Timothée Dhaussy

Bassam Jabaian

Fabrice Lefèvre

Radu Horaud

AUDIO-VISUAL SPEAKER DIARIZATION IN THE FRAMEWORK OF MULTI-USER HUMAN-ROBOT INTERACTION

Keywords: speaker diarization, multimodal, humanrobot interaction

The speaker diarization task answers the question "who is speaking at a given time?". It represents valuable information for scene analysis in a domain such as robotics. In this paper, we introduce a temporal audio-visual fusion model for multiusers speaker diarization, with low computing requirement, a good robustness and an absence of training phase. The proposed method identifies the dominant speakers and tracks them over time by measuring the spatial coincidence between sound locations and visual presence. The model is generative, parameters are estimated online, and does not require training. Its effectiveness was assessed using two datasets, a public one and one collected in-house with the Pepper humanoid robot.

INTRODUCTION

Matching speech signal to its emission source is a crucial task to perform an accurate analysis of a scene with different speakers. Commonly named speaker diarization, it consists in assigning audio segments to classes that correspond to speaker identities. This task brings an answer to the question "who spoke when?" [START_REF] Tranter | An overview of automatic speaker diarization systems[END_REF]. Early work on speaker diarization focused on audio modality [START_REF] Hershey | Audio vision: Using audio-visual synchrony to locate sounds[END_REF][START_REF] Reynolds | Approaches and applications of audio diarization[END_REF]. Nowadays, typical audio speaker diarization systems that use audio as input are composed of three components: (1) Speech segmentation, where the audio input is decomposed into short segments, each segment is supposed to have only one speaker, and the noise is filtered out. This component can be seen as a voice activity detection module. (2) Extraction of the audio embedding from the segmented sections through various methods. The most noticeable are MFCCs [START_REF] Kenny | Diarization of telephone conversations using factor analysis[END_REF], speaker factors [START_REF] Castaldo | Stream-based speaker segmentation using speaker factors and eigenvoices[END_REF] and ivectors [START_REF] Stephen | Unsupervised methods for speaker diarization: An integrated and iterative approach[END_REF]. And (3) Clustering module, where the extracted audio embeddings are clustered into speakers. For this task the number of speakers is determined. It is also possible to pre-process upstream using speech enhancement and denoising technics which lead to significant improvement through deep learning [START_REF] Gao | Densely connected progressive learning for lstm-based speech enhancement[END_REF]. Speaker representation has been largely improved with the arrival of neural networks and other new methods such as d-vector [START_REF] Wang | Speaker diarization with lstm[END_REF] and x-vector [START_REF] Villalba | State-of-the-art speaker recognition for telephone and video speech: The jhu-mit submission for nist sre18[END_REF]. An interesting alternative consists in the fusion of audio and visual data. The combination of these two modalities provides complementary information and, therefore, audiovisual approaches to speaker diarization are likely to be more robust than audio-only approaches. It can be associated with a face or mouth tracking through spatial coincidence on the image plane [START_REF] Israel | Audio-visual speech-turn detection and tracking[END_REF][START_REF] Israel | Audio-visual speaker diarization based on spatiotemporal bayesian fusion[END_REF], on a ground plane [START_REF] Wang | Prioritized target tracking with active collaborative cameras[END_REF] or in 3D [START_REF] Xiong | Multi-speaker tracking by fusing audio and video information[END_REF]. Methods for tracking person in 3D require spatially distributed camera networks and microphone arrays which are not tractable in the case of complex (real) scenarios. On the other hand, methods relying on plane or ground image may present a lack of information and suffer much more from occlusion but they offer the advantage of easier data collection and can be utilized in a wider range of scenarios. In addition of mouth position, several methods are based on the synergy between utterances and lip movements through different approaches such as mutual information [START_REF] Hershey | Audio vision: Using audio-visual synchrony to locate sounds[END_REF] and deep learning [START_REF] Joon | Who said that?: Audio-visual speaker diarisation of realworld meetings[END_REF].

In this study, we propose a method which models this fusion through the spatial coincidence of visual and sound source localization (SSL) and by combining this concordance model with a dynamic Bayesian formulation that tracks the identity of the active speaker. SSL provides several benefits in multi-user conversations such as the ability to handle overlapping speech segments, eliminating the need for a voice activation module. The proposed method can be applied in various acoustic conditions by leveraging spatial information from SSL and face location. This paper is organized as follows: in Section 2 we present our model for multi-user speaker diarization. Section 3 presents the experimental setup and the evaluation of the proposition.

PROPOSED METHOD

Problem definition

First, we introduce the notations and definitions of variables. Scalars are written in italic, vectors in italic bold and matrices are in italic and underlined. Upper-case letters denote random variables while lower-case letters denote their realization. We represent t the time-step index of both visual and audio frames, synchronized with each other. At frame t, there are at most N visual observations, X t = (X t1 , ..., X tn , ..., X tN ) ∈ R 2×N , where the random variable X tn corresponds to the mouth location of person n in image t. Then, a multi-person tracker provides a time series of N image locations, namely X 1:t = {X 1 , ..., X t } and associated visual-presence binary masks V 1:t , namely variable V tn associated with X tn such that V tn = 1 if person n is present in image t and 0 otherwise. N t = n V tn represents the number of persons that are observed at frame t. In practical, when V tn = 0, X tn = X t-i,n with t -i the most recent timeframe where V t-i,n = 1. We also consider a SSL module that provides the azimuth and elevation of the dominant sound source at each audio frame t. The soundsource location can then be mapped onto the image plane, such that an azimuth-elevation pair of observations is transformed into an image location modeled by a random variable Y t = (Y t1 , ..., Y tk , ..., Y tK ) ∈ R 2×K with K audio-visual observations for a visual frame at t and Y 1:t = {Y 1 , ..., Y t } its time serie. To these audio-visual observations we associate a speech-activity binary masks A 1:t = {A 1 , ..., A t }, such that A t = 1 if there is an active audio source at frame t or 0 otherwise. The objective is to track dominant speaker(s) at time t in associating audio responsibility over time the audio activity (if any) with one of the tracked persons. Audio sources out of the pictures are not taken into account. This is also referred to as audio-visual speaker diarization, addressed below in the framework of temporal graphical models. We introduce a time-series of discrete latent variables, S 1:t = {S 1 , ..., S t } such that S t = n, n ∈ 1, 2, ..., N if person n is both observed and speaks at frame t, and S t = 0 if none of the visible persons speaks at frame t. Notice that S t = 0 represent two different cases: firstly, there is at least one active sound-source at t (A t = 1) but its or their location cannot be associated with one of the visible persons and it can be interpreted as noise, secondly, there is no active sound-source at t, A t = 0. We will also use another latent variable Z t1:K = Z t1 , .., Z tK with Z tk = n which represents the attribution of the sound source k to the visual identity n. Z tk = 0 means the source k isn't assigned to any person in the image.

Speaker Diarization Model

The temporal speaker diarization problem can be formulated as finding a maximum-a-posteriori (MAP) solution, namely finding the most probable configuration of the latent state S t that maximizes the following posterior probability distribution. Also referred to as the filtering distribution it can be express in the following way:

ŝt = arg max st P (S t = n|y 1:t , x 1:t , v 1:t , a 1:t ) (1)
Following Bayes formula, the posterior probability (1) can be written and beyond developped as:

P (S t = s t |u 1:t ) = P (u t |S t = s t )G st N j=0 P (u t |S t = j)G j (2)
With G n = N i=0 P (S t = n|S t-1 = i)P (S t-1 = i|u 1:t-1 ) and u t = (x t , v t , y t , a t ). The evaluation of ( 2) is recursive and a reasonable number of person simultaneously tracked need to be considered [START_REF] Castaldo | Stream-based speaker segmentation using speaker factors and eigenvoices[END_REF][START_REF] Stephen | Unsupervised methods for speaker diarization: An integrated and iterative approach[END_REF][START_REF] Gao | Densely connected progressive learning for lstm-based speech enhancement[END_REF][START_REF] Wang | Speaker diarization with lstm[END_REF] in order to keep the calculation tractable. Computation of this equation requires the observed likelihood P (u t |S t = s t ) and the transition probabilities P (S t = j|S t-1 = i) explained in the next two subsections.

EM Audio-Visual Observation Model

The main feature of the proposed model is its ability to robustly associate the SSL at time t with a person. The expectation-maximization for Gaussian mixture model infers the posterior probability that a person utters speech from audio and visual observations that are mapped onto the same mathematical space. We distinguish two cases. The first one If there is no audio activity at time t(A t = 0), the posterior can be evaluated with the following formula, where c is a small positive scalar, e.g., c = 0.2:

P (S t = n|y t , x t , v t , A t = 0; c) = c/N t if 1 ≤ n ≤ N 1 -c if n = 0.
(3) If a sound-source is active at time t, (A t = 1), we assign it to a visual identity n such that Z tk = n plays the role of an assignment variable in a mixture model. Its location y tk is assumed to be drawn from the following Gaussian/uniform mixture:

P (y tk |x t , v t , A t = 1; θ t ) = N n=1 p tn v tn N (y tk |x tn , Σ tn ) + p t0 U(β) (4)
where θ t = ({p tn } N n=0 , {Σ tn } N n=0 , β) denotes the set of model parameters, namely the prior, N n=1 v tn p tn + p t0 = 1, the 2 × 2 covariance matrices Σ tn , and parameter β that characterizes the outlier component of the mixture, namely a uniform distribution. The parameter set θ t can be estimated via the EM algorithm for Gaussian mixtures.

The algorithm begin with E-step that evaluates the posterior probabilities r tkn using current parameters values θ t , Z t is our assignment variable, Z tk = n means y tk is generated by component n. We first compute r tkn ∀n, 1 ≤ n ≤ N which correspond that a sound source is associated with a visible person:

r tkn = p tn v tn N (y tk |x tn , Σ tn ) N i=1 p ti v ti N (y tk |x ti , Σ ti ) + p t0 U(β) ∀n, 1 ≤ n ≤ N (5)
We can also write the probability that a sound source is not associated with a visible person n = 0, either because it corresponds to a sound emitted by a non visible person or emitted by another type of source, i.e., the posterior of the uniform component of the mixture:

r tk0 = p t0 U(β) N i=1 p ti v ti N (y tk |x ti , Σ ti ) + p t0 U(β) (6)
M step re-estimates the parameters using the current responsibilities.

