Ariane Mallat
email: <ariane.mallat@hmn.aphp.fr

Sophie Lotersztajn

H Pital

Henri Mondor

Keywords: Arachidonic Acids, therapeutic use, Cannabinoids, therapeutic use, Fatty Liver, drug therapy, physiopathology, prevention & control, Humans, Insulin Resistance, physiology, Polyunsaturated Alkamides, therapeutic use, Receptor, Cannabinoid, CB1, drug effects, physiology, Receptor, Cannabinoid, CB2, drug effects, physiology, Receptors, Cannabinoid, drug effects

Cannabinoid receptors as novel therapeutic targets for

the management of non-alcoholic steatohepatitis.

Cannabinoid receptors as novel therapeutic targets for the management of non-alcoholic steatohepatitis

Ariane Mallat 1 2 * , Sophie Lotersztajn 1 2 IMRB, Institut Mondor de recherche biom dicale 1 Author Keywords fatty liver, non alcoholic steatohepatitis, endocannabinoids, cannabinoid receptors, liver fibrosis Preparations of the hemp plant have been used for medicinal purposes over centuries. THC was identified in 1964 as Cannabis Sativa the predominant cannabinoid compound responsible for psychoactive effects of marijuana. Thereafter, cloning of cannabinoid receptors CB1 and CB2 in the early nineties constituted a determinant milestone in the characterization of a novel biological system with a wide array of biological functions. Moreover, improvement in the understanding of the signaling mechanism responsible for cannabinoid actions has fostered research efforts in the development of therapeutic applications. Consequently, capsules of THC and its synthetic analog nabilone are approved in several countries for the management of chemotherapy-induced nausea and vomiting ( ) and rimonabant, a 1 selective CB1 receptor antagonist has been available since 2 years in Europe as an adjunctive treatment of obesity or overweight with associated type 2 diabetes or dyslipidemia ( ).

-4

In this context, accumulating experimental and clinical data have stressed the crucial role of the cannabinoid system in the pathogenesis of non alcoholic fatty liver disease (NAFLD). NAFLD is closely linked to the metabolic syndrome and the obesity epidemic ( ) and currently accounts for a rising cause of liver injury, with a 20 30 prevalence in Western countries. The spectrum of the disease 5 -% ranges from simple steatosis, a condition generally associated to a benign liver outcome, to steatohepatitis, an entity that associates steatosis, liver inflammation, hepatocellular injury. The latter stage is associated with an activation of fibrogenic pathways and carries a 10 20 risk of cirrhosis after 10 to 20 years. As shown in several recent studies, NASH is entailed with increased liver-related mortality, -% due to end-stage liver disease or development of hepatocellular carcinoma ( ). The present review will depict evidences suggesting that 6 cannabinoid receptor antagonism may offer novel therapeutic approaches for the management of NAFLD.

THE ENDOCANNABINOID SYSTEM

The endocannabinoid system comprises endogenous lipidic ligands, specific G-protein coupled receptors (CB1 and CB2) and proteins that are responsible for their biosynthesis, cellular uptake and degradation ( ). 7 -9

CB1 receptor was originally cloned from a rat brain library owing its high level of expression in the central nervous system ( ), and 10 subsequent studies have shown its presence at lower levels in many peripheral tissues. Expression of CB2 receptors predominates in the immune system, and although more restricted, is increasingly demonstrated in several cells ( , , ). Recent reports also suggest the 8 11 12 existence of additional cannabinoid receptors.

Endocannabinoids are hydrophobic fatty-acid derived compounds with predominantly autocrine/paracrine effects, among which anandamide (arachidonoyl ethanolamide) and 2-arachidonoyl glycerol (2-AG) are best known. Both compounds are synthetized on demand and are rapidly degraded by fatty acid amide hydrolase (FAAH) or monoacyglycerol lipase, following ligand binding and cellular reuptake ( , , ,

). Anandamide shows a higher affinity for CB1 than CB2 receptors and is therefore considered a major endogenous 8 9 11 12 CB1 ligand, whereas 2-arachidonoyl glycerol binds both receptors similar affinity ( ). In addition, both compounds also induce CB1 and 13

CB2-independent effects. Lipid mediators other than anandamide and 2-AG have been reported to bind CB receptors, but their biological significance remains undetermined.

MODULATORS OF CANNABINOID RECEPTORS AS THERAPEUTIC AGENTS

Rimonabant has been the first CB1 antagonist to reach the market in Europe ( ). The drug was initially developed for the treatment 2 -4 of obesity, in light of the positive impact of phyto-and endocannabinoids on central appetite regulating pathways. It soon became clear that CB1 anatagonism produces metabolic effects beyond those expected from weight loss alone, including improvement in dyslipidemia, insulin resistance or diabetes (

). In keeping with clinical data, experimental studies have established that multiple peripheral 14 mechanisms contribute to the beneficial effects of CB1 antagonism by enhancing energy expenditure, peripheral lipolysis and insulin sensitivity among others ( , ). Accordingly, trials are under way to further define the impact of CB1 antagonism on dyslipidemia, type 15 [START_REF] Kunos | Endocannabinoids and the control of energy homeostasis[END_REF] 2 diabetes and cardiovascular morbidity. Other therapeutic applications under evaluation also include management of alcohol and nicotine-dependence or neurodegenerative disorders ( ). The safety of CB1 antagonists in obesity has been questioned given the 9 occurrence of modest rates of anxiety and depression in susceptible individuals ( ). As a result, the FDA denied approval of rimonabant 14 pending additional data, whereas Merck recently suspended the development of taranabant in obesity, due to safety concerns. In this context, the development of peripherally restricted CB1 antagonists could prove of interest, by avoiding central adverse effects.

Although selective agonists and antagonists of CB2 receptors have not reached a clinical stage yet, preclinical studies nevertheless suggest meaningful therapeutic applications as anti-inflammatory, analgesic or anti-allergic compounds ( , ). Of interest, such 9 17 compounds should be devoid of central adverse effects.

Identification of cannabinoid receptors as potential therapeutic targets for the management of liver diseases ( ) has emerged recently 7 with the demonstration that CB1 receptors contribute to the pathogenesis of cirrhotic portal hypertension ( , ). Soon after, additional 18 19 studies uncovered a key role of cannabinoids in metabolic and ethanol-induced fatty liver, ischemia reperfusion, or the scarring process associated with chronic liver disease (

). 20 -25

PATHOGENESIS OF NAFLD

It is now admitted that metabolic steatosis and promoted by insulin resistance are in tight relationship ( ). Thus, rodent models have 26 shown that resistance to insulin promotes lipolysis in the adipose tissue, thereby increasing delivery of free fatty acids to the liver ( ). 26 Moreover, in the liver, hyperinsulinemia triggers de novo fatty acid and impairs -oxidation and lipid disposal. Conversely however, β steatosis may also contribute to hepatic insulin resistance ( ). The transition from steatosis to NASH is poorly understood and appears 26 multifactorial. Excessive accumulation of free fatty acids leads to increased oxidative stress and lipid peroxidation, thereby resulting in cellular injury. Moreover, enhanced cytokine production by infiltrating macrophages in adipose tissue and the liver are also incriminated in the progression of injury ( ). 5

CANNABINOID RECEPTOR ANTAGONISM REDUCES DEVELOPMENT OF NAFLD CB1 receptors promote metabolic steatosis and insulin resistance

Recent findings have shown that the hepatic cannabinoid system is activated in NAFLD. Thus, in the experimental model of diet-induced obesity, hepatic anandamide levels are increased following inhibition of its degradation by FAAH and CB1 receptor expression is strongly induced in hepatocytes ( ). 23

Accumulating experimental evidences indicate that CB1 receptors contribute to metabolic steatosis and the related insulin resistance ( , , ). CB1 receptor knockout mice are resistant to high fat-diet (HFD) induced obesity and steatosis and to the associated increase 23 24 27 in hepatic lipogenesis; moreover, HFD-fed CB1 ablated mice display reduced insulin resistance ( , ). In keeping, genetically-obese 23 28 fa/fa rats treated with rimonabant show reversal of hepatic steatosis and improved insulin sensitivity ( ). Interestingly, mice bearing a 27 selective deletion of CB1 receptors in hepatocytes become obese under a high-fat diet but are protected from hepatic steatosis and impaired glucose tolerance ( ). Finally, characterisation of functions of upregulated hepatic CB1 receptors during steatogenesis suggests 24 combined enhancement of lipogenesis and inhibition of fatty acid -oxydation ( , ). Collectively, these data indicate that peripheral

β 23 24
overactivation of the cannabinoid system promotes obesity-associated fatty liver and insulin resistance. Beyond its contribution to steatogenesis, CB1-dependent endogenous cannabinoid tone may also favour the inflammatory response associated to NASH. Thus, it has been shown that endogenous CB1 activation reduces secretion of adiponectin ( ), an adipocytokine with potent anti-inflammatory effects 29 in the liver ( ). In keeping with these observations, administration of rimonabant to genetically obese rats induces a significant 30 improvement in the hepatic inflammatory response ( ). Clinical studies also indirectly support the potential role of the endocannabinoids 27

and their receptors in the pathogenesis NAFLD. Analysis of pooled 1-year data from four pivotal trials in overweight patients indicates that rimonabant reduces alanine aminotransferase levels, a marker of NAFLD ( ). In addition, we recently investigated the impact of 14 cannabis use on steatosis grade in 307 patients with chronic hepatitis C and found that daily cannabis consumption is an independent predictor of severe steatosis ( ). Overall, these results provide a strong evidence for a steatogenic role of cannabinoids in humans.

