Discrimination between dominant and chaotic modes in species

Discrimination between dominant and chaotic modes in species

Abstract

The concepts of population and species play a fundamental role in biology. The existence and precise definition of higher-order hierarchies, such as division into species, is open to debate among biologists. Here, we present a new metric, small , that uses data from the natural environment to distinguish between the size of the population and various data values that are outside the range of neutral logarithmic populations and are specific to a given species. We define this metric by modifying the Price equation. We show that prime numbers may be related to speciation by considering discontinuities in the Riemann zeta function, including Bose-Einstein condensation, while prime closed geodesics of the Selberg zeta function may represent populations. Calculation of the prime closed geodesics shows that noninteracting adaptive species are in the mode , while interacting neutral populations are in the mode . The border between fluctuating populations and ordered species is , which is proven theoretically and by observation. The mod 4 of primes corresponding to , the zero points of the Riemann zeta function, reveal adaptive and disadaptive situations among individuals. Furthermore, our model has been partially successful at predicting transitions of biological phases. The time-dependent fitness function and the precise Hubble parameter of a fitness space can be predicted by the Schwarz equation. We thus introduce the metric that is useful for discrimination between population data set and species data set. The significance of biological hierarchy is also discussed. In our patch with zeta dominance (PzDom) model, calculations only require knowledge of the density of individuals over time.

Center for Anatomical, Pathological and Forensic Medical Researches, Graduate School of Medicine, Kyoto University, Konoe-cho, Yoshida, Sakyo-ku, Kyoto, Kyoto 606-8501 Japan Current Address: 146-2-4-205 Nakagawara, Makishima-cho, Uji 611-0041, Japan E-mail: f.peregrinusns@mbox.kyoto-inet.or.jp

1 Introduction

Living organisms encompass several levels of scaling and hierarchy. Inside cells, protein molecules are on the order of nanometers, and they cooperate or compete by activation or inhibition of specific biological activities. Normal eukaryotic cells are on the order of m, and their activities are the consequence of interactions between the molecules inside of them. In multicellular systems, each cell has its own role, and these combine to determine the interactions between the cells in the system. In nature, the number of individuals increases and decreases following the particular dynamics of that population, as characterized by their intrinsic physiology and the interactions between individuals. Molecular biology and ecology have already elucidated certain roles for the hierarchies that are inherent in living organisms. Therefore, in order to compare the dynamics of communities of various biological taxa, it is important to have a common definition for species. For example, population, which is ranked just below species in the taxonomic hierarchy, is often defined both ecologically and demographically. [54] defines a population as “a group of organisms of the same species occupying a particular space at a particular time.” This is obviously a qualitative definition, and researchers utilize definitions of population that are appropriate in a given context. In contrast, the definition of a species has a long history of clarification. John Ray produced a biological definition of species in his Historia Plantarum: … no surer criterion for determining species has occurred to me than the distinguishing features that perpetuate themselves in propagation from seed. Thus, no matter what variations occur in the individuals or the species, if they spring from the seed of one and the same plant, they are accidental variations and not such as to distinguish a species… Animals likewise that differ specifically preserve their distinct species permanently; one species never springs from the seed of another nor vice versa (translated by E. Silk; [79]). Although he considered species as a static creation, it is important to note his foresight in distinguishing between variations within species and differences between species. In this paper, we further expand this to form a definition based on the discontinuities in the spectrum of the Selberg zeta function and the zero points of the Riemann zeta function. In his Systema Naturæ, Carl von [101] presented systematic definitions of biological taxa in different hierarchies, such as species, genus, order, and class, and these were later followed by family, phylum, kingdom, and domain. He also established binomial nomenclature as a standardized way to write a scientific name. Although his idea of a species remained static, this enabled a systematic approach to a qualitative estimate of the relatedness among different living organisms. The static image of creationism itself was doubted following the Philosophie Zoologique of Jean-Baptiste de Lamarck [18], in which Lamarckian inheritance was proposed. Although this approach has been ignored until recently, it has been reevaluated in recent studies of transgenerational epigenetic inheritance [35]. The idea of evolution was further developed as natural and sexual selection by Alfred Russel Wallace [105] in “On the tendency of varieties to depart indefinitely from the original type” and by Charles Darwin [15] in On the Origin of Species. However, the actual cause of speciation was still not understood. In his “Versuche Pflanzenhybriden,” Gregor Mendel [63] proposed that genetic information might be important for plant hybridization and evolution; however, we now understand that the results of his experiments were achieved under hybridization of a plant species with different alleles, not actual hybrid species. The modern definition of species is both an evolutionary and reproductive concept, and was described by Ernst Mayr in Systematics and the Origin of Species [61] as follows: “Species are groups of actually or potentially interbreeding natural populations, which are reproductively isolated from other such groups”. Until now, the essence of the difference between the concepts of population and species has been whether the concepts themselves are ecological/physical or evolutionary/genetic in nature. However, complex situations may arise in which a combined definition is required, and it may be difficult to determine whether the obtained data originated from populations or species, due to ambiguities in the overlapping definitions [13] [109]. For example, in “ring species”, there are continuous phenotypic characteristics and reproductive viability between physical neighbors, in contrast to reproductive inviability between the physical edges of the species. This means that reproductive isolation alone cannot be used to properly define a species. In contrast, reports of sympatric speciation demonstrate the metaphysical existence of species and speciation with discontinuity, even in environmentally nonisolated situations (e.g., [5]). It has been claimed that a biological species is a mere concept that describes unification of individuals and their genotypes within the population and populations within the species, by gene exchange resulting from bisexual reproduction and migration. Thus, the degree of this unification should be correlated with the presence of bisexual reproduction, although in reality, this correlation is very low, and uniparental species often differ very little from their biparental relatives living in similar conditions. Thus, isolated populations of biparental species often retain their specific identity for a long time, in spite of the absence of gene exchange and efficiency of migration [78]. We partially solved this, at least for Japanese Dictyostelia, by decoupling the short-term ecological time scale from the long-term evolutionary time scale, which differ on the order of [1]. That is, populations can be “neutral” within themselves, but an assemblage of populations as a whole is not always neutral. Therefore, the time scale of species/gene flow is assumed to be entirely different from that of population/migration. In this manuscript, we would like to further clarify a metric that can be used to discriminate between the dynamics of populations/species. We will use the following definitions in our analysis, which is based on actual data obtained from a natural environment. We define a population as a group of individuals of a sexually reproducing species inhabiting the same area and time, and a species as a sum of populations with genetically close relationships distinguishable by discontinuity of genetic distances among different species specific to each niche. In this way, a species cannot be disentangled from the actual interactions that constitute a community. The history of social interactions in biology began with William Donald Hamilton’s “The genetical evolution of social behavior I” [31], “The genetical evolution of social behavior II” [32], John Maynard Smith’s “Group selection and kin selection” [88], and George Robert Price’s “Selection and covariance” [76]. They established that genetic relatedness is important for maintaining cooperative phenotypes and evolution of living organisms. For co-evolvable nonrelatives, it is important to consider how reciprocal altruism maintains cooperation, as proposed in Robert Trivers’s “The evolution of reciprocal altruism” [99]. Finally, multilevel selection theories have been proposed, such as those presented in “Reintroducing group selection to the human behavioral sciences” [111] and Unto Others: The Evolution and Psychology of Unselfish Behavior [89]. These theories can explain the actual cooperative selection process of individual genes that are distantly related within a cell or an individual as a reproduction unit. Biological hierarchies are thus regarded as an important idea for analysis of an evolutionally process. Despite the lack of agreement among biologists, information theory can be used to analyze the dynamics of the concomitantly observed scales. However, the extent of genetically close relationships between different species cannot easily be distinguished from that of populations. Therefore, we present a new metric for defining a species, especially for use in adaptive situations: we found that the geodesic successfully discriminates between adaptive species () and chaotic population/species ().