Σ new tn = 1 R tn K k=1
r tkn (y tk -x tn )(y tk -x tn ) T + εI (7)

p new tn = R tn K (8)
with ε > 0 is a scalar acting as a parameter to prevent empty clusters, and I is the 2 × 2 identity matrix and where we have defined:

R tn = K k=1 r tkn (9)
The algorithm can be easily initialized by setting all the priors equal to 1/N + 1 and by setting all the variances equal to a positive scalar. Because the component means are fixed, the algorithm converges in only a few iterations. We have N faces and K sources, which represent a combinatorial problem at each iteration. For each possible association at t, we have to consider all the possible cases for the next step. These computations between the faces and the sources will explode in the course of time. To address this issue and keep the audio-visual model tractable, instead of computing all combinations, we factorize all sources in one dominant source y tn * t . It first requires to choose the person with highest speaking probability represented by the prior:

n * t = max n p tn (10)
Therefore, the mean source y tn * t is the sound source location that is considered the most probable based on x tn * t :

y tn * t = K k=1 r tkn * t y tk R tn * t (11) P (S t = n|y tn * t , x t , v t , a t ) can be calculated given y tn * t , ∀n, 1 ≤ n ≤ N : P (S t = n|u tn * t ) = π tn v tn N (y tn * t |x tn , Σ tn ) N i=1 π ti v ti N (y tn * t |x ti , Σ ti ) + π t0 U(β t ) (12)
and for n = 0

P (S t = 0|u tn * t ) = π t0 U(β t ) N i=1 π ti v ti N (y tn * t |x ti , Σ ti ) + π t0 U(β t ) (13)
with u tn * = (y tn * t , x t , v t , a t ). Finally, by noting that the observed-data likelihood P (u tn * t ) does not depend on S t and by assuming a uniform distribution over the priors of visible person n (v tn = 1), i.e., π t0 = π tn = 1/(N t + 1), we obtain the following observation model:

P (u tn * t |S t = n) = P (S t = n|u tn * t )P (u tn * t )/P (S t = n) = P (S t = n|u tn * t )P (u tn * t )/π tn ∝ P (S t = n|u tn * t ). ( 14
)
This enables to replace the observed likelihood (left hand side of ( 14)) with the posterior (right hand side of ( 14)) in (2).

State Transition Model Audio-Visual

The state transition probabilities, p(S t = j|S t-1 = i), provide the temporal modality for tracking speech turns along timestep. p(S t = j|S t-1 = i) is computed through several cases based on the presence/absence of persons and on their speaking status (for convenience and without loss of generality we set v t0 = 1):

P (S t = j|S t-1 = i) =                p s if i = j and v t-1,i = v ti = 1 (1 -p s )/N t if i ̸ = j and v t-1,i = v tj = 1 0 if v t-1,i = v t-1,j = 1 and v tj = 0 1/N t if v t-1,i = 1, v ti = 0 and v tj = 1 1/N if v t-1,i = 0 and v ti = 0 (15)
The first case of ( 15) defines the self-transition probability, p s , e.g., p s = 0.8, of person i present at both t -1 and t. The second case defines the transition probability from person i present at t -1 to another person j present at t. The third case simply forbids transitions from person i present at t -1 to person j present at t -1 but not present at t. The fourth case represents the transition probability from person i present at t -1 but not present at t, to a person j present at t. The fifth case defines the transition probability from person i not present at t -1 to person j that is not present at t. One may easily verify that

N j=1 p(S t = j|S t-1 = i) = 1.

EXPERIMENTS

Data

The proposed method is evaluated on two corpora. The first considered corpus is CAV3D [START_REF] Qian | Multi-speaker tracking from an audio-visual sensing device[END_REF]. It contains 20 sequences with duration ranges from 15s to 80s. An evaluation is conducted on a subset SOT composed of 9 sequences with a single speaker and a subset SOT2 composed of 6 sequences with a single active speaker and a second interfering person (not speaking).

The second corpus is recorded by ourselves on Pepper, a humanoid robot from Softbank Robotics. It contains dialogs between two or three persons. A total of 9 different subjects participated in this experience: 2 women, 7 men. Participants were asked to speak one at a time and try to avoid overlapping. We brought variations to dialogs by asking participants to randomly move in and out the scene, face the robot or look at each others. Different positions in the room were used to obtain different acoustic configurations. In a three-person dialogue, the participant positioned in the middle was requested to remain silent, to act as a distractor. The total duration is around eleven minutes and dominant speaker is carefully labeled in each frame. Windows with a bounding box that is either undetected or inaccurately computed are put aside. It was observed that the Pepper SSL occasionally experiences issues with activation, and in the absence of calibration of the Pepper SSL module, windows containing speech but no SSL detection are discarded too.

Technical Specifications

For this experiment, speaker diarization model is implemented for CAV3D and the Pepper corpus with some differences. SSLs are extracted with [START_REF] Grondin | Lightweight and optimized sound source localization and tracking methods for open and closed microphone array configurations[END_REF], and interpolation is performed from 3D coordinate, given by SSL, to speaker mouth localization using SOT subset. Pepper SSL module retrieves the direction of the emitting source (azimuth and elevation angles) from the TDOAs measured on the different microphone pairs. The angles provided by the sound source localization engine match the real position of the source with an average accuracy of 10 degrees. Transition from angle to image plane localization is made by interpolation. We record SSL from a loudspeaker at different positions in the image and perform a regression to get a mapping angle into image position. Sound sources located out of the image are filtered. We calibrate and fine-tune the parameters of the micro configuration using the first SOT sequence for CAV3D and a training sequence for the Pepper corpus in preparation for the testing phase. Thus we set Σ = Diag[300, 800], β = 10 7 , ε = 100 for CAV3D and Σ = Diag[300, 500], β = 300000, ε = 200 for the Pepper corpus. The remaining parameters are shared between both experiences, c = 0.2, p s = 0.8.

Results

The diarization performance is evaluated by Diarization Error Rate (DER), the lower the better. It contains three terms: Missing Detection (MS), False Alarm (FA), and Speaker Error (SPKE).

To evaluate the acoustic conditions we investigate CAV3D dataset with and without an oracle VAD. Those results can be compared to those of the audio-visual speaker diarization state of the art (SOTA): WST [START_REF] Joon | Who said that?: Audio-visual speaker diarisation of realworld meetings[END_REF] and the audio only speaker diarization SOTA VBx [START_REF] Landini | Bayesian hmm clustering of x-vector sequences (vbx) in speaker diarization: theory, implementation and analysis on standard tasks[END_REF] on the AMI corpus [START_REF] Carletta | The ami meeting corpus: A pre-announcement[END_REF]. The AMI corpus is a collection of meetings which shows similarities with the Pepper corpus. WTS yields to 21.3% and 21.1% of DER on the two AMI subsets ES and IS and VBx to 38.65% for the whole AMI corpus. Both are computed without an oracle VAD. We denote similar results taking variation of DER between datasets into account. The theoretical complexity of the algorithm is O(n 2 ). The audiovisual observation model represents 99% of the running time. For 10 sound sources being detected, with 5 considered persons, the running time is 0, 0631 seconds, out of a total running time of 0, 0635 seconds.

CONCLUSION

We proposed a model for temporal speaker-diarization based on principled mathematical and algorithmic concepts coupled with two types of perception, SSL and plane image. This model shows good results with a capacity of adaptation to different acoustic conditions without training phase. Thus this diarization method is not biased towards a particular training dataset, hence it is applicable to a large number of practical human-robot interaction scenarios. The VAD function and Robustness are carried by the uniform component of the mixture, which collects sound source locations that are far from the Gaussian components, which are centered around the faces. However to get the audio-image fusion, an interpolation needs to be made between SSL and image plan for every micro configuration, thus removing this micro configuration dependency is a challenge for future work.

Table 1 .

 1 The use of an oracle significantly lowers the DER as it gives valuable information The performances (%) of our model for different experience set, A t is set with oracle VAD derived from diarization labels, or with presence or absence of SSL to reduce the number of FA. Results on CAV3D are promising, the model losses only 4.28% of DER between SOT and SOT2 without oracle VAD. With a DER of 19.27% on the Pepper corpus we can assume that our model fulfills its diarization goal in a standard robotic case. This method shows interesting results on SOT2 with a SPKE of 0%. The prediction only matches the right person when it detects a speaker. But it substantially decreases on the Pepper dataset. It comes from more complex scenarios and may also be related to Pepper micro quality.

	Experiences	MS	FA SPKE DER
	SOT (no sequence 6)	3.29 6.63	/	9.92
	SOT (no sequence 6) oracle 1.24	0	/	1.24
	SOT2	6.22 8.02	0	14.2
	SOT2 oracle	1.14	0	0	1.14
	Pepper corpus	5.35 0.1	13.82 19.27


Halima Mazouz

Abderrahmane Belghachi

Pierre-Olivier Logerais
email: pierre-olivier.logerais@u-pec.fr

Effect of solar cell structure on the radiation resistance of an InP solar cell

Effects of electron irradiation-induced deep level defects have been studied on both n/p and p/n Indium Phosphide (InP) solar cells with very thin emitters. The simulation results reveal that the n/p structure offers a somewhat better short-circuit current and that the p/n structure renders an improved opencircuit voltage, not only before electron irradiation but also after 1 MeV electron irradiation with 5×10 15 electrons per cm 2 fluence. Further, the calculated findings highlight that the n/p solar cell structure is more resistant than that of a p/n structure.

Introduction

Research into space solar cells has progressed rapidly over the last couple of decades. The fundamental objectives for developing space solar cells are to uplift their high efficiency and radiation resistance and in this framework, Gallium Arsenide (GaAs) cells have been found to be more promising than silicon (Si) cells for space applications [START_REF] Yamaguchi | IEEE Conference on Photovoltaic Specialists[END_REF][START_REF] Loo | [END_REF]. Developments have also been focused on Indium Phosphide solar cells (InP) for the reason that their radiation tolerance surpasses that of the GaAs and Si ones as reported in variant studies [START_REF] Yamaguchi | IEEE Conference on Photovoltaic Specialists[END_REF]3,[START_REF] Walters | Proc. of 1994 IEEE 6 th International Conference on Indium Phosphide and Related Materials (IPRM)[END_REF]. Moreover, they have several other advantages like a high-efficiency, a thin-film structure, a high-temperature operation and a simple cell structure [START_REF] Nubile | [END_REF]6]. InP cells appear then as very attractive materials for space solar cells as of late, elevated conversion efficiencies exceeding 29.2% (AM0 at 25°C) have been achieved [7]. Also, it has been demonstrated that InP solar cells exhibit relatively low levels of performance degradation for both proton and electron irradiation [START_REF] Yamaguchi | IEEE Conference on Photovoltaic Specialists[END_REF]7].

