START_REF] Lapeyre-Prost | Immunomodulatory Activity of VEGF in Cancer[END_REF][START_REF] Zitvogel | Immunogenic tumor cell death for optimal anticancer therapy: the calreticulin exposure pathway[END_REF].

The prognostic value of lymphocyte infiltrate has been demonstrated in CRC by several teams. However, no validated test is used in routine clinical practice. Previously, we described an automated and reproducible method for analysis of TIL [START_REF] Allard | Linear quantification of lymphoid infiltration of the tumor margin: a reproducible method, developed with colorectal cancer tissues, for assessing a highly variable prognostic factor[END_REF] and validated it for clinical use [START_REF] Emile | Prospective validation of a lymphocyte infiltration prognostic test in stage III colon cancer patients treated with adjuvant FOLFOX[END_REF]. Automated test (TuLIS) evaluating TIL are performed on virtual slides and has shown that, out of 1220 CRCs tested, 22% of MSS/pMMR tumours were highly infiltrated by CD3+ T cells. Patients presenting with a MSS/pMMR CRC with a high immune infiltrate had better PFS (HR=0.70; p=0.02). Another immunoscore ® described by Galon et al. also has a high prognostic value in CRC and is based on CD3+ and CD8+ T cell infiltration in the center and periphery of the tumour [START_REF] Galon | Type, density, and location of immune cells within human colorectal tumors predict clinical outcome[END_REF][START_REF] Pagès | International validation of the consensus Immunoscore for the classification of colon cancer: a prognostic and accuracy study[END_REF][START_REF] Pagès | Prognostic and predictive value of the Immunoscore in stage III colon cancer patients treated with oxaliplatin in the prospective IDEA France PRODIGE-GERCOR cohort study[END_REF], approximately 20% of tumours with a high immune infiltrate using this score have been found in patients with MSS/pMMR mCRC [START_REF] Emile | Prospective validation of a lymphocyte infiltration prognostic test in stage III colon cancer patients treated with adjuvant FOLFOX[END_REF][START_REF] Pagès | International validation of the consensus Immunoscore for the classification of colon cancer: a prognostic and accuracy study[END_REF]. This immunoscore ® has recently been shown to have a predictive value of treatment efficacy favoring 6 months of adjuvant FOLFOX in stage III CRC for patients with intermediate and high immunoscore, compared to 3 months of FOLFOX, from the International Duration Evaluation of Adjuvant Therapy (IDEA) France trial [START_REF] Pagès | Prognostic and predictive value of the Immunoscore in stage III colon cancer patients treated with oxaliplatin in the prospective IDEA France PRODIGE-GERCOR cohort study[END_REF].

Interestingly, the immune infiltrate seemed heterogeneous between the primary site and the metastatic sites (synchronous or metachronous) in mCRC, with high heterogeneity between different metastases from the same patient, making it difficult to interpret a single metastasis biopsy [START_REF] Van Den Eynde | The Link between the Multiverse of Immune Microenvironments in Metastases and the Survival of Colorectal Cancer Patients[END_REF][START_REF] Mlecnik | Metastasis immune-based scores predict patient survival[END_REF].

The POCHI trial was designed to evaluate the efficacy of pembrolizumab in combination with CAPOX and bevacizumab in patients with MSS/pMMR mCRC with a high immune infiltrate diagnosed in the primary tumor.

Study design

Population

The FFCD 1703-POCHI trial is a single arm, open-label, multicentre, phase II trial conducted in France, sponsored and coordinated by the Fédération Francophone de Cancérologie Digestive (FFCD), evaluating as first-line treatment of MSS/pMMR mCRC with a high immune infiltrate, pembrolizumab in combination with CAPOX and bevacizumab every three weeks. All French centres affiliated to the PRODIGE group (Partenariat de Recherche en Oncologie DIGEstive) could participate to the study.

Main inclusion criteria are patients ≥ 18 years old, with unresectable MSS/pMMR mCRC histologically proven, and a high immune response as defined by the 2 previously mentioned scores, assessed on the resected primary tumour (Table 1). Patients with synchronous or metachronous metastases and surgery of primary tumour are eligible. Patients with metachronous metastases could also be enrolled if they have a previous curative treatment of both primary tumour and metastatic disease and a recurrence more than 6 months after the end of this treatment. All patients must have been resected of their primary tumour and not have received any neo-adjuvant treatment that could have modified local immune infiltrates. Both MSI and MMR IHC status must have been determined before inclusion, either locally or in a centralised manner (Figure 1). This study (clinicaltrials.gov NCT 04262687) is performed in accordance with the declaration of Helsinki and Good Clinical Practice Guidelines. A French ethics committee approved the study. All patients have to provide written informed consent before entering the study.

Immune scores

The anti-tumour immune response will be evaluated with two different scores: Immunoscore ® described by Galon et al. [START_REF] Galon | Type, density, and location of immune cells within human colorectal tumors predict clinical outcome[END_REF] and TuLIS score described by Emile et al. [START_REF] Emile | Prospective validation of a lymphocyte infiltration prognostic test in stage III colon cancer patients treated with adjuvant FOLFOX[END_REF], on a surgical specimen of the primary tumour, containing at least 2 mm of tumour-free margin between the tumour and the non-tumour area, in centralised review by HalioDX (Marseille, France) and the pathology department of Ambroise Paré Hospital (Assistance Publique-Hôpitaux de Paris, France). For patient inclusion, one of these two scores must be considered as positive. Immunohistochemistry will be performed with anti-CD3 and anti-CD8 antibodies.

For TuLIS score, the intensity of the anti-tumour immune response will be determined based on the profile of the curve of variation of density of CD3 staining compared to the distance on either side of the tumour front [START_REF] Emile | Prospective validation of a lymphocyte infiltration prognostic test in stage III colon cancer patients treated with adjuvant FOLFOX[END_REF][START_REF] Allard | Linear quantification of lymphoid infiltration of the tumor margin: a reproducible method, developed with colorectal cancer tissues, for assessing a highly variable prognostic factor[END_REF]. For immunoscore ® , the intensity of the antitumour immune response will be determined based on the profile of density of CD3 and CD8 staining in the center and periphery of the tumour [START_REF] Galon | Type, density, and location of immune cells within human colorectal tumors predict clinical outcome[END_REF][START_REF] Pagès | International validation of the consensus Immunoscore for the classification of colon cancer: a prognostic and accuracy study[END_REF]. Immunoscore ® results were categorized using 5 sub-groups (IS-0, IS-1, IS-2, IS-3 or IS-4), as already published, and patients eligible in the study with high immune response were those with an Immunoscore ® IS-3 or IS-4.

Inclusion, treatment and follow-up

Clinical examination and laboratory tests for inclusion should be performed within 2 weeks prior to inclusion, CT-scan should be performed within 3 weeks prior to inclusion (Table 2). Eligible patients will be treated with pembrolizumab 200 mg at day 1 (IV infusion over 30 minutes) in combination with capecitabine 2000 mg/m²/day on days 1 to 14, oxaliplatin 130 mg/m² at day 1 (IV infusion over 2 hours) and bevacizumab 7.5 mg/kg at day 1 (IV infusion over 60 minutes). We chose capecitabine over 5-fluorouracil (5-FU) because of the rhythm of administration of pembrolizumab every 3 weeks, and a lack of arguments in the literature for lower efficacy of capecitabine associated with ICI, compared to 5-FU. Cycle 1 will be administered without pembrolizumab and without bevacizumab due to the time necessary to obtain the results of the immune scores (approximately 2 weeks), and not to impose a targeted therapy on patients who would ultimately not be eligible for the study.

Treatment will be repeated every three weeks, up until disease progression, unacceptable toxicity, refusal by the patient, withdrawal of consent, pregnancy or decision by the investigator.

The treatment must be administered on the day 1 of each new cycle, provided that the following criteria are satisfied: neutrophil count > 1500/mm 3 , platelets > 100,000/mm 3 , gastro-intestinal toxicity < grade 1. If the CAPOX bevacizumab regimen must be delayed due to toxicity, pembrolizumab will also be delayed and restarted at the same time.

Pembrolizumab is provided in the setting of the protocol up to disease progression or up to the maximum treatment duration of pembrolizumab, which is 35 cycles.

Safety

Toxicities will all be evaluated according to National Cancer Institute -Common Terminology Criteria for Adverse Events (NCI-CTCAE v4.0) scale. Immune-related AEs (irAEs) may occur shortly after the first dose or several months after the last dose of pembrolizumab treatment and may affect more than one body system simultaneously. Based on irAE severity, pembrolizumab will be withheld or permanently discontinued and corticosteroids will be administered.

Two analyses of safety will be performed. The first analysis will be done when 5 patients have been treated for 3 months and the second analysis when 10 patients have been treated for 3 months. The review of safety data will be carried out by an Independent Data Monitoring Committee (IDMC).

ACCEPTED MANUSCRIPT -CLEAN COPY 9

Endpoints and Assessments

The primary objective is to evaluate the efficacy of pembrolizumab in combination with CAPOX plus bevacizumab as first-line treatment of MSS/pMMR mCRC with a high immune infiltrate. Efficacy will be determined by analysis of the number of patients alive and without radiological and/or clinical progression at 10 months from the first chemotherapy course (according to the RECIST 1.1 criteria evaluated by the investigator).

Secondary objectives include overall survival (OS), serious adverse events evaluated according to NCI-CTCAE v4.0, secondary resection rate (R0 and R1), pathological response in case of secondary resection (TRG criteria) and outcome of tumour markers (CEA and CA19.9). Other secondary objectives, evaluated according to the investigator and centralised review according to the RECIST V1.1 criteria, will be: median PFS, time to radiological progression (TTP), time to objective response, best response during treatment, depth of response and early tumour shrinkage at 9 weeks; and, in those evaluated according to iRECIST criteria, the percentage of patients alive and without radiological progression at 10 months, median PFS, TTP and early tumour shrinkage at 9 weeks. Exploratory analyses of primary and secondary endpoints will also include investigation of prognostic and predictive values according to sub-classes of Immunoscore ® and TuLIS score for all patients, and the correlation between the two immune scores.

Thoraco-abdominal and pelvic CT-scan will be performed within 3 weeks prior to inclusion and every 9 weeks during treatment and in case of discontinuation of treatment with no disease progression (Table 2). The CEA and CA19.9 levels will be assessed with the same schedule.

Statistical analysis plan

ACCEPTED MANUSCRIPT -CLEAN COPY 10

The clinical hypotheses represent a percentage of patients alive and without radiological and/or clinical progression at 10 months greater than 50% (H1), but a 70% rate is expected. With an alpha risk of 5% (one-sided) and a power of 85%, using the exact binomial method, 50 patients are necessary. Taking into account the fact that 10% of patients will be not evaluable or lost to follow-up at 10 months, 55 patients will have to be included.