To evaluate the metric, we need a model system. A candidate model is that presented by Kimura (1964) in the theory of diffusion equations in population genetics. In the model, genetic characteristics are used to demonstrate the dynamics. Unlike the original genetic model, here, instead of the gene frequencies, we investigate the dynamics using the ratio of the number of individuals to the whole population. This means combining genetic information and environmental effects with the number of individuals. We note that spatially distributed models that assume knowledge of underlying stochastic processes, which are usually drawn from birth/death, immigration/emigration, mutation/speciation, and niche differentiation, were developed to further understand the nature of observed populations and species (e.g., [56]). In the unified neutral theory [36] [37] [12], a local community (an ecological unit composed of a group of organisms or a population of different species occupying a particular area, usually interacting with each other and their environment) dominates a population of a few species and results in extinction of rarer species, deviates from the neutral logarithmic distribution observed in dominant species and rare species . A metacommunity is a set of local communities that are linked by dispersal, and on this scale, there is greater biological diversity; a nearly logarithmic distribution is observed for the population abundance of ranked species. As with the diffusion equation model from population genetics, the distribution depends not on individual adaptations but on random ecological drift that follows a Markov process [36] [37]; that is, information entropy, as measured by the Shannon index, is maximized [85] [34] [4]. Idiosyncrasies seem to be involved in Hubbell’s theory ([77]; see also [1]). There is also a report that considers the MaxEnt (a model for a force that maximize entropy during time development) of geographic distribution [71]. Thus, the unified neutral theory is part of information theory. It has three characteristic parameters: population size, point mutation rate, and immigration rate. Please note that we only utilize the (a normalized fitness) part of Kimura’s theory without including the Hardy-Weinberg principle. Our model is unrelated to gene frequencies of different alleles as in the Hardy-Weinberg principle, because we utilize , not with gene frequency . Therefore, in our model, it is not necessary to assume genetic equilibrium. Please also note that we only utilize Hubbell’s theory as a basis for calculating the extent of the difference from an ecologically neutral situation. By modifying these theories, we were able to distinguish neutral populations with a shorter time scale from adaptive species with a longer time scale for a set of observed data of Dictyostelia; this data set had a significantly small immigration rate () [1]. Furthermore, in this model, we consider only population size and not mutation rate (as in [44], and [27]), or immigration rate; this is done because we are interested in the dynamics on the ecological time scale, not on the evolutionary time scale. We also consider the effects due to the randomness of population dynamics and the directionality of species dynamics in nature; this is in addition to the theoretical randomness discussed by [27] and the theoretical directionality (i.e., Bose-Einstein condensation) discussed in [6].

We propose an original model that uses a different definition of entropy than that used by [34] and a new definition of temperature: an integrative environmental parameter that determines the distribution of population/species deduced from the logarithmic distribution of populations/species. The units of this parameter are set to cells/g, and it is a half-intensive parameter, as described in the first parts of Results. We define new metrics that are based on statistical mechanics [28] [4] [95] in order to distinguish and interpret species and population counts in mixed communities; we apply this to an actual community of Eastern Japanese Dictyostelia [1]. The use of statistical mechanics to interpret biological systems began with [58], proceed to the disastrous complexity of the Hamiltonian described by [42], and continued with the Lotka-Volterra equations of N-interacting species in an artificially noisy environment [90]. [42] also showed an interesting model for a time-developing system; however, in this paper, we consider a static system. Kerner’s model and our model belong to different mathematical spaces, and a set of mathematically rigorous studies is required to precisely describe their interrelationship. This is different from a time-dependent ecosystem assembly model that is restricted to finite Markov chains, such as was proposed by [72] or [11]. Note that we will describe the nonrandom directionality of the model, which is based on number theory with . Importantly, we were able to calculate the different sets of critical temperatures and Weiss fields (with Bose-Einstein condensation) at which various natural first-order phase transitions take place among species or populations (where by ‘phase transitions’ we mean community moves from chaos that are caused by neutrality or nonadaptive situations and result in an increase of or domination by a particular species, or moves from that to domination by a particular population within a species). The order parameter in this model is large . This complex phase transition nature of different hierarchies in the wild was not fully explained by [34]; it was briefly mentioned by [4] as part of the relative entropy. The model shows that the populations of some highly adapted species are much more stable than those of others.