The development efforts of InP photovoltaic cells have been focused on the n/p structure under AM0 conditions and little has been published on the performance of the p/n structure [START_REF] Hoffman | Conference Record of the Twenty-Sixth IEEE Photovoltaic Specialists Conference 1997[END_REF][START_REF] Kilmer | Conference Record of the Twenty-Second IEEE Photovoltaic Specialists Conference 1991[END_REF][START_REF] Vilela | [END_REF][START_REF] Sharps | Conference Record of the Twenty-Fifth IEEE Photovoltaic Specialists Conference 1996[END_REF][START_REF] Jain | Comparison of np and pn Structures in Indium Phosphide Solar Cells[END_REF]. Table 1 summarizes the designs, the parameters of the emitters and bases, and the reported performances of three InP solar cells in space conditions (AM0, 1.35 kW.m -2 , 25°C). As can be seen, Sharps et al. found that n/p solar cells have heightened open-circuit voltage (Voc) and short-circuit current density (Jsc) regarding the p/n ones [START_REF] Sharps | Conference Record of the Twenty-Fifth IEEE Photovoltaic Specialists Conference 1996[END_REF]. Nevertheless, Jain et al. shed into light that p/n solar cells achieve greater efficiency in comparison with n/p solar cells, owing to more elevated Voc even though the value of Jsc is somewhat lower than for the n/p structure [START_REF] Jain | Comparison of np and pn Structures in Indium Phosphide Solar Cells[END_REF]. Since InP solar cells are promising for space applications, the objective of this paper is twofold: (i) compare the expected performance of n/p and p/n indium phosphide solar cells under space conditions, and (ii) model the effects of 1 MeV electron irradiation on InP solar cells.

Numerical modeling

Numerical simulations of InP solar cells have been carried out by using the finite element method in order to solve Poisson equation (1) coupled with the equations of continuity for both the electrons (2) and the holes (3):

-∇.( ε.∇ψ) = q(p-n+N) (1) dJn/dx = -q(Gopt -RSRH) (2) dJp/dx = q(Gopt -RSRH) ( 3
)
where the three dependent variables are ψ the electrostatic potential, q the elementary charge, n and p the electron and the hole concentrations respectively. Jn and Jp are the electron and the hole conduction current densities. ε is the permittivity and N is the fixed charge associated with ionized donors. In equations ( 2) and (3), RSRH describes the nonradiative recombination mechanisms such as Schockley-Read-Hall, Auger and surface recombinations. The InP cell is exposed to sunlight outside the atmosphere. The AM0 conditions corresponding to the characteristics of the sun radiation level before the passage of sunlight through the atmosphere of the earth with an integral power of 1.35 kW.m -2 have hence been considered along with an optical generation Gopt for the free carriers:

Gopt = (1-R).α(λ).ɸ.exp(-α(λ)x) ( 4
)
where R is the reflectivity of the front contact, α(λ) is the absorption coefficient dependent on the wavelength λ, ɸ is the photon flux and x is the depth in the semiconductor along the propagation direction. The parameters of the materials listed in Table 2 are used in the numerical simulations [START_REF] Weinberg | [END_REF][START_REF] Luo | LEOS 1992 Summer Topical Meeting Digest on Broadband Analog and Digital Optoelectronics, Optical Multiple Access Networks, Integrated Optoelectronics and Smart Pixels[END_REF]. The material properties of the n/p structure are detailed in Table 3. Alike properties are utilized in the case of the p/n configuration except for the back surface recombination velocity (l0 7 cm.s -1 ), the emitter diffusion length (2 µm) and the base diffusion length (5 µm). Values listed in Table 3 are reported from the literature [3,[START_REF] Kilmer | Conference Record of the Twenty-Second IEEE Photovoltaic Specialists Conference 1991[END_REF][START_REF] Sze | Physics of semiconductor devices[END_REF]. The geometries of both the n/p and the p/n junctions with their meshes were realized in the same manner. The example of the n/p structure is depicted in Figure 1. Both the meshes contain 860 nodes and 1592 elements. The minimum quality of the elements is of 0.8308.

Table 3. Parameters of the emitter and the base materials for

the optimal design of a n/p InP solar cell [3,[START_REF] Kilmer | Conference Record of the Twenty-Second IEEE Photovoltaic Specialists Conference 1991[END_REF][START_REF] Sze | Physics of semiconductor devices[END_REF]. Additionally, the solar cells are subjected to 1 MeV electron irradiation which produces six hole traps and five electron traps in the p-type layer of an InP cell, while 16 electron and five hole traps appear in the n-type one [START_REF] Tada | Solar Cell Radiation Handbook[END_REF][START_REF] Augustine | Proc. of Third International Conference Indium Phosphide and Related Materials[END_REF]. The attention is drawn here to dominant traps for which a specific atomic structure has been proposed. Accordingly, H2, H3, H4, H5 and E11 traps which are due to phosphorus atom-electron collisions are accounted for in the present model [START_REF] Tada | Solar Cell Radiation Handbook[END_REF]. Indeed, these centers act both as majority-carrier trap centers and recombination centers, and they also have higher introduction rates than the other centers. The trap levels, the capture cross sections and the introduction rates of these five electron and hole traps are shown in Table 4.

Numerical modeling is used as a means to simulate in 2 dimensions the operation modes of an InP solar cell with both p/n and n/p structures, and this is done with the presence of defects created by electron irradiation. The simulation is performed utilizing the electrostatic and diffusion modules of COMSOL Multiphysics.

Results and discussion

The numerical calculations were carried out to solve equations (1) to (4) in the case of both the p/n and n/p junctions, before irradiation and taking into consideration the modifications engendered by a 1 MeV electron irradiation at 5×10 15 electrons per cm 2 fluence on an InP solar cell. Figure 2 shows the calculated J(V) characteristics for the n/p structure before and after electron irradiation of 1 MeV whereas Figure 3 depicts them for the p/n junction. Table 5 expresses the various cell performance parameters of the two solar cell structures n/p and p/n before and after electron irradiation.

Table 5 clearly discloses that prior to irradiation, the n/p solar cell structure gives higher efficiencies compared to the p/n structure due to a raised short-circuit current density (Jsc). Thus, the usage of a n/p structure appears attractive to reduce the area and the mass of the solar arrays deployed for space power applications. Furthermore, it is found that the solar cell efficiencies are degraded for both the n/p and the p/n structure down to 22.19% and 16.73% respectively. Jsc is more affected by electron irradiation for the p/n structure with a decrease from 39.89 to 31.79 mA/cm 2 against 41.71 to 37.78 mA/cm 2 for the n/p structure. However, the opencircuit voltage Voc is not altered for the n/p solar cell structure while it is a bit more impacted for the p/n one with a lessening from 0.907 to 0.875 V. Besides, the fill factor FF remains in the same magnitude although it gets slightly lower for the p/n structure after electron irradiation.

In the n/p solar cell structure, the n-InP type layer is the emitter. The electron irradiation creates just one electron trap in this layer which could play the role of a doping level. Concerning the p/n structure, the p-InP type layer is the emitter. The electron irradiation creates four hole traps in this layer. The latter traps are considered as recombination centers. Thereby, they could be responsible for the serious degradation of the Jsc and they surely raise the barrier potential at the p/n interface. It is a fact that the open-circuit voltage of a solar cell is related to the barrier built-in voltage. Whereby, the elevation of the barrier potential compensates for the reduction of the Voc by electron irradiation.

For comparison, the J(V) characteristic for both the n/p and p/n solar cell structures after the electron irradiation are plotted in Figure 4. The n/p solar cell structure is more resistant to radiation at 5×10 15 electrons per cm 2 fluence than the p/n solar cell structure as Jsc is maintained at a 20% higher level. Obtained results are fairly in agreement with the findings in references [START_REF] Sharps | Conference Record of the Twenty-Fifth IEEE Photovoltaic Specialists Conference 1996[END_REF][START_REF] Wojtczuk | Proc. of 1994 IEEE 1 st World Conference on Photovoltaic Energy Conversion -WCPEC (A Joint Conference of PVSC, PVSEC and PSEC)[END_REF]. A possible explanation of the discrepancies is the type and the depth of the energy gaps of the defects created by irradiation for each solar cell structure. In the p/n structure, the emitter layer has four defects (H2, H3, H4 and H5), three of them happen to be deep level defects. In the n/p structure, the emitter layer is the n layer having only one defect (E11). The latter deep level defect has no significant influence on the initial J(V) characteristic. This phenomenon can equally be observed in a GaAs based solar cell irradiated by electrons of 1 MeV [START_REF] Mazouz | [END_REF].

Conclusion

Numerical simulation was used to compare the radiation resistance and the performances of n/p and p/n solar cell structures for an Indium Phosphide (InP) solar cell. It was noted that the optimum n/p solar cell structure was more efficacious than the optimum p/n solar cell structure on account of a larger short-current density which is in agreement with the outcomes of reference [START_REF] Sharps | Conference Record of the Twenty-Fifth IEEE Photovoltaic Specialists Conference 1996[END_REF]. The same electron irradiation effects are observed on both the n/p and the p/n structures for the shortcurrent density. The present solar cell modeling not only predicts that the n/p InP solar cell structure is more efficient than the p/n one but also that the former is more radiation resistant. As a perspective for space applications, a modeling study of InP solar cells can be developed in order to deal with a p + /nn + structure having a very thin emitter layer, by reason that a slim emitter enhances the collection of carriers generated by high energy incident photons from the solar spectrum.

Fig. 1 .

 1 Fig. 1. Geometry of the solar n/p junction with its mesh.

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. J(V) characteristics of the n/p solar cell structure before and after 1 MeV electron irradiation.

Table 1 .

 1 Designs and reported performances of InP solar cells[START_REF] Vilela | [END_REF][START_REF] Sharps | Conference Record of the Twenty-Fifth IEEE Photovoltaic Specialists Conference 1996[END_REF][START_REF] Jain | Comparison of np and pn Structures in Indium Phosphide Solar Cells[END_REF].

	Group	Vilela et al. [10]	Sharps et al. [11]	Jain et al. [12]
	Design	p/n	n/p	p/n	n/p	p/n
	Emitter doping (cm -3 )	2×10 18	5×10 16		1×10 18
	Emitter thickness (µm)	0.08	0.05		0.02	0.15
	Base doping (cm -3 )	5×10 17	5×10 16		1×10 17
	Base thickness (µm)	1.5	3		5
	Jsc (mA.cm -2 )	21.5	28.87	28.68	42.37	41.49
	Voc (V)	0.803	0.848	0.814	0.941	0.997
	FF	0.77	0.779	0.814	0.863	0.869
	η (%)	10.5	14.1	13.9	25.1	26.2

Table 2 .

 2 Parameters of the InP materials in the model[START_REF] Weinberg | [END_REF][START_REF] Luo | LEOS 1992 Summer Topical Meeting Digest on Broadband Analog and Digital Optoelectronics, Optical Multiple Access Networks, Integrated Optoelectronics and Smart Pixels[END_REF].

	Parameter	Symbol	Value
	Elementary charge	q	1.602×10 -19 C
	Room temperature	T	300 K
	Boltzmann constant	k	1.38×10 -23 J.K -1
	Gap energy	Eg	1.35 eV
	Relative permittivity	εr	12.6
	Intrinsic concentration	ni	1.2×10 7 cm -3
	Effective density of state in the conduction band	Nc	5.7×10 17 cm -3
	Effective density of state in the valence band	Nv	1.1×10 19 cm -3
	Electron mobility	µn	5900 cm 2 .V -1 .s -1
	Hole mobility	µp	150 cm 2 .V -1 .s -1
	Electron thermal velocity	Vthn	3.9×10 5 m.s -1
	Hole thermal velocity	Vthp	1.7×10 5 m.s -1
	Electron lifetime	τn	0.73 ns
	Hole lifetime	τp	151.5 ns
	Radiative recombination coefficient	τrad	1.2×10 -10 cm 3 .s -1
	Surface recombination coefficient	τs	9×10 -31 cm 6 .s -1

Table 4 .

