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Grossone  and the Adaptive Landscape -  A New Computational  Environment to do Population Level Social Selection In.

With a shaper ability to discriminate heritable phenotypic flexibility physiology (bluring Mayr's intellectual but not academic difference of proximate and ultimate) it is possible to see how social infrastructure selection leads to a better mating systems, prtental care and challenge to sexual selection itself.


One still ends up with Mayr’s somatic program but not his philosophical position on teleology.  To do this one can see that Shuker's geneotype – phenotype is like a heptagonal hyperbolic panbiogeography while social selection sensu Roughgarden et al is to homozygote heterzygote payoff evolution in a petagonal infinity of Galton (between even and odd). This is what shows that though Shuker took account of method he did not do the same for the logic of the stoichiology.

Fisher was mistaken to use soma where simple difference of exponential and tetratrional reside per population growth and with a double continuum phenotype physiology in genetic friability strata the difference is futhered no matter the ananlogy to density dependence since the volume may or may not change depending on the infinity the mating system used. Parental care can be communicated across generations genetically and heritable behavioral reaction to perturbation can be individually causal.

Thus Shuker is arguing from the classic infinty position that is shown in hexagonal symmetry where there is no even or odd difference and while the number is intermediate betweenthe even and odd (hetero and homo) of petagons which have a larger range of infinite expression. The claim of cooperative game theory is that evolution is more like a pentagon in it’s non binary biased division.  It is an empirical claim.  Shuker could be right but then payoff evolution would have little effect on fitness no matter the organism (plant or animal).


Differences in the grossone infinty numbers explain the difference and can obtain Fisher's fundamental theorm in Galton'slaw but with infintesimal differences due to elastic derivations of behaviroal effects on population numbers. The shape of the hyperbolic divisions express the evolution of the payoff matrix and thus different payoff matrices and have different amounts of infiniteswhich are classed as subtrations from 1 (in Galtons law as modified by Pearson and later Fisher). Shuker underestimated how the physiology of the difference of NBS and NCE is demonstrable genetically.  Sexual selection vs social infrastructure selection can be tested head to head and should competition for mates rather than cooperation amongst them be the normal institution then so be it firmly established – no soft economic analogies permitted!!

Social Selection sensu Roughgarden has power to shift Wright’s balance.

Here we show that social evolution provides a non-drift case wherein behavioral differences(stochastically acquired and intentionally prolonged) can cause the shifting balance of Wright’s orthogenesis to move between intrademe and interdeme selection levels but is accomplished wholly through individual selection.  Consequences for measures of heterozygosity and the evolution of dominance are displayed. Social infrastructure actually selected provide a means to differentiate Wright’s and Fisher’s views in population genetics.  A population level social selection example is provided.

Social selection sensu Roughgarden is a two tier process in which economic models are leverged through ESSs to combine a lower tier equilibrium with a higher tier equilibrium on the evolutionary time scale.

There has been doubt(s) about the role of ecology and lower level processes (to be able) to combine with higher (Gould). And there is a large variance in the way that Economic and Evolutionary equilibriA are to  (be) cognized (witness Frank -Darwin economy – discussions).

Grossone has been suggested as a new mathematical observation tool and here we suggest how to apply it to population level calculations of social selection. While some of social selection research has focused on the population level( Ackay etc) the defining work has remained within modeling for individuals (not summed up over populations) (Roughgarden and Roughgarden et al).  The use of a grossone environment to conduct social selection theory research enables the relation between verisimilitudes of individuals and groups (however probabilistic) to be better clarified and opens the way to determine if Fisher’s or Wright’s or Haldane’s mathematical approaches better qualify the population level computations of social selection sensu Rougharden.