CB receptors regulate liver fibrogenesis

As stated previously, transition from steatosis to NASH is associated with activation of fibrogenic pathways and exposes to the development of liver fibrosis ( ). We recently found that expression of CB1 and CB2 receptors is markedly upregulated in cirrhotic liver 32 samples, predominantly in liver fibrogenic cells and demonstrated that CB1 and CB2 receptors display potent pro and anti-fibrogenic properties, respectively ( , ). Antifibrogenic properties of CB2 receptors were established in CB2 knockout mice repeatedly exposed 22 25 to carbon tetrachloride, based on the findings that these mice show enhanced liver fibrosis and increased accumulation of liver fibrogenic cell compared to wild type animals ( ). The function of CB1 receptors in liver fibrogenesis was assessed in three different experimental 22 models (chronic carbon tetrachloride or thiacetamide administration and bile duct ligation). Administration of rimonabant or genetic inactivation of CB1 receptors significantly reduced progression of fibrosis ( ). Profibrogenic properties of CB1 receptors were ascribed 25

to the overactivation of CB1 receptors expressed by liver fibrogenic cells, leading to a combined enhancement of cell proliferation and decrease in apoptosis rate.

The clinical relevance of these experimental findings was confirmed in an epidemiological study of the impact of cannabis use on fibrosis severity in HCV-infected individuals. Daily cannabis use was documented as an independent predictor of fibrosis severity, suggesting that CB1 signaling dominates over CB2 during chronic hepatitis C ( ). A subsequent independent study in a canadian cohort 33 reported similar findings ( ). 34

Emerging role of CB2 receptors in the pathogenesis of NAFLD

CONCLUSION

Accumulating data indicate that the endocannabinoid system is upregulated in NAFLD and plays an important role in the pathogenesis of steatosis and insulin resistance via peripheral pathways. CB1 antagonism has proven efficient in the control of experimental NAFLD and liver fibrogenesis. Recent clinical trials have also established that inactivation of CB1 receptors not only reduces overweight but also improves several parameters of the metabolic syndrome, including insulin resistance and dyslipidemia ( ). These observations 14 undoubtedly provide a strong rationale for the evaluation of CB1 antagonists in the management of NASH, as currently underway in phase III clinical trials. Concern with potential adverse central effects of CB1 antagonists should stimulate ungoing efforts to develop peripherally restricted molecules.

é

  INSERM : U841 , Universit Paris XII Val de Marne é , H pital Henri Mondor 51, av du ô mal de lattre de tassigny 94010 CRETEIL CEDEX,FR Service d h pato-gastro-ent rologie 2

  Several studies have shown that obesity generates a low grade inflammatory state that contributes to the development of insulin resistance and NAFLD ( ). CB2 receptors are potent regulators of innate immunity ( ) and we recently investigated their potential 35 -37 38 role in the pathogenesis of NAFLD. Compared to wild type counterparts, mice invalidated for CB2 receptors are less prone to high fat diet induced obesity ( ). Moreover, CB2 knock out mice are resistant to steatosis and display improved glucose tolerance. The mechanism

-39 underlying steatogenic effects of CB2 receptors appears to involve proinflammatory effects of upregulated CB2 receptors in adipose tissue.


Blériot S Tchienkou Tchiengang

Israël Tankam

Jean Jules Tewa

A multi-seasonal model of the dynamics of the African maize stalk borer

Keywords: Multi-seasonal model, Semi-discrete model, Stability, Maize stalk borer, Pest management. 1

In this paper, we build a simple, multi-seasonal mathematical model describing the dynamics of the maize stalk borer Busseola fusca (Lepidoptera: Noctuidae). We consider in this model the main harmful stages of the pests and the control methods used by farmers to reduce pest populations from one maize cropping season to another. Firstly, immature stages are controlled during the maize cropping season by applying pesticides to destroy the eggs and immature larvae present on the plant leaves. secondly, crop residues harboring the larvae are burnt or buried in the soil after the harvest. The semi-discrete model obtained is studied and the basic reproduction number of the pests is computed. The numerical simulations carried out illustrate the theoretical results and also allow us to find the minimum quantity of larvae to be destroyed at the end of each maize cropping season to eradicate pest populations in the fields.

I INTRODUCTION

Maize (Zea mays L.), also known as corn, is one of the most cultivated crops in the world along with rice and wheat. The total world production of maize was estimated to be 1147.6 million tons in 2018, feeding a billion people on earth. Thus, maize production provides a safety valve against food insecurity in some parts of the world. In a large part of Cameroon regions, maize is the main staple food crop and represents an important part of the caloric intake of the population. It provides a sustainable and secure food supply in terms of high yields and a significant increase in farmers' incomes since the maize yield is often sold in local markets. However, this crop is threaten in particular by insects that bore its stalk and ear. Maize stalk borer insects in sub-Saharan Africa all belong to the Lepidopteran order, they are the only group of pests that have a real economic impact. Several species of maize pests have been listed in sub-Saharan Africa but only three of these species have appeared to be economically important: the European moths Eldana saccharina (Walker) and Mussidia nigrivenella (Ragonot) and the moth Busseola fusca (Fuller) [(9)]. The maize stalk borer (MSB) B. fusca is the most widespread and harmful pest to maize crops in Cameroon and the feeding habits of its larvae on maize lead to yield losses of up to 59% depending on several factors that include the maize cultivar, the development stage of the plant and the infestation rate [(5)]. Faced with the extent of the damage, several control programs and methods have been developed by maize producers, such as growing resistant varieties, improving cultivation practices, chemical and biological control and trapping [( 4)]. In this paper, we build a simple multi-seasonal model describing the dynamics of the African maize stalk borer B. fusca and taking into account two means of control used by farmers to manage the pests. The latter are based first on the application of pesticides on maize plants during their growth and secondly on the management of crop residues harbouring B. fusca's larvae after the harvest. Our aim is to show the importance of crop residue management in controlling pest populations. After the formulation of the model, we carry out its analysis and provide some numerical simulations to illustrate the theoretical results.

II BUSSEOLA FUSCA CYCLE AND MODELING

The maize stalk borer B. fusca is an African lepidoptera, an insect whose adult form is a butterfly, the larva, a caterpillar, and the immobile nymph is a pupa. Its life cycle (see [(4; 9; 10)]) includes the various classic stages of the development of moths, namely: eggs, caterpillars or larvae, pupae, imagos. The duration of these stages, in particular that of the larvae, depends on ecological conditions. In short, the life cycle starts with mating between a female and a male, then an oviposition takes place on the plant between the leaf sheath and the maize stalk. The eggs laid in a pile hatch about a week later and the young larvae go through a phyllophagous period of about ten days in the horn before entering the stem on which they keep feeding. At the last stage of their evolution they become pupae: this is the diapause phenomenon [( 12)]. Adul butterflies then raise 9 to 11 days after the beginning of the diapause and the cycle starts again. When weather conditions are unfavorable, the diapause phenomenon can last up to 6 months or more. A normal cycle lasts about 45 days for a female and 50 days for a male.

Based on the biological background, we build a multi-seasonal compartmental model illustrating the dynamics of B. fusca. We work on the scale of a whole maize growing area. The following hypotheses are considered.

• We assume that there host plants are always available for the pests during the maize cropping season and therefore we are only interested in the dynamics of the pests. • We summarize all the non-adult stages into one compartment named the compartment of immature stages, and we denote it by the state variable I. We also consider a compartment of female adults, that we denote by the state variable A. The units of these state variables are "larva" and "pest". This implies that all emerging immature individuals will become either males or females. For a matter of simplicity, we assume that when pests are presents, they are always males and females available in enough proportion for mating and reproduction to occur.

• We assume that the time needed for eggs emergence is 1 ν days during the cropping season, thus the transfer rate from I to A is ν.