 4 Electron and hole traps observed by Deep Level Transient Spectroscopy (DLTS) in an InP solar cell irradiated by 1 MeV electrons[START_REF] Tada | Solar Cell Radiation Handbook[END_REF][START_REF] Augustine | Proc. of Third International Conference Indium Phosphide and Related Materials[END_REF].

	Trap designation	Trap level (eV)	Capture cross section σ (cm 2 )	Introduction rate (cm -1 )
		H2	0.22	3×10 -17	0.02
	p-type	H3 H4	0.32 0.37	6×10 -16 8×10 -16	0.82 1.2
		H5	0.52	5.5×10 -15	0.04
	n-type	Ell	0.76	2.9×10 -12	0.2

Table 5 .

 5 InP cell performance parameters before and after 1 MeV electron irradiation at 5×10 15 electrons per cm 2 fluence.

	Output parameter	n/p Before	After	p/n Before	After
	Jsc (mA.cm -2 )	41.71	39.89	37.78	31.79
	Voc (V)	0.891	0.888	0.907	0.875
	FF	0.844	0.845	0.853	0.811
	ɳ (%)	23.24	22.19	21.66	16.73
	0,0

E3S Web of Conferences 191, 01005 (2020) REEE 2020 https://doi.org/10.1051/e3sconf/202019101005

The authors are highly thankful to Mrs. Wilhelmina Logerais, a native English speaker, for proofreading this paper.


Bilel Elbetch
email: elbetchbilal@gmail.com

Keywords: Mathematics Subject Classication: 37N25, 92D25, 34D23, 34D15 Population Dynamics, Migration rate, Logistic growth, Carrying capacity, Asymmetric dispersal, Perfect mixing, Slow-fast systems, Tikhonov's theorem

In this work, we consider the two models: source-source and source-sink, where the two patches are coupled by asymmetrical migration terms, the populations grows logistically on source patch and decay logistically on sink patch. We study the effect of asymmetric dispersal on the dynamic of the total populations in two patches and on the coexistence of species. First, used the theory of singular perturbation and theorem of Tikhonov, in the case of perfect mixing, i.e. when the asymmetric dispersal tends to infinity, we calculate the equilibrium of the two models considered and we give a good approximations of the solutions in this case. Second, for the source-source patch-model, we give the conditions under which fragmentation associated with asymmetric dispersal is either favorable or unfavorable to total population. We show the existence of a new behavior of the total equilibrium population when the asymmetric dispersal varies from zero to infinity and migration rate is fixed. In particular, the existence of two values of the asymmetric dispersal which the total equilibrium population equal to the sum of two capacities. This last situation does not exist in the case where the rate of migration varies. For the second model, we compare the total equilibrium population with the carrying capacity of the source patch. Complete analysis on the models demonstrates a mechanism by which the dispersal asymmetry can lead to either an decreased total size of the species population in two patches, increased for small values of the dispersal asymmetry and decreased for large values, or even extinction in both patches. We end by giving some numerical examples that affirm and illustrate our results. We also give some numerical examples of the total equilibrium population as a function of two variables, the rate of migration and the dispersal asymmetry.

Influence of dispersal intensity and asymmetry on total biomass in two-patch environment There are many factors affecting the growth and the general dynamics of population. One such important factor is the dispersal amounts and in random ways. These dispersal can cause disturbances to the various ecosystems as well as to the persistence or extinction of organisms.

Bibliographies can be found in the work of Allen [START_REF] Allen | Persistence and extinction in Lotka-Volterra reaction-diffusion equations[END_REF][START_REF] Allen | Persistence and extinction in single-species reaction-diffusion models[END_REF][START_REF] Allen | Persistence, extinction, and critical patch number for island populations[END_REF], Hanski [START_REF] Hanski | Metapopulation Ecology[END_REF], Holt [START_REF] Holt | Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution[END_REF]and Levin [START_REF] Levin | Dispersion and population interactions[END_REF][START_REF] Levin | Spatial patterning and the structure of ecological communities[END_REF].

In 1977, Freedman and Waltman [START_REF] Freedman | Mathematical Models of Population Interactions with Dispersal II: Differential Survival in a Change of Habitat[END_REF] considered a two-patch model with a single species in logistic population growth as follows:

       du 1 dt = r 1 u 1 1 - u 1 K 1 + ξ (u 2 -u 1 ), du 2 dt = r 2 u 2 1 - u 2 K 2 + ξ (u 1 -u 2 ), (1.1)
where u i represents the population density in patch i, the parameter r i is the intrinsic growth rate, K i is carrying capacity and ξ is the dispersal rate. Freedman and Waltman show that under certain conditions, the total population abundance can be larger than the total carrying capacities

K 1 + K 2 .
For more details and information on the existence and stability of (1.1), see [START_REF] Freedman | Mathematical Models of Population Interactions with Dispersal I: Stability of two habitats with and without a predator[END_REF]. Holt [START_REF] Holt | Population dynamics in two patch environments: some anomalous consequences of an optimal habitat distribution[END_REF] generalized these results to a source-sink system. In 2015, Arditi et al. [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF] gave a full mathematical analysis of the model (1.1) of Freedman and Waltman with symmetric dispersal.

In 2018, Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] extended the model (1.1) by considering asymmetric dispersal, i.e.

the model:

       du 1 dt = r 1 x 1 1 - u 1 K 1 + ξ (γ 2 u 2 -γ 1 u 1 ), du 2 dt = r 2 u 2 1 - u 2 K 2 + ξ (γ 1 u 1 -γ 2 u 2 ), (1.2)
where ξ γ 2 and ξ γ 1 with ξ γ i > 0, i = 1, 2 and ξ ≥ 0, are the migration terms which describe the flows of individuals from the patch 2 to the patch 1, and from the patch 1 to the patch 2 respectively. These flows can for example depend on the distance between the patches. By noting that the positive equilibrium (u * 1 , u * 2 ) of model (1.2) is the unique positive solution to

       r 1 u 1 1 - u 1 K 1 + r 2 u 2 1 - u 2 K 2 = 0, u 2 = 1 γ 2 γ 1 u 1 - r 1 ξ u 1 1 - u 1 K 1 ,
i.e. the intersection of an ellipse and a parabola. They used a graphical method to completely analyze model (1.2) in order to determine when dispersal is either favorable or unfavorable to total population abundance. In [START_REF] Elbetch | Effect of dispersal in Two-patch environment with Richards growth on population dynamics[END_REF], Elbetch generalizes the results of [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF][START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] to a Two-patch environment with Richards growth. For more details and information on the properties of Verhulst's and Lotka's formulations of the logistic model and the perfect mixing paradox, see the debate in [START_REF] Arditi | The perfect mixing paradox and the logistic equation: Verhulst vs[END_REF][START_REF] Arditi | The perfect mixing paradox[END_REF][START_REF] Ramos-Jiliberto | The perfect mixing paradox and the logistic equation: Verhulst vs[END_REF].

Recently, , Elbetch et al. [START_REF] Elbetch | The multi-patch logistic equation[END_REF][START_REF] Elbetch | The multi-patch logistic equation with asymmetric migration[END_REF] considered the following n patches model:

du i dt = r i u i 1 - u i K i + ξ n ∑ j=1, j =i (γ i j u j -γ ji u i ), i = 1, • • • , n. (1.3)
where n is the number of patches in the system, u i represents the population density in the ith patch. The parameters r i and K i are respectively the intrinsic growth rate and the carrying capacity of patch i. The parameter ξ represents the dispersion rate of the population, γ i j ≥ 0 denote the flux between patches j and i for i = j. We denote Γ the matrix Γ := (γ i j ) n×n with

γ ii = - n ∑ j=1, j =i γ ji . (1.4)
They studied the total population at equilibrium, as a function of the migration rate ξ . They gave conditions on the system parameters that ensure that symmetric and asymmetric migration is beneficial or detrimental for some particular cases.

In 2019, Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] studied the following two-patch source-sink model:

       du 1 dt = r 1 u 1 1 - u 1 K 1 + ξ (u 2 -γu 1 ), du 2 dt = r 2 u 2 -1 - u 2 K 2 + ξ (γu 1 -u 2 ), (1.5)
where u 1 and u 2 represent population densities of the species in patch 1 and 2, respectively.

Patch 1 is assumed to be the source (i.e., r 1 > 0) but patch 2 is the sink (i.e., r 2 > 0). α i = r i /K i the intraspecific competition degree. Parameter ξ represents the dispersal intensity and γ the dispersal asymmetry to the sink. The authors show that the dispersal asymmetry can lead to either an increased total size of the population in two patches, a decreased total size with persistence in the patches, or even extinction in both patches. They show also that for a large growth rate of the species in the source and a fixed dispersal intensity: (i) If the asymmetry is small, the population would persist in both patches and reach a density higher than that without dispersal and the population approaches its maximal density at an appropriate asymmetry, (ii)

If the asymmetry is intermediate, the population persists in both patches but reaches a density less than that without dispersal, (iii) If the asymmetry is large, the population goes to extinction in both patches.

Arino et al. [START_REF] Arino | Number of Source Patches Required for Population Persistence in a Source?Sink Metapopulation with Explicit Movement[END_REF] also studied a source-sink model of n patches, where the source patch follows a logistic growth rate, and the sink patch with exponential decay, i.e the model

     du i dt = r i u i 1 - u i K i + ξ ∑ n j=1, j =i γ i j u j , i = 1, . . . , s, du i dt = -r i u i + ξ ∑ n j=1, j =i γ i j u j , i = s + 1, . . . , n, (1.6)
where u i represent population densities of the specie in the patch i. The parameter r i > 0 is both the growth rate in the case of source patches and the death rate in the case of sinks, K i > 0 is the carrying capacity of source patches. For the model (1.6), the authors proved the existence of a threshold number of source patches such that the population potentially becomes extinct below the threshold and established above the threshold. For general information of the effects of patchiness and migration in both continuous and discrete cases, and the results beyond the logistic model, the reader is referred to the following works and these references: [START_REF] Elbetch | Effect of dispersal in single-species discrete diffusion systems with source-sink patches[END_REF][START_REF] Elbetch | On the effect of density-dependent dispersal on total biomass in multi-patch environment[END_REF] and [START_REF] Elbetch | Nonlinear diffusion in the multi-patch logistic model[END_REF][START_REF] Elbetch | How does nonlinear asymmetric dispersal affect the dynamics of a population and the coexistence of species in Two-patch source-sink heterogeneous environment[END_REF] for effect of linear and nonlinear diffusion on the total biomass respectively, [START_REF] Elbetch | Effects of rapid population growth on total biomass in Multipatch environment[END_REF] for the effects of rapid population growth on total biomass, [START_REF] Elbetch | Generalized logistic equation on Networks[END_REF] for some proprieties of generalized logistic equation in multi-patch and [START_REF] Gao | How does dispersal affect the infection size?[END_REF][START_REF] Gao | Fast diffusion inhibits disease outbreaks[END_REF][START_REF] Guo | Global stability of the endemic equilibrium of multigroup SIR epidemic models[END_REF] for susceptible-infected-susceptible (SIS) patch-model.

