Contribution to the theory of organismic sets,III

In line with previous studies on organismic sets, the division of all organismic sets intogeneral autotrophic and heterotrophic is introduced. The first produce their food themselves from some external source of energy, which in general may be an energy of any kind. The others use other organismic sets as the source of their food and energy. On earth we know only one kind of generalgeneral autotrophic organismic sets, namely, the autotrophic plants which use solar radiation as their source of energy and for production of their own food. It is shown why autotrophic animals do not exist on earth except as microorganisms like, e.g.,Euglena. A rigorous proof of the previously derived theorem that in an organismic set of ordern>1 no element can be completely specialized is given. It requires the introduction of new postulates. Finally, in considering the organic world as a whole, the notion of organismic sets ofmixed order is introduced.     

A remark on the course of development of organismic sets

The basic postulate la, which governs the development of organismic sets, introduced previously (Bulletin of Mathematical Biophysics,31, 159–198, 1969), is generalized so as to contain also the rates of changes of the number and variety of differentQ-relations which determine an organismic set. It is thus brought closer to the Lagrangian principle in physics. It is pointed out that the postulate also provides a criterion of stability of an organismic set.     

Outline of a unified approach to physics,biology and sociology

The theory of organismic sets, introduced by N. Rashevsky (Bulletin of Mathematical Biophysics,29, 139–152, 1967;30, 163–174, 1968), is developed further. As has been pointed out, a society is a set of individuals plus the products of their activities, which result in their interactions. A multicellular organism is a set of cells plus the products of their activities, while a unicellular organism is a set of genes plus the products of their activities. It is now pointed out that a physical system is a set of elementary particles plus the product of their activities, such as transitions from one energy level to another. Therefore physical, biological and sociological phenomena can be considered from a unified set-theoretical point of view. The notion of a “world set” is introduced. It consists of the union of physical and of organismic sets. In physical sets the formation of different structure is governed preponderantly by analytical functions, which are special type of relations. In organismic sets, which represent biological organisms and societies, the formation of various structures is governed preponderantly by requirements that some relations, which are not functions, be satisfied. This is called the postulate of relational forces. Inasmuch as every function is a relation (F-relation) but not every relation is a function (Q-relation), it has been shown previously (Rashevsky,Bulletin of Mathematical Biophysics,29, 643–648, 1967) that the physical forces are only a special kind of relational force and that, therefore, the postulate of relational forces applies equally to physics, biology and sociology. By developing the earlier theory of organismic sets, we deduce the following conclusions: 1) A cell in which the genes are completely specialized, as is implied by the “one gene—one enzyme” principle, cannot be formed spontaneously. 2) By introducing the notion of organismic sets of different orders so that the elements of an organismic set of ordern are themselves organismic sets of order (n−1), we prove that in multicellular organisms no cell can be specialized completely; it performs, in addition to its special functions, also a number of others performed by other cells. 3) A differentiated multicellular organism cannot form spontaneously. It can only develop from simpler, less differentiated organisms. The same holds about societies. Highly specialized contemporary societies cannot appear spontaneously; they gradually develop from primitive, non-specialized societies. 4) In a multicellular organism a specialization of a cell is practically irreversible. 5) Every organismic set of ordern>1, that is, a multicellular organism as well as a society, is mortal. Civilizations die, and others may come in their place. 6) Barring special inhibitory conditions, all organisms multiply. 7) In cells there must exist specially-regulatory genes besides the so-called structural genes. 8) In basically identically-built organisms, but which are built from different material (proteins), a substitution of a part of one organism for the homologous part of another impairs the normal functioning (protein specificity of different species). 9) Even unicellular organisms show sexual differentiation and polarization. 10) Symbiotic and parasitic phenomena are included in the theory of organismic sets. Finally some general speculations are made in regard to the possibility of discovering laws of physics by pure mathematical reasoning, something in which Einstein has expressed explicit faith. From the above theory, such a thing appears to be possible. Also the idea of Poincaré, that the laws of physics as we perceive them are largely due to our psychobiological structure, is discussed.     

