Nikolai Bessonov

Natalia Reinberg

Malay Banerjee

Vitaly Volpert
email: volpert@math.univ-lyon1.fr

The origin of species by means of mathematical modelling

Keywords: nonlocal reaction-diffusion equations, travelling waves, pulses, Darwin's diagram

Darwin described biological species as groups of morphologically similar individuals. These groups of individuals can split into several subgroups due to natural selection, resulting in the emergence of new species. Some species can stay stable without the appearance of a new species, some others can disappear or evolve. Some of these evolutionary patterns were described in our previous works independently of each other. In this work we have developed a single model which allows us to reproduce the principal patterns in Darwin's diagram. Some more complex evolutionary patterns are also observed. The relation between Darwin's definition of species, stated above, and Mayr's definition of species (group of individuals that can reproduce) is also discussed.

1 Evolution of species and Darwin's diagram Emergence and evolution of biological species continue to attract attention from both the biological point of view and from the modelling point of view. Modern theory distinguishes three main mechanisms of speciation. Allopatric speciation implies the existence of geographic or genetic isolation where there is no gene exchange between different taxa; parapatric speciation admits partial exchange; and sympatric speciation occurs without geographic or genetic barriers [6], [11]. It is generally accepted that allopatric speciation is biologically realistic, though the mechanism which leads to the appearance of isolating barriers may sometimes be unclear [6] (Chapter 2). Sympatric speciation continues to instigate intensive discussions. In spite of the big body of experimental data, observations in nature, and theoretical models, this question remains controversial [6] (Chapter 4). Various models of sympatric speciation are developed in [8], [9], [10], [12], [13], [14], [15].

Competition for resources and selection of the most successful in this competition are among the fundamental properties of biological systems. Once the elements of such a system begin to compete, they try to get as many resources as possible. For this purpose they consume resources not only at the spatial location, where they are located, but also in some areas around it. This results in nonlocal consumption of resources and intraspecific competition. Nonlocal consumption of resources, reproduction, and variation of functional traits (diffusion) determine the process of speciation [13].

In this work we will address the question about the emergence and evolution of biological species from the point of view of Darwin's diagram. It is the only figure featured in his famous book "The origin of species" [7], and it is the quintessence of his theory, which corresponds, in the modern terminology, to sympatric speciation. We will attempt to give a more precise mathematical interpretation of his theory and to point out the assumptions which are necessary in order to reproduce Darwin's diagram. He described this figure as follows. "Let A to L represent the species of a genus large in its own country; these species are supposed to resemble each other in unequal degrees, as is so generally the case in nature, and is represented in the diagram by the letters standing at unequal distance ... The little fan of diverging dotted lines of unequal length proceeding from (A), may represent its varying offspring. The variations are supposed to be extremely slight, but of the most diversified nature, they are not supposed all to appear simultaneously, but often after long intervals of time; not are they all supposed to endure for equal periods. Only those variations which are in some way profitable will be preserved or naturally selected. And here the importance of principle of benefit being derived from divergence of character comes in; for this will generally lead to the most different or divergent variations (represented by the outer dotted lines) being preserved and accumulated by natural selection. ... After a thousand generations, species (A) is supposed to have produced two fairly well-marked varieties, namely a 1 and m 1 . These two varieties will generally continue to be exposed to the same conditions which made their parents variable, and the tendency to variability is in itself hereditary, consequently they will tend to vary, and generally to vary in nearly the same manner as their parents varied. Moreover, these two varieties, being only slightly modified forms, will tend to inherit those advantages which made their common parent (A) more numerous than most of the other inhabitants of the same country" ([7], p. 104).

Fig. 1 Left: Darwin's diagram explaining the emergence and evolution of biological species (adapted from [7]). Right: A typical population density distribution with respect to a morphological characteristics. The coordinates of the peaks of such distributions on the (x, t)plane corresponds to the line in Darwin's diagram.

Thus, each line corresponds to one species or subspecies. The horizontal axis measures their resemblance; that is, some morphological characteristics or phenotype. The vertical axis is time. Different species are represented as lines (trajectories) in the (x, t)-space.

The appearance of new species is shown as several lines starting from the same point. In the beginning, these are small variations inside the same species. After some time they can become different species. Among many possible variations, only favorable ones are chosen and reinforced by natural selection. The others will disappear. One more important feature of the diagram is that the lines (species) can disappear when two of them approach each other. We can suppose that this happens because the corresponding species begin to compete for the same resources.

We will model the main elements of Darwin's diagram: existence of species as localized groups of individuals, their evolution (motion in the morphological space), emergence of new species, preservation of the most fitted species and disappearance of others due to natural selection, and competition of species for resources. We will introduce the model in the next section and will analyze it in Section 3. We will then discuss the results in Section 4.

2M o d e l

We consider the diagram in Figure 1 (left) from the bottom upwards (in the direction of growing time). It begins with 11 lines noted by the letters A, B, ..., L. We will mainly consider the left part of the diagram (A, B, ..., F ), though some other parts will also be mentioned when it is appropriate. Following Darwin's definition, we understand biological species as groups of morphologically similar individuals. A typical population density distribution for one species is shown in Figure 1 (right). The variable x here is a morphological characteristic or a functional trait of the population (height, weight, beak size, and so on) and the function u(x, t) describes the density distribution with respect to this characteristic at some given moment of time t. The species are characterized by narrow distributions, such that they can be described by the coordinate x m of the maximum of the function u(x, t). This coordinate x m (t), considered as a function of time, represents a line on the (x, t) plane. Darwin's diagram corresponds to the ensemble of such lines for different species.

Thus, our aim is to develop a model such that a) it has solutions in the form of pulses (or peaks), which describe population density distributions, and b) the coordinates of the maxima of these pulses on the (x, t)-plane form the patterns qualitatively similar to those in the diagram.

In [5] we reproduced some parts of the diagram independently of each other. In this work we will develop a new model which will allow us to describe these different parts together and to reproduce the left half of the diagram. We consider 6 species in accordance with the letters A, B, ..., F and introduce the corresponding population densities u i (x, t), i =1, 2, ..., 6. In order to describe the desired behavior of solutions, we consider the integro-differential system of equations:

∂u i ∂t = d i ∂ 2 u i ∂x 2 + a i u 2 i W i -σu i ,i =1, 2, ..., 6. (1)
Here, the space variable x corresponds to some morphological characteristics of the populations, t is time, and d i , i =1 , ..., 6 are the diffusion coefficients, a i and σ are some positive constants. The first terms in the right-hand side of these equations describe small variations of the morphology due to random mutations, the second terms describe the reproduction of the population and the last terms species' mortality. Let us note that the reproduction terms are proportional to the square of the population density (Supplementary materials) and to available resources W i , considered in the following general form:

W i =(1-J i (u))(1 -I i (u)),i =1, ..., 6,
where

J i (u)= 6 j=1 b ij J 0 j ,I i (u)= 6 j=1 s ij I 0 j , J 0 j = ∞ -∞ φ(x -y)u j (y, t)dy, I 0 i = ∞ -∞ u i (y, t)dy.
Here, b ij are non-negative numbers. The coefficients s ij ≥ 0 depend, in general, on the space variable x, though we will begin below with the simpler case of constant coefficients. The kernel φ(x) of the integral determines the efficacy of this consumption as a function of the distance x -y. We take it as a piece-wise constant function,

φ(x)= 1 2N 1 , |x| <N 0 , |x|≥N .
In the limit of small N , we get the case of local consumption. In the case of global consumption, the consumption of resources is proportional to the total population density (integrals I 0 i ) [5], [16]. In conventional models of population dynamics, available resources are considered in the form (1 -u i (x, t)/K), where K is carrying capacity. Consumption of resources is proportional to the population density u i (x, t)t a k e n at the space point x. In the case of nonlocal consumption of resources, it is replaced by the integral J i (u) [13]. This implies that the individuals of the population consume resources in some areas around their average location. If there are several species, then all of them (or only some of them) can contribute to the consumption of some particular resources.