Our aim in this work is to study the model of two patches coupled by asymmetric migration terms. In particular, we are interested in studying the effect of dispersal asymmetry on the dynamics of population and coexistence of species for the source-source and also for sourcesink patch-model. The results obtained by Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] and Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF], when they studied the total equilibrium population as function of the migration rate with fixed dispersal asymmetry, motivate us to know the changes and the differences that could be in the case when the dispersal asymmetry varies and the migration rate is fixed. We have answered our question in the theorems 2.5 and 3.6.

This paper is organized as follows: In Section 2, we introduce the model of two patches with source-source dynamic, and we recall some essential results of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] concerning the comparison between the total equilibrium population and the sum of the carrying capacities when the migration rate varies and the dispersal asymmetry is fixed. Next, in Subsection 2.2, we study the behavior of the system (2.1) in the case when the migration rate is fixed and dispersal asymmetry goes to 0. We study also the behavior of the model in the case when the dispersal asymmetry goes to infinity using singular perturbation arguments and Tykhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] ( see Theorem 2.4). Our first main result is stated in Theorem 2.5, which we compare the total equilibrium population with the sum of the two carrying capacities for all parameter space of the model (2.1) when the migration rate ξ is fixed and the dispersal asymmetry γ varies from 0 to ∞, by using the graphical method of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF]. In Section 3, we start by recalling some essential results of Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF], in particular, the global stability, the formula of perfect mixing and the comparison given in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]Proposition 5.11], between the total equilibrium population of (1.5) and carrying capacity of the source patch for fixed asymmetric dispersal γ. In Subsection 3.4, we study the behavior of the system (1.5) in the case when the migration rate ξ is fixed and dispersal asymmetry γ tends to infinity ( see Theorem 3.5). Our second main result is stated in Theorem 3.6, which we compare the total equilibrium population with the carrying capacity for all parameter space of the model (1.5), when the migration rate ξ is fixed and the dispersal asymmetry γ varies from 0 to ∞, by using the graphical method of Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]. In Section 4, some numerical examples which illustrate, confirm and extend our results are given. Section 5 concludes the paper and describes the future works suggestion.

2 Two-patch source-source model we consider the two-patch source-source model (1.2). If we denote by γ = γ 1 γ 2 , then the system (1.2) can rewritten as:

       du 1 dt = r 1 u 1 1 - u 1 K 1 + ξ (u 2 -γu 1 ) , du 2 dt = r 2 u 2 1 - u 2 K 2 + ξ (γu 1 -u 2 ) .
(2.1)

The system (2.1) admits unique equilibrium which is globally asymptotically stable (GAS), in the interior of positive cone. In all of this section, this positive equilibrium point is denoted by: E * (ξ , γ) = (u * 1 (ξ , γ), u * 2 (ξ , γ)) and the sum of

u * i (ξ , γ) for i = 1, 2, is denoted by U * T (ξ , γ)
for all ξ ≥ 0 and γ > 0. We recall some essential results of Arditi et al. [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF][START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] as function of the parameter γ and ξ . Note that, Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF], studied the model (2.1) as function of the parameters γ 1 and γ 2 and in [START_REF] Arditi | Is dispersal always beneficial to carrying capacity? New insights from the multi-patch logistic equation[END_REF], They studied (2.1) for the symmetric case γ 1 = γ 2 = 1. First of all, let's start by recalling the comparison between total equilibrium population and the sum of the carrying capacities K 1 + K 2 when the migration rate ξ varies and the dispersal asymmetry γ is fixed.

Total equilibrium population for fixed γ

In all of this section, we assume that γ is positive and fixed parameter and ξ varies from zero to infinity. The derivative of U * T (ξ , γ) with respect to ξ at ξ = 0 is given by:

dU * T dξ (0, γ) = (γK 2 -K 1 ) 1 r 1 - 1 r 2 . (2.2)
The perfect mixing formula of the model (2.1) (i.e when ξ → ∞) is given by:

U * T (+∞, γ) = (1 + γ) γr 1 + r 2 γ 2 α 1 + α 2 , (2.3)
where α i = r i /K i . We consider the regions in the set of the parameter γ denoted J 0 , J 1 and J 2 , defined by:

                               If r 2 > r 1 then            J 1 = γ : γ > α 2 α 1 J 0 = γ : α 2 α 1 ≥ γ > K 1 K 2 J 2 = γ : K 1 K 2 > γ If r 2 < r 1 then            J 1 = γ : γ < α 2 α 1 J 0 = γ : α 2 α 1 ≤ γ < K 1 K 2 J 2 = γ : K 1 K 2 < γ . (2.4)
We recall the following result of Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF], which gives the conditions for which patchiness is beneficial or detrimental in model (2.1).

Theorem 2.1. Assume that γ is fixed. The total equilibrium population of (2.1) satisfies the following properties

1. If r 1 = r 2 then U * T (ξ , γ) ≤ K 1 + K 2 for all ξ ≥ 0.
2. If r 1 = r 2 , let J 0 , J 1 and J 2 , be defined by (2.4). Then we have:

• if γ ∈ J 0 then U * T (ξ , γ) > K 1 + K 2 for any ξ > 0 • if γ ∈ J 1 then U * T (ξ , γ) > K 1 + K 2 for 0 < ξ < ξ 0 and U * T (ξ , γ) < K 1 + K 2 for ξ > ξ 0 , where ξ 0 = r 2 -r 1 γ α 2 -1 α 1 1 α 1 + α 2 . • if γ ∈ J 2 then U * T (ξ , γ) < K 1 + K 2 for any ξ > 0 • If γ = K 1 K 2 , then u * 1 (ξ , γ) = K 1 and u * 2 (ξ , γ) = K 2 for all ξ ≥ 0. Therefore U * T (ξ , γ) = K 1 + K 2 for all ξ ≥ 0.
The results proved by Arditi et al. [START_REF] Arditi | Asymmetric dispersal in the multi-patch logistic equation[END_REF] show that, there are only three cases that can occur:

the case where the total equilibrium population is always greater than the sum of carrying capacities, the case where it is always smaller, and a third case, the total equilibrium population is greater than the sum of carrying capacities for lower values of the migration rate ξ and smaller for the higher values.

Total equilibrium population for fixed ξ

In the remainder of this section, we assume that ξ is fixed, and γ varies from 0 to ∞. Our goal is to study the total equilibrium population as function of the asymmetry dispersal γ. First, let's we start by the following case:

2.2.1
The model when γ → 0

We have the following result Proposition 2.2. Consider the system (2.1). Then,

lim γ→0 E * (ξ , γ) =    (K 1 , 0), i f ξ ≥ r 2 , 1 2 K 1 + 1 2α 1 r 2 1 + 4ξ α 1 1 -ξ r 2 K 2 , 1 -ξ r 2 K 2 i f ξ < r 2 .
(2.5)

Proof. Denote E * (ξ , 0 + ) = (u * 1 (ξ , 0 + ), u * 2 (ξ , 0 + )) := lim γ→0 E * (ξ , γ).
When γ → 0, the equilibrium equations of (2.1) take the following form:

       0 = r 1 u * 1 (ξ , 0 + ) 1 - u * 1 (ξ , 0 + ) K 1 + ξ u * 2 (ξ , 0 + ), 0 = r 2 u * 2 (ξ , 0 + ) 1 - u * 2 (ξ , 0 + ) K 2 -ξ u * 2 (ξ , 0 + ), (2.6)
which implies

-α 1 (u * 1 (ξ , 0 + )) 2 + ξ u * 2 (ξ , 0 + ) + r 1 u * 1 (ξ , 0 + ) = 0, (r 2 -ξ )u * 2 (ξ , 0 + ) -α 2 (u * 2 (ξ , 0 + )) 2 = 0. (2.7)
If ξ ≥ r 2 , then the system (2.7) admits (0, 0) and (K 1 , 0) as solutions. Since (0, 0) is unstable for (2.1), then E * (ξ , γ) → (K 1 , 0) as γ → 0. If ξ < r 2 , the second equation in (2.7) gives

u * 2 (ξ , 0 + ) = 0 or u * 2 (ξ , 0 + ) = r 2 -ξ α 2 .
If we replace u * 2 (ξ , 0 + ) = 0 in the first equation of (2.7) we get u * 1 (ξ , 0 + ) = 0 or u * 1 (ξ , 0 + ) = K 1 , and if we replace u * 2 (ξ , 0 + ) = r 2 -ξ α 2 in the first equation of (2.7) we obtain the following equation:

-α 1 (u * 1 (ξ , 0 + )) 2 + r 1 u * 1 (ξ , 0 + ) + ξ (r 2 -ξ ) α 2 = 0, (2.8)
which admits as positive solution

u * 1 (ξ , 0 + ) = 1 2 K 1 + 1 2α 1 r 2 1 + 4ξ α 1 1 - ξ r 2 K 2 .
Therefore, if r 2 > ξ , then the system (2.7) admits three solutions: (0, 0), (K 1 , 0) and E * (ξ , 0 + )

given by the second equation of (2.5). Since, (0, 0), and (K 1 , 0) are unstable, so E * (ξ , γ) converge to E * (ξ , 0 + ) as γ → 0.

As corollary of the previous proposition we obtain the following result which describe the total equilibrium population U * T (ξ , γ) when γ → 0.

Corollary 2.3. we have:

lim γ→0 U * T (ξ , γ) := U * T (ξ , 0 + ) =    K 1 , i f ξ ≥ r 2 , 1 -ξ r 2 K 2 + 1 2 K 1 + 1 2α 1 r 2 1 + 4ξ α 1 1 -ξ r 2 K 2 , i f ξ < r 2 .
(2.9)

In the next theorem, we give the behavior of the model (2.1) when γ → ∞.

Theorem 2.4. Let (u 1 (t, γ), u 2 (t, γ)) be the solution of the system (2.1) with initial condition

(u 0 1 , u 0 2 ) satisfying u 0 i ≥ 0 for i = 1, 2. Let z(t) be the solution of the differential equation dX dt = r 2 X 1 - X K 2 , (2.10)
with initial condition z(0) = u 0 1 + u 0 1 . Then, when γ → ∞, we have

u 1 (t, γ) + u 2 (t, γ) = z(t) + o γ (1), uniformly for t ∈ [0, +∞) (2.11)
and, for any t 0 > 0, we have

u 1 (t, γ) = o γ (1), u 2 (t, γ) = z(t) + o γ (1), uniformly for t ∈ [t 0 , +∞).
(2.12)

Proof. Let X(t, γ) = u 1 (t, γ) + u 2 (t, γ)
. We rewrite the system (2.1) using the variables (X, u 2 ).