Organismic sets: Outline of a general theory of biological and social organisms

The discussion as to whether societies are organisms andvice versa has been going on for a long time. The question is meaningless unless a clear definition of the term “organism” is made. Once such a definition is made, the question may be answered by studying whether there exists any relational isomorphism between what the biologist calls an organism and what the sociologist calls society. Such a study should also include animal societies studied by ecologists. Both human and animal societies are sets of individuals together with certain other objects which are the products of their activities. A multicellular organism is a set of cells together with some products of their activities. A cell itself may be regarded as a set of genes together with the products of their activities because every component of the cell is either directly or indirectly the result of the activities of the genes. Thus it is natural to define both biological and social organisms as special kinds of sets. A number of definitions are given in this paper which define what we call here organismic sets. Postulates are introduced which characterize such sets, and a number of conclusions are drawn. It is shown that an organismic set, as defined here, does represent some basic relational aspects of both biological organisms and societies. In particular a clarification and a sharpening of the Postulate of Relational Forces given previously (Bull. Math. Biophysics,28, 283–308, 1966) is presented. It is shown that from the basic definitions and postulates of the theory of organismic sets, it folows that only such elements of those sets will aggregate spontaneously, which are not completely “specialized” in the performance of only one activity. It is further shown that such “non-specialized” elements undergo a process of specialization, and as a result of it their spontaneous aggregation into organismic sets becomes impossible. This throws light on the problem of the origin of life on Earth and the present absence of the appearance of life by spontaneous generation. Some applications to problems of ontogenesis and philogenesis are made. Finally the relation between physics, biology, and sociology is discussed in the light of the theory of organismic sets.     

A note on the development of organismic sets

It is suggested that the development of organismic sets is governed not by the maximalization of the integral survival value, as suggested previously (Bull. Math. Biophysics,28, 283–308, 1966;29, 139–152, 1967;30, 163–174, 1968), but by maximizing the number of new relations which appear as an organismic set develops.     

Organismic sets: II. Some general considerations

The theory of organismic sets, developed in previous papers (Bull. Math. Biophysics,29, 139–152; 389–393; 643–647) is further generalized. To conform better with some biological and sociological facts the basic definitions are made more general. The conclusion is reached that every organismic setS _o is in general the union of three disjoined subsetsS _o1 ,S _o2 andS _o3 . Of these the subsetS _o1 , called the “core” is equivalent to an organismic set defined in previous publications. Its functioning is essential for the functioning ofS _o . The subsetsS _o2 andS _o3 , taken alone, are not organismic sets. The first of them is responsible for such biological or sociological functions which are not necessary for the “immediate” survival ofS _o but which are important for adaptation to changing environment and are therefore essential for a “long range survival.” The second one,S _o3 , is responsible for biological or social functions which are irrelevant for the survival ofS _o . Biological and sociological examples ofS _o2 andS _o3 are given. In addition to the fundamental theorem established in the first of the above mentioned papers, three new conclusions are derived. One is that in organismic sets of order higher than zero not all elements are specialized. The second is that every organismic set of order higher than zero is mortal. The third is that with increasing specialization the intensities of some activities in some elements ofS _o are reduced. Again the biological and sociological examples are given. At the end some very general speculations are made on the possible relation between biology and physics and on the possibility of “relationalizing” physics.     

Sporadic sustained nonperiodic oscillations in the activities of organismic sets

In combining the author's theories of organismic sets (Rashevsky,Bull. Math. Biophysics,31, 159–198, 1969a) and Robert Rosen's theory of (M, R)-systems (Bull. Math. Biophysics,20, 245–265, 1958), a conclusion is reached that the number of either normal or pathological phenomena in organismic sets may occur. Those phenomena are characterized by occurring spontaneously once in a while but are not exactly periodic. Some epilepsies are an example of such pathological phenomena in the brain.     

Relational forces and structures produced by them

After giving a brief review of the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967;31, 159–198, 1969), in which the concept of relational forces, introduced earlier (Bull. Math. Biophysics,28, 283–308, 1966a) plays a fundamental role, the author discusses examples of possible different structures produced by relational forces. For biological organisms the different structures found theoretically are in general agreement with observation. For societies, which are also organismic sets as discussed in the above references, the structures can be described only in an abstract space, the nature of which is discussed. Different isomorphisms between anatomical structures, as described in ordinary Euclidean space, and the sociological structures described in an abstract space are noted, as should be expected from the theory of organismic sets.     

Organismic supercategores: II. On multistable systems

The representation of biological systems in terms of organismic supercategories, introduced in previous papers (Bull. Math. Biophysics,30, 625–636;31, 59–70) is further discussed. To state more clearly this representation some new definitions are introduced. Also, some necessary changes in axiomatics are made. The conclusion is reached that any organismic supercategory has at least one superpushout, and this expresses the fact that biological systems are multistable. This way a connection between some results of Rashevsky’s theory of organismic sets and our results becomes obvious.     