The model considered in this work contains two resources: one with nonlocal consumption and another with global consumption. We will see below that both are necessary in order to describe the patterns shown in the diagram. It is a generic model of the interaction of 6 species. It generalizes the previously considered models with local and nonlocal consumption of resources (see Supplementary materials).

Modelling of the diagram

Darwin's diagram illustrates different evolution patterns, such as speciation, survival of the better adapted, competition of species, among others. In the model presented above, we set b

11 = b 1 ,b 1j =0 ,j =2 , ..., 6, b ij = b 2 ,i = 2, ..., 6,j =1 , ..., 6, s 11 = s 1 ,s ii = s 2 ,i =2 , ..., 6, s ij =0f o ri = j.
These assumptions on the parameters signify that the first equation for the variable u 1 (x, t) is independent of the other equations but it does influence the other variables. Other values of parameters will be specified below.

Speciation. The main issue of Darwin's theory is the emergence of new species. This is shown in the diagram on many occasions: points A, a 1 ,a 2 , ..., m 1 , m 2 , ..., I, z 1 ,z 2 , ... For each point, there are several lines starting from a single point. These lines do not correspond to fully formed species, but only to small variations in the existing species, which can lead to new species in the future.

Among the species A, B, ..., F , only the first one manifests speciation and leads to the appearance of new species. The corresponding function u 1 (x, t) satisfies the equation

∂u 1 ∂t = d 1 ∂ 2 u 1 ∂x 2 + a 1 u 2 1 (1 -b 1 J 0 1 (u 1 ))(1 -s 1 I 0 1 (u 1 )) -σu 1 . (1)
In order to simplify the model and the analysis of the results, we chose this equation to be independent of other variables. Clearly, similar results can be obtained if the dependence on other variables is weak enough.

As discussed above, we characterized solutions of this equation by the positions of their maxima at the (x, t)-plane. They correspond to the lines (species) in Darwin's diagram. A typical branching pattern obtained in numerical simulations of equation ( 1) is shown in Figure 2 (the left part of the graph). The initial condition for the first species, u 1 (x, 0), is a piece-wise constant function different from zero in a narrow interval at the center of the computation domain. This initial population splits into two sub-populations (Figure 2, right), then again and again, expanding in space as a periodic travelling wave and leaving stationary localized peaks behind it. These peaks correspond to the new species.

Speciation can be described by the model with nonlocal consumption of resources (see Supplementary materials) and by the model with two resources: nonlocal and global. The question about the possibility of sympatric speciation was intensively debated in the literature, and various mathematical models were developed [1], [8], [9], [10], [13], [14]. Survival of the better adapted. We observe in the diagram that among several new sub-species only one or two survive, while the others disappear after a relatively short time. The model of speciation with nonlocal consumption does not describe this effect: once a species appears, it persists. This contradicts Darwin's description that only better adapted species survive. Usually, the outer lines (species) persist, while the species located inside become extinct (cf. Section 1).

If we accept the assumption that some species (or sub-species) are better adapted than the others, then their natality and mortality rates depend on the phenotype, that is on the space variable x. Hence the coefficients of the equations depend on the space variable. This approach was suggested in [5]. We will use it here in a more complete model.

We illustrate modelling with space-dependent coefficients for another part of the diagram starting with z 6 [5]. Figure 3 shows an example of numerical simulations of equation ( 1) with a space dependent function s 1 (x). The initial condition is a narrow pulse located at the center of the interval. Due to speciation, it gives six pulses. The outer two of these pulses reach the points where the function s 1 (x) decreases (step-wise constant function in the same figure). As a consequence, these two outer pulses begin to grow while the others disappear. A similar behavior repeats several times at each step of the function s 1 (x). We obtain a specific V -pattern, which is one of the main repeating patterns in the diagram. Similar patterns start at the points A, a 3 ,a 5 ,m 4 ,z 4 ,z 6 . Let us emphasize the total size of the population is limited due to global consumption. Therefore, if one of the species grows in size, then the others will decline. This is the mechanism which ensures development of the most adapted to the prejudice of the others. If we consider for example only non-local consumption, where the size of the population is not limited, then this mechanism does not work.

The absence of speciation. Species B, E, ..., F persist without speciation. Such behavior is observed in the case of global consumption (see Supplementary materials). Hence, the behavior of the variables u 2 , ..., u 6 is primarily determined by global consumption while nonlocal consumption takes into account competition with other species (the next paragraph).

Competition of species. Another repeating pattern in the diagram is extinction of one of the species when two of them approach each other. Among them, species B, C and D disappear when they meet the line m descendant from A. It also happens for G and u and in several other cases. Thus, we need to describe the situation where two species coexist if they consume different resources and one of them disappears when they begin to compete for the same resources. Since we associate species to the coordinates of the maxima of the corresponding density distributions, two species do not compete if these coordinates are different, and they begin to compete when they are close to each other.

Thus we need to describe a) the displacement of species in the morphological space, b) their competition when two species (coordinates of the maxima) are close and the absence of competition when they are far from each other in this space, c) the disappearance of some of the species due to this competition. Clearly, these properties are not described by the conventional model of competition of species which does not take into account their location in the space of phenotypes. L e tu sbe g i nw i t ht h ec a s ew h e r et h ec oe ffi c i e n t ss i are constant. The first component u 1 of the solution expands in space and begins to influence other components of the solution (Figure 2). First, it meets the second component u 2 , which is bounded from the other side by the third species and is rapidly going to extinction. Similarly, the third, fourth and fifth species disappear one after another when pushed from one side by the expanding first species and from the other side by the remaining species. The last, sixth species persists during the time interval shown in the diagram.

The introduction of space dependent coefficients allows us to take into account the survival of the better adapted discussed above. Figure 4 (upper left) shows the initial conditions and the function s 1 (x). The results of the numerical simulation are shown in Figure 4 (upper right).

Time-dependent coefficients. In order to finish modelling Darwin's diagram, we need to introduce time dependent coefficients. Let us explain why it is necessary. The pulse moves in such a way that it tends to improve living conditions of the population: decrease the mortality rate, increase natality and the total size of the population. These conditions determine the direction of its motion, and it cannot move in the opposite direction because it would be unfavorable for the population. These observations are in agreement with the biological meaning of the considered models, and they are confirmed by all obtained results. Consider the line from the point A to a 5 of the diagram which moves to the left. However, the line f , which starts at the point a 5 , turns to the opposite direction. If the parameters of the model remain the same, this should not happen because either the line a or the line f would deteriorate the life conditions of the corresponding species. Similar behavior is shown in other parts of the diagram (m 4 ,z 4 ,z 6 ). Hence we need to conclude that at the moment when the curve a reaches the point a 5 , the coefficients of the equation change. Biologically, this means that the new species possess new properties which are reflected through the corresponding changes in the coefficients of the model.

Discussion

Darwin's diagram represents the essence of his theory, and it contains several different evolutionary patterns. First of all, the emergence of new species, the question largely discussed in the modelling literature, and also several other models including survival of the adapted, existence of species without speciation, and competition of species. Even the question of existence of species, which might seem obvious, appears to be quite involved as far as mathematical models are concerned. All these questions stimulated the development of the new models that were brought together in this work. We discuss some of these below as well as some further possible developments of these models.

Existence of species. Biological species correspond to localized population density distributions in the form of pulses. Reaction-diffusion equation (1) (Supplementary materials) with a bistable nonlinearity has such stationary solutions. However, they are unstable. They become stable in the case of global consumption of resources (3). The bistable case corresponds to sexual reproduction where the reproduction rate is proportional to the square of the population density u 2 . The existence of such solutions is also proved for the system of two equations with different densities of males and females [17]. In the monostable case, where the reproduction rate is proportional to the first power of u, single stationary pulses do not exist. However, there are periodic travelling waves which can leave multiple pulses after propagation. Thus, the existence of biological species, considered as particular types of solutions, imposes certain conditions on the model and on the parameters.