One obtains:

       dX dt = r 2 u 2 1 - u 2 K 2 + r 1 (X -u 2 ) 1 - X -u 2 K 1 , du 2 dt = r 2 u 2 1 - u 2 K 2 + ξ (γ(X -u 2 ) -u 2 ) . (2.13)
When γ → ∞, (2.13) is a slow-fast system, with one slow variable, X, and one fast variable u 2 .

According to Tikhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] we consider the dynamics of the fast variable in the time scale τ = γt. One obtains

du 2 dτ = 1 γ r 2 u 2 1 - u 2 K 2 + ξ (X -u 2 ) - u 2 γ .
In the limit γ → ∞, we find the fast dynamics

du 2 dτ = ξ (X -u 2 ). (2.14)
The slow manifold is formed by the equilibrium points of the fast equation (2.14), which given by u * 2 = X. Since u * 2 is locally asymptotically stable (LAS) for the system (2.14), the theorem of Tikhonov ensures that after a fast transition toward the slow manifold, the solutions of (2.13) are approximated by the solutions of the reduced model which is obtained by replacing u * 1 into the dynamics of the slow variable, which gives the equation (2.10). The system (2.10) admits X * = K 2 as a positive equilibrium point, which is LAS in the positive axis. The approximation given by Tikhonov's theorem holds for all t ≥ 0 for the slow variable and for all t ≥ t 0 > 0 for the fast variable, where t 0 is small as we want. Therefore, let z(t) be the solution of the reduced model (2.10) of initial condition z(0) = X(0, γ) = u 0 1 + u 0 2 , then, when γ → ∞, we have the approximations (2.11) and (2.12).

According to previous theorem, when γ → +∞, we have E * (ξ , ∞) := (0, K 2 ) and U * T (ξ , +∞) = K 2 for all ξ .

2.2.3

The model when γ varies from 0 to ∞ In this section, Our aim is to compare the total equilibrium population U * T (ξ , γ) of (2.1) with the sum of carrying capacities K 1 + K 2 , when the dispersal asymmetry γ varies from zero to infinity, by analyzing the stable positive equilibrium E * (ξ , γ). We study the effect of asymmetric dispersal on the total equilibrium population for the two-patch (2.1) by describing the position affects the equilibrium E * (ξ , γ) when the asymmetric dispersal varies from zero to infinity, we will give the condition whether U * T (ξ , γ) is greater or smaller than the sum of carrying capacities.

We state our first main result as follows:

Theorem 2.5. Consider the model (2.1) with ξ fixed. Denote

γ * = K 2 K 1 and γ * * = α 1 α 2 + α 1 (r 1 -r 2 ) ξ (α 1 +α 2 ) .
We have:

1. If r 2 ≤ ξ , then (a) If r 1 = r 2 , then U * T (ξ , γ) ≤ K 1 + K 2 for all γ > 0. Moreover, U * T (ξ , γ) = K 1 + K 2 if and only if, γ = γ * . (b) If r 1 = r 2 then, U * T (ξ , γ) = < K 1 + K 2 , i f γ ∈]0, min {γ * , γ * * }] ∪ [max {γ * , γ * * }, +∞[, > K 1 + K 2 , i f γ ∈ [min {γ * , γ * * } , max {γ * , γ * * }].
(2.15)

2. If r 2 > ξ , then (a) If r 1 = r 2 , then U * T (ξ , γ) ≤ K 1 + K 2 for all γ > 0. Moreover, U * T (ξ , γ) = K 1 + K 2 if and only if, γ = γ * . (b) If r 1 > r 2 then, U * T (ξ , γ) = < K 1 + K 2 , i f γ ∈]0, γ * ] ∪ [γ * * , +∞[, > K 1 + K 2 , i f γ ∈ [γ * , γ * * ].
(2.16)

(c) If r 1 < r 2 , then i. if 1 2 K 1 + 1 2α 1 r 2 1 + 4ξ α 1 1 -ξ r 2 K 2 < α 2 K 1 +K 2 α 1 +α 2 , then U * T (ξ , 0 + ) < K 1 + K 2 and U * T (ξ , γ) = < K 1 + K 2 , i f γ ∈]0, γ * ] ∪ [γ * * , +∞[, > K 1 + K 2 , i f γ ∈ [γ * , γ * * ]. (2.17) ii. if 1 2 K 1 + 1 2α 1 r 2 1 + 4ξ α 1 1 -ξ r 2 K 2 = α 2 K 1 +K 2 α 1 +α 2 , then U * T (ξ , 0 + ) = K 1 + K 2 and U * T (ξ , γ) = > K 1 + K 2 , i f γ ∈]0, γ * ], < K 1 + K 2 , i f γ ∈ [γ * , ∞[. (2.18) iii. if 1 2 K 1 + 1 2α 1 r 2 1 + 4ξ α 1 1 -ξ r 2 K 2 > α 2 K 1 +K 2 α 1 +α 2 , then U * T (ξ , 0 + ) > K 1 + K 2 and U * T (ξ , γ) = > K 1 + K 2 , i f γ ∈]0, γ * ], < K 1 + K 2 , i f γ ∈ [γ * , ∞[. (2.19)
Proof. The equilibrium of the system (2.1) is the solutions of the following algebraic system:

       0 = r 1 u 1 1 - u 1 K 1 + ξ (u 2 -γu 1 ), 0 = r 2 u 2 1 - u 2 K 2 + ξ (γu 1 -u 2 ).
(2.20)

The sum of the two equations of (2.20) shows that the equilibrium points are in a ellipse noted E 1 , which its equation is given by:

E 1 : ψ(u 1 , u 2 ) := r 1 u 1 1 - u 1 K 1 + r 2 u 2 1 - u 2 K 2 = 0. (2.21)
The ellipse E 1 passes through the points (0, 0), (K 1 , 0), (0, K 2 ) and A := (K 1 , K 2 ). Note that, it is independent of dispersal intensity ξ and asymmetric dispersal γ (shown in red in Fig. 1 and in others figures).

When γ → 0, the equilibrium E * (ξ , γ) tend to A +0 := (K 1 , 0) if r 2 ≤ ξ , and tend to

A ++ := 1 2 K 1 + 1 2α 1 r 2 1 + 4ξ α 1 1 -ξ r 2 K 2 , 1 -ξ r 2 K 2 if ξ < r 2 .
In the case when γ → ∞,

the equilibrium E * (ξ , γ) tend to A ∞ := (0, K 2 ).
To facilitate comparison between the total equilibrium population U * T (ξ , γ) and

K 1 + K 2 ,
we consider the line ∆ with Cartesian equation u 1 + u 2 = K 1 + K 2 , of slope -1 and passing through the point A = (K 1 , K 2 ) (shown in blue in Fig. 1 and in others figures). If the equilibrium

(u * 1 (γ, ξ ), u * 2 (γ, ξ )), is on or below the line ∆, then U * T (γ, ξ ) ≤ K 1 + K 2 , whereas if the equilib- rium is above the line ∆, then U * T (γ, ξ ) ≥ K 1 + K 2 .
We see very simply that dispersal asymmetry is favorable to the total equilibrium population if E * (γ, ξ ) is above ∆, unfavorable if below ∆.

The equilibrium point E * (γ, ξ ) is always in ellipse, then, for γ → 0, the equilibrium point start at A +0 = (K 1 , 0) if r 2 ≤ ξ or at A ++ if r 2 > ξ , and when γ increases, E * (γ, ξ ) describes an arc of the ellipse and ends at point A ∞ = (0, K 2 ). Suppose that r 2 ≤ ξ , so, the equilibrium E * (γ, ξ ) start at A +0 . Let we prove the items (a) and (b) of the item 1.

1. The equation of the tangent line to the ellipse E 1 at the point A is given by:

(u 1 -K 1 ) ∂ ψ ∂ u 1 (A) + (u 2 -K 2 ) ∂ ψ ∂ u 2 (A) = 0, (2.22)
where the function ψ is given by the equation (2.21). Since

∂ ψ ∂ u 1 (A) = -r 1 and ∂ ψ ∂ u 2 (A) = -r 2 , Equation (2.22) becomes simply r 1 u 1 + r 2 u 2 = r 1 K 1 + r 2 K 2 .
(2.23)

If r 1 = r 2 in the equation (2.23), the tangent space to the the ellipse E 1 at A is the line ∆.

By the concavity of the ellipse, any point of ellipse lies below the line ∆. The equilibrium point E * (γ, ξ ) is always in ellipse, then, when γ increases, E * (γ, ξ ) describes an arc of the ellipse, start at A +0 and ends at point

A ∞ . Therefore E * (γ, ξ ) satisfies u * 1 (γ, ξ ) + u * 2 (γ, ξ ) ≤ K 1 + K 2 ,
for all γ > 0 ( see figure 1). At the point A, if we replace u * 1 (γ, ξ ) = K 1 and u * 2 (γ, ξ ) = K 2 in the system (2.20), we get γ * = K 2 K 1 , which completes the proof of item 1-(a).

2. We suppose now that r 1 = r 2 , then the line ∆ makes a second intersection with the ellipse

E 1 at a point noted C = α 2 K 1 +K 2 α 1 +α 2 , α 1 K 1 +K 2 α 1 +α 2 .
Therefore, we have two possibles cases according to the relative positions of the point C and the point A. In the case when Suppose that r 2 > ξ , so, the equilibrium E * (γ, ξ ) start at A ++ . Let we prove the items (a), (b) and (c) of the item 2. For the prove of items (a) and (b) is the same as in the previous items 1-a and 1-b, except, the equilibrium E * (ξ , γ) start at A ++ in stead of A +0 because r 2 > ξ (see In this section, we consider the two-patch system with source-sink dynamics given by (1.5).

r 1 < r 2 , O u 1 A +0 A ∞ u 2 ∆ A •
O u 1 A +0 A ∞ u 2 C ∆ • A • O u 1 A +0 A ∞ u 2 ∆ C • A •
figures 3). O u 1 A ++ A ∞ u 2 ∆ • A • O u 1 C • A ∞ u 2 ∆ A ++ • A •
This system is studied in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]. We recall some essential results of [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]. First of all, let's start by recalling the global dynamics of System (1.5).

Global dynamics

We consider the following regions denoted D 0 , D 1 and D 2 defined by:

               D 0 = γ : γ ≥ r 1 r 2 , D 1 = γ : r 1 r 2 < γ < r 1 r 2 + ξ r 1 r 2 ξ , D 2 = γ : γ ≥ r 1 r 2 + ξ r 1 r 2 ξ . (3.1)
The global dynamic of System (1.5) is described as follows.

Theorem 3.1 (Prop. 5.5 in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]). Consider Model (1.5).