The conclusion of the study will be defined according to the following rule: based on the 50 evaluable patients, if 32 patients or more are still alive and without radiological and/or clinical progression at 10 months, then the treatment will be considered as effective.

Analysis of the primary endpoint, the secondary endpoints and safety criteria will be performed on the modified intention to treat population (mITT): all the patients included, whatever their eligibility and who have received at least one dose of treatment (whatever the dose and the treatment). The primary evaluation endpoint and secondary efficacy endpoints will also be evaluated on the per-protocol population (mITT patients without any major protocol deviation).

Ancillary study

The ancillary study on biological samples (blood and stool) and tumour samples (primary tumours) will consist of analysis of circulating tumor DNA (at cycle 1, cycle 3 and disease progression), circulating lymphocyte phenotyping (at cycle 1 and cycle 3), pharmacokinetics of pembrolizumab and bevacizumab, microbiome study (before study and at week 9), tumor mutational burden, other immune scores, immunohistochemistry of PD-L1 and PD-L2 and expression and/or amplification of PD-L1 and PD-L2 (Table 2).

The objective of these studies is to identify prognostic biomarkers and/or predictive biomarkers of response to treatment (OS, response rate and PFS). For microbiome study, some studies suggest that the gut microbiota might be involved in the efficacy and toxicity of chemotherapies and immunotherapies [START_REF] Viaud | The intestinal microbiota modulates the anticancer immune effects of cyclophosphamide[END_REF][START_REF] Gopalakrishnan | Gut microbiome modulates response to anti-PD-1 immunotherapy in melanoma patients[END_REF][START_REF] Routy | Gut microbiome influences efficacy of PD-1based immunotherapy against epithelial tumors[END_REF]. We aim to evaluate the influence of gut microbiota composition on toxicity and efficacy of bevacizumab and pembrolizumab combination. We are seeking to detect an association between TMB, a high immune response and response to treatment, as described in other tumours. In the recently published NICHE trial [START_REF] Chalabi | Neoadjuvant immunotherapy leads to pathological responses in MMR-proficient and MMR-deficient early-stage colon cancers[END_REF], of the 15 patients with localized pMMR CCR treated with ipilimumab (anti-CTLA4) plus nivolumab (anti-PD1) in the neo-adjuvant setting, 3 had a major pathological response, including one with ≤ 10% residual viable tumour and 2 complete pathological responses, and one had a partial response with 50% residual viable tumor. The presence of T cells with co-expression of CD8 and PD-1 before starting treatment was a predictive biomarker of response in patients with pMMR tumor. In the same way, in the phase II CO.26 trial [START_REF] Chen | Effect of Combined Immune Checkpoint Inhibition vs Best Supportive Care Alone in Patients With Advanced Colorectal Cancer[END_REF], among 166 patients with refractory pMMR/MSS mCRC, treatment with durvalumab (anti-PDL1) plus tremelimumab (anti-CTLA4) was associated with significantly longer OS (HR=0.66; p=0.02) compared to best supportive care alone, with the greatest OS benefit for patients with TMB of 28 or more mutations by megabase (21% of pMMR/MSS patients).

Discussion

Nevertheless, at this stage, we do not have enough strong data to use ICI combination alone in MSS/pMMR mCRC with a high immune infiltrate. In addition, combination of ACCEPTED MANUSCRIPT -CLEAN COPY 12 CAPOX, bevacizumab and dual blockage with ICIs (anti-PD1 plus anti-CTLA4) seems to us to be toxic.

If we identify an immune score, which seems clinically relevant to predict sensitivity to ICI in MSS/pMMR mCRC, this will make it possible to plan a randomised phase III trial comparing chemotherapy and anti-angiogenic antibody versus chemotherapy and antiangiogenic antibody plus ICI in this mCRC sub-group.

  POCHI is a single arm phase II trial, evaluating in patients with MSS/pMMR mCRC and a high immune infiltrate, the efficacy of pembrolizumab in combination with CAPOX bevacizumab in a first-line setting. The study is open in April 2021 in about 70 centers in France and the end of inclusion is scheduled for end of 2023. The first patient was included on April 20, 2021.

Figure 1 .

 1 Figure 1. Study flowchart

Acknowledgments

We thank all physicians who are participating in the POCHI trial. We also thank all the cooperative PRODIGE group (FFCD -UNICANCER GI -GERCOR) for their contribution and participation in the present trial. Finally, we thank MSD, HalioDX and the "Ligue nationale contre le cancer" for their support. The authors thank Jeffrey Arsham for proofreading the manuscript.

Financial support

This study was supported in part by MSD. Fédération Francophone de Cancérologie Digestive (FFCD) is funding the bio-bank and molecular analysis.

Conflict of interest

CG has participated in consulting and/or advisory boards for Servier and Sanofi, and has received support for travel to meetings from Amgen. JFE, NL and CK are co-inventors of a patented method for evaluation of lymphocyte infiltration of tumor. JFE participated in compensated boards organized by HalioDX and MSD. AL has received honoraria as a speaker and/or in an advisory role from Amgen, Bayer, BMS, HalioDx, Merck, Pierre Fabre, Roche, Sanofi and Servier and has received support for travel to meetings from Bayer, BMS, Merck, MSD, Roche and Servier. PLP has received honoraria from MSD, Biocartis, Roche, Servier, Pierre Fabre, Lilly, MSD, AstraZeneca and Amgen. JMP has received honoraria as a speaker and/or in an advisory role from Merck, Amgen, Roche, MSD, Bayer, Servier, Pierre Favre and BMS and has received support for travel to meetings from Merck, Amgen, Roche, MSD, Bayer, Servier, Pierre Favre and BMS. TL has received honoraria as a speaker and/or in an advisory role from Amgen, Merck, AstraZeneca, Sanofi and Servier and has received support for travel to meetings from Amgen Ipsen and Servier. JT has received honoraria as a speaker and/or in an advisory role from Merck KGaA, Sanofi, Roche Genentech, MSD, Lilly, Astra Zeneca, Servier, Novartis, Pierre Fabre, HallioDx and Amgen. DT has received honoraria for education events and board from MSD. The other authors have no declared conflicts of interest.


Matthieu Alfaro
email: matthieu.alfaro@univ-rouen.fr

Dongyuan Xiao
email: dongyuan.xiao@umontpellier.fr

Lotka-Volterra competition-diffusion system: the critical competition case

Keywords: competition-diffusion system, traveling wave, Cauchy problem, large time behavior, bump phenomenon. AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35C07 (Traveling wave solutions), 35B40 (Asymptotic behavior of solutions)

We consider the reaction-diffusion competition system in the so-called critical competition case. The associated ODE system then admits infinitely many equilibria, which makes the analysis intricate. We first prove the non-existence of ultimately monotone traveling waves by applying the phase plane analysis. Next, we study the large time behavior of the solution of the Cauchy problem with a compactly supported initial datum. We not only reveal that the "faster" species excludes the "slower" one (with a known spreading speed), but also provide a sharp description of the profile of the solution, thus shedding light on a new bump phenomenon.

Introduction

We consider the Lotka-Volterra competition-diffusion system

∂ t u = u xx + u(1 -u -v), t > 0, x ∈ R, ∂ t v = dv xx + rv(1 -v -u), t > 0, x ∈ R, (1.1)
which is critical among systems in the form of (1.5). The main difficulty is that the underlying ODE competition system

u ′ = u(1 -u -v), t > 0, v ′ = rv(1 -v -u), t > 0, (1.2)
admits infinitely many (nontrivial) equilibria: the whole line u + v = 1. Because of that, there are very few available mathematical results on system (1.1). In the present paper, we fill this gap by proving the non-existence of ultimately monotone traveling waves, and giving a very precise description of the large time behavior of the solution starting from a compactly supported initial datum, thus revealing a new bump phenomenon.

In the absence of one species, system (1.1) reduces to the reaction-diffusion equation

∂ t u = du xx + ru(1 -u), t > 0, x ∈ R, (1.3)
introduced by Fisher [START_REF] Fisher | The wave of advance of advantageous genes[END_REF] and Kolmogorov, Petrovsky and Piskunov [START_REF] Kolmogorov | A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem[END_REF] as a population genetics model to investigate the propagation of a dominant gene in a homogeneous environment.

The KPP equation (1.3) has two main properties. Firstly, nonnegative traveling waves, corresponding to the ansatz u(t, x) = U (xct) and solving

dU ′′ + cU ′ + rU (1 -U ) = 0 in R, U (-∞) = 1, U (∞) = 0, (1.4)
exist if and only if their speeds c ≥ c * := 2 √ dr. Secondly, the solution of (1.3) starting from a nonnegative (nontrivial) compactly supported initial datum, satisfies lim see [START_REF] Aronson | Multidimensional nonlinear diffusion arising in population genetics[END_REF]. In other words, the minimal speed c * of traveling wave solutions corresponds to the spreading speed of the solution of the Cauchy problem with a compactly supported initial datum.

The general Lotka-Volterra competition-diffusion system is written

∂ t u = u xx + u(1 -u -av), t > 0, x ∈ R, ∂ t v = dv xx + rv(1 -v -bu), t > 0, x ∈ R. (1.5)
Here u = u(t, x) and v = v(t, x) represent the population densities of two competing species, d > 0 and r > 0 stand for the diffusion rate and intrinsic growth rate of v (while those of u have been normalized), and a > 0 and b > 0 represent the strength of v and u, respectively, as competitors. The parameters a and b determine the behavior of the underlying ODE system (see below) but, once fixed, the outcomes for system (1.5) are highly dependent on the parameters r, d and the initial datum. The situation is therefore very rich and we refer to the works mentioned below for more details and references. The so-called weak competition case corresponds to a < 1 and b < 1. Nontrivial solutions of the underlying ODE system tend to the co-existence equilibrium. For the diffusion system, it was proved by Tang and Fife [START_REF] Tang | Propagating fronts for competing species equations with diffusion[END_REF] that there exists a minimal speed c ⋆ > 0 such that a monotone traveling wave solution connecting the co-existence equilibrium to the null state (0, 0) exists if and only if c ≥ c ⋆ , which is comparable to the Fisher-KPP equation mentioned above. Concerning the large time behavior of the Cauchy problem, some first estimates were obtained by Lin and Li [START_REF] Lin | Asymptotic spreading of competition diffusion systems: the role of interspecific competitions[END_REF]. More recently, Liu, Liu and Lam [START_REF] Liu | Asymptotic spreading of interacting species with multiple fronts I: A geometric optics approach[END_REF][START_REF] Liu | Asymptotic spreading of interacting species with multiple fronts II: Exponentially decaying initial data[END_REF] obtained some rather complete results.