Emergence of species and survival of most adapted. One of the main issues of Darwin's theory concerns sympatric speciation. In our previous work [13] we have developed a minimal reaction-diffusion model that describes this process.

It includes nonlocal consumption of resources, which leads to intraspecific competition, reproduction, and small random mutations. In the case of sexual reproduction, the phenotypes, or the functional traits, related to consumption of resources of males and females are supposed to be the same. Their difference can influence the emergence of species (see the next paragraph).

The minimal model of the emergence of species is not sufficient to describe some properties of this process stressed by Darwin; namely, survival of the most adapted subspecies and disappearance of the others. In order to describe this effect, we introduce a model with two resources: nonlocal and global. It is interesting that, similar to Darwin's description, we observed that "the most different or divergent variations" survive while the others go to extinction.

Darwin versus Mayr. Darwin considered biological species as groups of morphologically similar individuals. According to Mayr, these are groups of individuals that can reproduce among themselves and have fertile offspring. We will briefly discuss here how these two definitions are related to each other from the point of view of the modelling presented in this work.

The reproduction terms in equations ( 1) are proportional to the second power of the population density. This term appears in the case of sexual reproduction where the reproduction rate depends on male and female densities which are supposed to be equal to each other (see Supplementary materials). Therefore, it is implicitly assumed that males and females having offspring possess the same functional trait that is responsible for consumption of resources.

Suppose now that males and females having common offspring can possess different phenotypes. Then, instead of u 2 , we have the nonlocal term u(x, t) ∞ -∞ ψ(x -y)u(y, t)dy, where the kernel ψ(x -y) shows the probability of phenotypes x and y to have common offspring. This model was studied in [4], and it was shown that the new species will appear if the support of ψ is small enough, such that the phenotypes x and y with common offspring belong to the same localized solution (species). If we allow inter-breeding, that is, the support of ψ is larger than the distance between the species in the space of phenotypes, then new species do not appear.

Hence, Mayr's condition is necessary for the emergence of species that are understood in the sense of groups of morphologically similar individuals. Indeed, if individuals from different species can have common offspring, then the groups of morphologically similar individuals will not emerge.

Multi-level speciation. Darwin's diagram contains many important elements of his theory. However, it is only an illustration, and many other cases and generalizations are possible. We will present here one of them in order to show an example of multi-level speciation.

Biological species are organized in genera, families, and so on. In the modelling presented above, we considered only species and not higher levels of this hierarchy. In order to introduce two levels of speciation, we need to consider double nonlocal consumption of resources with two different kernels. In this case, the modified model is as follows:

∂u ∂t = d ∂ 2 u ∂x 2 + au 2 (b 1 -c 1 J 1 (u))(b 2 -c 2 J 2 (u)) -σu, (1)
where

J i (u)= ∞ -∞ φ i (x -y)u(y, t)dy, φ i (x)= 1/(2N i ) , |x|≤N i 0 , |x| >N i ,i =1, 2.
Figure 6 shows an example of numerical simulations of equation ( 1). The maxima of solutions, which correspond to different species, are organized in groups that correspond to genera. Genera have a larger range of consumption of resources than species because of a greater variation of the total phenotype. Nonlocal prey-predator models. Competition of species leads to a quite simple dynamic where one of the species disappears or the two species coexist. It is interesting to note that the coexistence of species is not shown in Darwin's diagram. The situation is quite different if the interaction of species is determined by prey-predator dynamics with nonlocal consumption of prey by predator [2] or with nonlocal consumption of resources by prey [3]. In these cases, numerous stationary, time periodic, or aperiodic regimes are observed. search grant 16-01-00068, 2016-2018. V. Volpert was supported by the "RUDN University Program 5-100".

J(u)= ∞ -∞ φ(x -y)u(y, t)dy.
In population dynamics, nonlocal consumption of resources means that the individuals located at the space point y consume resources not only at this point, but also in some areas around it. The consumption rate depends on the distance to resources. Therefore, the kernel of the integral depends on (x -y). If we replace it by the δ-function, then we will return to the local equation.

From the modelling point of view, the main interest of this equation is that the homogeneous in space stationary solution u = b/c can be unstable. This is not the case for the local reaction-diffusion equation. As a result of this instability, spatial structures and periodic travelling waves can appear. In the bistable case, the corresponding equation becomes

∂u ∂t = d ∂ 2 u ∂x 2 + au 2 (b -cJ(u)) -σu. (3)
As in the monostable case, spatial structures and periodic travelling waves can be observed for this equation. Propagation of simple and periodic waves for the nonlocal reaction-diffusion equation was studied in [1], [4], [12], [13], [14], [9], [10], [12], [14] for k = 1 (monostable case) and in [1], [2], [3] for k =2 (bistable case).

Global consumption of resources. Along with the local and nonlocal consumption of resources, we will also consider the global consumption [16] where the integral J(u) in equation ( 3

In this case, individuals have equal access to resources independent of their location. This equation has pulse solutions. These are positive stationary solutions on the whole axis decaying at infinity. Such solutions exist for equation (1) with the function F 2 (u)if

u - 0
F 2 (u)du > 0. They can easily be found analytically. However, these solutions are unstable because the principle eigenvalue of the corresponding linearized operator is positive [17]. In the case of equation ( 4), existence of pulse solutions can also be easily verified. However, these solutions become stable. Their stability is observed in numerical simulations [16], but it is not yet proved analytically. If the support of the function φ(x) in equation ( 3) is sufficiently large, then it can also describe stable pulses or even multiple pulses where several pulses exist and slowly move from each other [START_REF] Volpert | Pulses and waves for a bistable nonlocal reaction-diffusion equation[END_REF]. Their number depends on the initial condition.

Periodic travelling waves, and simple and multiple pulses represent the main types of solutions used to describe the emergence and evolution of species. Periodic travelling waves describe the process of speciation [13], [14] while pulses describe stable species without speciation [5]. Their combination and the introduction of space dependence in the coefficients of the equations can describe the variety of patterns in Darwin's diagram.

Fig. 2

 2 Fig. 2 Left: Numerical simulations of system (1) with d i =0 .01,a i =1 ,b 1 =1 ,b 2 = 20,s 1 =0 .01,s 2 =0 .1,σ =0 .01,N = 5. The lines show positions of maxima of each component of the solution on the (x, t)-plane.The first species (variable u 1 (x, t)) shows a typical branching pattern, while the other species do not change until they meet the descendants of the first species and then become extinct. Different colors correspond to different components u i (x, t) of the solution. Right: Initially the population density had unimodal distribution. After some time, it split into two subpopulations (reprinted with permission from[13]).

Fig. 3

 3 Fig. 3 Modelling of the part of the diagram which begins at the point z 6 with equation (1). The green lines show the trajectories of the maxima of the function u(x, t), the red curve is a snapshot of solution, and the violet line shows the function s 1 (x). The values of parameters are as follows: d 1 =0 .1,a 1 =2 0 ,b 1 =0 .5,s 1 =0 .018 (lowest value of the function s 1 (x)), σ =0.1,N =3.

Fig. 4

 4 Fig. 4 Numerical simulations of system (1) with d 1 =0 .2,d 2 = ... = d 6 =0 .01,a 1 = 20,a 2 = ... = a 6 =1 ,s 2 = ... = s 6 =0 .1,b 1 =0 .5,b 2 =2 0 ,σ =0 .1,N = 3. The function s 1 (x) (magenta) changes in time and is shown for each time interval. The upper left image shows the initial conditions for all components of a solution and the initial form of the function s 1 (x). The result of this simulation at t = 167 is shown in the upper right image. The snapshots of solutions for t = 265 and t = 375 are shown in the lower figures. Each of the figures shows the form of the function s 1 (x) used at the next stage of the simulation.

Figure 4

 4 Figure 4 shows the results of numerical simulations of system (1). The initial functions u i (x, 0) and the initial function s 1 (x) are shown in the upper left image. The result of this first stage of the simulation is shown in the upper right image together with the function s 1 (x), which is used in the next stage. Further results and the function s 1 (x) are shown in the other images. The final result is shown in Figure 5 (left). It qualitatively repeats the main features of this part of the diagram (Figure 5, right).