Thus the confusion pointed out by Provine can be cleared and gene in the population and individual gene frequency contributions can be combined into one global optimization both in theory and application.  The use of a grossone global optimizations where the local minimization and are connected by tetrational asymptotes onto the evolutionary scale enables cross sections of the theoretical space through a Wrightian adaptive landscape both on the gene frequency of the individuals per population and population gene frequencies itself as the various tetrational asumyptotes converge or diverge between minima/maxima relatively.  The confusion of Provine is explained due to a lack of application of the grossone within the regions of two functions graphed convergences (Axiom 4 Transition to a limit Anatoly Zhiglijavsky Computing sums conditionally convergent and divergent series using the concept of grossone). The notion of Wright’s adaptive landscape, however named is thus not incomprehensible and further understood as that expanded theoretical place in which social evolution operates social selection.

The role that ESS plays in social selection is crucial not for the social evolution research per say but for the possible replacement of sexual selection with social evolutions.  The definition and first computer displays of ESSes (Price and Smith & Smith) indicate the use and need of computers in working with ESS and the game theory evolution concomitant.  The difference between interpretations of behavior within  the ESS (Roughgarden vs Sex and Sensibility, other paper) (behavior as game consequence vs behavior as intrinsic direction of game equilibiria) depends on how the individual behaviors are connoted with respect to approximations of the phenomena on computers simulating the action.  Thus a/the difference in the times for displays vs tournaments as to provocations does depend on how the computer is understood to represent series of various plays of the games.

Grossone applied in this situation opens up a new tool to grammetologically explain the lexos of the behavior to evolution connection.  It shows that levels of selection are not ontologically prior to the levels of organization that  develop within.

Let us take two populations that genetically have a tetrational asymptote internal to any given niche they exist in(gene vs individual).  Then the populational genetic approximation of one to the other in any F statitistic  of population structure moves in an infinitesimal field of structure that can only be described in Wrights as opposed Fishers or Haldanes frame.  If the equilibria of this structure match a macro extra niche pressure (mutation, migration, selection, drift) situation then it coordination provides a means to traverse the gene perpopulation  adapative landscape vs the population genes landscape as criticized as unworkable by Provine.  So grossone calculus can answer Provine criticism of Wright.

The best critcism of Yaro Sergeyev’s attempt to evolve Cantorian Infinity through a technological rupture of contingencies in the philosophy of mathematical analysis actually realizes the notion of selection of math psychology of Poincare intuitionism which was levied against Russell and others.  The Criticism amounts to a characterization as “Sergeyev’s idea is to introduce into arithmetic some infinitely large number – grossone, consider only the numbers less than grossone, and operate exclusively on these numbers using his grossone radix. “

The criticism asserts that “all linguistic and mathematical tools that are needed to Sergeyev are readily available within nonstandard analysis”.  This is not true.  Robinson THOUGHT he had understood the matrix use of Mendelian genetics but he failed to understand the relation of the tree representations that arise from the different mathematical uses for the same object of Fisher and Wright (under adaptability).  Wright’s path analysis contains a use of infinity different than Fisher which could not be related to Robinson’s work at all. No work has been done on  friably separating the differences for genetics between the math used by Wright and Fisher but in the psychology  of mathamtical discoveries presented by Poincare it is possible to use the new computational domain of grossone computing to demonstrate the graphic difference between Wright, Fisher and Haldane as presented by Wright once the network vision diagrammed is cognized macroevolutionarily as different relations less than grossone.  Social evolution is domain of macroevolution.

Provines View Of the Landscape

“Provine pointed out that Wright used two different interpretations of the 'landscape', which in Provine's view were inconsistent with each other: 'One of Wright's two versions of the fitness surface is unintelligible, and even if one were to escape this problem and put the gene combinations on continuous axes, the two versions would be mathematically wholly incompatible and incommensurable, and there would be no way to transform one into the other' (Provine, p.313).”