The results and discussions presented below are entirely original; to the best of our knowledge, no previous studies have considered the biological view proposed herein. In this manuscript, we introduce an index, small , to distinguish between populations and species; the value of this index is high for a species and low for a population. To begin, we modified the Price equation [76] in order to develop our index, small , which is based on the theorem and Weil’s explicit formula [106] [96] [110]. The Price equation describes evolution and natural selection, and here it is used to replace gene frequencies with the proportion of individuals in a given population or species. The small index is related via covariance and expectation to the Price equation. The nontrivial zeros of the Riemann function provide information about the bursts or collapses of a population. In this model, speciation is thus related to prime numbers; that is, a prime ideal indicates the status of a specific species in the system, and time-dependent multiplication of the fitness can be calculated by utilizing these primes. The border between fluctuating populations and ordered species is , which can be proven theoretically and by observation. We then calculate the unique equations of the model in the Maass form and examine the spectra of the data. Use of the Selberg zeta and Hasse-Weil to calculate the prime closed geodesics clarifies that the noninteracting adaptive species world (an integrative space of time and other dimensions) is in the mode , while the interacting neutral populations are in the mode . Combining these calculations with phylogenetical asymmetry, we determine whether the observed hierarchy of data represents chaotic populations/nonadaptive species or adaptive species in genuinely successful niches. Our model has been partially successful at predicting imminent transitions between biological phases (adaptation/disadaptation). By utilizing the Schwarz equation, we also determined the time-dependent fitness function that matches the observations. Additionally, web-based formalism [30] based on a combination of supersymmetry and an analogy to the transactional interpretation of quantum mechanics leads to a nine-dimensional model (three-dimensional nature three-dimensional fluctuations). The idea of a fitness space leads to a precise time-dependent Hubble parameter for that space, for an appropriate timescale. Finally, hybrid inviability attributes the scale of the population to the observed hierarchy. Recently Rodríguez and colleagues reported a physical framework for applying quantum principles to ecology (e.g., [81] [83]); however, this was based on thermodynamics and is different from the more mathematical/informational, nonthermodynamical approach described here. The proposed model combines information theory and observations from nature to bring new understanding to the biological ideas of population and species; this is different from the physical and theoretical thermodynamical approach used by [100]. Here, a patch is defined to be a small plot or piece of land, especially one that produces or is used for growing specific organisms. We call our model the patch with zeta dominance (PzDom) model, and it is only necessary to evaluate in order to determine whether a population is chaotic or dominated by species; the border is at . The model requires only the change in density of individuals over time. We will also discuss the significance of biological hierarchies. We propose an approach that will allow future researches to explore the nature of hierarchical systems.

2 Results

2.1 Universal equation for evolution based on the Price equation and logarithms

The neutral logarithmic distribution of ranked biological populations, for example, a Dictyostelia metacommunity [1], can be expressed as follows:

(1)

where is the population density or the averaged population density of species over patches, and is the index (rank) of the population. The parameters and the rate of decrease are derived from the data by sorting the populations by number rank. We also applied this approximation to an adaptive species in order to evaluate the extent of their differences from neutral populations [1]. We note that this approximation is only applicable to communities that can be regarded as existing in the same niche, and thus it is not applicable to co-evolving communities in nonoverlapping niches.

Based on the theory of diffusion equations with Markov processes, as used in population genetics [43], we assume that the relative abundance of the populations/species is related to the th power of ( in [43]) multiplied by the relative patch quality ( in [43]) (that is, ; see also [43]). In this context, represents the relative fitness of an individual; this varies over time and depends on the particular genetic/environmental background and the interactions between individuals. is a relative environmental variable and depends on the background of the occupying species; it may differ within a given environment if there is a different dominant species.

To better understand the principles deduced from Kimura’s theory, we introduce the Price equation [76]:

(2)

Note that , where and when ; we use this instead of the gene frequency in Price’s original paper. The relative distance between the logarithms of norms and the rank (where is the relative entropy from a uniform distribution as , and both logarithms are topological entropies) will be discussed below when we consider the Selberg zeta analysis [38]. Next, we assume that for a particular patch, the expectation of the individual populations/species is the averaged (expected) maximum fitness; this is to the power th, when is the average among all populations or the sum of the average over all patches among all species (). This is a virtual assumption for a worldline (the path of an object in a particular space). This assumption is because a population seems to be in equilibrium when it follows a logarithmic distribution [36] [37] [1] and species dominate [1]. We will prove below that the scale-invariant parameter small indicates adaptations in species in neutral populations. Under the assumptions in this paragraph, is and is approximately . If we set , and , with . When , but is removed from the calculation by an identical anyway, during introducing the equation below. Dividing the Price equation by , we obtain

(3)

We will consider the case when in a later subsections; see Eqns. (2.11) and (2.12). For simplicity, we will denote as and as . Note that , and and correspond to the -charges of the fermionic and bosonic functions, respectively. Note also that as a fermion, is a mutually exclusive real existence in the fitness space, is derived from , and values may be stacked as a boson.

2.2 Introducing allows us to distinguish types of neutrality

Zipf’s law is used to statistically analyze probability distributions that follow a discrete power law. For example, if the distribution of can be approximated by a logarithmic relation with a parameter , then Eqn. (2.1) holds. Zipf’s law is related to the Riemann zeta function as follows:

(4)

and this will normalize the th abundance by . We set absolute values of and for approximating both the (, ) and (, ) cases. If we set the density matrix to be , then the von Neumann entropy would be . Note that this model is a view from the first-ranked population/species , and either cooperation or competition is described by the dynamics/dominancy of . The partial trace of state over is . When , the logarithm of the von Neumann entropy can also be interpreted as the Rnyi entropy (the collision entropy of relative to a population/species): . To examine the difference between population and species dynamics, linearization of the model (4) leads to

(5)

Therefore, implies , implies , and implies . Each of the local extrema of thus represents a pole for the population/species, and a large (resp. small) value of represents a small (resp. large) fluctuation. Only those points of that are close to zero represent growth bursts or collapses of the population/species. According to the Riemann hypothesis, at these points, the following equation will approximately hold: and , where is a natural number independent of the population/species rank.