Fig. 5

 5 Fig. 5 Darwin's diagram (right) and the numerical simulations (left) show its main patterns: coexistence of several species, V -pattern, extinction of species due to competition, time dependent coefficients allowing the species f to return back in the space of phenotypes.

Fig. 6

 6 Fig. 6 Numerical simulations of equation (1) with double nonlocal consumption. The values of parameters are as follows: d =0 .2,a =1 ,b 1 =2 ,b 2 =1 0 ,c 1 =1 ,c 2 =1 5 ,N 1 =3 ,N 2 = 20,σ =0 .1. The red curve shows a snapshot of the solution and the green curve shows the positions of maxima at the (x, t)-plane.

  ) is replaced by the integral I(u):∂u ∂t = d ∂ 2 u ∂x 2 + au 2 (b -cI(u)) -σu, I(u)= ∞ -∞u(y, t)dy.

Acknowledgements

The authors are grateful to R. Penner for his help in the preparation of the manuscript. N. Bessonov was supported by Russian Foundation of Basic Re-

Supplementary materials 5 Reaction-diffusion equations with local, nonlocal and global consumption of resources

We present here some properties of local and nonlocal reaction-diffusion equations and their solutions used in this work.

Scalar reaction-diffusion equation. Let us recall that the usual (local) scalar reaction-diffusion equation

) can have travelling wave solutions, that is, solutions of the form y(x, t)= w(x -st), where s is the wave speed. Their existence and properties depend on the function F (u). Some typical examples are given by the functions:

The coefficients a, b, c, d,a n dσ are positive numbers. The number of parameters in this equation can be reduced. However, it is convenient to keep them in order to analyze their influence on the behavior of solutions. For the first function F 1 (u) (monostable case), monotone travelling waves exist for all s ≥ s 0 =2 √ abd. These waves are stable in the corresponding weighted spaces. If cσ/a < b 2 /4, then the second function F 2 (u) has three zeros: u + =0,u 0 and u -are solutions of the equation au(b -cu)=σ, u 0 <u -. The wave with the limits w(±∞)=u ± (bistable case) exists for a unique value of s. It is globally asymptotically stable. In population dynamics, equation ( 1) describes the distribution of the population density. The term au k (b -cu) is the reproduction rate proportional to the density u of the population in the first or second power and to the available resources (b -cu). The term -σu describes mortality of the population. Let us note that k = 1 corresponds to asexual reproduction or to sexual reproduction, where the reproduction rate depends only on the density of females. The value k = 2 corresponds to sexual reproduction where the reproduction rate depends on both densities, males and females. Assuming that these densities are equal to each other, we obtain the second power of the population density. A more general case without this assumption is studied in [START_REF] Volpert | On pulse solutions of a reactiondiffusion system in population dynamics Nonlinear Analysis[END_REF].

Nonlocal consumption of resources. In the case of nonlocal consumption of resources, equation (1) with the function F 1 (u) is replaced by the equation

where


Kevin Cox

Richard Robertson

Controlling

Controlling Failure of Polymer Skin/Foam Bilaminate Sheets

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

A line-shaped weakening is used in a number of products. Examples are the opening of soda pop cans (aluminum), soup cans (steel), and milk and soda pop bottles (plastic). The lines of weakening are visible in these products. In this work, attempts were made to keep the line of weakening hidden. Other requirements were (1) that the material be a polymer skin/foam bilaminate, (2) that its failure would occur only along the weakened line with no fragmentation, and (3) that the weakening be accomplished by mechanically scoring the skin on its backside, where it is bonded to the foam.

The immediate application of this technology is found in the weakening of an instrument panel to allow the passenger-side airbag to be deployed through it, a passenger side seamless airbag system (PSSAS), or in other words, a hidden airbag system. They are a recent trend in the automobile industry and Sun et al. has described a possible PSSAS design [1]. It consists of a hinged metal door bolted to the back of an instrument panel (IP). When the airbag is deployed, the hinged door breaks through a foam/polymer skin bilaminate, hopefully, in a predetermined location. This is achieved by the creation of a line-shaped weakening in the IP. One method used to create a line-shaped weakening is to mechanically score the polymer sheet that makes up the outer decorative skin of the IP. The skin is weakened on the class B surface. The class B surface is the surface of the skin that is not visible after the IP is manufactured. The class A surface is the one that is visible after the IP is manufactured. See Fig. 1 for a schematic of a PSSAS that is similar to the design described in Sun et al.'s patent. The bilaminate of the PSSAS is subjected to quasi-shear loading arising from the hinged door applying a load perpendicular to the sheet plane and the foam on the outside of the hinged door being adhered to the airbag housing. As described by Hersman et al. [START_REF] Hersman | Introduction of a Castable Thermoplastic Urethane Material for Instrument Panel Coverstock[END_REF], for a PSSAS deployment to be considered successful, the IP must fail at a predetermined location. In addition, no fragments that could cause injury to the passenger of the vehicle can be created during the airbag deployment. As described by Iannazzi and Daris [START_REF] Iannazzi | Seamless Passenger Side Air Bag Doors[END_REF], an additional product requirement of a PSSAS is to optimize the line-shaped weakening so it is weak enough to result in a predetermined failure, but strong enough to meet all product durability and cosmetic requirements of the IP.

EXPERIMENTAL AND RESULTS

The creation of the bilaminate samples was somewhat involved. A one mm thick TPO skin was vacuum-formed into the shape of an existing IP design [START_REF] Titow | PVC Plastics Properties, Processing and Applications[END_REF]. Grooves of various depths were produced on a flat section of the skin using a utility knife tool with an adjustable blade

Richard E. Robertson

University of Michigan length. The knife tool was run along a guide track that was placed on the class B surface of the skin. The result was the creation of a groove on the class B surface with a depth equal to the preset blade length. The scored skin was placed with a glass-filled styrene substrate into a reaction injection-molding machine. A layer of urethane foam was then injected between the skin and the substrate by reaction injection of isocyanate and polyol [START_REF] Klempner | Handbook of Polymeric Foams and Foam Technology[END_REF]. Rectangular samples were then cut from the IP shaped trilaminate so that the hidden groove crossed the width at the middle of the sample. Several samples were also produced that did not contain a hidden groove. A cut was then made through the substrate just below and parallel to the groove or across the middle of the sample if it did not have a groove.

A test was designed to show what leads to a locationcontrolled, fragmentation-free failure of a bilaminate when it is loaded in quasi-shear. Figure 2 shows a schematic of the sample loaded in quasi shear. It was discovered that for there to be fragmentation-free failure of the bilaminate, the failure needed to be initiated at both outer surfaces and the two cracks needed to meet in the middle of the foam. In other words, the skin had to fail prior to complete failure of the foam. This only happened if the groove was sufficiently deep. In other words, the remaining material thickness had to be sufficiently thin. Remaining material thickness is the amount of material left between the bottom of the groove and the class A surface. This remaining thickness is controlled by the skin's original thickness and the groove depth. If the foam completely failed prior to the skin, then the skin delaminated from the foam and foam fragments often formed.

A test was developed to observe the failure of the bilaminate when it was loaded in tension in the sheet plane perpendicular to the direction of the groove. It was learned from viewing the frames from different videos of these tensile tests that the same critical groove depth, between fragmentation-free and fragmentationprone failure, was observed whether the sample was loaded in quasi-shear or in tension. As with the tests in which the sample was loaded in quasi-shear, fragmentation-free failure of the bilaminate in tension involved the skin failing prior to complete failure of the foam. Frames from videos of four tensile tests that show how the foam and the skin failed when the bilaminates were loaded in tension and quasi-shear are shown in Fig. 3. The frames show that the same groove depth was required for the skin to fail prior to complete failure of the foam if the bilaminates were loaded in tension or quasi-shear.