• Females feed the immature compartment logistically at a rate r I 1 -I K where r I is the average number of eggs provided by fertilized females (the intrinsic egg laying rate) during the cultivation period and during the off season, K its carrying capacity. We assume this carrying capacity to be constant, even though it depends on the availability of the hosts, the amount of crop residues and the density of alternative hosts. • We term by u a parameter representing the efforts made to reduce larval populations in maize plantations. This can be seen as a chemical control through the use of pesticides that destroy the eggs and larvae that haven't yet penetrated the plant stem for feeding.

• We name p the probability that a larva gives an adult female, µ I and µ A respectively the mortality of immature pests and adult pests. • We name D the duration of a maize cropping season and τ the duration of the off-season, when maize plants are unavailable. We let T = D + τ the duration of maize cropping season plus the off-season and we set t n = nT, ∀n ∈ N. From the hypothesis above, during maize cropping seasons, i.e. t ∈ (t n , t n + D] , the dynamics of pests is given by the following equations:

   İ(t) = r I A(t) 1 - I(t) K -(ν + µ I + u)I(t), Ȧ(t) = pνI(t) -µ A A(t). (1)
The initial conditions at the beginning of the first season are:

I(0 + ) = I 0 ≥ 0, A(0 + ) = A 0 ≥ 0;
where the "+" superscript denotes the instant that directly follows.

At the end of a maize cropping season, after the harvest of corn, plant stems are either cut and buried in the ground or burnt to reduce pest reservoirs and therefore decrease the pest population. Depending on the effectiveness of these practices, we assume that a proportion q (q ∈]0, 1[) of larvae is be destroyed by the control efforts. We illustrate this switching in pest populations by the following difference equation when t = t n + D:

I(t n + D + ) = (1 -q)I(t n + D), A(t n + D + ) = A(t n + D). (2)
The case q = 0 stands for the lack of burning or burying operation.

During the off-season, i.e. when t ∈ (t n + D, t n+1 ], there are no maize plants and immature pests grow logistically, but nevertheless:

• Their limiting capacity K 0 is reduced as it depends only on the amount of residues and the density alternative hosts (K 0 ≤ K). • When climatic conditions are unfavourable for the development of immature pests, they enter in the diapause state [ [START_REF] Usua | Diapause in maize stemborer[END_REF]] that reduces the transfer rate of larvae to moths. We therefore note ν 0 ≤ ν the average transfer rate from larvae to moths. The dynamic of evolution of the pests during this period is given by the following system:

   İ(t) = r I A(t) 1 - I(t) K 0 -(ν 0 + µ I )I(t), Ȧ(t) = pν 0 I(t) -µ A A(t). (3)
At the beginning of the new maize cropping season, for t = t n+1 , the same quantity of adult pests is kept and we obtain the following switching rule between seasons:

I(t + n+1 ) = I(t n+1 ), A(t + n+1 ) = A(t n+1 ). (4)
The mathematical model formed by systems (1-2-3-4) represents our semi-discrete model of the dynamics of the African maize stalk borer B. fusca with a cultural control of pests, in a multi-seasonal framework.

III THEORICAL RESULTS

The well-posedness of our models relies on the existence, the uniqueness, the positivity and the boundedness of their solutions given the initial values.

Proposition III.1: Existence, uniqueness, positivity and the boundedness of the solutions.

1. The model (1-2-3-4) admits a unique solution that is continuous on (t n , t n + D] and on

(t n + D, t n+1 ]. 2. The positive set R 2
+ is positively invariant by the flow of the model (1-2-3-4). 3. For n ∈ N, the set

D = (I(t), A(t)) ∈ R 2 : 0 ≤ I(t) ≤ K, 0 ≤ A(t) ≤ pν µ A K is a positively invariant compact by the flow of model (1-2-3-4).
The proof is given in Appendix A.

In the following proposition, we find the transition laws of pest populations from one season to another around the pest-free solution (PFS). This finally allows us to compute the basic reproduction number of pests. In a pest-free context, I = 0, A = 0 thus (0, 0) is a pest-free solution of model (1-2-3-4).

Proposition III.2:

Basic reproduction number of pests.

1. For all n ∈ N, in the neighbourhood of the pest-free solution PFS= (I * (t), A * (t)) = (0, 0), the discrete pest dynamics of the linearized model are defined by:

I(t + n ) A(t + n ) = Φ 1,1 Φ 1,2 Φ 2,1 Φ 2,2 n I(0 + ) A(0 + ) , (5)
with the explicit values of Φ i,j (i, j ∈ {1, 2}) given in the proof.

Let

Φ = Φ 1,1 Φ 1,2 Φ 2,1 Φ 2,2
. The basic pest reproduction number for model (1-2-3-4) is given by:

R 0 = max |φ i |, i ∈ {1, 2}. (6)
where φ i are the eigenvalues of the matrices Φ.

The PFS of model (1-2-3-4) is locally asymptotically stable

if R 0 < 1.
The proof is given in Appendix B.

Remark III.3: 1. When the basic reproduction number R 0 is smaller than 1, the pests tend to disappear over seasons, and when it is larger than 1, they may persist. 2. The number R 0 can be defined as the absolute value of the largest factor of change in the size of a pest populations (pair of larvae and moths) from one season to the next in a pest-free context.

IV NUMERICAL SIMULATIONS

In this section, we perform numerical simulations to illustrate our theoretical results obtained above (Proposition III.2). Since we are in a multi-seasonal framework, the simulations are made over a period of 20 seasons. The length of a maize cropping season is D = 120 days and the length of the off-season is τ = 245 days (T + τ ≈ 1 year). We consider a large cropping field for which I(0 + ) = 150 larvae, A(0 + ) = 15 insects. The values of the different parameters are given in Table1. We illustrate in these simulations how, when the basic reproduction number of pests R 0 is strictly less than 1, pest populations gradually decrease over the seasons to negligible levels. Conversely, if these threshold are greater than 1, then pest populations persist throughout the seasons. Subsequently, through simulations we show how the basic reproduction number evolves as a function of the proportion q of larvae destroyed after each harvest. We find a linear relationship between R 0 and q. 0.5 day -1 [( 10)] 0,5 day -1 p 0, 3939 -0, 6078 [( 6)] 0.48 u / 0.0025 day -1 (3) q / 0.75 (1) K and K 0 are estimated in decent ranges according to what could be a field size.

(2) ν 0 is taken so that ν 0 ≤ ν. (3) u is taken such that chemical control destroys 30% of the larvae during the maize cropping season.

/ No data available in the literature. Figure 1 illustrates the case where the basic reproduction number of pests is less than 1. In this case, in addition to the larval control carried out during the maize cropping season, 75% of larvae are destroyed at the end of each maize cropping season and therefore pest populations gradually decrease over seasons. In Figure 2, no proportion of larvae is destroyed at the end of the maize cropping season and therefore the basic reproduction number is therefore greater than 1. The pests hence persist over seasons. These two figures illustrate the importance of crop residue management to control pest populations from one season to another. Figure 3 illustrates the linear relationship between the basic reproduction number R 0 and q : R 0 = -0.9050 × q + 1.6736 (without control u) and R 0 = -0.7628 × q + 1.4121 (with control u). This relationship allows us to find the minimum proportion of larvae to destroy after each harvest to have a decrease of the pest populations over the seasons. From this figure, it is noticable that if no effort is made during each maize cropping season to reduce larvae, at least 74.43% of larvae must be destroyed after each harvest for pest populations to decrease over the seasons. However, if larvae are regularly controlled by pesticides, only 54.02% would need to be destroyed.