Taking the logarithm of Eqn. (2.4), we obtain

(6)
(7)

Therefore,

(8)
(9)
(10)
(11)
(12)
(13)
(14)

is obviously scale invariant if is a fixed number in a particular system. Note that and can thus be approximated using data from the distribution of . When for a given species, can be calculated by an inverse function of . For convergence, it is necessary that , , , and . We will also assume that and when a single population/species was observed. In [1], we analyzed the values using both the relative abundances of the population and the species; we determined that they give significantly different results (see Figure 1 and Table 1). The population values are restricted to those between and , while those of species are often greater than . This proves that populations behave neutrally, while species are more likely to dominate; this will be discussed in more detail below. When is larger than , the dynamics correspond to that of species, as will be discussed below. In Table 1, some (6/54) values greater than 2 are indicated in red; this indicates that these were not observed in a population of 162 samples. In the following, the parameter is the small of this model. Note that when , the calculation of is the same for both a population and a species, and the border clarifies the distinction of a neutral population () versus a dominant species (); when , the distinction of population versus species only affects the calculation of as later described.

Figure 1: Variation of with time for species and populations in two quadrats. P. pallidum: Polysphondylium pallidum; D. purpureum: Dictyostelium purpureum; P. violaceum: Polysphondylium violaceum. The top two panels show the data for three different species, and the lower two panels show the data for nine different point quadrats.
P. pallidum (WE) D. purpureum (WE) P. violaceum (WE) P. pallidum (WW) D. purpureum (WW) P. violaceum (WW)
May 1 - 1 1 - 1
June 0.7693 1.8305 1.258 - 1 1
July - 1 1 1.2619 0.5752 1.7742
August - 1 1 3.4223 2.8795 1
September 3.5762 3.0777 1 1.7897 2.1411 0.3186
October 5.1648 4.9423 1 3.4417 1 2.9047
November 0.7481 1 2.056 - 1 1
December - 1 1 - 1 1
January - 1 1 - 1 1
P. pallidum (WE) D. purpureum (WE) P. violaceum (WE) P. pallidum (WW) D. purpureum (WW) P. violaceum (WW)
May 0 76 0 0 83 0
June 123 209 52 147 0 0
July 1282 0 0 80 215 320
August 1561 0 0 1330 181 0
September 901 107 0 809 77 649
October 1069 35 0 799 0 107
November 60 0 101 336 0 0
December 190 0 0 711 0 0
January 29 0 0 99 0 0
WE a b c d e f g h i
May 1.0336 1 1.9442 1 1 1 1.0821 1 1
June 0.5328 1 0.1545 0.0332 1.1928 1 0.1374 1.076 1.0071
July 1 1 0.6131 1.1497 0.3117 0.2016 1 1 1.148
August 1 1.4925 1.7167 0.6348 0.7075 0.3523 1 0.3502 1
September 1 1.1361 1 1 1 1.3035 1.0325 1.7248 1.085
October 1 0.6746 0.6937 0.6092 1 0.7836 0.9259 0.886 1.1746
November 1 1 1 1 1 0.7481 0.472 1 1
December 1 1 1 0.3429 1 0.2455 0.1712 0.5647 1
January 1 0.9516 1 1 1 1 1 0.8666 1.215
WW a b c d e f g h i
May 1 1 0.7125 0.8782 1 1 1 1 1.473
June 0.708 1.0735 0.7614 0.1056 1 1 0.7883 1 0.8612
July 0.3888 1 0.8635 1 1 0.8614 1 0.3629 0.263
August 1.0524 1 0.756 0.5644 1.3367 1.1911 1.0473 0.3985 1
September 0.4918 0.4236 0.4243 0.4427 0.4535 0.4969 0.5051 0.3985 1
October 1 0.8073 0.8982 0.6913 1 0.3219 1 0.9284 0.8523
November 1.1334 1 1 1 1 1 1 1.7225 1
December 1.1214 1.7164 1 1.1833 0.9208 1.0718 1 1.0594 1.1375
January 1 1.2501 1 1 1 1 1 1.5151 1.2228
WE a b c d e f g h i
May 680 0 94 0 0 1392 424 0 0
June 1120 0 2131 2580 221 2640 2270 384 372
July 0 0 1573 469 2613 3200 3680 0 580
August 0 331 170 1728 1800 3760 0 3267 4800
September 0 1240 0 4320 0 418 820 1307 960
October 0 1413 1680 2360 3600 1020 594 736 313
November 0 0 0 907 0 540 540 0 0
December 0 0 0 580 0 787 773 376 933
January 0 457 0 0 1300 0 0 391 560
WW a b c d e f g h i
May 840 0 384 457 0 0 0 0 109
June 1088 421 869 3160 0 0 1140 3400 1320
July 1680 0 613 0 0 720 2880 1933 2400
August 704 0 1627 2496 288 457 860 3520 4640
September 1760 1760 1627 1386 1440 2016 1147 2640 3480
October 3520 960 613 1350 0 2816 0 667 1380
November 760 0 0 0 0 0 2640 800 0
December 590 124 0 440 1600 784 4400 1013 2000
January 1307 331 0 0 0 0 0 160 560

WE: the Washidu East quadrat; WW: the Washidu West quadrat (please see [1]. Scientific names of Dictyostelia species: P. pallidum: Polysphondylium pallidum; D. purpureum: Dictyostelium purpureum; and P. violaceum: Polysphondylium violaceum. For calculation of , see the main text. is the cell number per 1 g of soil. Species names for Dictyostelia represent the corresponding values. a - i indicate the indices of the point quadrats. Red indicates values of species that were approximately integral numbers greater than or equal to 2.

Table 1: and values.