The so-called strong competition case corresponds to a > 1 and b > 1. Since the coexistence equilibrium is unstable and the equilibria (1, 0) and (0, 1) are both stable for the underlying ODE system, this case corresponds to a bistable situation. For the diffusion system, it was proved by Kan-On [START_REF] Kan-On | Parameter dependence of propagation speed of travelling waves for competition-diffusion equations[END_REF], see also [START_REF] Gardner | Existence and stability of traveling wave solutions of competition models: a degree theoretic approach[END_REF], that there exists a unique traveling wave solution connecting (1, 0) to (0, 1). The sign of the speed of this wave determines the "winner" between u and v, and thus is very relevant for applications, see the review [START_REF] Girardin | The effect of random dispersal on competitive exclusion-a review[END_REF]. We refer to [START_REF] Girardin | Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed[END_REF], [START_REF] Guo | The sign of the wave speed for the Lotka-Volterra competitiondiffusion system[END_REF] and [START_REF] Rodrigo | Exact solutions of a competition-diffusion system[END_REF] for some results on this delicate issue. As far as the large time behavior of the Cauchy problem is concerned, we refer to the recent work of Carrere [START_REF] Carrere | Spreading speeds for a two-species competition-diffusion system[END_REF] revealing the possibility of propagating terraces, see [START_REF] Ducrot | Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations[END_REF][START_REF] Ducrot | Spreading speeds for multidimensional reactiondiffusion systems of the prey-predator type[END_REF]. Very recently, Peng, Wu and Zhou [START_REF] Peng | Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition[END_REF] provided refined estimates of both the spreading speed and the profile of the solution.

Last, the so-called strong-weak competition case corresponds to a < 1 < b. Nontrivial solutions of the underlying ODE system tend to the state (1, 0) meaning that "u excludes v". For the diffusion system, the traveling wave solutions were constructed by Kan-On [START_REF] Kan-On | Fisher wave fronts for the Lotka-Volterra competition model with diffusion[END_REF]. Concerning the large time behavior of the solution of the Cauchy problem, Girardin and Lam [START_REF] Girardin | Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition-diffusion systems[END_REF] recently studied the spreading speed of solutions with an initial datum that is null (or exponentially decaying) on the right half line. They obtained a rather complete understanding of the spreading properties, revealing in particular the possibility of an acceleration phenomenon (see Appendix of the present paper for more details).

In the present paper, our goal is to complete the above picture by considering the issues of both traveling wave solutions and the Cauchy problem in the so-called critical competition case a = b = 1, corresponding to system (1.1).

Main results

A traveling wave solution of system (1.1) is defined as follows.

Definition 2.1 ((α, β)-traveling wave). Let 0 ≤ α, β ≤ 1 be given with α = β. Then an (α, β)-traveling wave solution (or traveling wave if there is no ambiguity) of (1.1) is a triplet (c, U, V ), where c ∈ R is the traveling wave speed and (U, V ) two nonnegative profiles, solving

           U ′′ + cU ′ + U (1 -U -V ) = 0, dV ′′ + cV ′ + rV (1 -V -U ) = 0, (U, V )(-∞) = (α, 1 -α), (U, V )(+∞) = (β, 1 -β).
(2.1) As mentioned above, for both the strong competition case and the strong-weak competition case, monotone traveling waves connecting (1, 0) to (0, 1) are known to exist. In the critical competition case under consideration, our first main result is that there is no ultimately monotone traveling wave connecting any two different nonnegative steady states on the line u + v = 1. Definition 2.2 (Ultimately monotone (α, β)-traveling wave). Let 0 ≤ α, β ≤ 1 be given with α = β. Then an ultimately monotone (α, β)-traveling wave solution is an (α, β)-traveling wave for which there further exist

-∞ < z 0 ≤ z * 0 < +∞ such that U ′ (z)V ′ (z) = 0, for all z ∈ (-∞, z 0 ] ∪ [z * 0 , +∞). (2.2) Remark 2.3. Obviously, if (c, U (z), V (z)) is an (ultimately monotone) (α, β)-traveling wave then (-c, U (-z), V (-z)) is a (ultimately monotone) (β, α)-traveling wave.
In other words, we do not require the traveling wave to be monotone on R, but only to be monotone in some neighborhoods of both -∞ and +∞. This reinforces our non-existence result which states as follows.

Theorem 2.4 (Non-existence of ultimately monotone traveling waves). Let 0 ≤ α, β ≤ 1 be given with α = β. Then, there is no ultimately monotone (α, β)-traveling wave for system (1.1).

The above theorem is proved in Section 3. The starting point consists in transforming system (2.1) into a first order system of four ODEs. Then, by a phase plane analysis, we prove that U + V -1 has to "oscillate" in a neighborhood of -∞ or +∞, from which we get a contradiction. Remark 2.5. As easily seen from the proof, to exclude the existence of a traveling wave with speed c > 0, c < 0, it is enough to assume that (2.2) holds in a neighborhood of -∞, +∞ respectively. In other words, there is no traveling wave for which the invading state is monotonically reached. The existence of a traveling wave for which the invading state is not monotonically reached remains an open issue. Last, as seen from subsection 3.1, the non existence of standing waves (c = 0) does not require any ultimately monotonicity assumption.

Our second main focus is concerned with the large time behavior of the solution of system (1.1) starting from a nonnegative (nontrivial) compactly supported initial datum. In both the strong competition case [START_REF] Carrere | Spreading speeds for a two-species competition-diffusion system[END_REF], [START_REF] Peng | Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition[END_REF], and the strong-weak competition case [START_REF] Girardin | Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition-diffusion systems[END_REF], the monotone traveling wave solutions of the entire system play a key role in studying the large time behavior of the solution of the Cauchy problem. However, for the critical competition case, such traveling wave solutions do not exist.

In order to state our result, we define the (minimal) Fisher-KPP traveling wave solution (c u , U KP P ) as

c u := 2, U ′′ KP P + c u U ′ KP P + U KP P (1 -U KP P ) = 0, U KP P (-∞) = 1, U KP P (∞) = 0, (2.3)
and, similarly, (c v , V KP P ) as

c v := 2 √ dr, dV ′′ KP P + c v V ′ KP P + rV KP P (1 -V KP P ) = 0, V KP P (-∞) = 1, V KP P (∞) = 0.
(2.4)

Let us recall that both U KP P and V KP P are uniquely defined "up to a shift". Note that, c u (resp. c v ) also represents the spreading speed of u (resp. v) in the absence of v (resp. u).

Theorem 2.6 (Propagation phenomenon). Let (u, v) = (u, v)(t, x) be the solution of system (1.1) starting from an initial datum (u 0 , v 0 ) = (u 0 , v 0 )(x) satisfying u 0 and v 0 are continuous, nontrivial, compactly supported, and 0 ≤ u 0 , v 0 ≤ 1.

(2.5)

Then the following holds.

(i) Assume dr > 1 (i.e. c v > c u ). Then lim t→∞ sup x∈R v(t, x) -V KP P |x| -c v t + 3d c v ln t + η * (t) + sup x∈R u(t, x) = 0, (2.6)
where η * is a bounded function on [0, ∞).

(ii) Assume dr < 1 (i.e. c v < c u ). Then

lim t→∞ sup x∈R u(t, x) -U KP P |x| -c u t + 3 c u ln t + η * * (t) + sup x∈R v(t, x) = 0, (2.7)
where η * * is a bounded function on [0, ∞).

The above theorem is proved in Section 4. Let us briefly comment on Theorem 2.6. First of all, for the case dr > 1 (or dr < 1), the "faster species", namely v, excludes the "slower one", namely u, and imposes its spreading speed, see (2.6). Furthermore, we find that the profile of the solution uniformly converges to the corresponding minimal KPP traveling wave solution, and this up to an identified logarithmic Bramson correction, see (2.6) again.

On the other hand, for the case dr = 1, it is difficult to decide which species leads the invading front, and the behavior of the solution is highly depending on both the parameters and the shape of the initial datum. For the case d = r = 1, for any initial datum satisfying (2.5), a coexistence phenomenon happens. However, for the case dr = 1 but d = 1, the behavior of the solution is much more intricate. Indeed, in this case, the logarithmic phase drifts for u and v are different and there is a narrow region of width O(ln t) where the behaviors of u and v are difficult to "anticipate". This may cause some subtle phenomena (both species driving the front or one excluding the other) and makes the mathematical analysis quite involved. We hope to address these issues in a future work.

Our second result on the Cauchy problem deals with the region "|x| ≤ ε * t", where the profile of the solution is more of the "Heat equation type". Theorem 2.7 (Bump phenomenon). Let (u, v) = (u, v)(t, x) be the solution of system (1.1) starting from an initial datum (u 0 , v 0 ) = (u 0 , v 0 )(x) satisfying (2.5). Denote

k * := min 1 2d , d 2 , d * := max(1, d).
Then the following holds.

x 1 0 v u -2 √ drt + 3 2 d r ln t ← → 2 √ drt -3 2 d r ln t BUMP -ε * t ← → ε * t Figure 1:
The asymptotic profile of the solution, in the case dr > 1.

(i) Assume dr > 1 (i.e. c v > c u ). Then for ε * > 0 small enough and 0 < θ < 1 2 , there exist C 2 > C 1 > 0 and T > 0 such that both

C 1 t -1 2 e -x 2 4t ≤ u(t, x) ≤ C 2 t -k * e -x 2 4d * t , (2.8) max C 1 t -1 2 e -x 2 4t -t -(1+θ) , 0 ≤ 1 -v(t, x) ≤ C 2 t -k * e -x 2 4d * t , (2.9)
hold for any t ≥ T , |x| ≤ ε * t.

(ii) Assume dr < 1 (i.e. c v < c u ). Then for ε * * > 0 small enough and 0 < θ < 1 2 , there exist C 4 > C 3 > 0 and T > 0 such that both

C 3 t -1 2 e -x 2 4t ≤ v(t, x) ≤ C 4 t -k * e -x 2 4d * t , (2.10) max C 3 t -1 2 e -x 2 4t -t -(1+θ) , 0 ≤ 1 -u(t, x) ≤ C 4 t -k * e -x 2 4d * t , (2.11)
hold for any t ≥ T , |x| ≤ ε * * t.

The above theorem is proved in Section 5. Let us briefly comment on Theorem 2.7, say in the case dr > 1. As revealed by (2.8) and (2.9), the solution converges to (0, 1) exponentially in regions of the form |x| ≥ εt with ε > 0, but only algebraically in "sublinear regions" of the form |x| √ t. We call this a bump phenomenon, see Figure 1. Such a phenomenon does not occur in the strong competition case [START_REF] Peng | Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition[END_REF]. As far as the strong-weak competition case is concerned, the results as stated in [START_REF] Girardin | Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition-diffusion systems[END_REF] are not sufficient to decide if it occurs or not, but we assert it does not, as proved in the forthcoming work [START_REF] Wu | Lotka-Volterra diffusion systems: the strong-weak type[END_REF]. Therefore, the present paper is the first one revealing a bump phenomenon in the context of competition-diffusion systems. We believe such a phenomenon is reserved for the critical case a = b = 1, and is rare to happen in the context of homogeneous reaction-diffusion equations.