Table 1 shows whether the skin of a bilaminate failed prior to complete failure of the foam or the foam prior to the skin when there were different remaining material thicknesses in the skin and the bilaminate was loaded in tension or quasi-shear. The table also shows that the same groove depth was required for the skin to fail prior to complete failure of the foam if the bilaminates were loaded in tension or quasi-shear Figure 4 shows load per unit width vs. strain for three TPO skin/foam bilaminates. It shows curves for one bilaminate composed of a skin with no groove, one with a 0.30 mm deep groove, and one with a 0.40 mm deep groove. All three of these bilaminates had the hard plastic substrate completely removed prior to testing. The figure shows a much larger average general strain in the bilaminate without a groove as compared to the bilaminates that had a 0.30 mm or 0.40 mm deep groove.

DISCUSSION

The directions of the stresses in a bilaminate loaded in quasi-shear were analyzed in an attempt to explain how the bilaminate failed.

Because the bilaminate is constrained by the substrate and pushed by the hinged door, two bending moments are generated and the bilaminate is stretched between them. A schematic of the stress directions when a bilaminate is loaded in quasi-shear is shown in Fig. 5.

The directions of the stresses in a bilaminate loaded in tension were analyzed in an attempt to explain why, if the line-shaped weakening in the skin was not weak enough, it often resulted in delamination and fragmentation.

To prevent delamination and foam fragments, the skin must fail prior to complete failure of the foam. The directions of the stresses were analyzed in a bilaminate loaded in tension rather than quasi-shear in order to simplify the problem. This was thought to be valid for the bilaminate discussed in this work because the same groove depth in the skin resulted in it failing prior to complete failure of the foam, whether the bilaminate was loaded in tension or quasi-shear, and the bilaminate appeared to fail in tension in the sheet plane even when loaded in quasi-shear.

Prior to delamination, the average general strain in the foam, skin, and bilaminate are equal over a large area. If the general strain of the bilaminate exceeds the ultimate tensile strain of the foam, the foam will fail and delaminate from the skin. If the local strain of the skin at the groove reaches the ultimate tensile strain of the skin prior to the bilaminate reaching the ultimate tensile strain of the foam, the foam will not fail and delaminate from the skin. Figure 6 shows a schematic of the general and local strain in a bilaminate at the point delamination begins and at the point the skin fails without delamination. The relationship between the average local strain in the skin, the ultimate tensile strain of the skin, the general strain in the bilaminate, the ultimate tensile strain of the foam, and delamination is illustrated.

CONCLUSION

If the general strain in the bilaminate reaches the ultimate strain of the foam before the local strain in the skin along the groove reaches the ultimate strain of the skin, the foam will fail and delaminate from the skin. This often leads to foam fragments. To prevent foam fragments, it is critical for the skin to fail prior to complete failure of the foam.

It is important to minimizing the groove depth because at a constant skin thickness, minimizing the groove depth could also minimize cosmetic and durability problems. At a constant skin thickness, the groove depth required for fragmentation-free failure may be minimized by minimizing the ratio of the load required for the skin to fail at the weakened line over the load required for the bilaminate to delaminate. This could be accomplished by minimizing the ratio of the ultimate tensile stress of the skin over the average general stress in the skin at the point of delamination. The average general stress in the skin at the point of delamination is reached when the skin's average general strain equals the ultimate strain of the foam. This ratio is equivalent to the average slope of the skin's stress vs. strain curve between the ultimate strain of the foam and the ultimate strain of the skin. It is important to choose a skin material with the flattest possible stress vs. strain curve between these ultimate strains, in order to minimize the critical groove depth. In other words, skin material that exhibits minimal strain hardening should be chosen. Critical groove depth is the minimum groove depth at which delamination does not occur.

Choosing foam with a higher ultimate strain could also reduce the critical groove depth. If the foam had an ultimate strain equal to the skin, the foam will not delaminate from the skin regardless of groove depth.

If the skin thickness is allowed to vary, then the remaining skin thickness between the bottom of the groove and the class A surface becomes the important value to maximize. By maximizing the remaining skin thickness, cosmetic and durability problems could be minimized.

For a given remaining skin thickness, increasing the original skin thickness will reduce the general strain in the bilaminate at the skin failure load. This would make it possible to take a bilaminate in which the foam completely failed prior to the skin, leave the remaining material thickness unchanged, and increase the original skin thickness to a point where the skin fails prior to complete failure of the foam. So choosing a thicker skin can increase the critical remaining material thickness.

The quasi-shear and tensile test on the bilaminate could be used to predict if the groove is sufficiently deep to result in a location-controlled, fragmentation-free failure of a bilaminate. These tests could be conducted at the strain rates and temperatures required by specific applications.

TABLES

Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 1 <Figure 6 1 ε

 12345161 Figure 1 Schematic of seamless airbag system (not drawn to scale): (A) top view of passenger and horizontal cross section through IP and (B) side view of vertical cross section through IP

Controlling Failure of Polymer Skin/Foam Bilaminate Sheets Kevin R. Cox

	Automotive Components Holding, LLC, Saline Plant

Table 1

 1 Skin and foam failure in bilaminates when loaded in tension or quasi-shear

		Remaining	Loaded In Tension	Loaded In Quasi-Shear
	Groove	Material	Failure:	Failure:
	Depth	Thickness	Foam Failed Prior To Skin	Foam Failed Prior To Skin
	(mm)	(mm)	Skin Failed Prior to Foam	Skin Failed Prior to Foam
	0.51	0.13	S	S
	0.38	0.26	S	S
	0.25	0.39	F	F
	0.13	0.51	F	F
	0.00	0.64	F	F

ACKNOWLEDGMENTS

The work reported is based on the Doctor of Engineering thesis of one of us (Kevin Cox) at the University of Michigan, Ann Arbor. We wish to thank the thesis committee, Frank Filisko, Ellen M. Arruda, and Mark De Mott, for all the time and effort they spent helping with this work. Special thanks are also due to the Visteon Corporation for its support of this research.


Hui Peng
email: davidpeng1749@yahoo.com

Double Slit Experiment Still Has Much to Offer (2): Extended-Which-Way Experiments---Violating Complementarity Principle

Keywords: which way double slit experiment, which way cross-double slit experiment, wave-particle duality, complementarity principle, cross-double slit experiment, quantum mechanics, quantum experiment

Young's 1D-dimension-double-slit experiments represent the basic mystery of quantum world and was extended to which-way-1D-double-slit experiment. To explore the mystery, 2D-cross-double-slit diaphragm have been proposed, which consists of more than one double-slits intersecting to each other at the same spot. In this article, the extended which-way 2D cross-double slit experiments are presented to test the normal interpretation of the which way experiments. The experimental results show that photons passing through the detected double-slit behave as particle, while photons passing through other undetected double-slit(s) distribute as wave in the same experiment with the same diaphragm. Namely the particle nature and wave distribution coexist, which violates Bohr's complementarity principle. It is a challenge to interpret those observations consistently.

Introduction

Young's 1D-double slit experiment was performed, the first time, in 1801 [START_REF] Ananthaswamy | Through Two Doors at Once[END_REF] [2] and interpreted as that each photon has arrived by both slits at the same time and created interference pattern. 100 years later, 1D-double-slit experiment led to wave-particle duality. Feynman called it "a phenomenon […] has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics]" [START_REF] Feynman | The Feynman Lectures on Physics[END_REF]. J. Taylor states: "And yet we have no clue how any of the fundamental facts of quantum mechanics including wave-particle duality, […], or the double-slit experiment -that disarmingly simple setup -actually work." [START_REF] Taylor | [END_REF]. Moreover, the nature of photons really puzzled Einstein.

He wrote to M. Besso: "All these 50 years of conscious brooding have brought me no nearer to the answer to the question: What are light quanta?" [START_REF] Rashkovskiy | Is a rational explanation of wave-particle duality possible?[END_REF].