If , there are two possibilities: (i) (where is the function) must be true in order for to converge, and for to converge, we need ; (ii) For to converge, we need when . In this case, we have . When (i) holds, we have true neutrality among the patches. Both cases (i) and (ii) can be simply explained by a Markov process for a zero-sum population, as described in Hubbell (2001). In both cases, , and the populations are apparently neutral for . When there is true neutrality in both the populations and the environment, , we say that there is neutrality. When there is apparent neutrality of with , we say that there is harmonic neutrality. The value of can thus represent the characteristic status of a system. We now consider situations in harmonic neutrality. If the average fitness is , the individuals are more adaptive than those individuals in harmonic neutrality with . Therefore, is an indicator of adaptation beyond the effects of fluctuation from individuals with harmonic neutrality. Also note that represents the bosonic with an even number of prime multiplications, the fermionic with an odd number of multiplications, or as the observant, which can be divided by . This occurs when two quantized particles interact. Note that in about 1400 CE, Madhava of Sangamagrama proved that

This means that the expected interactions of a large number of fermions can be described as . The interaction of the two particles means multiplication by , which results in , as is discussed in the next subsection.

2.3 Introducing explains adaptation/disadaptation of species

Next, we need to consider theory, Weil’s explicit formula, and some algebraic number theory to define precisely. theory is based on an ordinary representation of a Galois deformation ring. If we consider the mapping to that is shown below, they become isomorphic and fulfill the conditions for a theta function for zeta analysis. Let (), where is complex. First, we introduce a small that fulfills the requirements from a higher-dimensional theta function. Assuming , , , , as a higher-dimensional analogue of the upper half-plane, a complex , (Hecke ring, theorem), dual with [106], , could be set on and is a part of . Thus, , and the functions described here constitute a theta function. The series converges absolutely and uniformly on every compact subset of [68], and this describes a (3 + 1)-dimensional system. This is based on the theorem and Weil’s explicit formula (correspondence of zeta zero points, Hecke operator, and Hecke ring); for a more detailed discussion, see [106], [96], [110], and [45]. This model can be partially summarized as an adjunction:




Since is real, and . Thus, is related to the absolute value of an individual’s fitness, and is the time scale for oscillations of and is the argument multiplied by the scale. Therefore,

(15)
(16)

Note that when , is indefinable because there is no expectation of population increase/decrease, even though it is possible to calculate some ill-defined values for .

When and , and we usually have harmonic neutrality. This case was often prominent in the Dictyostelia data. When and , and we usually have neutrality. When , the population/species can diverge when , that is, when it equals the imaginary part of a nontrivial zero of . Thus, the population/species can diverge when . We also note that

(17)

so for quantization (compactification of , which is a natural number; generally, quantization refers to the procedure of constraining something from a continuous state to a discrete state), assuming that the distribution of population/species numbers is in equilibrium and is dependent on interactions between them, as described in the previous subsection; thus, . With the Riemann-von Mangoldt formula [102], the number of nontrivial zero points is

(18)

so that . Note that from Stirling’s approximation, , indicating that the population/number of species is equal to the sum of the relative entropies. On the other hand, . Therefore, for populations/species as a whole, . Since the axis and the axis are orthogonal and the scale of the latter is times that of the former, (Table 2) gives a good fit to a highly adaptive population/species growth burst or collapse for an entire population or species and can be calculated as

(19)

If we set a particular unit space for calculation of population density, is obviously a scale invariant for the case of species, where is a scale invariant to system size, is the order of the ratio of the sum of population densities of a particular species to the number of patches, and is the ratio of the sum of the population densities to the number of patches. For a given population, if is the order of the population density of a particular patch, it is also a scale invariant to the sampling size, assuming that a sufficiently large number of samples are collected. Nontrivial zeros of are prime states (those related to prime numbers), and they are indicators of imminent growth bursts or collapses of the population/species. Note that can also be expressed as follows [80]:

(20)

In order to avoid a discontinuity at a zero of , is 1/2 or an integer. Zero points of thus restrict both and to a particular point. Note that consists of the imaginary parts of the zeros, which are not integers themselves in the quantization. This model is found to be consistent with the results for some species, as shown in Table 1; these are shown in red, as follows: () = (3.078, 14.99, 0.01003), (4.942, 38.74, 0.01723), (2.056, 275.5, 2.994), (2.8795, 13.80, 0.009451), (2.1411, 115.9, 0.05094), (2.9047, 13.93, 0.004941). Thus, this model gives a logical explanation for the observed quantization in some situations for the Dictyostelia species, and for a population, the data do not seem to be at a zero point, according to the value. Except for the case (2.056, 275.5, 2.994), they are in a situation similar to a Bose-Einstein condensate; this is discussed in the later sections.