Non-existence of traveling waves

This section is devoted to the proof of Theorem 2.4 on the non-existence of ultimately monotone traveling waves for system (1.1).

Preliminary results and observations

In this subsection, the ultimately monotonicity assumption is not required, and thus a traveling wave is understood in the sense of Definition 2.1. We start with the following a priori estimates for a traveling wave.

Lemma 3.1. Any traveling wave has to satisfy

0 < U < 1, 0 < V < 1, and U ′ (±∞) = V ′ (±∞) = 0.
Proof. The positivity of the profiles follows from the strong maximum principle. If U ≤ 1 is not true, then U has to reach a maximum value strictly larger than 1 at some point, and evaluating the U -equation at this point gives a contradiction. Hence U ≤ 1 and, from the strong maximum principle, U < 1. Similarly, one has V < 1.

We now prove the limit behavior U ′ (+∞) = 0, the other ones being proved similarly. Denote the set of accumulation points of

U ′ in +∞ by A. Since U is bounded, 0 ∈ A. Let ℓ ∈ A. Then there exists a sequence z n → +∞ such that U ′ (z n ) → ℓ as n → +∞. Then (U n , V n )(z) := (U, V )(z + z n ) solves U ′′ n + cU ′ n = -U n (1 -U n -V n ).
Since the L ∞ norm of the right hand side term is uniformly bounded with respect to n, the interior elliptic estimates imply that, for all R > 0 and 1 < p < ∞, the sequence

(U n ) is bounded in W 2,p (-R, R). From Sobolev embedding theorem we have that, up to a subsequence, U n converges to some U ∞ in C 1 loc (R). The boundary condition U (+∞) = β thus enforces U ∞ ≡ β and U ′ ∞ ≡ 0. As a result, U ′ (z n ) = U ′ n (0) → U ′ ∞ (0)
= 0, and thus ℓ = 0. Hence A = {0}, which concludes the proof.

We conclude this subsection by showing the non-existence of traveling wave solutions for two special cases.

Proposition 3.2 (Non-existence of standing waves).

There is no standing wave, i.e. traveling wave with speed c = 0, for system (1.1).

Proof. Assume c = 0. By adding the both sides of the U -equation and the V -equation, we find that W := U + V satisfies

W ′′ + U + r d V (1 -W ) = 0, W (±∞) = 1.
If W ≡ 1, then W reaches either a maximum value strictly larger than 1 or a minimum value in (0, 1), which is impossible from the above equation (recall that U + r d V > 0). As a result W = U + V ≡ 1. Going back to the original equations we have U ′′ = V ′′ = 0. Since U and V are bounded, U and V must be constant, which cannot happen since α = β. Proposition 3.3 (Non-existence of traveling waves when d = 1). Assume d = 1. Then there is no traveling wave for system (1.1).

Proof. Again, by adding the both sides of the U -equation and V -equation, we see that

W := U + V satisfies W ′′ + cW ′ + (U + rV )(1 -W ) = 0, W (±∞) = 1,
so that, as in Proposition 3.2, we have W ≡ 1, and thus U ′′ + cU ′ = 0. From Lemma 3.1, by integrating both sides from -∞ to +∞, we have c(βα) = 0, which yields a contradiction since c = 0 and α = β.

Proof of Theorem 2.4

We now consider the case of ultimately monotone traveling waves.

Lemma 3.4. Let 0 ≤ α, β ≤ 1 be given with α = β. Let (c, U, V ) be an ultimately monotone (α, β)-traveling wave solution. Then (2.2) is refined in

U ′ (z)V ′ (z) < 0, for all z ∈ (-∞, z 0 ] ∪ [z * 0 , +∞). (3.1)
Proof. We only deal with the behavior around -∞. If the conclusion is false, we may assume that U ′ (z) > 0 and

V ′ (z) > 0 for all z ∈ (-∞, z 0 ], the case U ′ (z) < 0 and V ′ (z) < 0 being treated similarly. From the boundary conditions (U, V )(-∞) = (α, 1 -α) and (U, V )(+∞) = (β, 1 -β), there must exist a point z 1 > z 0 such that U ′ (z) > 0, V ′ (z) > 0, for all z < z 1 , U ′ (z 1 ) = 0 or V ′ (z 1 ) = 0. Assume w.l.o.g. that U ′ (z 1 ) = 0. In particular U ′′ (z 1 ) ≤ 0. From the U -equation, this enforces (U + V )(z 1 ) ≤ 1 which contradicts to (U + V )(-∞) = 1 and (U + V ) ′ > 0 on (-∞, z 1 ).
We now prove, in the case d = 1, the non-existence of ultimately monotone traveling waves with speed c = 0. In view of subsection 3.1, this is enough to complete the proof of Theorem 2.4.

Completion of the proof of Theorem 2.4. For d = 1, let us consider (c, U, V ) an ultimately monotone (α, β)-traveling wave with c = 0. In the sequel we only deal with the case c > 0 for which we perform a phase plane analysis around -∞ (for the case c < 0, one has to perform a phase plane analysis around +∞ with similar arguments). We define W := α -U , P := U ′ , R := V -1 + α, Q := V ′ . Then we can rewrite (2.1) as

                         W ′ = -P, P ′ = -cP -(α -W )(W -R), R ′ = Q, Q ′ = - c d Q - r d (R + 1 -α)(W -R), (W, P, R, Q)(-∞) = (0, 0, 0, 0), (W, P, R, Q)(+∞) = (α -β, 0, α -β, 0). (3.2) Assume that W -R is ultimately nonnegative, that is ∃z * < z 0 , ∀z ≤ z * , (W -R)(z) ≥ 0. (3.3)
From (3.1), we know that it holds either P > 0 or Q > 0 on (-∞, z 0 ]. Moreover, from Lemma 3.1, α -W ≥ 0 and R + 1α ≥ 0. If P > 0 on (-∞, z 0 ], then from the P -equation in (3.2), we have P ′ < 0 on (-∞, z * ], which contradicts to P > 0 on (-∞, z 0 ] and P (-∞) = 0. If Q > 0 on (-∞, z 0 ], we similarly get a contradiction from the Q-equation. Hence (3.3) does not hold. Assume that W -R is ultimately nonpositive, that is

∃z * < z 0 , ∀z ≤ z * , (W -R)(z) ≤ 0. (3.4)
From (3.1) again, we know that it holds either P < 0 or Q < 0 on (-∞, z 0 ]. If P < 0 on (-∞, z 0 ], then from the P -equation in (3.2), we have P ′ > 0 on (-∞, z * ], which contradicts to P < 0 on (-∞, z 0 ] and P (-∞) = 0. If Q < 0 on (-∞, z 0 ], we similarly get a contradiction from the Q-equation. Hence (3.4) does not hold. As a result, since W -R is neither ultimately nonnegative nor ultimately nonpositive, we can find a local maximum point z 1 < z 0 and a local minimum point z 2 < z 0 such that

(W -R)(z 1 ) > 0, (W -R) ′ (z 1 ) = 0, (W -R) ′′ (z 1 ) ≤ 0; (W -R)(z 2 ) < 0, (W -R) ′ (z 2 ) = 0, (W -R) ′′ (z 2 ) ≥ 0. Note that (W -R) ′′ = cP + c d Q + α -W + r d (R + 1 -α) (W -R). From (3.1), it holds either Q > 0 or Q < 0 on (-∞, z 0 ]. Let us first consider the case d < 1. If Q > 0 on (-∞, z 0 ], since (W -R) ′ (z 1 ) = 0 means (P +Q)(z 1 ) = 0, we have (cP + c d Q)(z 1 ) > 0. Therefore, from the above equation, (W -R) ′′ (z 1 ) > 0, which is a contradiction. On the other hand, if Q < 0 on (-∞, z 0 ], since (P + Q)(z 2 ) = 0, we have (cP + c d Q)(z 2 ) < 0.
Therefore, from the above equation, (W -R) ′′ (z 2 ) < 0, which is a contradiction. Last, the case d > 1 can be treated similarly.

Therefore, we conclude that system (1.1) does not admit any ultimately monotone traveling wave.

The Cauchy problem

In this section, we consider system (1.1) with a compactly supported initial datum, and prove the propagation result, namely Theorem 2.6.

Preliminaries

Let us start by briefly recalling the competitive comparison principle. Define the operators

N 1 [u, v] := u t -u xx -u(1 -u -v) and N 2 [u, v] := v t -dv xx -rv(1 -v -u). Consider a domain Ω := (t 1 , t 2 ) × (x 1 , x 2 ) with 0 ≤ t 1 < t 2 ≤ +∞ and -∞ ≤ x 1 < x 2 ≤ +∞. A (classical) super-solution is a pair (u, v) ∈ C 1 (t 1 , t 2 ), C 2 ((x 1 , x 2 )) ∩ C b Ω 2 satisfying N 1 [u, v] ≥ 0 and N 2 [u, v] ≤ 0 in Ω. Similarly, a (classical) sub-solution (u, v) requires N 1 [u, v] ≤ 0 and N 2 [u, v] ≥ 0.
Proposition 4.1 (Comparison Principle). Let (u, v) and (u, v) be a super-solution and subsolution of system (1.1) in Ω, respectively. If

u(t 1 , x) ≥ u(t 1 , x) and v(t 1 , x) ≤ v(t 1 , x), for all x ∈ (x 1 , x 2 ),
and, for i = 1, 2,

u(t, x i ) ≥ u(t, x i ) and v(t, x i ) ≤ v(t, x i ), for all t ∈ (t 1 , t 2 ), then, it holds u(t, x) ≥ u(t, x) and v(t, x) ≤ v(t, x), for all (t, x) ∈ Ω. If x 1 = -∞ or x 2 =
+∞, the hypothesis on the corresponding boundary condition can be omitted.

Denote (u, v) = (u, v)(t, x) as the solution of (1.1) starting from (u 0 , v 0 ) = (u 0 , v 0 )(x) satisfying (2.5). Obviously, (1, 0) is a super-solution while (0, 1) is a sub-solution. It thus follows from (2.5), the comparison principle and the strong maximum principle that 0 < u(t, x) < 1 and 0 < v(t, x) < 1, for all t > 0, x ∈ R.