“Evolution would be quicker, and more beneficial to the species, if there were some meanls of shifting populations away from these suboptimal local peaks. According to the shifting balance theory in its original form, the only way of moving a population from a peak, other than a large and permanent change in environmental conditions, is by genetic drift, which enables a population to cross 'valleys' of relatively low fitness”

Now we must make sure that we understand that group selection in this form where it is for the benefit of the species is not comprehended under social selection.  This does not mean that a shifting balance can occur socially.  Drift may even cause social selection but with tetrational crossing the social population has more options towards the maxima than non-behavioral cooperating subsets of the total population not doing so and more thus variance in the path on up. Social evolution occurs in species since they get “stuck” less in the upper atmosphere of the landscape. Thus Wright’s original view of drift caused subpopulational change may infact be caused by social selections and thus vindicate his original conception that drift can yield significant evolutionary change.  If the multidimensional holes preclude this then lifetimes fitnesses themselves can also be drivers across the grossone functions.

So it the use of the grossone infinitesimals and infinites are used NOT on the genetic axes (or phenotypic ones in the Simpson version and that dealing with what phenotype subsets of the genetic field are on the surface of selective values prominanently) but for the fitness that links the behavioral equilibria  and ess equilibria. The grossone environment radically polarizes the “landscape” such that “topography” really does not fit analogically since the actual structure is string and spaggetti like rather than rolling hills. The topological connectivity is the same however the surface is just increadily more elastically streached than otherwise visualized.  This streaching is in the surface of the selective values NOT in the correlation of the genetic combination fields (either in a individual or in the gene frequencies in the population) to the phenotypes that map to the places in the surface of selective values which also change. ESS locations are stable with respect to changes in mutation.


The tetrational crossings permit one to move between the individual landscape and population frequency landscape in a unitary one to one and onto system.  Thus Wright’s vision is perfectly comprehensible. It only lacked the grossone frame or such in which to be computed.  Wright’s preference for a relative measure of fitness rather than an absolutist one of Fisher works well with Grossone in which the numeral system is relative to the grossone itself. It remains only to definte the divergences and convergences of the tetrations as the cross the finite numbers between the infinitesimal and the infinite as team fitnesses sum differently than competitive individuals (teams/cooperations converge  , individual competitions diverge).  Populational summed indiivudal lifetime fitnesses provide the “set of conditions” at least those relative to social behavior that apply in Wright’s cases more generally. Payoff matrix evolution thus becomes a matter of the shifting balance under restricted/constrained conditions (assuming no extra terrestrestrial bolide impact for instance)

“hyperlink op cit  The real objection, it seems to me, is not that the surface is not strictly continuous, but that the necessary correspondence between fitness and distance does not exist. Genotypes which differ only in a single allele may differ widely in fitness, for example if the heterozygote at a given locus has above-average fitness, whereas the recessive homozygote is lethal. I do not see any basis for an assumption that differences in fitness correspond, even loosely, to the number of genetic differences between two genotypes.” Here we use divergeneces where the distance does not exist and convergences where it does and thus find that motion in this “discontinuous” and continuous space is possible as individuals either compete or cooperate as individuals or even possibly their genes.  That will depend on the orthogonals again.


“I suggest that the following picture is more plausible. A very large part of the 'genotype space' must correspond to zero fitness, since it would involve combinations of rare disadvantageous alleles which are unlikely ever to be combined in reality. Only a small 'corner' of the space is inhabited by actual genotypes. Most of these will have rather similar average fitness, equivalent to producing around two surviving offspring (by sexual reproduction), since, on average, this is what most genotypes actually achieve under their normal circumstances. (If they did not, the population would soon die out.) Among these mediocre genotypes there will be a scattering of super-fit types, and a larger scattering of low-fitness types. The geometrical picture is that most of the landscape would be flat, with uniformly zero fitness, rising gently up to a small inhabited plateau of mediocre fitness, in which there are numerous 'holes' corresponding to genotypes with low fitness (e.g. lethal recessives) compared to their immediate neighbours. [Note 4] There will also be scattered pimples or wrinkles of modest height representing clusters of genotypes containing advantageous genes that are still in the process of selection, and shallow depressions representing mildly disadvantageous genes. But because it contains numerous 'holes' - isolated genotypes or groups of genotypes with fitness much lower than their neighbours - the landscape is not even approximately a continuous surface.”