WE P. pallidum WE D. purpureum WE P. violaceum WW P. pallidum WW D. purpureum WW P. violaceum WE P. pallidum WE D. purpureum WE P. violaceum WW P. pallidum WW D. purpureum WW P. violaceum
May May
June 147.4228 30.4249 June 8.1822 148.6187 31.1005
July 40.9187 174.7542 July 39.3062 5.3315 174.5203
August 21.0220 14.1347 August 22.6267 13.7962 2.4878
September 21.0220 14.1347 52.9703 116.2267 September 23.2403 14.9897 2.4101 53.1153 115.8800 2.0281
October 48.0052 37.5862 21.0220 14.1347 October 45.6795 38.7450 2.0958 22.6675 2.4764 13.9291
November 14.1347 275.5875 November 7.7262 15.3777 275.5449
December December
January January
WE P. pallidum WE D. purpureum WE P. violaceum WW P. pallidum WW D. purpureum WW P. violaceum WE a WE b WE c WE d WE e WE f WE g WE h WE i WW a WW b WW c WW d WW e WW f WW g WW h WW i
May 1 1 1 1 1 1 1 1
June 1 239 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
July 17 1 317 1 1 1 1 1 1 1 1 1 1 1 1
August 3 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
September 3 2 1 31 157 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
October 23 13 1 3 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
November 1 2 677 1 1 1 1 1 1
December 1 1 1 1 1 1 1 1 1 1 1 1 1
January 1 1 1 1 1 1 1 1 1
for WE P. pallidum WE D. purpureum WE P. violaceum WW P. pallidum WW D. purpureum WW P. violaceum WE a WE b WE c WE d WE e WE f WE g WE h WE i WW a WW b WW c WW d WW e WW f WW g WW h WW i
May 1.021 0.885 1.017 1.029 1.015 0.997 1.009 0.958
June 0.938 0.946 0.863 0.971 0.993 0.999 0.989 0.997 0.994 0.991 0.996 0.973 0.994 0.995 0.995 1.001 0.999 0.997
July 0.884 1.000 0.972 0.989 0.988 0.991 0.990 0.989 0.983 1.002 1.002 0.999 0.985 0.990
August 0.647 0.673 0.900 0.902 0.965 0.979 0.987 0.987 0.992 0.995 0.981 0.976 0.931 0.976 0.995 0.985 0.993
September 0.639 0.644 0.874 0.922 1.000 1.025 1.112 0.970 1.047 0.782 1.013 1.007 1.008 1.008 1.008 1.008 1.007 1.007 1.007 0.988
October 0.679 0.657 0.646 0.665 0.981 0.989 0.962 1.000 1.000 1.002 1.000 0.974 0.999 1.001 0.996 0.969 0.988 0.997 0.998
November 0.946 1.008 0.969 0.985 1.007 0.999 0.968
December 0.983 0.990 0.994 1.005 0.999 0.988 0.876 0.987 1.005 0.986 0.998 0.986 0.988
January 0.994 1.008 1.008 0.999 1.037 1.017 1.057 1.044 1.053

WE: the Washidu East quadrat; WW: the Washidu West quadrat (please see [1]). Scientific names of Dictyostelia species: P. pallidum: Polysphondylium pallidum; D. purpureum: Dictyostelium purpureum; and P. violaceum: Polysphondylium violaceum. consists of the theoretical imaginary parts of the Riemann zero points corresponding to and . a - i indicate the indices of the point quadrats. The of populations are not shown because the are so small that and do not correspond to each other. In this case, is set at 1. For calculation of and , see the main text. Blank values are undefinable. Red indicates species for which was approximately 2/3.

Table 2: , , , and values.

When and , the population/species distribution is always structured without neutrality, since there is no zero point. On the other hand, when and , and harmonic neutrality usually occurs. This is true for Dictyostelia. Usually, is the equilibrium state. Therefore, populations/species reach either neutrality or harmonic neutrality.

We note that if we assume , the information entropy (), which is the same as the Shannon index, can be written as

(21)

That is, the expected output of information entropy for the th population/species is . Therefore, maximizing for is the expected result from maximizing the information entropy. The populations/species thus usually fluctuate for . If there is no force against entropy, this is the expected future. Additionally, as zero points of are approached, information is minimized and approaches negative infinity; this is the opposite what occurs when , and it indicates ordering/domination. The concept described here is analogous to in [2], where , , and are the population entropy, growth rate, and reproductive potential, respectively. That is, is analogous to , and the reproductive potential is analogous to .

Interestingly, according to [33],

(22)
(23)

Since ,

(24)

gives an indicator of the distance from the speciation line on . Since , is the sum of (the probability that an individual in the top populations/species replaces an individual of the th population/species) (the number of combinations/entanglements, assuming the constituents are equivalent). This represents the expected number of entanglements and approaches when the community progresses far beyond the speciation phase. Ideally, the development of a community begins when and eventually arrives at the expected state.

A virtual world of adaptation of a particular species/population is thus represented on a purely imaginary axis of small . If we want to calculate the synthetic fitness, , we can simply evaluate , where is the manifold of an space and is the Lefschetz operator [84]. The hard Lefschetz theorem indicates whether the dimension of the space is decreasing or increasing with . The Hasse zeta function is

(25)

which is the zeta function of when is a one-variant polynomial ring.

2.4 Selberg zeta-function and Eisenstein series reveal Maass wave form as a function of probability of population number distribution and genetic information

Once we have obtained the small for a system, we then apply the automorphic -function in order to calculate the Eisenstein series. This allows us to understand the relation of small to the diffusion equation in neutral theory and to obtain further information about the prime closed geodesics, which are used to further analyze the intra-population/species interacting mode [66]. The prime closed geodesics on a hyperbolic surface are a primitive closed geodesic that traces out its image exactly once. The expression prime obeys an asymptotic distribution law similar to the prime number theorem. For this application, we must discriminate between the discrete spectrum and the continuous spectrum of a Selberg zeta-function. We can then proceed to calculate the Eisenstein series that corresponds to the discrete spectrum.

The Selberg zeta function is defined by

(26)

where is a prime closed geodesic. The determinant of the Laplacian of the complete Selberg zeta-function is

(27)
(28)

where , and and denote discrete and continuous spectra, respectively [66]. It is evidently true both in populations and species, that the discrete spectrum dominates the continuous spectrum by (populations) or (species). When is assumed to be the dimension of a compact oriented hyperbolic manifold, the number of prime closed geodesics in a Selberg zeta-function is [17]

(29)

Table 3 lists the calculated determinants, Magnus expansion/Eisenstein series , and other parameter values. is defined as follows:

(30)
(31)
(32)

where is the modified Bessel function of the second kind, and is the divisor function [66].