(4.1)

Actually, the comparison principle also holds for the so-called generalized sub-and supersolutions. This is a rather well-known fact, and we refer to the clear exposition in [9, subsection 2.1] for more details. In particular, if (u 1 , v) and (u 2 , v) are both classical sub-solutions, then (max(u 1 , u 2 ), v) is a generalized sub-solution. Also, if (u, v 1 ) and (u, v 2 ) are both classical sub-solutions, then (u, min(v 1 , v 2 )) is a generalized sub-solution.

We now start the proof of Theorem 2.6. Observe that, by changing the variables as x = d r y and t = 1 r s, system (1.1) can be rewritten as

∂ s v = v yy + v(1 -v -u), s > 0, y ∈ R, ∂ s u = d -1 u xx + r -1 u(1 -u -v), s > 0, y ∈ R.
Therefore, without loss of generality, we assume from now that dr > 1, that is c v > c u , and shall prove the statement (i) in Theorem 2.6.

Since c v > c u and u cannot propagate faster than c u , the behavior of the solution in the region |x| > c u t is rather well-understood. and

lim t→∞ sup c 1 t≤|x|≤c 2 t |1 -v(t, x)| = 0, for all c u < c 1 < c 2 < c v . (4.4)
Proof. Without loss of generality, we only deal with the case x ≥ 0. Define

U (t, x) := C 1 e -cu 2 (x-cut) and V (t, x) := C 2 e -cv 2d (x-cvt) ,
where C 1 > 0 and C 2 > 0 are chosen large enough so that U (0, •) ≥ u 0 and V (0, •) ≥ v 0 . We can easily check that (U, 0) is a super-solution while (0, V ) is a sub-solution. As a result, we have

0 < u(t, x) ≤ min 1, C 1 e -cu 2 (x-cut)
and 0 < v(t, x) ≤ min 1, C 2 e -cv 2d (x-cvt) , (4.5) from which (4.2) and (4.3) follow.

Next, let c u < c 1 < c 2 < c v be given. Select 0 < a < 1 < b and consider (u * , v * ) the solution of the strong-weak competition system

∂ t u * = u * xx + u * (1 -u * -av * ), ∂ t v * = dv * xx + rv * (1 -v * -bu * ), (4.6)
starting from (u 0 , v 0 ). Obviously, (u * , v * ) is a super-solution for system (1.1), and thus v * (t, x) ≤ v(t, x) ≤ 1 for all t ≥ 0 and x ∈ R. Since the statements ( 2) and ( 3) in [9, Theorem 1.1] imply lim

t→∞ sup c 1 t≤x≤c 2 t |1 -v * (t, x)| = 0,
the same conclusion holds for v.

Construction of the super-solution

The goal of this subsection is to construct an adequate super-solution in, roughly speaking, the region |x| < c u t. More precisely, let 1 d < r 1 < r be given and define c * v := 2 √ dr 1 < c v . In the sequel, we introduce V 1 as a traveling wave solution with speed

c * v = 2 √ dr 1 solving dV ′′ 1 + c * v V ′ 1 + r 1 V 1 (1 -V 1 ) = 0, V 1 (-∞) = 1, V 1 (∞) = 0. ( 4.7)
As well-known, V ′ 1 < 0 and there are λ 1 > 0 and

M 1 > 0 such that 1 -V 1 (ξ) ∼ M 1 e λ 1 ξ as ξ → -∞. (4.8)
Let us fix c u < c 1 < c * v . For T > 0, we will work in the domain (which is "expanding in time")

Ω 1 (T ) := {(t, x) ∈ (T, ∞) × R : |x| < c 1 t}. (4.9)
It turns out that the construction of the super-solution is highly dependent on the value of d.

• The case d ≤ 1. We introduce s = s(t, x) as the solution of the Cauchy problem

∂ t s = s xx , s(0, x) = s 0 (x) := B 1 e -q|x| , (4.10)
and look for a super-solution ( Ũ , Ṽ ) in the form

   Ũ (t, x) := t 1-d 2 (1 -e -τ t )s(t, x), Ṽ (t, x) := V 1 (x -c * v t) + V 1 (-x -c * v t) -1 -Ũ (t, x). (4.11)
All parameters that will be determined below (namely B 1 , q and τ ) are positive, and q < 1.

• The case d ≥ 1. We introduce s = s(t, x) as the solution of the Cauchy problem

∂ t s = ds xx , s(0, x) = s 0 (x) := B 1 e -q|x| , (4.12)
and look for a super-solution ( Ũ , Ṽ ) in the form

   Ũ (t, x) := t d-1 2d (1 -e -τ t )s(t, x), Ṽ (t, x) := V 1 (x -c * v t) + V 1 (-x -c * v t) -1 -Ũ (t, x). (4.13)
All parameters that will be determined below (namely B 1 , q and τ ) are positive, and q < 1 d . Obviously, (4.10)-(4.11) and (4.12)-(4.13) coincide when d = 1.

Proposition 4.3 (Super-solutions). The following holds.

(i) Assume d ≤ 1. Let 0 < q < 1 and 0 < τ < λ 1 (c * vc 1 ) be given. Then there exists T * > 0 such that, for all B 1 > 0, ( Ũ , Ṽ ), given by (4.10)-(4.11), is a super-solution in the domain Ω 1 (T * ) as defined in (4.9).

(ii) Assume d ≥ 1. Let 0 < q < 1 d and 0 < τ < λ 1 (c * vc 1 ) be given. Then there exists T * > 0 such that, for all B 1 > 0, ( Ũ , Ṽ ), given by (4.12)-(4.13), is a super-solution in the domain Ω 1 (T * ) as defined in (4.9).

Proof. Since our super-solutions are even functions, it is enough to deal with x ≥ 0. In other words, we work for t ≥ T (with T > 0 to be selected) and 0 ≤ x < c 1 t, with c u < c 1 < c * v . For ease of notations, we shall use the shortcuts

ξ ± := ±x -c * v t. Since ξ -≤ -c * v t and ξ + ≤ -(c * v -c 1 )
t, it follows from V ′ 1 < 0 and (4.8) that there exist C -> 0 and C + > 0 such that, for T > 0 large enough,

1-V 1 (ξ -) ≤ C -e -λ 1 c * v t and 1-V 1 (ξ + ) ≤ C + e -λ 1 (c * v -c 1 )t , for all (t, x) ∈ Ω + 1 (T ), (4.14)
where Ω + 1 (T ) := Ω 1 (T ) ∩ (T, ∞) × [0, ∞). Moreover, up to enlarging T > 0 if necessary, there exists 0 < ρ < 1 3 such that

0 < 1 -V 1 (ξ ± ) ≤ ρ, for all (t, x) ∈ Ω + 1 (T ). ( 4

.15)

We first assume d ≤ 1. Some straightforward computations combined with (4.10) yield

N 1 [ Ũ , Ṽ ] = t 1-d 2 (1 -e -τ t )s 1 -d 2 t -1 + τ e -τ t 1 -e -τ t -2 + V 1 (ξ + ) + V 1 (ξ -) .
In view of (4.14), by choosing τ < λ 1 (c * vc 1 ), we deduce that, for T > 0 large enough,

N 1 [ Ũ , Ṽ ] ≥ 0 in Ω + 1 (T ).
On the other hand, some straightforward computations combined with (4.10) and (4.7) yield

N 2 [ Ũ , Ṽ ] = J 1 + J 2 + J 3 ,
where

J 1 := t 1-d 2 (1 -e -τ t )s r 2 -V 1 (ξ + ) -V 1 (ξ -) - τ e -γt r(1 -e -τ t ) - 1 -d 2 t -1 -(1 -d) ∂ t s s , J 2 := (r 1 -r)V 1 (ξ + )(1 -V 1 (ξ + )), J 3 := (1 -V 1 (ξ -)) ((r 1 -r)V 1 (ξ -) + r(2 -2V 1 (ξ + ))) .
Since r 1 < r and 0 < V 1 < 1, we have J 2 ≤ 0. Next, from (4.15), we have

J 3 ≤ (1 -V 1 (ξ -)) ((r 1 -r)(1 -ρ) + r(2 -2V 1 (ξ + ))) ,
which, in view of (4.14), is nonpositive up to enlarging T > 0 if necessary. Last, from the "Heat kernel expression" of s(t, x), namely

s(t, x) = (G(t, •) * s 0 ) (x), where G(t, x) := 1 √ 4πt e -x 2 4t ,
we can check that ∂ t s(t, x) ≥ -1 2t s(t, x). As a result, since d ≤ 1, we have

J 1 ≤ t 1-d 2 (1 -e -τ t )sr 2 -V 1 (ξ + ) -V 1 (ξ -) - τ e -τ t r(1 -e -τ t )
.

In view of (4.14) and τ < λ 1 (c * vc 1 ), we have J 1 ≤ 0 up to enlarging T > 0 if necessary. As a result, N 2 [ Ũ , Ṽ ] ≤ 0 in Ω + 1 (T ). Next, we assume d ≥ 1. Some straightforward computations combined with (4.12) yield

N 1 [ Ũ , Ṽ ] = t d-1 2d (1 -e -τ t )s d -1 2d t -1 + d -1 d ∂ t s s + τ e -τ t 1 -e -τ t -2 + V 1 (ξ + ) + V 1 (ξ -) . As above, since d ≥ 1, ∂ t s(t, x) ≥ -1 2t s(t, x) implies N 1 [ Ũ , Ṽ ] ≥ t d-1 2d (1 -e -τ t )s τ e -τ t 1 -e -τ t -2 + V 1 (ξ + ) + V 1 (ξ -) .
In view of (4.14) and τ < λ 1 (c * vc 1 ), we deduce that, for T > 0 large enough, N 1 [ Ũ , Ṽ ] ≥ 0 in Ω + 1 (T ). On the other hand, some straightforward computations combined with (4.12) and (4.7) yield

N 2 [ Ũ , Ṽ ] = J 1 + J 2 + J 3 ,
where

J 1 := t d-1 2d (1 -e -τ t )s r(2 -V 1 (ξ + ) -V 1 (ξ -)) - τ e -τ t 1 -e -τ t - d -1 2d t -1 , J 2 := (r 1 -r)V 1 (ξ + )(1 -V 1 (ξ + )), J 3 := (1 -V 1 (ξ -)) ((r 1 -r)V 1 (ξ -) + r(2 -2V 1 (ξ + ))) .
By applying the same argument as that for d ≤ 1, we get

N 2 [ Ũ , Ṽ ] ≤ 0 in Ω + 1 (T ).
Note that, time T * in Proposition 4.3 is independent on B 1 > 0, which leaves "some room" to enlarge B 1 so that the "initial order" and the "order on the boundary of the domain" are suitable for the comparison principle to be applicable.