The grossone frame permits one to work with this topographic but where those regions of increasing fitness are presented as ESS strategies under full selection connected by tetrational crossings, just as Wright envisoned motion on the surface.  The continuity is really in these much smaller areas where the pimples and wrinkles may tetrationally (quickly) become something very \ (either infintesmal  behahvorail equilibrated minima/saddles  or evolutioarny maxima/slopes) It is continuous “” really only in that the topology is preserved not that the texture is smooth.  Provines attempt to relate this transition as a phase transition was the problem for him in recognizing a different organon for Wright’s organic understanding.

Using grossone to compute relatively different infinite tetrational exponentiations.

Let us say that one wishes to calculate the tetrational limit starting from different steps towards infinity.


2^2^2^2… VS 4^2^2^2…..  IS THE FIRST DIFFERENT THAN THE 2ND? What happens when the tetration does not converge?

It makes a difference with respect to the individual alleles vs the alleles in the population since 2x2 need both behavioral and evolutionary equilibria while 4 is already a larger population size subject to the same selections.

The use of the tetrations which very quickly approach asymptotes is suggested to be able to trace the empirical discontinuities suffiently that continuous motion in a purely discontinuous space (Cantor) need not be framed.  This obviates Kaplan’s (2008) criticism (that motion in the adaptive landscape as pictured below the title) of an interpretation of the “landscape”, field or surface.  Kaplan wrote, “Here, the shape of the landscape is simply a description of the population dynamics, not an explanation of those dynamics.”  The explanation rather IS that cooperation rather than competition is part of this dynamics and social evolution IS simply the population dynamics equilibrated at two levels(epistemologicaql) (which gets groups) but not selection on two levels (ontological) (see Hierarchy Theory in Biology Salthe). Social selection IS about understanding the genotype-fitness mapping functions (objective functions) under social evolution.  Criticism of social selection sensu Roughgarden can only be that stabilization of behaviors within lineages is not due to this optimization but to some other global optimization not that the genotype-fitness mapping function can not be dissected out of the possible functions (competitive path to cooperative goal, cooperative path to a cooperative goal etc).  Grossone appears to be able to pack many dimensions into a few (Grossone rather than Grosstwo) etc such that discussion of the “surface”(topologically nearest neighbor is retained) for the field is possible as a heuristic for the Grossone infinity artithmetic (algebra vs geometry). Whether the large flucutations from infinitesimal to  infintite should be called a topography or landscape rather than a spaggetti or trenchs probably not).


Kaplan writes that it is even uclear if evo-eco can be incorporated into the high-dimensional landscape models.  Grossone frameworks permits this for social selection. Social evolution modeled with grossone optimizations is one way to do EVO-ECO relative to team work vs individual action for the most offspring in the next generation.

When Potochnik contends “changes in strategy are taken to result from the changing composition of the population instead of the same individuals performing different actions” this fails cognize the two different aspects of Wright’s selective surface.  If the different actions of individuals work by expressing different allelelic substitutions which in interaction with those actions of other individuals the behavior itself can effect changes in the demes gene frequency through a change in individual options.  This asks us to view both of Wright’s representations as one – through the game theory.

This is possible when we realize that Smith and Price’s categorical on intensity distinction was only necessary for the kinds of the computer simulations they used. They had no way to change the intensity into the category except via a threshold into the different behaviors.  But behavior is not likely to be like this. They are multi dimensional and integrade into each both in kind and intensity and this can be expected to arise from many kinds of genetic backgrounds per phenotype.  This is possible to simulate with Grossone. Pricre and Smith made this mistake because they attempted to see persistence of a display and a tournament as of the same temporality.  With variations in the grossone infinite plays per quantity played one can see that all different kinds of temporalities can grade or not into different behaviors. How the games divide these up strategically probably require multiple means NCE NBS etc. Without the use of grossone Price and Smith necessarily limited the variance possible in nature through the threshold between limited war and dangerous weapons.