Natural transformations of organismic structures

The mathematical structures underlying the theories of organismic sets, (M, R)-systems and molecular sets are shown to be transformed naturally within the theory of categories and functors. Their natural transformations allow the comparison of distinct entities, as well as the modelling of dynamics in “organismic” structures.     

Organismic sets and biological epimorphism

It is shown that the principle of biological epimorphism (Rashevsky,Mathematical Principles in Biology and Their Applications, Springfield, Ill.: Charles Thomas, 1960) is contained in the theory of organismic sets (Bull. Math. Biophysics,29, 139–152, 1967) if an additional postulate not directly connected to mappings is made.     

Contributions to the theory of organismic sets: Leadership

The theory of organismic sets (Bull. Math. Biophysics,31, 159–189, 1969) is applied to the theory of leadership in human society. The ability of making decisions, required for leadership, is a product of the activities of the cells of the cerebral cortex, which are elements of the subsetS ₀₂ of the organismic set “man” (loc. cit.). Products of the activities of the elements of an organismic set do not need to be of a material nature. Such things as thoughts, feelings, attitudes, etc., are also products of the activitiesa ₁ of the elements. An individual can makeall necessary decisions for adaptation in a changing environment, when his subsetS ₀₂ contains as a proper subset a set {a ₁₂ ^′ ∼ ⊂S ₀₂ of activities. It is shown that such individuals are rare. If none exist, then the one who possesses a subset {a ₁₂ ^* ∼ ⊂ {a ₁₂ ^′ ∼ of higher cardinalityc _m than any other individual, will be the leader. The possibility is discussed that fromN individualsN′ 〈N possess subsets {a ₁₂ ^* ∼ ⊂ {a ₁₂ ^′ ∼ all of the same cardinalityc _m but differing in the type of their elements, thus resulting in several leaders. It is then discussed what determines which of theN −N′ individuals will choose a particular oneN′ individuals as leader. Cooperation and competition between leaders is discussed.     

Some consequences of Henri Poincaré's hypothesis about the biological nature of the three-dimensionality of our space

Henri Poincaré (Derniere Pansées, Paris, Flamerion, 1920) makes the interesting suggestion that our space is three-dimensional because ourvoluntary movements are those of quasi-rigid bodies in three dimensional space. Inasmuch as according to the theory of organismic sets (Rashevsky,Bulletin of Mathematical Biophysics,31, 159–198, 1969) organisms are conceivable, perhaps in some remote parts of the universe, for which the primary voluntary changes may be not spatial movements but changes of other physical qualities, it is pointed out that the acceptance of Poincaré's hypothesis will require an invariance of the physical laws in an abstractn-hyperspace with respect to the choice ofm<n coordinates as the basic frame of reference.     

Some considerations on relational equilibria

A previous study (Bull. Math. Biophysics,31, 417–427, 1969) on the definitions of stability of equilibria in organismic sets determined byQ relations is continued. An attempt is made to bring this definition into a form as similar as possible to that used in physical systems determined byF-relations. With examples taken from physics, biology and sociology, it is shown that a definition of equilibria forQ-relational systems similar to the definitions used in physics can be obtained, provided the concept of stable or unstable structures of a system determined byQ-relations is considered in a probabilistic manner. This offers an illustration of “fuzzy categories,” a notion introduced by I. Bąianu and M. Marinescu (Bull. Math. Biophysics,30, 625–635, 1968), in their paper on organismic supercategories, which is designed to provide a mathematical formalism for Rashevsky's theory of Organismic Sets (Bull. Math. Biophysics,29, 389–393, 1967;30, 163–174, 1968;31, 159–198, 1969). A suggestion is made for a method of mapping the abstract discrete space ofQ-relations on a continuum of variables ofF-relations. Problems of polymorphism and metamorphosis, both in biological and social organisms, are discussed in the light of the theory.     