To explore the nature of photon further, Young's 1D-double slit experiments were extended to which-way-double slit experiments by observing which slit a photon would pass through. Technically feasible realizations of which-way experiment were proposed in the 1970s [START_REF] Greenstein | The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics[END_REF] [START_REF] Bartell | Complementarity in the double-slit experiment: On simple realizable systems for observing intermediate particle-wave behavior[END_REF]. A photon is observed near a slit by utilizing a photoelectric-detector to register the photon.

Practically, to register a photon absorbs it, i.e., blocks the path of photons, and is equivalent to cover the slit [START_REF] Frabboni | Ion and electron beam nanofabrication of the which-way double-slit experiment in a transmission electron microscope[END_REF] [START_REF] Müller-Kirsten | Introduction to Quantum Mechanics[END_REF]. Once which slit a photon passing through is determined/detected, the wave-like interference pattern disappears.

The operational definition of "wave/particle" stands for "ability/inability to create interference". [START_REF] Greenstein | The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics[END_REF] [10] [START_REF] Ionicioiu | Proposal for a Quantum Delayed-Choice Experiment[END_REF]. The 1D-double-slit experiment has also been performed with varying numbers of slits, e.g., parallel-triple-slit and parallel-multi-slits [START_REF] Eibenberger | Matter-wave interference of particles selected from a molecular library with masses exceeding 10 000 amu[END_REF].

Based on the which-way-1D-double-slit experiment, it is concluded that photons can behave as either particles or waves, but cannot be detected as both at the same experiment. The which way experiments were considered as the confirmation of Bohr's complementarity principle. Namely, the particle nature and wave nature cannot manifest in the same time in the same experiment. However, the light is always found to be absorbed at the screen at discrete points, as individual particles; the interference pattern appears via the varying density of these particle hits on the screen.

Recently, to studying both the mystery of double-slit experiments and nature of photon further, the 1D-double slit experiments are extended to 2D-cross-double-slit experiment [START_REF] Peng | Cross-Double slit Experiment and Extended-Mach-Zehnder Interferometer[END_REF] [14] [START_REF] Peng | Comprehensive Double Slit Experiments---Exploring Experimentally Mystery of Double Slit[END_REF], in which more than one double slit cross to each other at the same spot. The experimental results challenge the standard interpretation of 1D-double-slit experiments.

The which-way-1D-double-slit experiments show the particle nature of photon only. It is nature to ask that, when one performs which-way experiment by utilizing 2D-cross-double-slit, what patterns would we expect? Namely, since a 2D-cross-double-slit apparatus consists of two or more double-slits, when we observe at one slit of one double-slit photons passing through, would we detect particle nature of photon passing through all double-slits, or would we detect wave distribution of photons passing through other un-observed double-slit(s)? According to the complementarity principle, we can detect only one nature of photon, either particle or wave, but not both. If we observe both natures, it would be a paradox.

The varieties of which way double slit experiments have been proposed and performed at classical level and quantum level. At the classical level, i.e., using a laser light source, the which way 2D-cross-double slit experiments have been proposed (Figure 1a) [START_REF] Peng | Cross-Double slit Experiment and Extended-Mach-Zehnder Interferometer[END_REF] and performed [START_REF] Peng | Observation of Which-Way-2D-Cross-Double-Slit Experiments: Violation of Bohr's Complementarity Principle[END_REF]. The dashed slit-A (Figure 1a) represent an observer is behind the slit-A. At the quantum level, the double-double slit apparatus (two separate double slits) is utilized to perform which way double slit experiments with quantum entangled photons (Figure 1b) [START_REF] Kaur | Quantum double-double-slit experiment with momentum entangled photons[END_REF]. Note that at both the classical and quantum levels, the cross-double slit and double-double slit are macroscopic devices.

Figure 1 Two which way experiments

In this article, more classical which-way-2D-cross-double slit experiments are presented.

2D-Cross-Double Slit

The diaphragms of the 2D-cross-double slit contain the following: double slit crossing double slit/triple-slit, single-slit crossing double slit/triple-slit, two single slit crossing double slit/triple slit.

Hereafter, denoted all as "cross-double slit" (Figure 1). We argue that, if we shift the focus from detecting photons at one slit to observe the final patterns (to see whether the pattern show the particle nature or wave distribution), then detecting a photon behind the slit of the double slit is equivalent to using a single slit, the logic chain is the following (Figure 5): A) To observe photons behind/in front of a slit is equivalent to block photons after/before passing through the slit;

A (A)
B) To block photons after/before passing through the slit is equivalent to cover that slit; C) To cover the slit is equivalent to using a single slit instead of using a double slit and covering one slit. Thus, to perform which-way experiments with the diaphragm of the cross-double slit, we made one of the double slits as a single slit (Figure 2).

Which-Way Cross-double slit Experiments: Same Spacing

Figure 6 shows the standard which-way experiment with an "observer" at slit-A and its pattern.

Figure 7 shows the cross-double slit with an "observer" at slit-A, and the right is the pattern.

The significant difference is that, for the standard which-way experiment, remaining photons must go through slit-B, while for the which-way cross-double slit experiment, remaining photons can go through either slit-B or slit-C or slit-D.

Since it is known that the "observation" would make the interference pattern disappear, we are not focus on observation of "which way". So, in the following experiments, one can use an object to block the propagation of photons coming out from slit-A (represented by dotted slit-A in Fig. 7).

We observe the following.

(A) The interference pattern due to slits-C and slit-D still exists, which implies that photons acting as wave goes through slit-C and slit-D.

(B) However, at the same time, the photons acting as particles pass through slit-B and are blocked at

A B C slit-A.
In this situation, photons are manifesting both nature of particle (passing through slit-B) and wave (passing through Slit-C and slit-D) simultaneously in the same experiment, which is a paradox.

Note that in optics, the single slit creates the diffraction pattern, it is explained that the diffraction pattern shows the wave behavior of photons. However, in the which way double slit experiment, one slit opens and other slit is covered. So, the photons cannot pass through the covered slit and reach the screen, the photons passing through the open slit create the diffraction pattern. There is no interference pattern as that created by the double slit.

The operational definition of "wave/particle" stands for "ability/inability to create interference" [START_REF] Greenstein | The Quantum Challenge: Modern Research on the Foundations of Quantum Mechanics[END_REF] [10] [START_REF] Ionicioiu | Proposal for a Quantum Delayed-Choice Experiment[END_REF]. According to this definition, it is stated that the which the way experiments, i.e., with one covered slit of the double slit experiments, present the particle nature.

Therefore, in the following which way cross-double slit experiments, one slit of the double-slit is closed, i.e., the single slits instead of the double slits (with one slit covered) are utilized to perform the which way cross-double slit experiments. Now the which way experiments are easily performed.

Experiment-1 (Figure 8): which way double slit experiment (single-slit crossing double slit).

Figure 8 Figure 9 The pattern of the single-slit crossing double slit experiment (Figure 8) is the same as that of Figure 7.

According to the operational definition of "wave/particle" standing for "ability/inability to create interference", the "particle nature" and "wave distribution" of photons coexist in the same experiment.

The horizontal pattern shows the particle nature, while the vertical interference pattern shows the wave distribution.

Experiment-2 (Figure 9): which way double slit experiment (single-slit crossing tilt-double slit).

This experiment leads to the same conclusion of the experiment-1. The photons have to "measure" the angle of the tilt-double slit to form the interference pattern accordingly. And the photons need to judge which slit they will pass through and behave accordingly.

Experiment-3 (Figure 10): which way cross-double slit experiment (single-slit crossing two double slits of the same spacing).

Experiment-4 (Figure 11): which way cross-double slit experiment (single-slit crossing two tilt-double slits of the same spacing).

Experiment-5 (Figure12): which way cross-double slit experiment (single-slit crossing three double slits of the same spacing). Experiment-6 (Figure 13): which way cross-double slit experiment (single-slit crossing five double slits of the same spacing).

The cross-double slit of more double slits cross together makes more difficult for the photons to "make" decision on how to behave:

(1) the single slit or double slit they are passing through; (2) which double slit they pass through; [START_REF] Feynman | The Feynman Lectures on Physics[END_REF] what are the angles of those double slits.