V CONCLUSION AND PERSPECTIVES

In this paper, we formulated a simple model for the dynamics of the African maize stalk borer B. fusca. Since the host of the pest is available seasonally, we have implicitly integrated into the model its seasonal availability and the means of control used by farmers to control pest populations from one season to another. The semi-discrete model obtained was analyzed and the basic reproduction number of pests was calculated. Numerical simulations were carried out over several seasons to illustrate the theoretical results. Namely, when the basic reproduction number of pests is less than 1, pest populations gradually decrease over the seasons and when this number is greater than 1, pests persist. Using numerical simulations, we also illustrated the importance of larval control during the maize cropping season, and the importance of crop residue management after harvest to control pest populations.We found the minimum proportion of larvae to be destroyed at the end of each maize cropping season for the control based on crop residue management to be effective. This work can therefore open the debate on the possibilities for biologists to carry out field experiments, and these could help us to calibrate the model. It is desirable that field experiments can be done in order to estimate some precise parameters for these pests and to be able to validate our model.

exp t tn Bdt = e B(t-tn) = Γ 1,1 (t -t n ) Γ 1,2 (t -t n ) Γ 2,1 (t -t n ) Γ 2,2 (t -t n ) Where Γ 1,1 (t) = 1 λ 2 -λ 1 (e λ 1 t (λ 2 + α) -e λ 2 t (λ 1 + α)) Γ 1,2 (t) = r I λ 2 -λ 1 (e λ 2 t -e λ 1 t ) Γ 2,1 (t) = pν λ 2 -λ 1 (e λ 2 t -e λ 1 t ) Γ 2,2 (t) = 1 λ 2 -λ 1 (e λ 1 t (λ 2 + µ A ) -e λ 2 t (λ 1 + µ A ))
Therefore ( 8) has a solution:

I(t) = Γ 1,1 (t -t n )I(t + n ) + Γ 1,2 (t -t n )A(t + n ) A(t) = Γ 2,1 (t -t n )I(t + n ) + Γ 2,2 (t -t n )A(t + n ) (9)
For t = t n + D in (9) we have the following relation:

I(t n + D) = Γ 1,1 (D)I(t n ) + Γ 1,2 (D)A(t n ) A(t n + D) = Γ 2,1 (D)I(t n ) + Γ 2,2 (D)A(t n ) (10)
For all t ∈ (t n + D, t n+1 ], we considere the system (4). The jacobian matrix in the neighbourhood of PFS (0, 0) of system (3) is given by:

J PFS = -(ν 0 + µ I ) r I pν 0 -µ A
If we pose Y (t) = (I(t), A(t)) T , in a neighbourhood of (0, 0), the system (3) is equivalent to a following linearised system :

Ẏ (t) = J PFS Y (t) (11)
As previously, we can extract from (11) the following subsystem:

İ(t) = r I A(t) -(ν 0 + µ I )I(t), Ȧ(t) = pν 0 I(t) -µ A A(t). (12)
The solutions of the system (12) are given by:

I(t) = Π 1,1 (t -(t n + D))I(t n + D + ) + Π 1,2 (t -(t n + D))A(t n + D + ) A(t) = Π 2,1 (t -(t n + D))I(t n + D + ) + Π 2,2 (t -(t n + D))A(t n + D + ) (13)
Where the Π i,j are given by:

Π 1,1 (t) = 1 λ 2 -λ 1 (e λ 1 t (λ 2 + α 0 ) -e λ 2 t (λ 1 + α 0 )) Π 1,2 (t) = r I λ 2 -λ 1 (e λ 2 t -e λ 1 t ) Π 2,1 (t) = pν 0 λ 2 -λ 1 (e λ 2 t -e λ 1 t ) Π 2,2 (t) = 1 λ 2 -λ 1 (e λ 1 t (λ 2 + µ A ) -e λ 2 t (λ 1 + µ A ))
and λ 1,2 are the eigenvalues of the matrix B 0 = -α 0 r I pν 0 -µ A , α 0 = ν 0 + µ I .

For t = (n + 1)T = t n+1 , taking into account the system (2), the relation (13) gives:

I(t n+1 ) = (1 -q)Π
Considering [START_REF] Ntahomvukiyea | Study of the Population Dynamics of Busseola fusca[END_REF] we obtain from (14) the following relation:

I((n + 1)T ) = Φ 1,1 I(nT ) + Φ 1,2 A(nT ) A((n + 1)T ) = Φ 2,1 I(nT ) + Φ 2,2 A(nT ) (15)
where Φ 1,1 = (1 -q)Π 1,1 (τ )Γ 1,1 (D) + Π 1,2 (τ )Γ 2,1 (D) Φ 1,2 = (1 -q)Π 1,1 (τ )Γ 1,2 (D) + Π 1,2 (τ )Γ 2,2 (D) Φ 2,1 = (1 -q)Π 2,1 (τ )Γ 1,1 (D) + Π 2,2 (τ )Γ 2,1 (D) Φ 2,2 = (1 -q)Π 2,1 (τ )Γ 1,2 (D) + Π 2,2 (τ )Γ 2,2 (D)

By adopting the following notations

I(t n ) = I n , X n = (I n , A n ) T , Φ = Φ 1,1 Φ 1,2 Φ 2,1 Φ 2,2 ,
the relation (15) gives X n+1 = ΦX n =⇒ X n = Φ n X 0 and relation ( 5) is thus obtained. 2. From relation (15), we obtain the following discrete dynamic system illustrating the pest dynamics in the neighbourhood of the pest-free solution:

I n+1 = Φ 1,1 I n + Φ 1,2 A n A n+1 = Φ 2,1 I n + Φ 2,2 A n ( 16
)
The system of equations defined by relation ( 16) admits (0, 0) as a unique fixed point. Let's pose φ i the eigenvalues of the matrix Φ. According to the Jury criteria (1), (0, 0) is locally asymptotically stable if and only if max|φ i | < 1 for i = 1, 2. Hence the basic reproduction number of pests for model (1-2-3-4) is R 0 = max|φ i |.

Figure 1 :

 1 Figure1: Long-term dynamics of pest for model (1-2-3-4) with 20 cropping seasons, when q = 0.75. This parameter means that 75% of pest larvae are destroyed at the end of each maize cropping season. This leads to a basic reproduction number R 0 = 0.84.

Figure 2 :

 2 Figure 2: Long-term dynamics of pest for model (1-2-3-4) with 20 cropping seasons, when q = 0. This means that no pest larvae are destroyed at the end of each maize cropping season. This leads to basic reproduction number R 0 = 1.41.

Figure 3 :

 3 Figure 3: Linear relationship between the proportion q of larvae destroyed at the end of each maize cropping season and the basic reproduction number of pests R 0 .

Table 1 :

 1 Table of parameters and values.

	Parameters	Literature values	value(s)
	r

I 4 -116 pests.day -1 [(3)] 20 pests day -1 K / 5000 pests (1) K 0 / 2500 pests (1) ν 1/45 day -1 [(2)] 1/45 day -1 ν 0 / 0.0177 day -1 (2) µ I 0.35 day -1 [(10)] 0.35 day -1 µ A

  1,1 (τ )I(t n + D) + Π 1,2 )A(t n + D) A(t n+1 ) = (1 -q)Π 2,1 (τ )I(t n + D) + Π 2,2 (τ )A(t n + D)

,

,

APPENDIX A : Proof of proposition III. [START_REF] Lettj.-C | Modélisation Mathématique en Ecologie[END_REF] 1. The system (3) with initial condition given by (2) satisfies the Cauchy-Lipschitz conditions. Then it admits a unique continuous solution in the interval (t n + D, t n+1 ]. The system (1) has (I 0 , A 0 ) as initial condition when n = 0 and for n ≥ 1 the initial condition are given by (4). This system satisfies also the Cauchy-Lipschitz conditions so it admits a unique continuous solution in the interval (t n , t n +D]. Therefore the model is well-posed. 2. First, we suppose n = 0 and let X(t) = (I(t), A(t)). Since these variables represent biological quantities then X(0 + ) ≥ 0. We consider the system (1) with the initial condition X(0 + ). We have:

Therefore , ∀t ∈]0; D] all solutions of system (1) with initial positive conditions stay in the non-negative set R 2 + , so R 2 + is positively invariant by the flow of system (1). Now, if we consider the system (3), it has as initial conditions X(D + ). Using discrete equation ( 2)

The same reasoning as above leads us to conclude that R 2 + is positively invariant by the flow of system (3). Thus R 2 + is positively invariant by the flow of the model (1-2-3-4) at season 1. Furthermore, if R 2 + is positively invariant by the flow of the model (1-2-3-4) at season n then the initial conditions for season n + 1 given by (4) will always be positive and therefore R 2 + will be positively invariant by the flow of the model (1-2-3-4) for season n + 1. 3. We first consider the model (1-2-3-4). We use the barrier theorem to bound the solutions of the continuous systems of our model. Consider the system (1), ∀n ∈ N, ∀t ∈]t n , t n +D] .

Thus, according to the above, ∀t ∈]t n , t n + D] all solutions of system (1) are bounded. At time t = nT + D, all these solutions remain bounded. In fact, according to (2) we have:

For t ∈ (t n + D, t n+1 ], according to (3) we show in the same way as before, using the barrier theorem that:

For t = t n+1 , according to (4), we have:

So for t = t n+1 , the solutions of the model (1-2-3-4) also remain bounded and therefore, from the above we finally get: ∀t ∈]t n , t n+1 ],

This allows us to show that all solutions of model (1-2-3-4) are bounded, hence the construction of the set D.