Then, if γ ∈ D 0 ∪ D 1 , System (1.5)
admits unique equilibrium in R 2 \{0} denoted E (γ, ξ ), which is GAS, and if γ ∈ D 2 , then the origin is GAS.

Total population abundance for fixed γ

In this section, we recall the comparison given in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]Proposition 5.11], between the total equilibrium population

U T (γ, ξ ) = u 1 (γ, ξ ) + u 2 (γ, ξ ), E (γ, ξ ) = (u 1 (γ, ξ ), u 2 (γ, ξ )),
of (1.5) and carrying capacity K 1 , by analyzing the stable positive equilibrium E (γ, ξ ). Note that, when there is no dispersal (i.e., ξ = 0), the total equilibrium population is U T (γ, 0) = K 1 .

We consider the regions defined by:

                           If r 2 ≥ r 1 then      L 0 = γ : γ ≥ r 1 r 2 , L 1 = γ : γ < r 1 r 2 . If r 2 < r 1 then            L 2 = γ : γ ≥ r 1 r 2 , L 3 = γ : r 1 r 2 > γ > r 1 -r 2 α 2 (K 1 +K 2 ) , L 4 = γ : γ ≤ r 1 -r 2 α 2 (K 1 +K 2 ) . (3.2)
Theorem 3.2. The total equilibrium population described by (1.5) satisfies the following properties 1. If r 2 ≥ r 1 , let L 0 and L 1 be defined by (3.2). Then we have:

• if γ ∈ L 0 then U T (γ, ξ ) ≤ K 1 for all ξ ≥ 0.
More over, there is ξ * = r 1 r 2 γr 2 -r 1 , such that:

0 < U T (γ, ξ ) ≤ K 1 If ξ < ξ * , U T (γ, ξ ) = 0 If ξ ≥ ξ * . (3.3)
• if γ ∈ L 1 then 0 < U T (γ, ξ ) ≤ K 1 for all ξ ≥ 0.
2. If r 2 < r 1 , let L 2 , L 3 and L 4 be defined by (3.2). Then we have:

(a) if γ ∈ L 2 then U T (γ, ξ ) > K 1 for ξ < ξ 0 and U T (γ, ξ ) < K 1 for all ξ > ξ 0 , where

ξ 0 = (r 1 -r 2 ) (K 1 + K 2 ) ((r 1 -r 2 ) + γα 1 (K 1 + K 2 )) (α 1 -1 + α 2 -1 )
, with α i = r i /K i .

(3.4)

Moreover, there is ξ * ≥ ξ 0 such that U T (γ, ξ ) = 0 for all ξ ≥ ξ * . (b) if γ ∈ L 3 then we have U T (γ, ξ ) ≥ K 1 If ξ ≤ ξ * , 0 < U T (γ, ξ ) < K 1 If ξ > ξ * . (3.5) (c) if γ ∈ L 4 , then U T (γ, ξ ) ≥ K 1 for all ξ ≥ 0.
Proof. All the results were established by Wu at al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF]Proposition 5.11]. Note that, the explicit expression (3.4) of ξ 0 was not given in [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF].

Perfect mixing

In the case of perfect mixing (i.e when ξ → ∞), we have the following result [32, Proposition 5.10]:

Proposition 3.3. We have:

U T (γ, ∞) =    (1 + γ) r 1 -γr 2 r 1 /K 1 + γ 2 r 2 /K 2 i f γ < r 1 /r 2 , 0 otherwise. (3.6)
Wu et al. [START_REF] Wu | Dispersal asymmetry in a two-patch system with source-sink populations[END_REF] proved that large dispersal intensity (i.e., ξ → ∞), the intermediate asymmetry γ can lead to population density higher than that without dispersal, and extremely small asymmetry is still favorable, while extremely large asymmetry is unfavorable: (i) When the dispersal asymmetry is small, the species can approach a density larger than that without dispersal, while it reaches its maximum value at an intermediate asymmetry γ = r 1 -r 2 2α 2 (K 1 +K 2 ) . (ii) When γ is extremely large, the species goes to extinction in both patches. Mathematically speaking, we can rewrite the following result [32, Proposition 5.10]:

Proposition 3.4. [32, Proposition 5.10] Assume that γ < r 1 r 2 . Consider the total equilibrium population for ξ → ∞ given by (3.6). We have:

U T (γ, +∞)        > K 1 i f γ < r 1 -r 2 α 2 (K 1 +K 2 ) , = K 1 i f γ = r 1 -r 2 α 2 (K 1 +K 2 ) , < K 1 i f γ > r 1 -r 2 α 2 (K 1 +K 2 ) . (3.7)
Moreover, U T (+∞) approaches its maximum value r 2 K 1

K 1 +K 2 4(r 1 K 2 +γ 2 r 2 K 1 ) r 1 -r 2 α 2 (K 1 +K 2 ) 2 at γ = r 1 -r 2 2α 2 (K 1 +K 2 ) .

Total population equilibrium for fixed ξ

In this section, we assume that ξ is fixed. Note that, when γ → 0, the equilibrium E (ξ , γ) tend to (K 1 , 0).

The equilibrium point start at point A +0 and when γ increases from 0 to ∞, E (ξ , γ) moves along the ellipse E 1 ,passes through the point C for a γ and ends at the point A ∞ = O. To find the value of γ , just substitute the coordinates of C in one of the equations of the system (3.15).

The value of γ * deduced from Theorem 3.1, which completes the proof ( see figure 5).

O u 1 u 2 E 1 Case r 1 ≤ r 2 ∆ O u 1 E 1 C Case r 1 > r 2 • u 2 ∆ Figure 5:
The illustration of items 1 and 2 of Theorem 3.6. The ellipse E 1 is shown in red for some values of parameters and the straight line ∆ in blue. Left: case where r 1 ≤ r 2 and in the right case where r 1 > r 2 . The equilibrium E (ξ , γ) moves clockwise from (K 1 , 0) to O with increasing γ from 0 to ∞.

Speaking biologically, the result of the previous theorem show that, (i) the small values of asymmetric dispersal γ can lead to either an increased total size of the population in two patches, and the large values of γ can lead a decreased total size, with extinction in both patches, (ii) asymmetric dispersal can lead a decreased total equilibrium population, with extinction in both patches.

Numerical illustrations

In this section, first, we plot, for fixed ξ , the total equilibrium populations γ → U * T (ξ , γ) of (2.1) and γ → U T (ξ , γ) of (1.5), as a function of dispersal asymmetry γ for the sets of parameters choosing in Tables 1, 2 and 3. We give some examples of all the cases proven in Theorems 2.5 and 3.6.

In the figures 13 and 14, we plot, for some parameters of the model (2.1), the total equilibrium population (ξ , γ) → U * T (ξ , γ) ( surface shown in red) when the dispersal asymmetry γ and dispersal intensity ξ varies from 0 to ∞. In figure 15, we plot the total equilibrium population (ξ , γ) → U T (ξ , γ) of the model (1.5) for the parameters given in Table 3. Note that, the total equilibrium population U * T satisfies:

U * T (ξ , γ) = K 1 + K 2 + ξ (u * 2 (ξ , γ) -γu * 1 (ξ , γ)) 1 α 1 u * 1 (ξ , γ) - 1 α 2 u * 2 (ξ , γ) , (4.1)
and U T :

U T (ξ , γ) = K 1 + ξ (u 2 (ξ , γ) -γu 1 (ξ , γ)) 1 α 1 u 1 (ξ , γ) - 1 r 2 + α 2 u 2 (ξ , γ) . (4.2)
Table 1: Parameters values of the three case used in Fig. 6, 7 and 8 for the case r 2 ≤ ξ in Theorem 2.5.

Figure Case

r 1 K 1 r 2 K 2 ξ U * T (ξ , 0 + ) U * T (ξ , ∞) Case A 1 1 1 3 2 1 3 Fig. 6 Case B 1 2 1 2 2 2 2 Case C 1 3 1 2 2 3 2
Case A 1 1 0. 6: Illustration of items 1 and 2 of Theorem 2.5. Total equilibrium population U * T (shown in red) of system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table 1. The horizontal line shown in blue is K 1 + K 2 and in green is K 2 . For the three cases, the small values of asymmetric dispersal γ ≤ 3 can lead to either an increased total size of the population in two patches, and the large values of γ > 3 can lead a decreased total size. The point [START_REF] Allen | Persistence, extinction, and critical patch number for island populations[END_REF][START_REF] Arditi | The perfect mixing paradox and the logistic equation: Verhulst vs[END_REF] represents that the total equilibrium population U * T equal the sum of carrying capacities K 1 + K 2 = 4 at asymmetric dispersal γ = 3. The dispersal asymmetry is unfavorable for all γ > 0.

γ γ γ U * T U * T U * T Case A: K 1 < K 2 Case B: K 1 = K 2 Case C: K 1 > K 2 Figure
Case A: T (shown in red) of system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table 1. The horizontal line shown in blue is K 1 + K 2 and in green is K 2 . For the three cases, the small values of asymmetric dispersal γ can lead to either an increased total equilibrium population U * T in two patches, and the large values of γ can lead a decreased U * T . There are two points γ 1 and γ 2 of the asymmetric dispersal, which the total equilibrium population U * T equal the sum of carrying capacities K 1 + K 2 = 4. The dispersal asymmetry is unfavorable for all γ < γ 1 and γ > γ 2 , while is favorable when γ 8: Illustration of item 1-b of Theorem 2.5 with r 1 < r 2 . Total equilibrium population U * T (shown in red) of system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table 1. The horizontal line shown in blue is K 1 + K2 and in green is K 2 . There are two points γ 1 and γ 2 of the asymmetric dispersal, which the total equilibrium population U * T equal the sum of carrying capacities K 1 + K 2 = 4. The dispersal asymmetry is unfavorable for all γ < γ 1 and γ > γ 2 , while is favorable when γ

K 1 < K 2 Case B: K 1 = K 2 Case C: K 1 > K 2 γ γ γ U * T U * T U * T
1 ≤ γ ≤ γ 2 . γ γ γ U * T U * T U * T Case A: K 1 < K 2 Case B: K 1 = K 2 Case C: K 1 > K 2 Figure
1 ≤ γ ≤ γ 2 .
Table 2: Parameters values of the three case used in Fig. 9, 10 and 11 for the case r 2 > ξ in Theorem 2.5. 9: Illustration of item 2-a of Theorem 2.5. Total equilibrium population U * T (shown in red) of system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table 2. The horizontal line shown in blue is K 1 + K2 and in green is K 2 . Table 3: Parameters values of the two cases used in Fig. 12. Case A: T (shown in red) system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table 2. The horizontal line shown in blue is

r 1 K 1 r 2 K 2 ξ U * T (ξ , 0 + ) U * T (ξ , ∞) Case A 1 1 5 3 2
γ γ γ U * T U * T U * T Case A: K 1 < K 2 Case B: K 1 = K 2 Case C: K 1 > K 2 Figure

Figure

r 1 K 1 r 2 K 2 ξ U * T (ξ , 0 + ) U * T (ξ , ∞)
K 1 < K 2 Case B: K 1 = K 2 Case C: K 1 > K 2 γ γ γ U * T U * T U * T
K 1 + K2 and in green is K 2 . γ γ γ U * T U * T U * T Case (i)
Case (ii) Case (iii) Figure 11: Illustration of item 2-c of Theorem 2.5. Total equilibrium population U * T (shown in red) of system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table 2. The horizontal line shown in blue is K 1 + K 2 and in green is K 2 . For the cases (i) and (ii), the small values of asymmetric dispersal γ can lead to either an increased total equilibrium population U * T in two patches, and the large values of γ can lead a decreased U * T . In the case (iii), the total equilibrium population decreased for all γ. γ γ

U T U T

Case A

Case B Figure 12: Illustration of items 1 and 2 of Theorem 3.6. Total equilibrium population U T (shown in red) of the system (1.5) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table 3. The horizontal line shown in blue is K 1 . Case A, the dispersal asymmetry is unfavorable for all γ < γ 2.11 and U T = 0 for all γ ≥ 2.11. Case B, the point (3.14, 1) represents that the total equilibrium population U T equal the carrying capacity K 1 = 1 at asymmetric dispersal γ = 3.14. Therefore, the dispersal asymmetry is favorable as γ < γ * 3.11, while it is unfavorable as γ ≥ 3.11 and U T = 0 for all γ ≥ γ 7.11.