Which Way Double Slit Experiments: Different Spacing

Experiment-7 (Figure 14): which way cross-double slits experiment (single-slit crossing two double slits with different spacings).

Figure 14 Figure 15 Experiment-8 (Figure 15): which way cross-double slits experiment (single-slit crossing two double slits with different spacings).

Experiment-9 (Figure 16): which way cross-double slits experiment (single-slit crossing three double slits with different spacings).

Figure 16 Now the photons also need to "know" the spacing of the double slits they passing through to behave.

Which Way Triple Slit Experiments

Experiment-10 (Figure 17): which way triple slit experiment (single-slit crossing triple slit with same narrower spacing)

Figure 18 Experiment-11 (Figure 18): which way triple slit experiment (single-slit crossing triple slit with same wider spacing)

Experiment-12 (Figure 19): which way triple slit experiment (single-slit crossing triple slit with nonuniform spacing).

Figure 20 Experiment-13 (Figure 20): which way cross-triple slits experiment (single-slit crossing two triple slits with different/nonuniform spacing).

The photons' task becomes more and more difficult.

Which Way Double Slit Crossing Triple Slit Experiments

Experiment-14 (Figure 21): which way cross-double/triple slit experiment (two single-slits crossing double slit and triple slit with nonuniform spacing).

Figure 21 Figure 22 Experiment-15 (Figure 22): which way cross-double/triple slit experiment (two single slits crossing double slit and triple slit with same spacing).

Which Way Cross-Double Slit Experiments: Blocking Half of Patterns

Experiment-16 (Figure 23): blocking the regular which way double slit experiment.

A blocker is placed between the diaphragm and the detector/screen.

4.Discussion and Conclusion

Base on the practical definition of "wave/particle behavior" and the fact that each of single/double slit/triple slit on the diaphragm of cross-double slit creates its own pattern, the single slits present the particle nature of the photons. The which way cross-double slit experiments manifest particle mature and wave distribution simultaneously in the same experiment and thus, violate Bohr's complementarity principle. Which-way-1D-double-slit experiment shows particle nature of photon and thus, confirms the complementarity principle. On the contrary, as shown in this article, the which-way 2D-cross-double slit experiments show the coexistence of wave distribution and particle nature of photons in the same experiment with the same apparatus and thus, the complementarity principle is violated (Table 1).

Moreover, even we observe wave distribution, photons are always found to be absorbed at the detection screen at discrete points, as individual particles; the interference pattern appears via the varying density of these particle hits on the screen.

Moreover, a thought-experiment (Appendix) would show that, even before passing the cross-double slit, photons behave as particle. On the other hand, this conclusion has been confirmed by completely different experiments [START_REF] Peng | Comprehensive-Double Slit Experiments ---Exploring Experimentally Mystery of Double slit[END_REF] [20] [START_REF] Peng | Experimental Study of Mystery of Double Slit ---Comprehensive Double Slit Experiments[END_REF].

Thus, we conclude that the particle nature is primary nature of photon, while wave distribution is secondary nature.

The question is: How can photons adjust their behaviors simultaneously when passing through different slits correspondingly? It would be interesting to perform above experiments by emitting photons one at the time.

Appendix: Photoelectric Which Way Cross-double Slit Thought-Experiment and Photoelectric Cross-double Slit Thought-Experiment

The regular interpretation of the double-slit experiment contains a statement that before arriving a double-slit apparatus, the photons behave as wave from the time of its emission to the time of its detection.

To test the statement, let's combine the apparatus of the cross-double-slit experiments and the apparatus of the photoelectric effect. We propose two combined apparatuses, denoted as Photoelectric (3) the photons whether passing through slit-A and being blocked or there is no slit-A, the photons passing through slit-B and arriving at the photoelectric-detection-screen behave as particles, not as wave. This is a paradox.

Figure 2

 2 Figure 2 Diaphragms of 2D-cross-double slit

Fig. 3 . 4 .

 34 Fig. 3. 1D-double-slit Experiment Fig. 4. Which way 1D-double-slit Experiment [18]

Figure 5

 5 Figure 5 Observing (A) = blocking (B) = single slit (C)

Figure 12 Figure 13

 1213 Figure 12Figure13

Figure 23 Figure 24 Experiment- 18 (

 232418 Figure 23 Regular which way double slit experiment: blocking half pattern Figure 23c shows that half of the pattern of Figure 23b is blocked. For Young's double slit experiment, if trajectory theory holds, e.g., trajectories do not cross, one knows which way a photon passes through without observing and destroying the interference pattern. Experiment-17 (Figure 24): Which way cross-double slit experiments: blocker(s)

Figure 25

 25 Figure 25 Which way 2D cross-double slit experiments: blocker-CD Observations: Figure 25b shows that block-CD blocks the bottom half of the interference pattern created by the double slit-CD, but it does not affect the top half of the interference pattern. The pattern created by the double slit-AB is not affected.Experiment-19 (Figure26a): Placing both blocker-AB and blocker-CD between the cross-double slit and the screen.

Figure 26

 26 Figure 26 Which way 2D cross-double slit experiments: blocker-AB and blocker-CD Observations: Figure 26b shows that block-CD blocks the bottom half of the interference pattern created by the double slit-CD, while the block-AB blocks the left-half pattern. The block-CD does not affect either the top half of the interference pattern or the right half of the pattern created by the double slit-AB. Block-AB does not affect either the right half of the pattern or the top half of the interference pattern. These phenomena are the consequences of photons propagating along trajectories. Next let us perform the which way tilt-cross-double slit experiments with blocker(s).

Figure 27 Figure 27

 2727 Figure27shows the which way tilt cross-double slit experiments without blocking.

Figure 28a :

 28a Figure 28a: Placing blocker-AB between the tilt cross-double slit and the screen.

Figure 28

 28 Figure 28 Which way tilt cross-double slit experiment: blocker-AB Observations: Figure 28b shows that block-AB blocks the right-half of the pattern created by double slit-AB with "observation at slit A", while it does not affect either the left half of the pattern or the interference pattern created by the double slit-CD.Experiment-21 (Figure29a): Blocker-CD was placed between the tilt cross-double slit and the screen.

Figure 29

 29 Figure 29 Which way tilt cross-double slit experiment: blocker-CD Observations: Figure 29b shows that block-CD blocks the bottom half of the interference pattern created by the tilt double slit-CD, while it does not affect either the top half of the interference pattern or the pattern created by the double slit-AB.Experiment-22 (Figure30a): Place both blocker-AB and blocker-CD between the tilt cross-double slit and the screen.

Figure 30

 30 Figure 30 Which way tilt cross-double slit experiment: blocker-AB and blocker-CD Observations (Figure 30b and 30c): Figure 30b shows that the block-CD blocks the bottom half of the interference pattern created by the tilt double slit-CD, while the block-AB blocks the left half of the pattern created by the double slit-AB. The block-CD does not affect both the top half of the interference pattern and the right half of the pattern created by the double slit-AB. The block-AB does not affect either the right half of the pattern or the top half of the interference pattern created by the tilt double slit-CD.

Figure 30c shows that

  Figure 30c shows that block-CD blocks the top half of the interference pattern created by tilt double slit-CD, the block-AB blocks the right half of the pattern created by double slit-AB. The block-CD does not affect either the bottom half of the interference pattern or the left half of the pattern created by double slit-AB. The block-AB does not affect either the left half of the pattern or the bottom half of the interference pattern created by tilt double slit-CD.

It is a challenge

  to consistently interpret those which-way-2D-cross-double-slit experiments. All of above experiments suggest that the which-way-2D-cross-double-slit experiments open a new door for re-studying basic quantum phenomena.

  photoelectric cross-double slit (Figure Aa) or the photoelectric which way cross-double slit (Figure Ab).