APPENDIX B :

Proof of Proposition III.2 1. For all t ∈ (t n , t n +D], the jacobian matrix in the neighbourhood of PFS=(I * (t), A * (t)) = (0, 0) of system ( 1) is given by:

In a neighbourhood of PFS, the system (1) is equivalent to a following linearised system:

Where Ĩ(t) = I(t) -I * (t) = I(t) and Ã(t) = A(t) -A * (t) = A(t).

i.e

let's name X(t) = (I(t), A(t)) T , and B = -α r I pν -µ A , α = ν + µ I + u,

.

the system [START_REF] Castelj | From elaborate to compact seasonal plant epidemic models and back: is competitive exclusion in the details?[END_REF] gives

The matrix B is a Metzler matrix and therefore it admits two distinct real eigenvalues λ 1,2 = tr(B) 2 ± 1 2 tr 2 (B) -4det(B), and therefore we have:


E Yu Beliayev

Dianela Osorio

P Contreras

Coalescence of Impurity Strontium Atoms in Lightly-Substituted Samples of Lanthanum Cuprate

Keywords: Coalescence, unitary limit, reduced elastic scattering phase space, line nodes, lightly-substituted lanthanum cuprates

By comparing experimental results obtained in the study of the electron transport and magnetic properties of samples of lightly strontium-substituted lanthanum cuprates and the results of a previously published numerical analysis in the reduced elastic scattering space (RPS) of the elastic scattering cross-section, we construct a phenomenological classical self-consistent strong-binding energy density profile in the superconducting La2-xSrxCuO4 cuprate. It is found that the imaginary part of the RPS can be zero, almost constant, or strongly selfconsistent, depending on the Sr substituted concentration. If the Sr atomic concentration is dilute, a constant imaginary part of the elastic scattering cross-section brings a sub-melted coalescent superconducting metallic profile similar to the one with a constant scattering lifetime in the metallic state of normal metals, except for the unitary narrow, sharp peak at zero frequency. A self-consistent broadening of this peak occurs for a higher concentration of Sr atoms, and finally, for very small amounts of non-magnetic dirt, only the narrow unitary peak prevails around the zero frequency.

1.

Introduction.

This work is dedicated to understanding at experimental and phenomenological levels some classical properties in the unusual low-temperature behavior of unconventional, lightly-strontium-substituted lanthanum cuprate samples (La2-xSrxCuO4). We compare experimentally obtained evidence for the lightly-doped superconductor with a phenomenological analysis in the reduced elastic scattering phase space (RPS) to understand the relation between classical and quantum physical phenomena in lightly-doped cuprates.

The RPS is a physical self-consistent elastic scattering space based on the imaginary and real parts of the elastic scattering cross-section SCCS given by the pair of coordinates (ℜ(𝜔), ℑ (𝜔)), i.e., energy is conserved in the presence of an external non-magnetic potential U0, but momentum "k" is not conserved. This happens if the fermionic meanfree path "l" is comparable to the inverse Fermi momentum "kF -1 ", or to the lattice parameter "a" following the physical kinetics relation l kF ~ l a -1 ~ 1, which corresponds to the unitary limit (c = 0), where c is an inverse parameter to the external potential U0 firstly proposed by Pethick and Pines [START_REF] Pethick | Transport processes in heavy-fermion superconductors[END_REF] to explain low temperature transport properties in unconventional superconductors. The RPS resembles the "phase space distribution probability in quantummechanics" introduced by Wigner in 1932 [START_REF] Wigner | On the quantum correction for thermodynamic equilibrium[END_REF].

In references [START_REF] Wigner | On the quantum correction for thermodynamic equilibrium[END_REF][START_REF] Carruthers | Quantum collision theory with phase space distributions[END_REF], the main feature of this approach is pointed out: "the method is useful in providing easy reductions from quantum theory to classical physics kinetic regimes under suitable conditions". Therefore, in this work, we use this method to compare the interval (ℜ(𝜔), ℑ (𝜔)) as a function of the doping "xtheory" with previously reported evidence of experimental doping "xexp" for some low concentration phases in strontium-substituted lanthanum cuprate samples on the threshold between the superconducting phase of the lightly-doped La2-xSrxCuO4 compound and antiferromagnetic phase of the Mott-Hubbard dielectric compound La2CuO4. With this tool, we explain part of Dalakova and co-workers' experimental findings [5 -17].

This work is organized as follows. The section 2 shows the experimental results for low strontium doping in La2-xSrxCuO4, remarking some physical insights due to the strontium substitution in the original Mott-Hubbard dielectric. The section 3 proposes an energy self-consistent density profile without entering in details of how to derive the equations (previously explained and derived in [START_REF] Contreras | Scattering Due to Nonmagnetic Disorder in 2D Anisotropic d-wave High Tc Superconductors[END_REF][START_REF] Contreras | Dressed behavior of the quasiparticles lifetime in the unitary limit of two unconventional superconductors[END_REF]), sketching only the results for the SCCS ℜ(𝜔) and ℑ (𝜔) parts. The self-consistent phenomenological density energy profile 𝑛 (𝜔 ̃), is based on the unitary limit analysis of a 2D tight-binding [START_REF] Harrison | Electronic Structure and Properties of Solids[END_REF] La2-xSrxCuO4 Scalapino nodal line model [START_REF] Scalapino | The case for dx2 -y2 pairing in the cuprate superconductors[END_REF]. The 𝑛 (𝜔 ̃) analysis helps phenomenologically explain the origin of three phases for the Sr substituted HTSC compound. Finally, the Section 4, elaborates conclusions and recommendations, particularly on observations of the scattering lifetime and theoretical fits of the metallic coalescence state for unconventional superconductors.

2.

Experimental low-temperature pieces of evidence for coalescence in lightly-dopped La2-xSrxCuO4 samples with strong AFM correlations.

The experimental material presented in this article is based on a review of already published works by N. Dalakova and E. Beliayev (who is also one of the coauthors of this work) and their colleagues [5 -7, 11 -16]. For those studies, at St. Petersburg State University (SPbSU), a series of La2-xSrxCuO4 samples with strontium content x = 0.001, 0.002, 0.005, 0.010 were prepared by the solid-phase synthesis reaction by N. Bobrysheva and A. Selyutin in their laboratory. Synthesis details and the results of structural studies, which were carried out at St. Petersburg State University using SEM Cam Scan electron microscope, EDS LINK AN-10000, and WDS MIKROSPEC spectrometers, are given in [START_REF] Dalakova | Suppression of antiferromagnetic order in low-dopped ceramic samples La2-xSrxCuO4," Материалы 16-го Международного симпозиума[END_REF][START_REF] Dalakova | Low-temperature nonlinear effects in the conductivity of lightly doped cuprates La2-xSrxCuO4 in antiferromagnetic state[END_REF][START_REF] Далакова | Подавление антиферромагнитного порядка в слабодопированных керамических образцах La2-xSrxCuO4[END_REF][START_REF] Далакова | Влияние Композиционного Беспорядка в Системе Сверхпроводящих Гранул на Сверхпроводящие Свойства Керамических Образцов La2-xSrxCuO4[END_REF]. The strontium content was determined in five areas for each sample obtained using a high-resolution WDS spectrometer. The magnetic properties (by PPMS-5M) and electron transport properties (using the homemade helium cryostat with Kapitsa-design electromagnet and measurement system based on Keithley 2182 nanovoltmeter and Keithley 2000 multimeters) were carried out at B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine.