WE P. pallidum WE D. purpureum WE P. violaceum WE P. pallidum WE D. purpureum WE P. violaceum WE P. pallidum WE D. purpureum WE P. violaceum
May
June 0.1166-3.721 -2907+9002 75.36+6.341 107.9-246.1 -1.676E+293-4.636E+293 1.014E+50-1.183E+50 -1.553E+24-3.218E+24 -4.658E+134-3.981E+134
July
August
September -5.049E+4-5.802E+4 -4019+1061 5.121E+31+2.8968E+31 3.702E+14+2.907E+15 -9.081E+95-8.357E+95 1.940E+56-1.809E+56
October 7.881E+7-3.648E+8 -6.713E+7+2.216E+7 4.255E+75+4.292E+75 6.913E+61+4.055E+61 1.352E+201+5.734E+200 1.292E+168+5.540E+167
November -5.847-0.1007 -8.654E+4+5762E+4 -25.17-86.36 -2.837E+22-3.789E+22
December
January
WW P. pallidum WW D. purpureum WW P. violaceum WW P. pallidum WW D. purpureum WW P. violaceum WW P. pallidum WW D. purpureum WW P. violaceum
May
June
July 100.7-20.98 -4.174+1.950 -6908+6514 8.135E+66+5.148E+65 0.0008612-0.0004888 6.966E+173-8.522E+172 -1.315E+11+5.805E+9
August 4754+4.299E+4 149.7-1908 2.338E+30+3.249E+30 -4.876E+12-2.196E+12 6.660E+91+2.055E+92 2.747E+51+8.049E+50
September -1067-599.3 -1.533E+4-2.132E+4 -0.6369-0.1195 1.669E+95+5.284E+94 9.423E+224-1.419E+224 -6.086E-5+7.396E-6 3.359E+239-1.062E+238 -9.681E-6+1.044E-5
October -2671+4.615E+4 1118-1781 -2.835E+30-2.7966E+30 -1.536E+13-6.848E+12 5.946E+92+6.116E+92 9.392E+50+2.948E+51
November
December
January
WE population a b c d e f g h i
May 1.689-0.4811 1.362+2.347 1.684-0.5220 1.695-0.4159
June 0.7967-0.2722 1.489-0.6494 1.482-0.5900 -3.637-1.471 -3.265+1.460 1.491-0.6431 -3.888+0.8620 -3.335+1.431
July 1.206-0.4145 -1.224+1.828 1.479-0.6824 1.494-0.6664 -0.1460+1.035 -1.209+1.820
August -5.434+0.4162 -3.909-6.883 1.122-0.3407 0.9469-0.1481 1.459-0.6731 1.460-0.6736 -0.3406+1.133
September 0.8091+0.6884 1.093+0.1722 0.2645+1.505 1.035+0.2829 -3.165+3.610 0.9290+0.4783
October 0.5892+0.01517 0.5005+0.09150 0.8522-0.2121 -1.806+1.614 0.004016+0.5061 -1.085+1.269 -0.73946+1.055 -3.740+1.533
November 0.4751+0.6602 1.147-0.2378 1.444-0.6489
December 1.108-0.5486 1.373-0.6452 1.464-0.6530 -0.4479+0.2800 -1.182-2.214
January 1.390-0.2006 1.352-0.08658 1.443-0.3694 1.092+0.5930
WE population a b c d e f g h i
May -0.0007441+0.0015797 5.899E-5+0.0002192i -0.0008437+0.001642 -0.0006145+0.001484
June -0.0003174+0.0001383 -0.003930-0.0004962 -0.006792-0.002592 -7.3801931E-8+1.306E-7 -1.751E-06+2.008E-06 -0.004276-0.0006772 -5.566E-7+7.463E-7 -1.5805E-06+1.839E-6
July -0.0006207+0.0003732 -1.225E-5+02092E-5 -0.002707+0.00037618 -0.004148-0.0001904 -4.487E-5+5.822E-5 -1.246E-6+2.119E-5
August -1.520E-7+6.738E-7 6.547E-9+2.447E08 -0.0004982+0.0003134 -0.0003172+0.0002353 -0.002185+0.0004423 -0.002205+0.0004393 -3.666E-5+4.691E-5
September -9.872E-5+0.0001962 -0.0002168+0.0003280 -3.217E-5+9.374E-5 -0.0001812+0.0002923 9.017E-9+8.620E-6i -0.0001342+0.0002402
October -0.0001907+0.0001242 -0.0001636+0.0001110 -0.0003148+0.0001763 -8.884E-06+1.060E-5 -7.641E-5+6.192E-5 -1.960E-5+2.050E-5 -2.931E-5+2.858E-5 -1.053E-6+1.764E-6
November -9.076E-5+0.0001247 -0.0004263+0.0003626 -0.0016388+0.0006494
December -0.0006964+0.0001186 -0.001613+4.996E-5 -0.002808-0.0002560 -6.020E-5+2.657E-5 1.141E-08+2.077E-08
January -0.0004369+0.0006407 -0.0003490+0.0005662 -0.0006340+0.0007780 -0.0001126+0.0002949
WE population a b c d e f g h i
May 0.1655-0.2418 0.2766+0.05377 0.1543-0.2511 0.1805-0.2277
June 0.001801-0.1721 -0.2118-0.1539 -0.2727-0.07898 0.05182+0.01284 0.07611-0.02970 -0.2218-0.1457 0.06921-0.009994 0.07565-0.02774
July 0.01072-0.2132 0.1224-0.06005 -0.1478-0.2161 -0.2099-0.1794 0.1130-0.1072 0.1223-0.06059
August 0.08196+0.03297 0.03305+0.05488 0.02127+2.796 0.04785-0.1872 -0.1213-0.2216 -0.1225-0.2213 0.1107-0.1005
September 0.1568-0.1298 0.1318-0.1715 0.1731-0.07667 0.1388-0.1618 0.1510+0.04875 0.1487-0.1457
October 0.04334-0.1625 0.04902-0.1571 0.02119-0.1795 0.09455-0.06120 0.07109-0.1302 0.09101-0.08467 0.08721-0.09754 0.08633-0.01128
November 0.1211-0.1331 0.05783-0.2045 -0.06664-0.2427
December -0.06560-0.1717 -0.1281-0.1727 -0.1802-0.1571 0.03012-0.1168 0.03424+0.007115
January 0.1250-0.2098 0.1385-0.1962 0.09781-0.2320 0.1815-0.1306
WW population a b c d e f g h i
May 1.499-0.2584 1.552-0.6434 1.535-0.4636 1.037+1.278
June 0.6953+0.005382 -1.772+1.771 0.4713+0.2177 1.490-0.6277 0.3437+0.3362 -1.080+1.434 -0.05846+0.6924
July 1.340-0.6114 -0.8316+1.041 -0.8131+1.030 -2.186+1.656 1.374-0.6322 1.461-0.6683
August -0.7692+1.442 0.7818+0.02336 1.242-0.4748 -3.727+2.059 -2.023+2.073 -0.7302+1.415 1.430-0.6536 -0.3950+1.159
September -2.852+0.9236 -1.675+0.4108 -1.690+0.4171 -2.040+0.5713 -2.238+0.6590 -2.917+0.9497 -3.009+0.9845 -1.197+0.2043 4.910+4.391
October -1.080+1.434 0.2468+0.4246 -0.2961+0.8878 0.7576-0.05433 1.452-0.6691 -0.5079+1.051 -0.004859+0.6467
November 1.804-0.5315 1.757-0.6573 1.943+0.5431
December -0.3324+1.410 -6.858-1.056 -0.7473+1.747 0.6101+0.4127 -0.04755+1.143 0.3004+0.7754 0.01739+1.077 -0.4338+1.498
January 1.683-0.5205 1.701-0.1286 1.669+0.5323 1.701-0.1815
WW population a b c d e f g h i
May -0.0004866+0.0008370 -0.001444+0.001296 -0.0007981+0.001050 -2.963E-5+0.0002245
June -0.0002117+0.0001521 -8.412E-6+1.179E-5 -0.0001416+0.0001132 -0.005478-0.001234 -0.0001147+0.00009658 -1.773E-5+2.177E-5 -6.296E-5+6.057E-5
July -0.001241+0.0003091 -2.753E-5+2.544E-5 -2.814E-5+2.590E-5 -6.121E-06+7.211E-06 -0.001449+0.00030657 -0.002520+0.0001481
August -2.193E-5+3.069E-5 -0.0002230+0.0001822 -0.0007333+0.0003824 -1.135E-6+2.897E-6 -5.782E-6+1.052E-5 -2.296E-5+3.184E-5 -0.001741+0.0004760 -3.468E-5+4.423E-5
September -3.498E-6+1.108E-6 -1.801E-5+4.372E-6 -1.770E-5+4.313E-6 -1.163E-5+3.065E-6 -9.034E-6+2.485E-6 -3.061E-6+9.868E-7 -2.465E-6+8.207E-7 -3.106E-5+6.698E-6 -0.8425+0.4974
October -1.773E-5+2.177E-5 -9.851E-5+8.592E-5 -4.566E-5+4.691E-5 -0.0002390+0.0001657 -0.002193+0.0003354 -3.484E-5+3.774E-5 -6.789E-5+6.428E-5
November -0.0006908+0.002217 -0.001149+0.002449 0.0001674+0.0009833
December -2.921E-5+4.922E-5 -1.297E-9+3.961E-7 -1.748E-5+3.322E-5 -0.0001276+0.0001491 -4.315E-5+6.628E-5 -7.359E-5+9.922E-5 -4.742E-5+7.121E-5 -2.563E-5+4.454E-5i
January -0.0008408+0.001635 -0.0002792+0.001140 -2.042E-5+0.0006735 -0.0003226+0.001194
WW population a b c d e f g h i
May 0.1437-0.2153 0.03308-0.2796 0.1036-0.2473 0.2223-0.06741
June 0.05090+1.831 0.1046-0.05581 0.06576-0.1535 -0.2498-0.1277 0.07229-0.1455 0.1024-0.07878 0.08685-0.1230
July -0.07951-0.2060 0.08284-0.09623 0.082594-0.09693 0.09030-0.05283 -0.09435-0.2059 -0.1547-0.1934
August 0.1142-0.08209 0.06316-0.1729 -0.008232-0.2156 0.1045+0.0006674 0.1160-0.03936 0.1139-0.08372 0.1139-0.08372 0.1100-0.09872
September 0.03142-0.05401 0.01886-0.08188 0.01906-0.08157 0.02353-0.07398 0.02575-0.06955 0.03193-0.05197 0.03261-0.04876 0.01136-0.09211 6.670E+17+2.805E+18
October 0.1024-0.07878 0.07653-0.1397 0.09252-0.1112 0.04573-0.1731 -0.1302-0.2108 0.1024-0.07878 0.09631-0.1016 0.08532-0.1258
November 0.2069-0.2351 0.1619-0.2742 0.3064-0.04207
December 0.1316-0.08667 0.07125+0.06142 0.1345-0.06719 0.1048-0.1494 0.1275-0.1024 0.1187-0.1250 0.1262-0.1063 0.1326-0.08157
January 0.1542-0.2509 0.2222-0.1755 0.2623-0.08933 0.2164-0.18421