Figure A

 A Figure A Experimental setups: Photoelectric-Cross-Double-Slit and Photoelectric-Which-Way-Cross-Double-SlitIt is reasonable to expect that the photoelectric effect, the pattern of cross-double-slit and the pattern of which way cross-double slit would independently exist simultaneously.First let us study the apparatus of Figure Aa. Some of photons strikes on the photoelectric-board of cross-double-slit-ABCD. Electrons are ejected. Collecting and measuring those electrons disclose the particle-nature of photons, regardless the existence of narrow slit A, slit B, slit C and slit D on the board. Namely before passing through the cross-double-slit, photons behave as particle, not as wave.Rest of the photons passing through cross-double-slit-ABCD behaves mysteriously, that: (1) the photons as particles knock electrons out of the photoelectric-detection-screen (a photoelectric effect again), regardless how the photons distribute on photoelectric-detection-screen, and thus, considered as particle; (2) the photons form interference distribution on the photoelectric-detection-screen, regardless material of the detection-screen, and thus, considered as wave. This is a paradox.Next, let us study the apparatus of Figure Ab. To test the statement, we propose the Photoelectric which way cross-double slit apparatus (FigureAb) that consists either double slit-AB with an "Observer" behind slit-A or slit-A is blocked or there is only single slit-B. Some of the photons strikes on the photoelectric-board of the which way cross-double-slit.Electrons are ejected. Collecting and measuring those electrons disclose the particle-nature of the photons. Namely before passing through the photoelectric board of the which way cross-double-slit, the photons behave as particles, not as waves.Rest of the photons passing through the slit-B, slit-C and slit-D and arriving at the photoelectric detection-screen behaves mysteriously, that: (1) the photons as particles knock electrons out of the photoelectric-detection-screen (photoelectric effect again), regardless how photons distribute on photoelectric-detection-screen, and thus, considered as particles; (2) the photons passing through

Table 1

 1

		Comparison of 1D-double-slit and 2D-cross-double-slit experiments
		Young's 1D-double-slit experiments	2D-cross-double-slit experiments
	Standard	Mystery of quantum world	Challenge standard interpretations of
		(Feynman)	1D-double-slit experiment
			More mysterious
	Which-way	Confirm	Violate
		Bohr's complementary principle	Bohr's complementary principle


Benjamin Wetzel
email: b.wetzel@sussex.ac.uk

Mikko Närhi

Cyril Billet

Jean-Marc Merolla

Shanti Toenger

Thibaut Sylvestre

Roberto Morandotti

Goëry Genty

Frederic Dias

John M Dudley
email: john.dudley@univ-fcomte.fr

Real-Time Measurements of Ultrafast Spontaneous Modulation Instability in Optical Fiber

Keywords:

We report on the real-time observation of highly-localized structures and rogue waves spontaneously emerging from noise during fiber propagation. Experimental results, obtained via a time-lens magnifier, are in good agreement with theory and numerical simulations.

Introduction

Modulation instability (MI) constitutes one of the most fundamental processes of nonlinear science, and develops in fiber optics when a weak perturbation on a continuous wave field experiences gain and evolves into highly-localized "breather" structures. Although MI has been widely studied over the last decades, there is still an intense interest in understanding its dynamics because, when triggered from noise, it generates high amplitude and statistically-rare "rogue waves" of importance in hydrodynamics and optics [1]. Due to experimental limitations, however, directly observing the ultrafast instability dynamics of noise-driven spontaneous MI in optics is extremely challenging. Although real time observations of this fundamental and spontaneous process have been reported via spectral measurements [2,3], experimentally accessing the temporal properties of the instability is usually limited by the bandwidth capability of current electronic detection systems. In this paper, we use an ultrafast time-lens magnifier system to perform direct measurements of real time temporal structures in MI. Our results show an extended series of transient high intensity breather pulses emerging from noise, and our statistical analysis allows the presence of long tails (rare events) to be readily observed and compared with numerical and analytic predictions.

Results

Fig. 1(a) presents the typical optical field evolution obtained from numerical simulations of fiber propagation and showing the evolution of noise-driven MI in the temporal domain. The simulations use our experimental parameters where a narrowband CW laser at 1552 nm was injected in up to 20 km of single mode fiber (β2 = -21.4 ps 2 km -1 , β3 = 0.012 ps 3 km -1 , γ = 1.2 W -1 km -1 ). Here, we see the development from noise of a distinct series of temporal breather pulses that develop into a quasi-regular breather train at an initial compression point (z ~11 km) before evolving chaotically with subsequent propagation. With input power P0 = 0.7 W, peak MI gain occurs at frequency Ω/2π ~ 43 GHz where Ω 2 = 2γP0/|β2| yielding (unstable) breather structures of ~ 2 ps duration. To observe these fluctuations experimentally, we use a time-lens magnification technique similar to that used to reveal rogue wave fluctuations in optical fiber turbulence [4]. In our setup, 2 ps MI breathers are stretched to 0.15 ns and readily captured in real-time using a 38 GHz photoreceiver and ultrafast oscilloscope (see Ref. [5] for further details). Temporal intensity profiles extracted at two propagation distances typical of these qualitatively different breather localization dynamics are respectively presented in Fig. 1(b) and (c), comparing experimental results obtained from our time-lens characterization technique (top panel) with numerical simulations (bottom panel). One can readily observe the excellent qualitative agreement between the transient pulse trains recorded from sequential time-lens measurements and numerical predictions.

Analysis

In order to provide a more quantitative comparison with numerical and analytical predictions, we analyzed the experimentally recorded temporal traces by extracting individual pulse profiles and associated peak intensities and pulse durations. In Fig. 2, we present the intensity histogram of the chaotic MI pulse train at 17 km and typical temporal structures profiles retrieved from experimental measurements (red lines). The results clearly show long tails in the peak intensity histogram (see Fig. 1(a)), associated with extreme event formation and in good agreement with analysis from stochastic numerical simulations (black). Fig. 2(b) shows a subset of the detected peaks with normalized intensity P/<P> ~ 9 which are superimposed and compared with the theoretical Peregrine soliton (PS) structure (dashed black line) [6], showing remarkable agreement with breather theory over the main intensity lobe region. An example of a temporal structure with higher peak intensity extracted from our experiment (red) is also illustrated in Fig. 2(c). The profile fit, obtained assuming a second-order breather solution and using the Darboux transformation (black line) [7], also provide excellent agreement with the pulse retrieved using our time-lens technique, thus illustrating the ability of the breather theory to describe the emergence of localized structures beyond their fundamental solutions. In addition, we observed that the relationship between the duration and power of spontaneously emerging temporal structures retrieved from experiment agrees well with simulations. They both cluster strongly around the theoretical prediction based on the analytic properties of NLSE elementary breathers and their collisions.

Conclusions

In conclusion, our results show how the optical time-lens can be used to provide direct insights into the properties of ultrafast modulation instability and to enable the direct capture of transient breather pulses emerging from noise. Besides this particular implementation, the technique has tremendous potential to shed light into the study of nonlinear instabilities and/or stochastic phenomena in a broader physical context.

Fig. 1 .

 1 Fig. 1. (a) Single-shot simulation showing temporal evolution of spontaneous MI. (b,c) Results obtained at (a) 11.7 km and (b) 17.3 km, showing intensity profiles from simulations (bottom, black) and experiments (top, red) as indicated. Plotted intensities P are normalized with respect to the average output background power <P> and plotted against time after rescaling to account for the effect of magnification. The experimental traces in (b,c) are reconstructed form sequential measurement windows (see dashed lines between measurements) for visual clarity.

Fig. 2 .

 2 Fig. 2. (a) Histogram of the peak intensities (shown as normalized probability density) from the chaotic pulse trains at 17.3 km, comparing experiment (red) and simulation (black). The inset plots the histogram on semi-logarithmic axes. (b) Results extracting 10 intensity profiles at with P/<P> ~ 9 from experiment (red lines) are compared with the analytic Peregrine soliton (black dashed line). (c) A peak of higher intensity P/<P> ~ 13, extracted from experiments (red lines), is compared with a higher-order breather fit (black dashed line).