Wright suggests that variables be visualized at the termini of unit vectors with their correlation being the orthogonal projection of one on the other (thus independent variables are 90* separated and negative correlations >90* but related by altering the sense of one).  One seeks to find a transcendental function that is diagonal to a set of variables’ orthogonalities.  This diagonal will map to both the individual genes  per individual and population gene frequency representations of the surface of selective values (it is a surface insofar as the normal to the surface is defined).  The diagonal function will move from unit quaternions to others but sustains the visualization scheme by extending  the distance metrics added to the units and coded into the transcendental that make up the function.  This should be computable and adapative topographies creatable especially if the added metric is lengthened by a larger span between used infintesimals and infinites.


“All this would be swallowable, just, if some use could be demonstrated.”

Social selection sensu Roughgarden as an effect is a modeling framework for social evolution in which pleasure based team work operating on economic analogies sums to higher individual fitneses than those surviving sans team works. It is suggested that these more likely evolutionary states are due to some function operating NBS equilibria over NCE.  It was proposed that pure team work arises simply with teams that happen to have the same pleasure gradient that correlates with the higher sums. Thus pleasure is coordinated with the NBS modeling.

Ernst Mayr introduced the notion of the behavior computer in which goal directed process much like a computer are speculated to operate as somatic programs. These teleonomic goal directions are not yet described as to how they relate or not to actual computer programs.  There is no doubt that brains an behavior may indeed possess such kinds of information flows but what material histogeny is responsible is not known.

It seems possible that rather than having the pleasure gradient effectively cause the NBS effect in social selection it could be that somatic behavior computers morphogenically affect NBS equilibria by avoiding NCEs as the goal direction or telos.

Furthermore  relation of organ size to genetic background in these brains (how big the behavior as computer is) can be surfaced from the game frame but must be orthogonal to some surface of fixing genes.  The use of grossone as a new computational paradigm appears capable to provide both since it specifies a new way to define the evolutionary stable strategy within a Wrightian network landscaped in a global optimization of Wright orthogonals within a common expression made of infinitesimal and infinite parts.

Grossone permits Kant's notion of density (often criticized as circular) to apply to volumes (wherein the attractions are limited comparted to the repulsions) of mathematical vs dynamical filling (Metaphysics of Natural Science) of Ubigraph spaces.  These are representational/systematic and not real spaces however but are demonstrable with graph databases being rational rather than fully empirical. Building such software is certainly possible.

Kaplan, Jonathan, The end of the adaptive landscape metaphor? Biol Philos (2008) 23:625-638 DOI 10.1007/s10539-008-9116-z

Evolution as a four-parameter family gradient system (body (phenotype)in 3D and genetic change (genotype) in 1D)  The seven elementary catastrophes are recoded through gene expression where dominance and recessive sequences between them are reordered into different phenotypes.

The continuous changes of the grossone infinitesimal subtraction from Galton’s law displays both Mendelian alternatives in population and successful fixation of genes in individuals.

The continuous system parameters are the regression to the mean but a path analysis of the infinitesimal inverted to its infinity across deme structures can result in changes in the default recessive- dominant opposite motions to a different alleomorph series and with this pushed Galtonian pool cue the faceted polygon alters its base. Quaternions  (rather than vectors) can be used to track changes in the path analysis so as to circumscribe the the singularites between the genetic  separation and the phenotypic friability (under compression of a given penetration).

Thus the regression is onto the future genertation of offspring no matter the reverted deviation it will in the future come to regress onto the mean value of the most immediate.  Regression is a damping effect that prevents ancestral energy from necessarily being utilizable in the current generation. It is an entropy like  or heat dissipation effect. The interesting cases are catastrophes that through infinitesimal subtractions from the mean the infinite inverse of that is able to find a different catastrophe surface for the dominants and recessives in the SAME FORM(catastrophe surfaces around the ridges and critical points) (repulsion and attraction effect (possible “netural” amino acid substittions which are not strict active forces guanosine neutral) but be in the same logical sorting of combined alternative and blending inheritance.  Goldshmidts?  This would vary from the “natural endowments”.