WE: the Washidu East quadrat; WW: the Washidu West quadrat [1]. Scientific names of Dictyostelia species: P. pallidum: Polysphondylium pallidum; D. purpureum: Dictyostelium purpureum; and P. violaceum: Polysphondylium violaceum. Blank values are either infinity, undefinable, or overflows.

Table 3: Eisenstein series.

In the diffusion equation of the neutral theory of population genetics [43],

(33)

where and are an eigenvalue and an eigenfunction, respectively, and . If we let , then is a Morse function because the Hessian of is assumed to be nonzero. We would like to know in order to analyze the conditions under which the system is at equilibrium. When is a function of genetic information, and the Dirac operator is . In adapted/collapsed positions of the Riemann zero values, the most promising virtual adaptation of is on the purely imaginary axis of the Dirac operator if the Riemann hypothesis is true. Indeed, in our physical model, the hypothesis is very likely to hold, as will be discussed below.

2.5 Geodesics of zeta-functions elucidate the mode of interaction within the systems and its expansion

Let be an elliptic curve over a rational of a -approximated conductor , defined above in the first part of Results when discussing the Price equation. Let be the corresponding prime for each value, including when , and consider the Hasse-Weil -function on :

(34)
(35)

Note that an ideal is a conductor of when is a finite extension of a rational . When , the global Artin conductor should be 1 or 1, and the system will fluctuate [68]. Note that if , converges as expected from the border between populations and species [41].

Considering that the geodesic in the Selberg zeta function, the Hasse-Weil -function on is