Proposition 4.4 (First estimate on (u, v)

). There exist 0 < q < min(1, Proof. We aim at applying the comparison principle in Ω 1 (T ), as defined in (4.9), with a well-chosen T > 0. Select 0 < q < min(1, 1 d ) small enough so that max(qc 1q 2 , qc 1dq 2 ) < c 1c u . (4.16)

From Proposition 4.3, for any T ≥ T * , we are equipped with a super-solution ( Ũ , Ṽ ) for which B 1 > 0 is arbitrary. We only deal with the case d ≤ 1, the case d ≥ 1 being similar. We first focus on x = c 1 t, t ≥ T * (the case x = -c 1 t, t ≥ T * being similar). Let us prove that, up to enlarging T * > 0 if necessary, it holds u(t, c 1 t) ≤ Ũ (t, c 1 t), for all t ≥ T * .

(4.17)

From the proof of Proposition 4.2, we know that u(t, c 1 t) ≤ C 1 e -(c 1 -cu)t (recall that c u = 2). Recalling s 0 (x) = B 1 e -q|x| , we have

s(t, x) = B 1 √ 4πt 0 -∞ e -(x-y) 2 4t e qy dy + +∞ 0 e -(x-y) 2 4t
e -qy dy , which can be recast, after some elementary computations,

s(t, x) = B 1 √ π e q 2 t-qx +∞ 2qt-x 2 √ t e -w 2 dw + e q 2 t+qx +∞ 2qt+x 2 √ t e -w 2 dw . (4.18)
In particular, since 2q < 2 < c 1 , we have, by enlarging

T * > 0 if necessary, Ũ(t, c 1 t) ≥ 1 2 s(t, c 1 t) ≥ B 1 4 e -(qc 1 -q 2 )t ≥ B 1 4 e -(c 1 -cu)t .
The last inequality holds from the choice (4.16). Thus B 1 > 4C 1 is enough to get (4.17).

Let us recall that v ≥ v * where (u * , v * ) is the solution of the strong-weak competition system (4.6) with the same initial datum (u 0 , v 0 ). From Lemma A.1 (ii) (see Appendix), up to enlarging T * if necessary, there exist µ > 0 and K > 0 such that v * (t, c 1 t) ≥ 1 -Ke -µt for all t ≥ T * . On the other hand, the construction of Ṽ implies that Ṽ (t, c 1 t) ≤ 1 -Ũ (t, c 1 t). Therefore, by choosing qc 1q 2 < µ, up to enlarging T * > 0 if necessary, we have

Ṽ (t, c 1 t) ≤ v(t, c 1 t), for all t ≥ T * . (4.19)
Now, q > 0 and T * > 0 are fixed from the above discussion. We focus on the initial datum, namely t = T * , |x| ≤ c 1 T * . As above, we deduce from (4.18) that

inf |x|≤c 1 T * Ũ (T * , x) ≥ 1 2 s(T * , c 1 T * ) ≥ B 1 4 e -(qc 1 -q 2 )T * ≥ 1 ≥ sup t>0,x∈R u(t, x),
provided that B 1 > 0 is large enough. On the other hand, sup

|x|≤c 1 T * Ṽ (T * , x) ≤ 1 -inf |x|≤c 1 T * Ũ (T * , x) ≤ 1 - B 1 4 e -(qc 1 -q 2 )T * ≤ 0 ≤ inf t>0,x∈R v(t, x),
provided that B 1 > 0 is large enough. As a consequence, the comparison principle can be applied in Ω 1 (T * ), which concludes the proof of Proposition 4.4.

Proof of Theorem 2.6

From the discussion above, we are now in the position to obtain the following spreading speed result.

Proposition 4.5 (Spreading speed). Let (u, v) = (u, v)(t, x) be the solution of (1.1) starting from an initial datum (u 0 , v 0 ) = (u 0 , v 0 )(x) satisfying (2.5). Then the following holds.

(i) Assume dr > 1 (i.e. c v > c u ). Then, for any 0 < c 1 < c v < c 2 , lim t→∞ sup x∈R u(t, x) + sup |x|≤c 1 t |1 -v(t, x)| + sup |x|≥c 2 t v(t, x) = 0. ( 4
0 < c 1 < c v , we select c 1 < c * v < c v . Then, Proposition 4.4 yields sup |x|≤c 1 t |1 -v(t, x)| → 0 as t → ∞.
The last part of (4.20) is nothing else than the estimate (4.3).

We are now in the position to complete the proof of Theorem 2.6.

Proof of Theorem 2.6 (i). Since the proof for x ≤ 0 follows from the same argument, we only deal with x ≥ 0. Let us prove (2.6). For a given m ∈ (0, 1), we define E m (t) as the m-level set of v(t, •), namely E m (t) := {x > 0 : v(t, x) = m}.

We claim that there exist M > 0 and T > 0 such that

c v t - 3d c v ln t -M ≤ min E m (t) ≤ max E m (t) ≤ c v t - 3d c v ln t + M, for all t ≥ T. (4.22)
The upper bound in (4.22) is obtained by using the solution of

∂ t v = dv xx + rv(1-v), starting from v(0, x) = v 0 (x)
, as a super-solution. We refer to [START_REF] Peng | Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition[END_REF]Lemma 4.1], see also [START_REF] Bramson | Convergence of solutions of the Kolmogorov equation to travelling waves[END_REF] and [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF].

As for the lower bound in (4.22), it follows from [START_REF] Peng | Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition[END_REF]Lemma 4.5] which is based on an idea of [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF]. A sketch of the proof is as follows. Let us a fix a small ε > 0 (this is necessary because of the bump phenomenon). By combining (4.5), Proposition 4.4 and (4.18) 3 , we see that there exist C > 0, µ > 0 and T > 0 such that

sup |x|≥εt u(t, x) ≤ Ce -µt , for all t ≥ T. (4.23)
As a result, by setting C 0 = rC, we have

∂ t v ≥ dv xx + v(r -rv -C 0 e -µt
), for all t > 0, x > εt.

3 from which one can straightforwardly deduce that sup x≥εt s(t, x) = s(t, εt

) = O e -ε 2
The key idea, borrowed from [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF], is then to linearize the above equation, and to consider

∂ t w = dw xx + w(r -C 0 e -µt ), t > 0, x > Γ(t) := c v t - 3d c v ln(t + t 0 ), (4.24)
together with the Dirichlet boundary conditions w(t, Γ(t)) = 0 and a compactly supported initial datum w(0, •). Then, one can exactly reproduce the technical arguments of [20, Lemma 4.3 and 4.4], mainly borrowed from [START_REF] Hamel | A short proof of the logarithmic Bramson correction in Fisher-KPP equations[END_REF], to obtain the lower bound in (4.22). Last, by applying (4.22), we can reproduce the argument of [12, Section 4], see also [20, Proof of Theorem 2], to conclude that there exists a bounded function

η * : [0, ∞) → R such that lim t→∞ sup x≥0 v(t, x) -V 1 x -c v t + 3d c v ln t + η * (t) = 0, (4.25)
which, combined with (4.20), concludes the proof of (2.6).

The bump phenomenon

In this section, we will provide a lower estimate for the solution of system (1.1) starting from a compactly supported initial datum, and prove Theorem 2.7 on the bump phenomenon.

Construction of the sub-solution

The goal of this subsection is to construct an adequate sub-solution in, roughly speaking, the region |x| < c v t. More precisely, let r 2 > r be given and define

c * * v := 2 √ dr 2 > c v . Let us fix c v < c 2 < c * * v .
For T > 0, we will work in the domain (which is "expanding in time")

Ω 2 (T ) := {(t, x) ∈ (T, ∞) × R : |x| < c 2 t}. ( 5.1)
A key observation for the construction is the following: from Proposition 4.2, in the region c u t < |x| < c v t, we have u + v ≈ 1, and therefore

∂ t u ≈ u xx , ∂ t v ≈ dv xx .
We thus introduce f = f (t, x) and h = h(t, x) the solutions of the Cauchy problems

∂ t f = f xx , f (0, x) = f 0 (x) := B 2 1 (-1,1) (x), ∂ t h = h xx , h(0, x) = h 0 (x) := B 3 e -k|x| , (5.2)
and look for a sub-solution (U, V ) in the form

   U (t, x) := g(t)f (t, x) -h(t, x), V (t, x) := V 2 (x -c * * v t -ζ 0 ) + V 2 (-x -c * * v t -ζ 0 ) -1 -U (t, x) + 1 t 1+θ , (5.3) where g(t) := exp 1 δ(1 + t) δ .
Here, all parameters that will be determined below (namely

B 2 , B 3 , k, ζ 0 , θ, δ) are positive, B 2 < 1, B 3 < 1, while V 2 is the traveling wave solution with speed c * * v = 2 √ dr 2 satisfying dV ′′ 2 + c * * v V ′ 2 + r 2 V 2 (1 -V 2 ) = 0, V 2 (-∞) = 1, V 2 (∞) = 0.
(5.4)

It is well-known that V ′ 2 < 0 and there exist λ 2 > 0 and

M 2 > 0 such that 1 -V 2 (ξ) ∼ M 2 e λ 2 ξ as ξ → -∞. (5.5)
Next, we shall provide some estimates which are based on the "Heat kernel expressions" of the solutions f and h of (5.2). Note that, in Lemma 5.1, 0 < B 2 < 1 and 0 < B 3 < 1 can be relaxed to B 2 > 0 and B 3 > 0.

Lemma 5.1. Let δ > 0 and k > 0 be given, and set B 3 = γB 2 with some γ > 0. Then the following holds.

(i) For any given 0 < j < k, h(t, x) ≤ B 3 √ π k k 2 -j 2 t -1 2 e -x 2 4t
, for all t > 0, |x| ≤ 2jt.

(ii) For any given 0 < j < k and T > 0, there exists

γ 1 > 0 such that, if 0 < γ ≤ γ 1 , then, for all B 2 > 0, g(t)f (t, x) -h(t, x) > 0, for all t ≥ T, |x| ≤ 2jt.
(iii) There is T 0 > 0 such that, for all B 2 > 0,

g(t)f (t, x) -h(t, x) ≤ 0, for all t ≥ T 0 , |x| = 2kt.
Proof. Since f (t, •) and h(t, •) are even functions, it is enough to deal with x ≥ 0. Recalling that h 0 (x) = B 3 e -k|x| , we have

h(t, x) = B 3 √ 4πt 0 -∞ e -(x-y) 2 4t e ky dy + +∞ 0 e -(x-y) 2 4t
e -ky dy , which can be recast, after some elementary computations,

h(t, x) = B 3 √ π e k 2 t-kx +∞ 2kt-x 2 √ t e -w 2 dw + e k 2 t+kx +∞ 2kt+x 2 √ t e -w 2 dw . ( 5.6)
Now, recalling that +∞ X e -w 2 dw ≤ e -X 2 2X for any X > 0, the above expression implies that, for any 0

≤ x ≤ 2jt < 2kt, h(t, x) ≤ B 3 √ π e -x 2 4t √ t 2kt -x + √ t 2kt + x ≤ B 3 √ π k k 2 -j 2 t -1 2 e -x 2 4t ,
which proves (i).