For all the samples studied, the electron conduction properties were essentially nonlinear up to the appearance of N-shaped Voltage-Current Characteristics (VCC) [START_REF] Dalakova | Low-temperature nonlinear effects in the conductivity of lightly doped cuprates La2-xSrxCuO4 in antiferromagnetic state[END_REF][START_REF] Далакова | Нелинейные эффекты в проводимости слаболегированных купратов La2-xSrxCuO4 в антиферромагнитном состоянии при низких температурах[END_REF]. This assumed a strongly inhomogeneous phase state of the samples under study. Moreover, the degree of nonlinearity of VCC increased with a decrease in the degree of doping level xexp. Fig. 1. Micrographs of thin sections of La2-xSrxCuO4 samples with x = 0.001 (left) and x = 0.002 (right) [START_REF] Dalakova | Suppression of antiferromagnetic order in low-dopped ceramic samples La2-xSrxCuO4," Материалы 16-го Международного симпозиума[END_REF] Additional XRD, SEM, and microprobe studies confirmed that despite all the efforts of the technologists, the samples turned out to be inhomogeneous on a microscopic scale. Microprobe analysis invariably showed that in the mixture with a strontium concentration of x = 0.001 after completing the solid-state synthesis, there were areas with either a Sr concentration of xexp = 0.002 or a Sr concentration being close to zero (Fig. 1) [START_REF] Dalakova | Suppression of antiferromagnetic order in low-dopped ceramic samples La2-xSrxCuO4," Материалы 16-го Международного симпозиума[END_REF]. For all other concentrations, there were no such problems. For Sr concentrations of xexp = 0.005 or xexp = 0.01, the strontium concentration in any sample part corresponds to the prescribed one. However, at very low xexp = 0.001, the sample was prone to phase separation into two phases with almost zero strontium content and a strontium concentration close to xexp = 0.002. Also, the Neel temperature values TN, measured for all the prepared low-substituted La2-xSrxCuO4 samples, which were supposed to increase monotonically with decreasing strontium concentration according to the known study by B. Keimer and co-workers [START_REF] Keimer | Magnetic excitations in pure, lightly doped, and weakly metallic La2CuO4[END_REF] at the concentration of xexp = 0.001, suddenly dropped sharply (Fig. 2 taken from [START_REF] Dalakova | Low-temperature nonlinear effects in the conductivity of lightly doped cuprates La2-xSrxCuO4 in antiferromagnetic state[END_REF] for this article has been supplemented by data taken from [START_REF] Dalakova | Suppression of antiferromagnetic order in low-dopped ceramic samples La2-xSrxCuO4," Материалы 16-го Международного симпозиума[END_REF] at xexp = 0.002, and by the commonly known TN value for stoichiometric La2CuO4). Fig. 2 demonstrates that free carriers (holes) that appear upon strontium doping level xexp = 0.001 destroyed the strict AFM ordering in La2CuO4 Mott-Hubbard dielectric, leading to a sharp drop in TN values and severe deterioration in its antiferromagnetic characteristics. All the lines in Fig. 2 are only guides to the eye. The coalescence phenomenon can also be explained using another physical language based on the "hot centers" picture [START_REF] Чирков | Самолокализованные колебания в высокотемпературных сверхпроводниках[END_REF][START_REF] Чудинов | Влияние ангармонизма на динамику решетки системы La-Sr-Cu-O (метод молекулярной динамики[END_REF]. The substitution of a heavy La for Sr atomthe atom with a smaller mass but having a much larger interaction potential with neighboring atoms in the crystal lattice [START_REF] Чирков | Самолокализованные колебания в высокотемпературных сверхпроводниках[END_REF][START_REF] Чудинов | Влияние ангармонизма на динамику решетки системы La-Sr-Cu-O (метод молекулярной динамики[END_REF], led to an increase in the frequency of local vibrations near a single localized strontium atom, thus significantly increasing the local temperature in that part of the crystal lattice. However, with an increase in the degree of doping, some beats have inevitably arisen between such hot impurity centers that disrupt the high-frequency resonance vibrations.

For this reason, at a specific concentration, the accumulation of strontium atoms in one place turns out to be energetically favorable since it lowers the local temperature. This is the physical basis for the coalescence phenomenon when single strontium atoms tend to merge into clusters. Having approached close enough distance, the "hot centers" began to interact through harmonics and lowered their energy of resonant vibrations, giving it to the lattice.

3.

Numerical self-consistent evidence for coalescence in the unitary scattering limit for La2-xSrxCuO4.

In this section, we numerically propose a coalescence self-consistent energy density function profile 𝑛 (𝜔 ̃) with the help of a computational model based on the SCCS analysis for dressed normal quasiparticle carriers in strontiumsubstituted lanthanum cuprate already developed theoretically in our previous works [START_REF] Contreras | Scattering Due to Nonmagnetic Disorder in 2D Anisotropic d-wave High Tc Superconductors[END_REF][START_REF] Contreras | Dressed behavior of the quasiparticles lifetime in the unitary limit of two unconventional superconductors[END_REF], aiming at comparing the results from this section using a dimensionless xtheory concentration parameter defined by the relation

𝑥 𝑡ℎ𝑒𝑜𝑟𝑦 = Γ + Δ 0 ⁄ (1),
where  + (that will be defined below) is the concentration parameter in the RPS space in meV, and 0 is the zero temperature ARPES value in meV of the superconducting gap. The aim of this section is to compare xtheory with the previous experimental section analysis for the strontium substituted concentration xexp using the RPS SCCS method.

The reduced scattering phase space (RPS) with the pair of coordinates (ℜ(𝜔 ̃), ℑ(𝜔 ̃)) is built with several parameters. The first group of three TB anisotropic parameters for a particular model: the superconducting gap at zero temperature meV, the Fermi energy 𝝐F = 0.4 meV, and the first neighbour hoping t = 0.2 meV, accordingly to the experimental ARPES data [START_REF] Yoshida | Coexisting pseudo-gap and superconducting gap in the High-Tc superconductor La2-xSrxCuO4[END_REF], using the Scalapino linear model [START_REF] Scalapino | The case for dx2 -y2 pairing in the cuprate superconductors[END_REF][START_REF] Tsuei | Pairing symmetry in cuprate superconductors[END_REF]. The normal state TB first neighbors energy is given by the expression 𝜉(𝑘 𝑥 , 𝑘 𝑦 ) = 𝜖 𝐹 + 2 𝑡 [cos(𝑘 𝑥 𝑎) + cos ( 𝑘 𝑦 𝑎)], with electron-hole symmetry 𝜉(𝑘 𝑥 , 𝑘 𝑦 ) = 𝜉(-𝑘 𝑥 , -𝑘 𝑦 ). This is the expression for a band centred at the corners (±π/a, ±π/a) of the first Brillouin zone as is sketched in Fig. 3, where the nodal lines are the straight belts drawn in brown color and the anisotropic FS in clear blue color, using an implicit plot for the normal state energy and the singlet gap. In references [START_REF] Momono | Evidence for nodes in the superconducting gap of La2-xSrxCuO4. T 2 dependence of electronic specific heat and impurity effects[END_REF][START_REF] Sun | Transport Properties of D-Wave Superconductors with Impurities[END_REF], it was theoretically and experimentally found that La2-xSrxCuO4 is in the unitary strong scattering limit if the Fermi surface is isotropic. They employed a specific superconducting electronic heat analysis at very low temperatures. The constant pressure electronic specific heat was measured by a conventional adiabatic heat pulse from 2 K to 10 K in order to extract the dimensionless residual density of states "N(0)/NF" which is proportional to the superconducting specific electronic heat jump "s/n". By comparing the relations "s/n ~ N(0)/NF where hold the relation cp-elect = sat very low temperatures, using the equation for suppression of Tc due to non-magnetic disorder, proposed by Larkin [START_REF] Larkin | Vector pairing in superconductors of small dimensions[END_REF]. Using this procedure, Momono and Ido [START_REF] Momono | Evidence for nodes in the superconducting gap of La2-xSrxCuO4. T 2 dependence of electronic specific heat and impurity effects[END_REF] extracted the residual DOS from very low temperatures electronic specific heat measurements at constant pressure, concluding that La2-xSrxCuO4 is in the unitary strong-scattering limit (c=0).

The original idea of parametrizing the SCCS based on the strength c and the impurity concentration  + for isotropic HTSC with line nodes was taken from the work by Schachinger and Carbotte [START_REF] Schachinger | Residual absorption at zero temperature in d-wave super-conductors[END_REF]. Additionally, in [START_REF] Schachinger | Residual absorption at zero temperature in d-wave super-conductors[END_REF], the disorder parameters for non-magnetic parameters are defined as follows: for the nonmagnetic impurity concentration  + = nimp ( 2 NF) -1 , and the atomic nonmagnetic Sr atom strength c = ( NF U0) -1 . This allowed us to build a complicated numerical procedure where we linked the TB parameters taken from Arpes measurements with the two nonmagnetic disorder parameters in a self-consistent way for very low frequencies needed to properly track the unitary regimen of the elastic SCCS, which is defined as

 ̃( + 𝑖 0 + ) =  + 𝑖 𝜋 Γ + 1 𝑔(𝜔 ̃),
and

𝑔(𝜔 ̃) = 〈  √ ̃2-∆ 2 (𝑘 𝑥 ,𝑘 𝑦 )
〉 𝐹𝑆 is a tight-binding numerical averaged. The whole set of microscopic parameters that can numerically be controlled in the fittings are five in this case, namely (𝜖 𝐹 , 𝑡, Δ 0 , 𝑐, and Γ + ).

Furthermore, following Scalapino linear nodal idea, the first neighbours' superconducting gap terms in the irreducible tight-binding representation B1g corresponds to an even paired anisotropic singlet order parameter basis function 𝜙(𝑘 𝑥 , 𝑘 𝑦 ) = 𝜙(-𝑘 𝑥 , -𝑘 𝑦 ) = cos(𝑘 𝑥 𝑎) -cos( 𝑘 𝑦 𝑎), with the gap equation Δ(𝑘 𝑥 , 𝑘 𝑦 ) = Δ 0 𝜙(𝑘 𝑥 , 𝑘 𝑦 ) = Δ 0 [cos(𝑘 𝑥 𝑎) -cos( 𝑘 𝑦 𝑎)]. The superconducting gap for this symmetry corresponds to the one-dimensional irreducible representation B1g of the tetragonal point symmetry group D4h (as seen in Fig. 3). In this model, nonmagnetic disorder induced by the atomic external strontium potential "U0", strongly quenches superconducting ordering leading to suppression of Tc. The Ginzburg-Landau coefficients correspond in this case to (1,0).