Population regulation of brook trout (Salvelinus fontinalis) in Hunt Creek,Michigan: a 50‐year study

1. Fisheries models generally are based on the concept that strong density dependence exists in fish populations. Nonetheless, there are few examples of long‐term density dependence in fish populations. 2. Using an information theoretical approach (AIC) with regression analyses, we examined the explanatory power of density dependence, flow and water temperature on the per capita rate of change and growth (annual mean total length) for the whole population, adults, 1+ and young‐of‐the‐year (YOY) brook trout (Salvelinus fontinalis) in Hunt Creek, Michigan, USA, between 1951 and 2001. This time series represents one of the longest quantitative population data sets for fishes. 3. Our analysis included four data sets: (i) Pooled (1951–2001), (ii) Fished (1951–65), (iii) Unfished (1966–2001) and (iv) Temperature (1982–2001). 4. Principle component analyses of winter flow data identified a gradient between years with high mean daily winter flows, high daily maximum and minimum flows and frequent high flow events, and years with an opposite set of flow characteristics. Flows were lower during the Fished Period than during the Unfished Period. Winter temperature analyses elucidated a gradient between warm mean, warm minimum and maximum daily stream temperatures and a high number of minimum daily temperatures >6.1 °C, and years with the opposite characteristics. Summer temperature analyses contrasted years with warm summer stream temperatures vs years with cool summer stream temperatures. 5. Both YOY and adult densities varied several‐fold during the study. Regression analysis did not detect a significant linear or nonlinear stock–recruitment relationship. AIC analysis indicated that density dependence was present in 15 of 16 cases (four population segments × four data sets) for both per capita rate of increase (w_i values 0.46–1.00) and growth data (w_i values 0.28–0.99). The almost ubiquitous presence of density dependence in both population and growth data is concordant with results from other trout populations and other studies in Michigan.     

The approximation method,relational biology and organismic sets

It is pointed out that the approximation method for diffusion problems, developed by N. Rashevsky in 1937 and successfully used since then by many authors, was in a sense a precursor of relational biology. The connection between the approximation method, relational biology, and the theory of organismic sets, developed in a series of recent papers by N. Rashevsky, is discussed. A number of conclusions known to hold experimentally, are then derived from relational considerations and some of them are applied to organismic sets.     

The effects of hydropsychid colonization on algal response to nutrient enrichment in a small Michigan stream, U.S.A.

1. We tested the hypothesis that the indirect effects of colonization by Hydropsyche spp. (Trichoptera: Hydropsychidae) may be greater than direct effects of nutrients on the benthic algal community growth. Two sets of nutrient-releasing substrates (a total of twenty-four) were deployed into a small pristine stream in northern Michigan. Each set was composed of four treatments replicated three times: (i) no nutrient enrichment (C), (ii) 0.5 M phosphate-P enrichment (P), (iii) 0.5 M nitrate-N enrichment (N) and (iv) 0.5 M phosphate-P plus 0.5 M nitrate-N enrichment (P + N). All hydropsychids colonizing on the substrate in one set (twelve substrates) were removed regularly and the other set (twelve substrates) with undisturbed hydropsychids served as the controls. 2. Algal biomass and gross primary productivity were estimated as chlorophyll a (chl a) concentration, algal biovolume, and carbon fixation rate, respectively. There was a significant interactive effect of hydropsychid colonization and P enrichment on algal biomass measured as chl a concentration. With removal of hydropsychids, chl a concentration increased 11-fold in the P enrichment treatments relative to the controls. The effects of P on chl a was, however, not significant in the presence of hydropsychids. Such interactive effects were not observed when algal responses were measured as biovolume and carbon fixation rate (GPP). 3. It is recommended that algal responses to nutrient enrichment should be measured as biovolume or carbon fixation rate in small streams where hydropsychids are commonly present.     

Hazard Assessment of Deer–Vehicle Collisions in Michigan

The relationship between deer–vehicle collision counts and vehicle miles traveled (VMT) is studied in this article using the Michigan (USA) crash database and categorized exposure data under different levels of data disaggregation. Negative binomial regression models were developed to establish the association between (human and deer) exposure and the frequency of deer–vehicle collisions. It is shown that VMT is nonlinearly correlated with the collision counts for most sets of circumstances except in rural areas. Observation of the association between the deer–vehicle collision count and the annual VMT across counties in Michigan reveals the unique characteristic of heteroskedasticity (i.e., variances of collision count are amplified with the increasing VMT). The regression results demonstrate that deer density stands out as the most significant exposure predictor in describing deer–vehicle collisions at the state level. It is recommended that differentiated exposure measurements for deer–vehicle collisions should be employed for three ecoregions in Michigan. Exposure predictors for deer–vehicle collisions appear to be more dependent on deer density in Michigan's Upper Peninsula and North Lower Peninsula ecoregions, while in the South Lower Peninsula ecoregion deer and human populations are equally important.