Recalling that f 0 (x) = B 2 1 (-1,1) (x), we have

f (t, x) = B 2 √ 4πt 1 -1 e -(x-y) 2 4t dy = B 2 √ π x+1 2 √ t x-1 2 √ t e -w 2 dw. ( 5.7)
Hence, from g(t) ≥ 1, B 3 = γB 2 , (5.7) and (i), we deduce that, for all 0 ≤ x ≤ 2jt and t ≥ T ,

g(t)f (t, x) -h(t, x) ≥ B 2 √ π t -1 2 e -(x+1) 2 4t -γ k k 2 -j 2 t -1 2 e -x 2 4t ≥ B 2 √ π t -1 2 e -x 2 4t e -1 4T e -j -γ k k 2 -j 2 , ( 5.8)
which is enough to prove (ii). From (5.6), we have h(t, 2kt)

≥ B 3 √ π e -k 2 t +∞ 0 e -w 2 dw = B 3 √ π e -k 2 t √ π 2 .
Hence, from B 3 = γB 2 and (5.7), we deduce that, for t ≥ T 0 := 1 2k ,

g(t)f (t, 2kt) -h(t, 2kt) ≤ B 2 √ π g ∞ t -1 2 e -(2kt-1) 2 4t -γ √ π 2 e -k 2 t ≤ B 2 √ π e -k 2 t g ∞ e k t -1 2 -γ √ π 2 ,
which is nonpositive, up to increasing T 0 if necessary. The proof of (iii) is complete.

(t, x)| ≤ G(t, •) ∞ f 0 1 ≤ C G t -1 2 f 0 1 , with some C G > 0. Also, we have ∂ t f (t, x) = (∂ t G(t, •) * f 0 )(x)
, and thus

|∂ t f (t, x)| ≤ ∂ t G(t, •) ∞ f 0 1 ≤ C ′ G t -3 2 f 0 1 ,
with some C ′ G > 0. Note that, f 0 1 = 2B 2 , which implies C is independent on B 2 < 1 in (5.9) and (5.10).

Since h 0 ∈ L 1 (R), similar estimates hold for h(t, x) and ∂ t h(t, x), and

C = C(k) since h 0 1 = 2B 3 k ≤ 2 k .
We are now in the position to complete the construction of the sub-solution (U, V ) in the form (5.3). Proposition 5.4 (Sub-solutions). Let 0 < δ < θ < 1 2 be given. Let us fix k > 0, and set B 3 = γB 2 with 0 < γ < 1.

Then there exists T * > 0 such that, for all 0 < B 2 < 1 and ζ 0 > 0, (U, V ) is a sub-solution in the domain Ω 2 (T * ) as defined in (5.1).

Proof. Since U (t, •) and V (t, •) are even functions, it is enough to deal with x ≥ 0. In other words, we work for t ≥ T (with T > 0 to be selected) and 0 ≤ x < c 2 t, with c v < c 2 < c * * v . For simplicity of notations, we shall use the shortcuts

ξ ± := ±x -c * * v t -ζ 0 . Since ξ -≤ -c * * v t and ξ + ≤ -(c * * v -c 2 )
t, it follows from V ′ 2 < 0 and (5.5) that there exist C -> 0 and C + > 0 such that, for T > 0 large enough,

1 -V 2 (ξ -) ≤ C -e -λ 2 c * * v t and 1 -V 2 (ξ + ) ≤ C + e -λ 2 (c * * v -c 2 )
t , for all (t, x) ∈ Ω + 2 (T ), (5.11) where Ω + 2 (T ) := Ω 2 (T ) ∩ (T, ∞) × [0, ∞). Moreover, up to enlarging T > 0 if necessary, there exists 0 < ρ < 1 3 such that 0 < 1 -V 2 (ξ ± ) ≤ ρ, for all (t, x) ∈ Ω + 2 (T ).

(5.12) Some straightforward computations combined with (5.2) yield

N 1 [U, V ] = g ′ f -(gf -h)(2 -V 2 (ξ + ) -V 2 (ξ -) -t -(1+θ) ) ≤ -(1 + t) -(1+δ) gf + gf t -(1+θ) + h 2 -V 2 (ξ + ) -V 2 (ξ -) -t -(1+θ) , since 2 -V 2 (ξ + ) -V 2 (ξ -) > 0.
Thus, it follows from (5.11) that

N 1 [U, V ] ≤ gf -(1 + t) -(1+δ) + t -(1+θ) + h C -e -λ 2 c * * v t + C + e -λ 2 (c * * v -c 2 )t -t -(1+θ) .
Since δ < θ, it follows that, for T > 0 large enough, N 1 [U, V ] ≤ 0 in Ω + 2 (T ). Next, some straightforward computations combined with (5.2) and (5.4) yield

N 2 [ Ū , V ] = r 2 V 2 (ξ + )(1 -V 2 (ξ + )) + r 2 V 2 (ξ -)(1 -V 2 (ξ -)) + (1 + t) -(1+δ) gf +(d -1)(g∂ t f -∂ t h) -(1 + θ)t -(2+θ) -r(V 2 (ξ + ) + V 2 (ξ -) -1 -gf + h + t -(1+θ) )(2 -V 2 (ξ + ) -V 2 (ξ -) -t -(1+θ) ) = I 1 + • • • + I 5 ,
where

I 1 := rgf 2 -V 2 (ξ + ) -V 2 (ξ -) + 1 r (1 + t) -(1+δ) , I 2 := (1 -V 2 (ξ -)) (r 2 -r)V 2 (ξ -) + r 2 -2V 2 (ξ + ) -t -(1+θ) -h , I 3 := (1 -V 2 (ξ + )) ((r 2 -r)V 2 (ξ + ) -rh) , I 4 := (d -1)(g∂ t f -∂ t h), I 5 := rt -(1+θ) V 2 (ξ -) + 2V 2 (ξ + ) -2 + t -(1+θ) -gf + h - 1 + θ r t -1 .
Since 0 < V 2 < 1, we have I 1 ≥ 0. From r 2 > r, (5.9) and (5.12), we have

I 2 ≥ (1 -V 2 (ξ -)) (r 2 -r)(1 -ρ) -rt -(1+θ) -rCt -1 2 ≥ 0,
up to enlarging T > 0 if necessary. Similarly, we obtain I 3 ≥ 0. Last, from (5.12) and Lemma 5.3, we obtain

I 4 + I 5 ≥ rt -(1+θ) 1 -3ρ -g ∞ Ct -1 2 - 1 + θ r t -1 -C|d -1|( g ∞ + 1)t -3 2 .
Since θ < 1 2 and 0 < ρ < 1 3 , we have I 4 + I 5 ≥ 0 up to enlarging T > 0 if necessary. As a result, for T > 0 large enough, N 2 [U, V ] ≥ 0 in Ω + 2 (T ), and the proof of Proposition 5.4 is complete.

Note that, time T * in Proposition 5.4 is independent on 0 < B 2 < 1 and ζ 0 > 0, which leaves "some room" to reduce B 2 and to enlarge ζ 0 so that the "initial order" is suitable for the comparison principle to be applicable. We shall also need the suitable "order on the boundary of the domain", which will be obtained by choosing an appropriate k and Lemma 5.1 (iii). More precisely, the following holds. Proposition 5.5 (Second estimate on (u, v)). Let 0 < δ < θ < 1 2 be given. Let us fix k := c 2 2 > 0, and set B 3 = γB 2 with 0 < γ < 1. Then there exist T * * > 0, 0 < B 2 < 1 and ζ 0 > 0 such that

U (t, x) ≤ u(t, x) and v(t, x) ≤ V (t, x), for all t ≥ T * * , |x| ≤ c 2 t,
where (U, V ) is given by (5.3).

Proof. We aim at applying the comparison principle in Ω 2 (T ), as defined in (5.1), with a wellchosen T > 0. From Proposition 5.4, for any T ≥ T * , we are equipped with a sub-solution (U, V ) for which 0 < B 2 < 1 and ζ 0 > 0 are arbitrary. We now focus on |x| = c 2 t, t ≥ T , that is, with a slight abuse of language, the boundary of Ω 2 (T ). We now set T * * := max(T * , T 0 ), where T 0 > 0 is provided by Lemma 5.1 (iii). In particular this implies (recall the choice k = c 2 2 ) that U (t, ±c 2 t) ≤ 0 ≤ u(t, ±c 2 t), for all t ≥ T * * .

Next, recalling that c v < c 2 < c * * v , it follows from Proposition 4.2 that v(t, ±c 2 t) → 0 as t → ∞. On the other hand, for any t ≥ T * * ,

V (t, ±c 2 t) ≥ V 2 (-(c * * v -c 2 )t) + V 2 (-(c 2 + c * * v )t) -1.
As a result, up to enlarging T * * if necessary, one has

v(t, ±c 2 t) ≤ V (t ± c 2 t), for all t ≥ T * * .
Last we focus on the initial data, namely t = T * * and |x| ≤ c 2 T * * . From (4.1), there exists ε > 0 such that

min inf |x|≤c 2 T * * u(T * * , x), inf |x|≤c 2 T * * 1 -v(T * * , x) ≥ ε > 0.
We now select 0 < B 2 < ε 2 g ∞ . From this choice, we have

U (T * * , x) ≤ g ∞ B 2 ≤ ε 2 ≤ u(T * * , x), for all |x| ≤ c 2 T * * ,
and, for all |x| ≤ c 2 T * * ,

V (T * * , x) ≥ 2V 2 (-ζ 0 ) -1 -g ∞ B 2 ≥ 2V 2 (-ζ 0 ) -1 - ε 2 ≥ 1 -ε, by selecting ζ 0 > 0 large enough, which implies v(T * * , x) ≤ V (T * * , x), for all |x| ≤ c 2 T * * .
As a consequence, the comparison principle can be applied in Ω 2 (T * * ), which concludes the proof of Proposition 5.5.

Proof of Theorem 2.7

It remains to prove the bump phenomenon which, as explained in Section 2, is reserved to the critical competition case under consideration. Let 0 < ε < c u be given and let us prove (2.8) and (2.9). Let us set T * * > 0 such that both Proposition 5.5 and Proposition 4.4 apply. In the sequel, we always consider t ≥ T * * and 0 ≤ x ≤ εt.