Guanosine microtuble dynamics can give a measure of the division between the total repulsion and attraction to the

penetrative surface as a distribution of the heritage to the gametes. Latent and Patent gems of aperiodic heritability are dependent of the ability to reach a catastrophe surface in the repulsion and attractions for any distance stretched across the empty space passed from the generation (parents) prior where rescessive and dominant are alike to latent and patent with correlations as discussed by Wright, Fisher and Haldane.



This is how a gene fixation can move out the right and stiffen ala wright wherein an alleomorph series becomes a different sequence (recessive – dominance) and thus a different genetic locus mappable!!

Grossone derived subtractions from Galton’s law are cases of Galton’s faceted polygon in which discontinuous evolutionary changes are manifested by infintitesimal energetic contributions of remote ancestors

Galton’s Polygon, Grossone Computing and Catastrophe Theory: A modeling environment for successful /beneficial gene mutation acquisition through discontinuous (atavistic reversion follow through) alteration.

Wright’s adaptive landscape has had a storied history.  It has been criticized as being incomprehsible possessing allegedly two different incompatible perspectives (individuals’ gene frequency and frequency of genes in a population).  Here the new grossone numeral system is applied to this problem and a manifold functionality of infinite numbers are described in which the individuals involved are enumerated by the finite part of the positional number decomposition.

The intuition and the concept of the adaptive landscape are two different things.  It was wrong to criticize the Wright’s contribution on the basis of ostensibly containing two different incompatible interpretations since this commentary actually applies to illustrative technigques used in genetics papers throughout the discipline at the time.  The notion of a mutant phenotype, 2 zygotic homozygotes, a heterozygote, and the wild type suffers from the same aggregation of moving forces not classified so it is unfair to conceptually argue against Wright’s landscape extension of his method when it is the intuition and its math rather than its philosophy which was lacking. 

This problem is solved by using the infinite grossone numerals recently described. An elucidation of the Wright’s network is provided and contrasted with the neutral theory within the infinitesimal manifold  while the Fisher and Haldane versions are observed in the infinite side of the finite decomposition per individual.

A new tool for meta community analysis – grossone numerals and facilitation.

Although it is true that IN GENERAL increased complexity tends to beget diminished stability. With respect to the relation of ecological interactions and evolutionary fitness that natural selection operates genetically on however,  clarity in the meaning of stability and the understanding of complexity IS required.  It is certainly possible that complex coordinations of coadapted complexes can be seen and understood simply in particular but generalizing these exemplars is difficult.  This does not mean that one’s intuition must be shelved for the ordinations that follow from a given model.  So whether increased species numbers in communities begets stability or depends crucially on what levels of selection are operative and how the levels of organizations so being selected are divisible into movable forces or not.  Forces at rest relative to species ecological persistence may lead to either stability or unstability but depend alternatively on gene vs population selection and thus general statements on the complexity-stability question can not be gain said without clarity as to how PHYSICAL the mathematical model can leverage.  Some modeling situations though dynamic are not able to explore more detailed relations of the force attraction and repulsions that lay beneath the phenomenology used to compare experiment to observation.

One species may facilitate another through an interaction but the time dependence of these interactions may be critical as many species are added to a community.  (Thus) it has been argued that there must be a decrease in connectance as more species are added(that intensity  per interaction Is limited).  Under this analysis adding more species makes a community more unstable.  Here a modeling approach is presented in which in some cases adding more species  both increase in connectance and increase stability.  These different results are derived by modeling the community matrix with grossone numerals.  Time to reach equilibria is encoded in the size of the infinity represented by a given species.  Intensity is determined by the coeffients on the terms that make up the infinite grossone number.  These terms may be connected in many different ways. Some ways actually add connectivity with intensity and also increase the number of species in the matrix.  Other combinations of the terms give the standard result.  Intepretation of the grossone terms are discussed in terms of K and gamma. Intra and interspecific interactions become exchangeable between species when the intra specific interactions exactly decreases the time to K equilbria for double the connectance of interspecfic interacionts.