Based on Fig. 4 and the theory and calculations elaborated in references [START_REF] Contreras | Scattering Due to Nonmagnetic Disorder in 2D Anisotropic d-wave High Tc Superconductors[END_REF][START_REF] Contreras | Dressed behavior of the quasiparticles lifetime in the unitary limit of two unconventional superconductors[END_REF], we propose the empirical relation (2) following the classical theory of active collisions in physical kinetics, but taking into account, that the physical parameter 𝐿 of the problem is proportional to the lattice constant 𝑎 (unitary limit condition), i.e., 𝐿 ∼ 𝑎, and for the coalescence metallic region we notice that ℑ

[ℴ] ~ ℑ[𝜔 ̃] ~(𝑛 𝑑𝑟𝑒𝑠𝑠𝑒𝑑 (𝜔 ̃) -𝛿 ℑ(𝜔 ̃) ) (𝑎 𝑛 𝑐𝑜𝑛𝑑𝑒𝑛𝑠𝑎𝑡𝑒 ) ⁄
where a is the lattice parameter, ndressed refers to fermion concentration, and ncondensate to boson concentration (readers interested in those details can follow the aforementioned works and references therein).

Aiming at helping to understand the coalescence region observed by N. Dalakova and co-workers in a series of experimental works [START_REF] Dalakova | Suppression of antiferromagnetic order in low-dopped ceramic samples La2-xSrxCuO4," Материалы 16-го Международного симпозиума[END_REF][START_REF] Dalakova | Low-temperature nonlinear effects in the conductivity of lightly doped cuprates La2-xSrxCuO4 in antiferromagnetic state[END_REF][START_REF] Далакова | Подавление антиферромагнитного порядка в слабодопированных керамических образцах La2-xSrxCuO4[END_REF][START_REF] Keimer | Magnetic excitations in pure, lightly doped, and weakly metallic La2CuO4[END_REF] introduced and explained in Section 2, the function which we call the energy density selfconsistent profile 𝑛 𝑑𝑟𝑒𝑠𝑠𝑒𝑑 (𝜔 ̃)~ Γ + [ 𝑔(𝜔 ̃)] -1 ~ 𝜏 𝑠𝑢𝑝 -1 is proportional to the imaginary part of the self-consistent elastic scattering cross-section ℑ[ℴ], (in rationalized Planck units, it is given by meV and is suitable for an experimental contrast with the theory). The classical analysis elaborated for the SCSC can qualitatively separate three different phases based on a dimensionless concentration parameter xtheory in the following way:

𝑛 𝑑𝑟𝑒𝑠𝑠𝑒𝑑 (𝜔 ̃)~{ → 0, Γ + = 0.01 𝑚𝑒𝑉 (Mott Hubbard dielectric) 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 + 𝛿 ℑ(𝜔 ̃), Γ + = 0.05 𝑚𝑒𝑉 (𝑏 -lighty subtituted Coalesc. 𝐿𝑎 2-𝑥 𝑆𝑟 𝑥 Cu𝑂 4 ) ℑ(𝜔 ̃), Γ + > 0.05 𝑚𝑒𝑉 (𝑐 -𝑠𝑡𝑟𝑎𝑛𝑔𝑒 𝑆𝐶 𝑚𝑒𝑡𝑎𝑙 𝐿𝑎 2-𝑥 𝑆𝑟 𝑥 Cu𝑂 4 )

In relation [START_REF] Wigner | On the quantum correction for thermodynamic equilibrium[END_REF], we can distinguish and observe three phases that can be compared with the experimental results from Section 2 where the most important feature observed in this unitary computational model, that attracted our attention when we discussed our findings with the experimental Kharkiv group was that at very low concentration Γ + , only the central unitary resonance peak at zero frequencies seems to remain, i.e., the imaginary part of the elastic scattering cross-section ℑ (𝜔 ̃) 𝑦𝑖𝑒𝑙𝑑𝑠 → 0 and 𝛿 ℑ(𝜔 ̃) ≪ 1 , for most of the superconducting frequency interval, that in the language of physical kinetics means that there are no dressed quasiparticles outside this narrow zero frequency interval, Therefore, it is proposed in this section that this resonance could be a signature of a phase transition to the antiferromagnetic phase of the Mott Hubbard dielectric La2CuO4 as Sr non-magnetic atoms concentration is reduced to zero since xtheory 𝑦𝑖𝑒𝑙𝑑𝑠 → 0.

It is worth noticing that in a previous work [START_REF] Contreras | Non-magnetic tight binding effects on the  sheet of Sr2RuO2[END_REF] (see Fig. 4, the yellow line in that paper), we observed that in the compound Sr2RuO4 at a very low concentration, Γ + , the same resonance persists for the unitary strong scattering regime (c = 0). In [START_REF] Contreras | Non-magnetic tight binding effects on the  sheet of Sr2RuO2[END_REF], the computational calculation of the SCCS gives a larger mean averaged width of the imaginary part, nevertheless limiting the reduced scattering space to a small energy peak interval around the zero energy superconducting frequency.

In order to conclude this short but instructive numerical section analysis, we give Table 1 based on the calculation using the anisotropic TB parameters in agreement with the experiments mentioned and the posteriorly numerical calculation: We attribute this region to being a signature of the Mott-Hubbard dielectric La2CuO4 antiferromagnetic phase, which N. Dalakova, E. Beliayev and co-workers reported in their experimental studies on the results of strontiumsubstituted xexp doping parameter as being in essence La2CuO4. In the previous numerical work with the same model, we reported this phase as the one where 𝜏 -1 (𝜔 ̃) → 𝑐𝑜𝑛𝑠𝑡 ≪ 1, without comparing with experimental data since we did not have it at that moment [START_REF] Contreras | Dressed behavior of the quasiparticles lifetime in the unitary limit of two unconventional superconductors[END_REF].

Region (b) in the left part of Fig. 3 is given by the clear blue curve ( + = 0.05 meV). It represents the almost constant imaginary scattering cross-section ℑ(𝜔 ̃) ~ 𝜏 -1 (𝜔 ̃) → 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡, for most of the entire  interval with a small amount of dressed by non-magnetic impurity scattering quasiparticles at the impurity concentration given by the slightly higher value of the concentration parameter Γ + = 0.05 meV except for a small resonance 𝛿 ℑ(𝜔 ̃). It corresponds to 𝑥 𝑡ℎ𝑒𝑜𝑟𝑦 = 0.002 following the empirical relation [START_REF] Pethick | Transport processes in heavy-fermion superconductors[END_REF] and this represents the metallic coalescing phase for a lightly doped La2-xSrxCuO4.

To explain coalescence, we propose that since linear momentum "k" is not conserved in the unitary limit for the elastic scattering cross-section, it is proposed that the dressed quasiparticles' momentum k is transferred to strontium atoms in the crystal lattice. Those Sr atoms migrate through the lattice and stick together in a coalescing metallic state, but only for a minimal concentration of Sr atoms, as was observed experimentally by Dalakova, Beliayev and co-workers in their results. Therefore, near the superconducting-antiferromagnetic transition in lightly strontiumsubstituted lanthanum cuprate, a robust metallic coalescing phase with an almost constant scattering lifetime can be interpreted theoretically and experimentally from a classical point of view and it is supported by the theory of active collisions in physical kinetics.

Region (c) represents, what we call the strong self-consistent imaginary phase of the SCCS for most of the entire frequency region when ℑ(𝜔 ̃) ~ 𝜏 -1 (𝜔 ̃), and Γ + = ( 0.05, 0.10, 0,20) meV. It is observed for the values of the empirical relation (1) 𝑥 𝑡ℎ𝑒𝑜𝑟𝑦 = (0.003, 0.004, 0.010), shown in Fig. 4 shadowed gray. Undoubtedly, the value  + = 0.20 meV and 𝑥 𝑡ℎ𝑒𝑜𝑟𝑦 = 0.010. It represents the so-called in the current literature strange metal phase in cuprates, where RPS self-consistent effects in the imaginary part of the SCCS mix the nodal Fermi dressed quasiparticles and superconducting pairs, making it very difficult to explain the transport properties based in parameters such as the mean lifetime "" and free path "l" with strongly self-consistent frequency-dependency.

From the point of view of the experimentally observed electronic transport properties of lightly-doped lanthanum cuprates, attention may be drawn to Fig. 1. There, one can see a resonance at near-zero xexp concentration. Thus, the hypothesis arises that it is precisely this resonance that can explain the fact that in a series of aforementioned articles the team led by Dalakova and co-workers failed to prepare homogeneous samples of strontium-substituted lanthanum cuprate with an ultra-low concentration of Sr atoms (x = 0.001) and instead found evidences of the Mott Hubbard dielectric La2CuO4 as is shown in Table 1 and discussed in this paper.