In particular, one has

g(t)f (t, x) -h(t, x) = U (t, x) ≤ u(t, x) ≤ Ũ (t, x) = t -(k * -1 2 ) (1 -e -τ t )s(t, x).
This estimate and (5.8) (with j = ε 2 ) yield the lower estimate in (2.8). On the other hand, Lemma 5.1 (i) (with j = ε

2 ) provides an upper bound of the form t -1 2 e -x 2 4t in the case d ≤ 1 (since then ∂ t s = s xx ) and of the form t -1 2 e -x 2 4dt in the case d ≥ 1 (since then ∂ t s = ds xx ). This gives the upper estimate in (2.8).

Similarly, one obtains

U (t, x)-t -(1+θ) ≤ 1-V (t, x) ≤ 1-v(t, x) ≤ 1-Ṽ (t, x) ≤ 2-V 1 (-c * v t)-V 1 (-(c * v -ε)t)+ Ũ (t, x),
which gives the lower estimate in (2.9). On the other hand, we deduce from (4.8) that there exist C -> 0 and C + > 0 such that, for all 0 ≤ x ≤ εt,

1 -Ṽ (t, x) ≤ C -e -λ 1 c * v t + C + e -λ 1 (c * v -ε)t + Ũ (t, x) ≤ Ct -k * e -x 2 4d * t + Ũ (t, x),
with some C > 0 provided ε > 0 is chosen sufficiently small. This gives the upper estimate in (2.9) and concludes the proof of Theorem 2.7.

A A result on the strong-weak competition system

In this Appendix, we consider the strong-weak competition system (0 < a < 1 < b)

∂ t u = u xx + u(1 -u -av), ∂ t v = dv xx + rv(1 -v -bu), (A.1)
supplemented with an initial datum (u 0 , v 0 ) satisfying (2.5), for which we need a technical result, namely Lemma A.1, which is inspired by [START_REF] Peng | Sharp estimates for the spreading speeds of the Lotka-Volterra diffusion system with strong competition[END_REF]Lemma 2.8] and the forthcoming work [START_REF] Wu | Lotka-Volterra diffusion systems: the strong-weak type[END_REF].

When c v = 2 √ rd > c u = 2, as proved in [START_REF] Girardin | Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition-diffusion systems[END_REF], the spreading properties are rather subtle: the rapid competitor v invades first at speed c v and is then replaced by the strong competitor u at a speed C which can take two different values. To make this clear, we quote the following from [START_REF] Girardin | Invasion of an empty habitat by two competitors: spreading properties of monostable two-species competition-diffusion systems[END_REF] to which we refer for more details. First, the strong-weak system admits a minimal monotone traveling wave solution (c LLW , U, V ) with speed 2 √ 1a ≤ c LLW ≤ 2, defined as

           U ′′ + c LLW U ′ + U (1 -U -aV ) = 0,
dV ′′ + c LLW V ′ + rV (1 -V -bU ) = 0, (U, V )(-∞) = (1, 0), (U, V )(∞) = (0, 1),

U ′ < 0, V ′ > 0.
Next define the decreasing function f :

c nlp := f -1 (2 √ rd) = √ rd - √ a + 1 -a √ rd - √ a ∈ (c LLW , 2).
Then we can precise

C = c nlp if 2 < c v < f (c LLW ), (acceleration phenomenon), c LLW if c v ≥ f (c LLW ).
We now state the result used in the proof of Proposition 4.4.

Lemma A.1. Assume dr > 1 (i.e. c v > c u ). Let (u, v) = (u, v)(t, x) be the solution of the strong-weak competition system (A.1) starting from an initial datum (u 0 , v 0 ) = (u 0 , v 0 )(x) satisfying (2.5). Then the following holds.

(i) For any c > C , there exist C 1 > 0, ν 1 > 0, T 1 > 0 such that sup |x|≥ct u(t, x) ≤ C 1 e -ν 1 t , for all t ≥ T 1 .

(ii) For any c 1 and c 2 with C < c 1 < c 2 < c v , there exist C 2 > 0, ν 2 > 0, T 2 > 0 such that sup c 1 t≤|x|≤c 2 t v(t, x) ≥ 1 -C 2 e -ν 2 t , for all t ≥ T 2 .

Proof. Let us briefly start with (i). If c > c u then the conclusion is clear by the same argument as in Proposition 4.2. Since c v > c u , it thus suffices to consider the case C < c < c v . In this case, the conclusion is already included in [9, Proposition 1.5] and the proof of [9, Section 3.2.3, Theorem 1.1]. We do not present the full details but only emphasize that a key tool is, for any small δ > 0, the minimal monotone traveling wave of the perturbed system

           U ′′ + cU ′ + U (1 + δ -U -aV ) = 0,
dV ′′ + cV ′ + rV (1 -2δ -V -bU ) = 0, (U, V )(-∞) = (1 + δ, 0), (U, V )(∞) = (0, 1 -2δ),

U ′ < 0, V ′ > 0.
Let us now turn to (ii) (which is the estimate we need in the proof of Proposition 4.4) for which the above perturbation argument seems unapplicable. Let C < c 1 < c 2 < c v be given. We only deal with x ≥ 0. From this and (i), one can choose ε > 0 small enough and T 0 ≫ 1 such that 0 < u(t, x) ≤ C 1 e -ν 1 t , v(t, x) > 1ε, for all t ≥ T 0 , x ∈ [c 1 t, c 2 t].

From the v-equation in (A.1), we have ṽt ≥ dṽ xx + cṽ x + r(1ε)(1 -ṽ) -rbC 1 ṽe -ν 1 t , for all t ≥ T 0 , x ∈ [-c 3 t, c 3 t],

v t ≥ dv xx + r(1 -ε)(1 -v) -rbC 1 ve -ν 1 t ,
where c 3 := c 2 -c 1 2 . To estimate ṽ, for any T ≥ T 0 , we define α(t) := 1 + bC 1 1ε e -ν 1 (t+T ) , for all t ≥ 0.

Up to enlarging T > 0 if necessary, we may assume α(0) < Up to a rescaling, we may assume d = 1 so that (A.5) is very comparable to [15, problem (3.12)] on which we now rely. Denoting G 1 (t, x, z) the Green function of [15, page 53] (with obvious changes of constants), we obtain the analogous of [15, (3.14)], namely

Φ(t, x) ≥ r(1 -ε) t 0 e Q(s) (1 -(1 -ε)α(s)) c 3 T -c 3 T
G 1 (ts, x, z)dz ds, in virtue of (A.7). On the other hand, we know from the comparison principle that ṽ(t + T, x) ≥ ϕ(t, x) for t ≥ 0 and |x| ≤ c 3 T , which together with (A.9) implies that ṽ δ 2 c 3 T + T, x ≥ 1 -(K 1 + K 2 )e -r(1-ε)δ 2 c 3 T , for all |x| ≤ (1δ)c 3 T.

We further take t = (δ 2 c 3 + 1)T , which yields ṽ(t, x) ≥ 1 -C 2 e -ν 2 t , for all large t and |x| ≤ (1-δ)c 3 1+c 3 δ 2 t, where C 2 := K 1 + K 2 and ν 2 := r(1-ε)δ 2 c 3 1+c 3 δ 2 > 0. Recalling that ṽ(t, x) = v(t, x + ct) with c = c 1 +c 2 2 , that c 3 = c 2 -c 1 2 and since δ > 0 can be chosen arbitrarily small, the above estimate completes the proof of (ii).

  u(t, x)| = 0, for all c < c * ,

Proposition 4 . 2 (

 42 Estimates in the region |x| > c u t). We have lim t→∞ sup |x|≥ct u(t, x) = 0, for all c > c u , (4.2) lim t→∞ sup |x|≥ct v(t, x) = 0, for all c > c v , (4.3)

[ 2 √ 1 - 2 √

 212 a, +∞) → (2 √ a, 2( √ 1a + √ a)] f (c) := cc 2 -4(1a) + 2 √ a so that f -1 (c) rd ∈ (2, f (c LLW )), then define

From [ 9 ,

 9 Theorem 1.1] we know lim t→∞ sup c 1 t≤x≤c 2 t u(t, x) + |1v(t, x)| = 0.

Φ

  t = dϕ xx + cϕ x + r(1ε)[1α(t)ϕ], t > 0, -c 3 T < x < c 3 T, ϕ(t, ±c 3 T ) = 1ε, t > 0, ϕ(0, x) = 1ε, -c 3 T ≤ x ≤ c 3 T. (A.4) Letting Φ(t, x) := e Q(t) [ϕ(t, x) -1 + ε], Q(t) := r(1ε)t -rbC 1 ν 1 e -ν 1 (t+T ) , so that Q ′ (t) = r(1ε)α(t), it follows from (A.4) that t = dΦ xx + cΦ x + r(1ε)e Q(t) (1 -(1ε)α(t)), t > 0, -c 3 T < x < c 3 T, Φ(t, ±c 3 T ) = 0, t > 0, Φ(0, x) = 0, -c 3 T ≤ x ≤ c 3 T.(A.5)

  .20) (ii) Assume dr < 1 (i.e. c v < c u ). Then, for any 0 < c 3 < c u < c 4 ,

	lim t→∞	sup x∈R	v(t, x) + sup |x|≤c 3 t	|x|≥c 4 t |1 -u(t, x)| + sup	u(t, x) = 0.	(4.21)

Proof. Let us prove (i). The result on u in (4.20) is obtained by combining (4.2) and Proposition 4.4. Next, for a given

  for all t ≥ T 0 , x ∈ [c 1 t, c 2 t].

					(A.2)
	Defining	ṽ(t, x) := v(t, x + ct), c :=	c 1 + c 2 2	,	(A.3)
	it follows from (A.2) that

t √ t.

Acknowledgement. M. Alfaro is supported by the ANR project DEEV ANR-20-CE40-0011-01. D. Xiao is supported by the LabEx Solutions Numériques, Matérielles et Modélisation pour l'Environnement et le Vivant (NUMEV) of the University of Montpellier.

for all t > 0, -c 3 T < x < c 3 T . Next, for any small 0 < δ < 1, we define

From the same process used in [15, pages 54-55], there exist C 3 , C 4 > 0 such that the following lower estimate holds

where

Inserting this into (A.6) and using e y ≥ 1 + y for all y ∈ R, we have, for all

Letting

with some K 1 , K 2 > 0. The last inequality holds since we can always choose ν 1 < C 4 . As a conclusion, we have

provided that δ > 0 is sufficiently small for (A.7) to hold and T > 0 is sufficiently large. In particular (A.8) implies that, for all |x| ≤ (1δ)c 3 T , ϕ δ 2 c 3 T, x ≥ 1 -K 1 e -ν 1 T -K 2 e -r(1-ε)δ 2 c 3 T ≥ 1 -(K 1 + K 2 )e -r(1-ε)δ 2 c 3 T , (A.9)