We are going to create a new mathematical regime in which to discuss community stability and diversity.  We will introduce species persistence in terms of different amounts of infinity  that can be pressed out (deleting a speices from a grosssone system) or pulsed by infinitesimal addition or subtractions.  Thus we do not need to have an unstable equilibrium point on which the backbone of a stable but non-stationary strucuter may exist since thiswill exist due to the nature of grossone addition subtraction mulitiplicaton an d division and the function that relates the infinitesimal to the infinite is what the investigation (whether more speices makes a community more stable or less stable).  The strength of the connectionwill be in th efininte parts of the infinite representation while the equilinrium point far out is in the ininfite and small changes are in the infinitesimal parts.  The modes of oscillation are not a function of the simple choice of stable vs unstable but rather on the how the infintesmals are related to the infinities which can be structured differently  for moving forces based on repulsion s or attractions.  Thus it will matter if gravity or em is involved.  One can approach this my trying out some kind of stability analysis on top of grossone system used and then get rid of the stable unstable interms of the vector analysis.  This is because  the coordinate frame is quaternionic  rather than vectors in terms of visualization of stability and untstability.  There are some kinds of paths that are topologically catatroshpoic which are difined as quaternions in grossone terms but nether simply definied as stable or unstable (there is no sense of flowing into or out of the point but only on the composability of higher order catastrophes to larger infinites and more defined infintesimals.

With this new modeling method and stoichiological organon one does not need to reach May’s conclusion that the dynamic consequence of competition (+ -) and cooperation (mutualism (++)) are less kinematically likely than effective commensalism and amensalism(0-)(0+).  Here he simply did not separate the ontological from the epistemological use of the notion of complexity.  Chaitin’s deductive synthesis of complexity for biology (one that does not imply “progress”  in the traditionally negative sense accepted by many 20th century analyzing biologists) does show that stability might rather be found in systems where differential equations themselves do not represent the kinematics.  This use of the infinity computer puts one in a position to begin to model this new discrete math also without using difference equations – it is done numerically with a very particular language/semantics.  The complex is simple and stable but uses a very intricate mechanical device not used before.  Either competition or cooperation can be understood to give rise to faciliative stability and thus the limits that 70s and 80s thinking put on 60s and 70s ideas is not longer reasonable especially given 90s and 2000s data.

By using infinitesimals it is possible model very very weak interactions and thus this new framework will permit various kinds of types of interaction ( no longer defined as zero vs some influence but rather zero plus an infintesimal) and thus make the result below availlable for actual empricial test.  Thus depending on how many and a how large the infinite persistances compose the grossone numeration before a grosstwo or three (higher order catastrophes) are used the infintesimal differences from a menasalisum become a predator-prey or mutualism etc.  The categories are too simplistic to the relations that limit growths.

"The authors reexamined May’s results that showed that weak interactions made large systems more likely to be stable. In particular they examined how the distribution of interactions strengths, rather than the mean value alone, affected system stability. In contrast to accepted ideas, they found that when there were many weak interactions, predator-prey systems tended to become less stable, suggesting that weak interactions destabilize predator-prey systems. In contrast, weak interactions tended to stabilize competitive and mutualistic systems. The authors concluded, “Our analysis shows that, all other things being equal, weak interactions can be either stabilizing or destabilizing depending on the type of interactions between species. "

Eating something is more topologically involved  on the surfaces being used in the models but no less selectable genetically than behaviors that compete or cooperate when increasing rs serve as  fitness proxy.  The new framework uses the dame game plan but levels the "playing" field.

Grossone permits a Medelian population model that uses Cantor's idea of cotinuous motion in a discontinuous space.  This is likely most empirically supported within social evolution.  Here the discontinuity in the long range horizon of the indiviudal and it's ecology and evolution is modeled using a function with compenents in the finite and infinite range while the relation of the gene to the level of organization i tis contained in in the above is based on infintesimal continua probabalistically which appear as zero or Mendel numbers across generations but provided the link in which the continuous motion from ancestor to descendent is in hertiage.  Thus the divergence of  the kinetic  motions converge in the actual path.  Deriving these integrals beyond Galton but by avoiding Fisher's difference in sex is still needed.