4.

Conclusions and recommendations.

In this work, we qualitatively explain experimental data and build a phenomenological model of La2-xSrxCuO4 for which three x-doping phases are distinguished as Dalakova and co-workers experimentally found on lightly strontium-substituted lanthanum cuprate samples. Using the idea of an energy concentration profile (𝑛 (𝜔 ̃)~ Γ + 𝑔(𝜔 ̃) ⁄ )

of normal state dressed quasiparticles based on the imaginary and real parts of SCCS, for different values of experimental and theoretical doping, a phenomenological energy density profile 𝑛 𝑑𝑟𝑒𝑠𝑠𝑒𝑑 (𝜔 ̃) can be zero, almost constant for most of the frequency interval, or strongly self-consistent and frequency-dependent in the whole energy (frequency) interval, depending on the impurity atoms concentration  + , for the strong scattering unitary limit.

In there were distinguished: one almost antiferromagnetic phase where 𝑛 (𝜔 ̃) → 0, without dressed quasiparticles; another metallic coalescent lightly-substituted Sr phase with 𝑛 𝑑𝑟𝑒𝑠𝑠𝑒𝑑 (𝜔 ̃) → 𝑐𝑜𝑛𝑠𝑡 with a small amount of non-magnetic impurities; and, finally, the strange metallic phase in dopped La2-xSrxCuO4, where the scattering lifetime is strong self-consistently dependent.

According to this model, as the antiferromagnetic-superconducting phases approach each other, the substitution of heavy lanthanum for lighter strontium atoms led to an increase in the frequency of vibrations near a single local Sr atom, thus significantly increasing the local temperature of the crystal lattice, therefore, at a particular concentration, the accumulation of strontium atoms in one place is energetically favorable, since it lowers the local temperature, which is the basis of coalescence phenomenon (Fig. 2). Thus, single non-magnetic strontium atoms tend to merge into metallic clusters, and, since the linear momentum of dressed quasiparticles k is not conserved in the unitary metallic limit, it is transferred to strontium atoms in the crystal lattice, which migrate through the lattice and stick together forming the coalescing metallic cluster, for the lowest concentrations of non-magnetic strontium atoms. This finding of the metallic coalescent phase with a constant scattering "s"agrees with a hypothesis firstly proposed by Schmitt-Rink, Miyake and Varma [START_REF] Schmitt-Rink | Transport and thermal properties of heavy-fermion superconductors: A unified picture[END_REF] to fit low temperatures transport superconducting data in unconventional superconductors. They proposed that the superconducting scattering lifetime "s" is almost constant and almost the same as in the normal metallic state "normal". Additionally, to this amendment, in the past, we were able to fit the low temperature superconducting data from directional superconducting ultrasound attenuation [START_REF] Lupien | UltrasoundAttenuation in Sr2RuO4: An Angle-Resolved Study of the Superconducting Gap Function[END_REF][START_REF] Contreras | Determining the superconducting gap structure in Sr2RuO4 from sound attenuation studies below Tc[END_REF] and electronic superconducting thermal transport [START_REF] Tanatar | Thermal conductivity of superconducting Sr2RuO4 in oriented magnetic fields[END_REF][START_REF] Contreras | Electronic heat transport for a multiband superconducting gap in Sr2RuO4[END_REF] experiments in clean samples of the triplet superconducting Sr2RuO4 using their proposal and a TB anisotropic model.

On the other hand, recently, using a similar method, we found three phases in the triplet compound strontium ruthenate, two phases with quasinodal points (one of them very inhomogeneous that corresponds to a tiny gap triplet model) and one phase with point nodes and the ground state similar to the HTSC line nodes model but with smaller 0, being all phases in the unitary scattering regime [START_REF] Contreras | Quasi-point versus point nodes in Sr2RuO4, the case of a flat tight binding  sheet[END_REF].

Therefore, we may conclude that the analysis of the elastic scattering cross-section in the RPS with a selfconsistent procedure that accounts for the effects of the variations in the set of TB and disorder parameters (𝜖 𝐹 , 𝑡, Δ 0 , 𝑐, Γ + ), is very instructive and deserves further attention (see Fig. 1 in Contreras, Osorio and Beliayev paper [START_REF] Contreras | Tight-Binding Superconducting Phases in the Unconventional Compounds Strontium-Substituted Lanthanum Cuprate and Strontium Ruthenate[END_REF] for a general diagrammatic sketch used in this method). It serves as a recent proposed tool aiming at understanding some phenomenological aspects of unconventional superconductors, such as the different phases which depend on the impurity strength c and nonmagnetic disorder concentration   , but in the TB approach, extensions can be made to the flat band case [START_REF] Contreras | Quasi-point versus point nodes in Sr2RuO4, the case of a flat tight binding  sheet[END_REF][START_REF] Contreras | Tight-Binding Superconducting Phases in the Unconventional Compounds Strontium-Substituted Lanthanum Cuprate and Strontium Ruthenate[END_REF][START_REF] Contreras | The Tight-Binding Flat Band Case in the Strontium-Substituted Lanthanum Cuprate[END_REF]. We would like to clarify that some other aspects of Dalakova and coworkers work could be addressed in the future, following the ideas about the activation energy presented in the review by Prof. B. I. Beletsev [START_REF] Belevtsev | Superconductivity and localization of electrons in disordered two-dimensional metal systems[END_REF], and also in the work by I. Lifshitz and collaborators, for example that for a system of low impurities concentration but strongly scattering impurities, the use of a one-site approximation and the first power in impurities concentration  + must be quite effective [START_REF] Lifshitz | Introduction to the theory of disordered systems[END_REF] as the formalism briefly presented in this work points out.

Fig. 2 :

 2 Fig. 2: Dependence of Neel temperature TN in strontium-substituted La2-xSrxCuO4 samples on Sr concentration. Lines are guides to the eye.

Figure. 3 :

 3 Figure. 3: The Scalapino linear nodal model for singlet HTSC such as strontium substituted La2-xSrxCuO, showing in the brown lines the behavior of the nodal lines implicit plot, and in clear blue the tight-binding FS implicit plot.

  i.e., their inverse scattering lifetime 𝜏 -1 𝑦𝑖𝑒𝑙𝑑𝑠 → 0 𝑚𝑒𝑉 (see Fig 4, left region shadowed very light blue color).

Fig. 4

 4 is as follows. Region (a) shadowed very clear blue in the left part of Fig. 4 represents an almost zero imaginary self-consistent scattering cross-section ℑ(𝜔 ̃) ~ 1 𝜏(𝜔 ̃) ⁄ → 0 for most of the entire energy window interval 𝜔 ∈ ( -40, +40) meV. It means there are no dressed by non-magnetic impurity scattering (due to a U0 potential) quasiparticles, except a few at the unitary resonance region around the zero frequency ( + = 0.01 meV → 𝑥 𝑡ℎ𝑒𝑜𝑟𝑦 = 0.0003 ~ 𝑥 𝑒𝑥𝑝 ).

Fig. 4 :

 4 Fig. 4: The reduced scattering phase space RPS -(ℜ(𝜔 ̃), ℑ(𝜔 ̃)) plot showing the different phases explained in Section 3, the imaginary part of the scattering cross-section as function of the non-magnetic impurities parameters  + is calculated for 5 values, a new sixth smallest value f  + = 0.005 meV have been very recently in reference [36] .

Table 1 :

 1 Phases as a function of doping for antiferromagnetic La2CuO4 and lightly-strontium substituted metallic samples of La2-xSrxCuO4 where the doping is given by 𝑥 𝑡ℎ𝑒𝑜𝑟𝑦 = Γ + Δ 0 ⁄ and 𝑥 𝑒𝑥𝑝

	Phases	a -MH dielectric	b-Lightly-doped	c -Strange	idem	Idem
		La2CuO4	coalescent	metal phase
			La2-xSrxCuO4
	Γ + (meV)	0.01	0.05	0.10	0.15	0.20
	𝑥 𝑡ℎ𝑒𝑜𝑟𝑦	0.0003	0.002	0.003	0.004	0.010
	𝑥 𝑒𝑥𝑝	0.001	0.002	-	0.005	0.010
	A detailed analysis of Table 1 and

Acknowledgments

The authors did not receive any financial support for this article's research, authorship, and/or publication. Experiments were performed by Drs. Dalakova and Beliayev and their co-workers at B. Verkin Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine. Numerical calculations are attributed to Profs. Contreras and Osorio, and the physical analysis and visualization of this particular problem belongs to the three participants of this research.