Case A Case B Case C Figure 13: Some example of Total equilibrium population U * T (ξ , γ) ( surface shown in red) of system (2.1) as a function of the dispersal asymmetry γ and dispersal intensity ξ . The horizontal plane shown in blue is K 1 +K 2 and in green is K 2 . Case A plotted for r 1 = 1, K 1 = 1, r 2 = 1, K 2 = 3, case B: r 1 = 1, K 1 = 1, r 2 = 0.5, K 2 = 3 and case C: r 1 = 0.5, K 1 = 3, r 2 = 1, K 2 = 1.

Case (i)

Case (ii) Figure 14: Total equilibrium population U * T of the system (2.1) as a function of the dispersal asymmetry γ and dispersal intensity ξ . The horizontal plane shown in blue is K 1 + K 2 and in green is K 2 . Case (i) plotted for the parameters r 1 = 7, r 2 = 4, K 1 = 1, K 2 = 3 and case (ii) for r 1 = 3, r 2 = 7.5, K 1 = 1, K 2 = 3.

Case A

Case B Figure 15: Total equilibrium population U T (ξ , γ) (shown in red) of the system (1.5) as a function of the dispersal asymmetry γ and dispersal intensity ξ for the sets of the parameter values given in Table 3. The horizontal plane shown in blue is K 1 .

The goal of this paper is to study the model of two patches coupled by asymmetric migration terms. In particular, we are interested in studying the effect of dispersal asymmetry on the dynamics of population and coexistence of species for the two-patch source-source and also for two-patch source-sink model. The ecological problem that has motivated this study is to find the conditions for which asymmetrical dispersal can lead to higher total equilibrium population than the sum of carrying capacities.

Rigorous analysis on the models demonstrates a mechanism by which the dispersal asymmetry can lead to either an increased total size of the species in two patches, a decreased total size with persistence in the patches, or even extinction in both patches. The analysis also provides new insights. Numerical simulations confirm and extend our results.

Finally, some question remains open: how do our results generalize to situations with more than two patches? If we consider a more general growth dynamic than logistic, this has an effect on the total equilibrium population. I think these questions are difficult to answer, and require a lot of work and mathematical tools.

3. 1

 1 Global dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 Total population abundance for fixed γ . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Perfect mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.4 Total population equilibrium for fixed ξ . . . . . . . . . . . . . . . . . . . . . 13 3.4.1 The model when γ → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4.2 The model when γ varies from 0 to ∞ . . . . . . . . . . . . . . . . . . 15

Figure 1 :

 1 Figure 1: The illustration of item 1-(a) of Theorem 2.5. The ellipse E 1 is shown in red for some values of parameters and the straight line ∆ in blue.The equilibrium E * (ξ , γ) moves clockwise from A +0 to A ∞ with increasing γ. The total equilibrium population is always smaller than K 1 + K 2 for all γ, because it belongs to the ellipse.

Figure 2 :

 2 Figure 2: The illustration of item 1-(b) of Theorem 2.5. The ellipse E 1 is shown in red for some values of parameters and the straight line ∆ in blue. Left: case where r 1 > r 2 and in the right case where r 1 < r 2 . The equilibrium E * (ξ , γ) moves clockwise from A +0 to A ∞ with increasing γ.

Figure 3 :

 3 Figure 3: The illustration of items 2-(a) and 2-(b) of Theorem 2.5. The ellipse E 1 is shown in red for some values of parameters and the straight line ∆ in blue. Left: case where r 1 = r 2 and in the right case where r 1 > r 2 . The equilibrium E * (ξ , γ) moves clockwise from A ++ to A ∞ with increasing γ.

2 Figure 4 :

 24 Figure 4: The illustration of item 2-(c) of Theorem 2.5 where r 1 < r 2 . The ellipse E 1 is shown in red for some values of parameters and the straight line ∆ in blue. The equilibrium E * (ξ , γ) moves clockwise from A ++ to A ∞ with increasing γ.

Figure 7 :

 7 Figure 7: Illustration of item 1-b of Theorem 2.5 with r 1 > r 2 . Total equilibrium population U *T (shown in red) of system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table1. The horizontal line shown in blue is K 1 + K 2 and in green is K 2 . For the three cases, the small values of asymmetric dispersal γ can lead to either an increased total equilibrium population U * T in two patches, and the large values of γ can lead a decreased U * T . There are two points γ 1 and γ 2 of the asymmetric dispersal, which the total equilibrium population U * T equal the sum of carrying capacities K 1 + K 2 = 4. The dispersal asymmetry is unfavorable for all γ < γ 1 and γ > γ 2 , while is favorable when γ 1 ≤ γ ≤ γ 2 .

Fig. 12

 12 Fig. 12-A 3 1 7.5 3 2 3 0 Fig. 12-B 7.5 1 3 3 2 3 0

Figure 10 :

 10 Figure 10: Illustration of item 2-b of Theorem 2.5. Total equilibrium population U *T (shown in red) system (2.1) as a function of the dispersal asymmetry γ for the sets of the parameter values given in Table2. The horizontal line shown in blue is K 1 + K2 and in green is K 2 .

In the next theorem, we give the behavior of the model (1.5) when γ → ∞. Theorem 3.5. Let (u 1 (t, γ), u 2 (t, γ)) be the solution of the system (1.5) with initial condition (u 0 1 , u 0 2 ) satisfying u 0 i ≥ 0 for i = 1, 2. Let z(t) be the solution of the differential equation

with initial condition z(0) = u 0 1 + u 0 1 . Then, when γ → ∞, we have

and, for any t 0 > 0, we have

. We rewrite the system (2.1) using the variables (X, u 2 ).

One obtains:

When γ → ∞, (3.11) is a slow-fast system, with one slow variable, X, and one fast variable u 2 .

According to Tikhonov's theorem [START_REF] Banasiak | A Note on the Tikhonov Theorem on an Infinite Interval[END_REF][START_REF] Lobry | On Tykhonov's theorem for convergence of solutions of slow and fast systems[END_REF][START_REF] Tikhonov | Systems of differential equations containing small parameters in the derivatives[END_REF][START_REF] Wasow | Asymptotic Expansions for Ordinary Differential Equations[END_REF] we consider the dynamics of the fast variable in the time scale τ = γt. One obtains

In the limit γ → ∞, we find the fast dynamics

The slow manifold is formed by the equilibrium points of the fast equation (3.12), which given by u * 2 = X. Since u * 2 LAS for the system (3.12), the theorem of Tikhonov ensures that after a fast transition toward the slow manifold, the solutions of (3.11) are approximated by the solutions of the reduced model which is obtained by replacing u * 2 into the dynamics of the slow variable, which gives the equation (2.10). The system (2.10) admits X * = 0 as a equilibrium point, which is LAS in the positive axis. The approximation given by Tikhonov's theorem holds for all t ≥ 0 for the slow variable and for all t ≥ t 0 > 0 for the fast variable, where t 0 is small as we want. Therefore, let z(t) be the solution of the reduced model (3.8) of initial condition z(0) = X(0, γ) = u 0 1 + u 0 2 , then, when γ → ∞, we have the approximations (3.9) and (3.10).

According to previous theorem, when γ → ∞, we have E * (ξ , ∞) = (0, 0) and U * T (ξ , +∞) = 0 for all ξ .

In this part, we compare the total equilibrium population U T (γ, ξ ) of (1.5) and carrying capacity K 1 , by analyzing the stable positive equilibrium E (γ, ξ ) when the dispersal asymmetry varies from 0 to ∞.

Theorem 3.6. Consider the model (1.5). We have,

, such that:

Moreover, U T (γ, ξ ) = 0 for all γ ≥ γ * .

Proof. The equilibrium of the system (1.5) is the solutions of the following algebraic system:

(3.15)

The sum of the two equations of (3.15) shows that the equilibrium points are in a ellipse noted E 1 , which its equation is given by:

The ellipse E 1 passes through the points (0, 0), A := (K 1 , 0), (0, -K 2 ) and (K 1 , -K 2 ). Note that, it is independent of dispersal intensity ξ and asymmetric dispersal γ (shown in red in Fig. 5).

When γ → 0, the equilibrium E (ξ , γ) tend to A +0 := (K 1 , 0). In the case when γ → ∞, the equilibrium E (ξ , γ) tend to, A ∞ = O.

To facilitate comparison between the total equilibrium population U * T (ξ , γ) and the carrying capacities K 1 , we proceed as in the case of two-patch source-source model. Indeed, we consider the line ∆ with Cartesian equation u 1 + u 2 = K 1 , of slope -1 and passing through the point A = (K 1 , 0) (shown in blue in Fig. 5). If the equilibrium (u 1 (γ, ξ ), u 2 (γ, ξ )), is on or below the line ∆, then U T (γ, ξ ) ≤ K 1 , whereas if the equilibrium is above the line ∆, then U T (γ, ξ ) ≥ K 1 .

The equilibrium point E (γ, ξ ) is always in ellipse, then, for γ → 0, the equilibrium point start at A +0 = (K 1 , 0) and when γ increases, E (γ, ξ ) describes an arc of the ellipse and ends at point

Suppose r 1 ≤ r 2 . Then the slope of the space tangent of the ellipse E 1 at A +0 equal -r 1 r 2 ≥ -1, which means that the ellipse E 1 is below the line ∆ in the first quadrant and U T (γ, ξ ) ≤ K 1 for all γ as shown in Fig. 5. In the case when r 1 > r 2 , direct calculation finds that the ellipse E 1 and the line ∆ have two intersections: