On the theory of competition.Annidation as a fifth evolutionary factor

by Wilhelm Ludwig, Heidelberg

Ludwig, W. 1950. Zur Theorie der Konkurrenz. Die Annidation (Einnisching) als fünfter Evolutionsfaktor. In: "Neue Ergebnisse und Probleme der Zoologie" (Klatt-Festschrift)

Translation Notes: Author names in the original text are referenced in upper case, and the same is done hereEmphasis in the original text is indicated by increased the spacing between letters; here it is indicated by bold typeface.Text that is in small type in the original is represented in small type (11 point font size) here.

p. 516 I. Introduction.

In the VOLTERRA theory of communities, competition for resources among species or races2 (2 for the sake of simplicity, we imply either races or species) in their occupied living areas has up to now almost only been investigated in a very schematic sense, where two species with different numbers of offspring live on and compete for the same nutrients. Then clearly the species with the lower reproductive rate will be eliminated by the other. But it is the more general and more realistic case where the resource used by the two species is not exactly the same that leads to important results, especially of a theoretical and evolutionary kind, that have not been presented completely clearly up to now .

Modern selection theory works with four evolutionary factors: mutability, selection, deviation from panmixia (inbreeding, homogamy, isolation), and chance. This is correct in so far as the HARDY WEINBERG law of the constancy of genotypic composition ("hereditary constancy") of populations is based on four independent premises: 1. Absence of mutations, 2. Absence of fitness differences and therefore selection, 3. Panmixia and 4. Infinite population size and therefore the absence of chance effects. Tchetwerikoff's "life fluctuations", later called "population fluctuations", as well as REINIG's "elimination" are special cases of chance3 (3 with the latter, compare HENKE (1938), LUDWIG (1939a)).

p.517 However, the law of hereditary constancy depends on several other assumptions which

are mostly considered as implicit, such as for example equal use of the living space by the types of individuals living there. Where there are differences in the use of the living space - for example, only some members of a species can use a food-type A, but others both A and B - as will soon be shown, fitness differences cannot be equated, and thereby do not in general support the law-like behavior of selection.

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The existence of this kind of difference and the outcome of the resulting competition shed light onto important and long-standing questions of evolution: the origin of species in the same location, the stable occurrence of related species in the same living space, and the origin of maladaptive or neutral characteristics. These considerations will lead to the addition of a fifth evolutionary factor "Einnischung", or Annidation to those we already know.

II. The concepts of "fitness" and "selection"We speak of selection when out of two competing races or species, the one with the

lower relative fitness is gradually eliminated. Here, fitness is defined as follows: If two races R and R' with N and N' numbers of individuals live in the same infinite and isolated living space and if the ratio N:N' decreases from one generation to the next on average to N(1-k):N', where k>0, then race R' will have the higher fitness. The quantity k is called the relative fitness of R' with respect to R, or fitness difference for short4 (4 because of the small size of k, we have the relation (1-k):1 ≈ 1:(1+k).). It is easy to see that the ratio of the two population sizes will become always more extreme. Now because the finite and limited resources in the long term only support a certain number of individuals, so N+N' cannot exceed a certain high value N*, then the race with the lower fitness will be gradually eliminated.

The fitness difference k can come about in two ways. Either R' has a higher reproductive rate (multiplication rate) than R, and otherwise both races do not differ.

p. 518Or both have the same reproductive rate, but R' has an advantage in some way over R in the struggle for existence. In the latter sense, thus considering two races with the same reproductive rate, one says that R' has a selective advantage ofer R of k5 (5 this definition of advantage corresponds with that of Haldane. With regard to other definitions of advantage cf. Ludwig (1939b). Obviously both can be the case, and thus it follows right away that on one hand a positive k has an effect through a higher reproductive rate and on the other hand that it can be compensated for by a lower number of descendants. Three things are therefore characteristic of selection:1. Selection results in the elimination of races of lower fitness. A stable equilibrium between races of differing fitness is only possible when other forces (mutation pressure, immigration) work against selection.2. Each positive selective advantage is equal in its effect on increasing the multiplication rate and can only be compensated for by a reduction in the latter.3. A newly arisen selectively neutral trait can on average, i.e. without chance effects, only increase in a population if it is absolutely linked with an advantageous trait, such as for example when both characters are affected by the same allele; this is without considering rare mutation pressure. The same is true for deleterious characters.

III. Accidental, eaten, and shortage fractions

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From the young individuals of a species, only a small proportion come to successful reproduction, the rest dying beforehand. The latter proportion of all offspring we call the probability of mortality rate or the mortality rate v. It is, if the population number remains constant, and n is the number of offspring, (n-1)/n.

As presented briefly by the author in another place (1939b), mortality can be classified into three groups. Mortality from:a) Accident, i.e by abiotic influences (accident especially, cold, drought, etc.), through disease (here distinguished from what in b is classified as cases of parasitism), failure to survive metamorphosis, birth, etc. This proportion of juvenile forms dying due to chance effects we will call the accident fraction.b) Being Eaten, i.e. mortality due to natural enemies (predators; macroparasites, microparasites with intermediate hosts): eaten fractionc) Shortage of resources, e.g. food, oxygen, habitat (reproductive space, mating, nest or overnight resting sites, territories ["reviere"] (for example for the large mammals in Africa), being a host for parasites, heating material for people (Peru) etc.; shortage fraction.

The sum of the accident, eaten and shortage fractions is the mortality rate.The accident fraction of a species can be independent of the population size, and,

because overpopulated habitat will hardly be considered in the following, also independent of the population density, and therefore from now on it will be considered as constant.

With the eaten fraction, the only consideration is that a prey lives in equilibrium with its enemies, or a predator with its prey. After the considerations of Volterra it can be assumed that the population size of each species will vary about a mid-point, so that the eaten fraction can also be seen as constant overall. Furthermore, in general in Volterra's considerations only the population sizes N are included, not the densities, so the eaten fraction can be considered as independent of the population density.

The role of the shortage fraction will be explained in the next section. In contrast to accidental and eaten fractions, it certainly depends on the population density.

IV. The role of the shortage fraction. The shortage factor.With the shortage factor the critical "life factor" in the living area is that which is the

scarcest. It is called the shortage factor. As a result of its limited quantity, in the living space over a period of time, only a certain maximum number N* of individuals of a particular species can exist. We say that the habitat provides this species with N* existence spaces.

If a population finds itself in equilibrium in a habitat, its average population size N will remain constant. For this number N, the accident, eaten, and shortage fraction can be determined. Thus at the outset it is improbable that the accident fraction prevents the further increase in N, except (mathematically speaking; infinitely) in the improbably case when it is as big as to on average leave exactly two progeny per pair. For the schematic case of goat-wolf it means that the population number of goats Nz will increase after removal of the wolves, till through a shortage of resources it will increase further to a certain value of Nz

*. Considering there is an additional limiting factor for sheep, let us call it "cabbage", leads to a consideration of the three

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species system cabbage-goat-wolf, even without mathematical considerations to the result that the numbers of goats (Nz) as well as wolves (Nw) are determined by the shortage factor of cabbages (Nk): with declining Nk, Nz and Nw also decline, till finally Nw = 0, i.e. till cabbage still only is enough for the existence of the goats but not the wolves. This situation is also dealt with in Volterra's theory (cf. D'ANCONA, Ch. XX): in a habitat, wolves, sheep and goats live in certain resource conditions, and depending on the relationships of particular constants, this can maintain all three species, or wolves die out, or finally the two animal species. For what follows it is relevant that for population number and density all the inhabitants of a habitat are eventually dependent on the available shortage ratio. The strengths of predatory as well as all species that have no natural enemies6 (not including disease causing bacteria etc. whose effects occur in the accidental and not in the scarcity proportion) would be direct , which the prey species or other species with natural enemies would be indirectly determined or co-determined by the scarcity fraction.

Even so the VERHULST-PEARL equation, which describes population growth of a species under limited resources and has been frequently verified, is based on the assumption of a limited number of existence places and a shortage fraction. The equation expands the meaning given by the author (1928) and later by GAUSE (1932):

{Population growth during dt} is proportional to {number of yound individuals produced during dt} X {number of existence places still free} (Eq.1)

where the latter factor in the brackets (by inclusion of the constant c) can be converted to "probability of finding a new existence place".

Relationship Eq. 1 can be expressed as

dN = c . N ε dt . (N* - N). (Eq. 1a)

Thus for N = N* when all the existence places are occupied, dN = 0, and c = 1/N*, and 1(a) becomes the ususal form of the VERHULST-PEARL equation.

p. 521.(1/N) (dN/dt) = ε (1-(N/N*)) (Eq. 1b)

Thus the role of resource shortage can be shown to be critically important. A few further remarks can also be added.

Assume for example that one is dealing with a herbivore such as a rodent with very important enemies, and that food is an important factor. The number of existence places N*will probably be determined mainly by the resources available during the winter. Moreover one can suggest that when all the existence places are occupied, also all winter food will be exhausted, so that to a certain degree N* would be as if in the harshest times the available food for each animal would be the minimum necessary to sustain it. Many more would die of this because they would by chance find themselves without food,

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and conversely for the same reasons a certain amount of food will remain unused. If the living space in food density very small then often there no individual would be able to survive for the duration - N* becomes 0 - as well the total food quantity for some individuals would be exhausted.

It can be further added that the VERHULST-PEARL also applies to humans (PEARL). N* is the number of "places" that an individual requires to provide living needs. For example, that the reproductive rate of man responds to any decline in "free spaces", such as with unemployment, is clearly manifest and widespread in the post-war statistics, in spite of the many sided complications by secondary factors.

V. EcomutationsIn the animal kingdom we know of many ecological differences between closely related

species. One thinks of the food specialization of many insects and their larva or of the host specialization of gall forming insects and above all of parasites. These differences are at least mostly inherited and must have arisen, in terms of what one knows about speciation, via mutations. That he mutations themselves - we will call them ecomutations - have hardly been observed is understandable. After all inherited differences between races are already known, e.g. in plants (Turesson-Lyssenko) and in some situations its Mendelian basis is established, as for the preferred temperature regime by warm blooded animals (Herter) or for the duration of dormancy in tent moths [gypsy moths?] (Goldschmidt). Above all it is feasible that many morphological differences have ecological effects. Longer tongues in insects [Schwaermen] makes it possible for them to visit deeper nectaries. Increase in size or changes in the shape of mouth parts will have the effect of changing the available resources; with a more pronounced proboscis

p. 522thick surfaces, whether in plants or animals, can be penetrated. Smaller body sizes makes new niches and hiding places accessible and thereby also new food sources, while other advantages may accrue from smaller body sizes. Smaller hosts are good for smaller parasites, such as an aphid for the smallest parasitoid wasps. An increase in the hemoglobin concentration - it would be the morphological mutation - would enable chironomid larvae to reach areas of lower oxygen concentration and thereby establish new existence places. These examples could be repeated.

We consider now ecomutations that act on the shortage factor. It is useful to consider a schematic case. A species is living in a closed habitat; if the number of individuals is N* all existence places would thus be occupied. One is dealing with a predator that itself has no enemies, or a "social" insect. The major factor determining N* will be food. Now a mutant could occur that makes available another food source such that for this race (if it alone is present) the number of existence places is increased in the habitat under consideration from N* to N*+R*. For notational convenience, we will call the mutant diphagous, and the wild type monophagous. From this various things follow that are more or less self evident:

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1. Both food types, the new and the old, can be accessed by any of the diphagous species. This race can therefore live alone on the new food. Moreover it has no preference for one food type over the other.

2. Because in the animal kingdom food preferences are generally determined instinctively, the diphagous individual must have the instinct to eat the new food as well as the ability to use it. Because the species can already have one or the other the mutation needs only to occur in one trait or the other.

3. It follows that because the R* new existence places actually increases the number of individuals from N* to N*+R*, it follows that if another scarcity factor does not come into play, that the indeed more than N* but smaller than N*+R* existence places remain for each, which so to say they occupy second place in the hierarchy of shortage factors [need to work on this sentence]. In such cases the population size can only increase to the values described by this factor. Because this only means a modification but no basic change to the question being considered here, such possibilities can be excluded from consideration.

4. The new food shall be sufficiently specific that it will not bring the new mutant in competition with any other species (this premise will be relaxed later).

p. 523Such a mutation results in a change from stenophagy to euryphagie. For all other

ecological characters, it is possible to think of mutations that lead from steno- to euryphagie. In this regard we speak from now on of the "monoicous" species and the "dioicous" mutants. The dioicous individuals thus occupy an up to now unused niche of the habitat - this word being understood in the sense of ELTON. A symbolic representation is given in Fig. 1a: scattered among the pre-existing existence places are found R* more which up to now have not been filled. A species which in this sense through a mutation becomes euryoicous does not increases its living space from the outside inwards, in that its living area becomes larger, but from the inside. Mone things of , e.g., of a chironomus larva which because of a certain increase in the carrying capacity of hemoglobin can work its way deeper int ot he mud and so can also live on the food found there (shortage factors could equally be food or space. Fig. 1 b).

What will be shown in the following is above all that those mutations that increase the number of existence places do not follow the rule-like laws of selection. These mutations will usually spread more quickly, also when they have an arbitrary selective advantage, as long as they are at minimally viable. The advantage of dioicous races can thus not be compensated for by a selective advantage (more exactly: never completely). This can be explained by a brief example. In a habitat there are wolves, goats and sheep. Wolves only like to eat goats, and moreover sheep and goats do not compete. A new wolf race now arises by mutation which eats sheep in addition to goats. Even when this diphagous wolf reproduces much less than the original monophagous race, if it for example is less cold-tolerant or by is less able to catch prey as well s the other species, it will, in so far as

p. 524 it is present and viable, in that it can reproduce itself and not die out by itself, increase steadily in numbers, because the sheep will not be taken as a food source by anything else, to a specific value determined by food shortage. Additionally, these diphagous wolves, even though they less

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capable than the monophagous one, will still like to kill and eat a goat, and so the population numbers of the original race will fall to an equilibrium between it and the new race in which all the living places are occupied. The spread of the mutants will in general be faster than in the case of a selectively advantageous mutation, because the mutant here in part will be colonizing empty space (R*), in which there is no action of competition.

VI. Types of CompetitionIn nature there are no doubt many forms of competition between a dioicous and

monoicous species (or race) for a habitat resource. However it is easy to classify these in an ordered series according to the degree to which the dioicous species acquires the existence places N* that are available for both species.

In Fig. 2 a-c, the existence places are indicated in the middle: N* = 4, which can be acquired by both species, and R*=2 which are only available to the dioicous species. (In reality these numbers would be much higher). Within this quantitative series, the progeny of the monoicous species are presented, totaling Nmnm, where Nm is the population number and nm the reproductive rate. These monoicous progeny strive among each other for the two existence places (4). The young individuals of the dioicous species are represented at the top, in total Ndnd and these strive in varying intensities for the two types of existence places. One can distinguish three types, of which the first and certainly the last hardly represent a distinct case. Practically the only important thing is the overall range.

1. The "unregulated" type (Fig. 2a). Here the dioicous progeny occupy the existence places that are there (thus for N* + R*), unaware that

p. 525the N* existence places are also being "stormed" by the monoicous type. This type is, if the existence places are searched for actively, is barely achievable because (in our symbolic representation) one individual cannot use all the existence places at the same time. In the schematic example above, this would be approach the situation of the case where both di- and monophagous wolves feed on whatever they comes across (sheep or goats), and in that they are never full nutritionally to want to avoid any prey. In this type numbers increase purely by passive occupancy of existence places (see below).

2. The "weakly regulated" type (Fig 2b). Here the dioicous young individuals all occupy the existence places, regardless of what the monoicous type does. This type might often be found in nature. To this belongs the example of two wolf races, where an individual after capture and eatingof a prey animal is satisfied. (The biological fact that many young individuals indeed casually find prey, but not enough to live to become reproductive is identical to the schematic formulation, "no existence places can be found").

3. The "strongly regulated" type (Fig. 2c). Here the young individuals are strongly "restricted" in that as far as possible they cannot compete as strongly for the usable monoicous existenz places, that in all places eqeual densities are present (in Fig. 2c, four per place). As a schematic example, one thinks of a slim and a plump antelope race and water as the limiting

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factor (for drinking). The thin species can also look for drink that is hard to reach, that is unreachable for the plump ones, as there there are fewer crods. This type is very unlikely to be realized.

p. 526A final type would eb the last one in the series where the dioicous individuals willingly only

strive for the R* existence places that they can use, and leave the N* places entirely for the monoicous one. This case is unlikely to occur in nature and therefore does not need to be considered further.

For each of these type we now ask what the population equilibrium would be, that is after each of the population numbers Ňm and Ňd that result if the two species are allowed to freely compete with each other. Then the number of existence places occupied will be: Ňm + Ňd = N* + R*, (Eq.2)and there will be three possibilities for the ratio Ňm:Ňd (Fig. 3): either the dioikous species will occupy all the existence places (Fig 3a), or as well as R* some of the N*-places (Fig. 3b), or it will be restricted to the R*-places (Fig. 3c). The individual calculations are simply presented here visually:

Fig. 3 Type I Type II Type III

a) case: nd/nm ≥N*/(N*+R*) nd/nm ≥1 nd>nm

b) case: nd/nm <N*/(N*+R*) nd/nm<1 nd=nm

c) case: (only where nm = ∞) (only where nm = ∞) nd<nm

1. "Unregulated type" (Fig. 2a). Here the existence places are divdied according to the available young individuals, while R*-places are available exclusively for the dioikous species. It follows Nm = N*. Nmnm/(Nmnm+Ndnd) , Nd = R* + N*. Ndnd/(Nmnm+Ndnd)

so giving the equilibrium values:

Ňm =N* - R*:( nm/nd -1), Ňd= R*:( 1-nd/nm). (Eq. 3)

p. 527.One sees right away that condition (2) is fulfilled. Moreover, in a constant habitat (N8, R*) the result depends on the ratio of the reproductive rates (nd/nm), in which according to Section II can be thought of as the result of character differences. That the dioikous form is would be restricted to the R*-food places (Fig. 3) will only be possible when nm = ∞ and therefore can be excluded. Conversely, it will exterminate the monoicous form (Fig. 3 a) when nd/nm ≥N*/(N*+R*), even in

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the case in which it has a lower reproductive rate or a lower fitness than the form that is eliminated. This will occur when the dioikous species occupies an especially large R* so that it produces so many progeny here that these flood the N*-habitat, there to gradually to overwhelm the progeny of the monoicous form. - For the example shown in Fig. 2a, Ňm=2, and Ňd=4.

2. "Weakly regulated type." Here the dioikous form puts a fraction g of its progeny in the N*, and the fraction (1-g) in R*-places. It follows that

Ndg:Nd(1-g)=N*:R* or g=N*:(N*+R*). (Eq. 4)The size of the fraction of monoikous individuals that survive, i.e. themselves mature to reproduction, will be determined in the N*-region, and the survival probability is proportional to the ratio of the number of their progeny relative to the total number of progeny in N*. At equilibrium the number of these surviving must equal the original number Nm. It follows

N*.Nmnm/(Nmnm+gNdnd) = Nm (Eq. 5)

and from (2), (4) and (5) one obtains the equilibrium values

Ňm =N*(N*+R*)/(N*+R*:(1-nd/nm)), Ňd = N*+R*-Ňm (Eq. 6)

From these it follows: The monoikous form will be eliminated when nm ≤ nd, in which case it is selectively equal or disadvantaged to the dioikous form. On the otherhand, the dioikous form can only fully occupy the N*-region when nm = ∞. The reserve R* thus has the effect that the monoikous form can never completely be eliminated from the common area (N*), and that it can only eliminate the second form if it is equal in its characteristics, but not when it is disadvantaged . - In our example the result is Ňm=Ňd =3.

p. 5283. "Strongly regulated type." Here the density of occupants in N*- and R*-regions the

same, thus

(Nmnm+gNdnd) : N* = Ndnd (1-g) : R* (Eq. 7)

Biologically this only has meaning for values of g between 0 and 1. From (7), (2) and (5) it follows that in the case where all existence places are occupied and the individual numbers of the monoikous species are considered, their number of individuals in the next generation is

N'm = (N*+R*) Nm/((nd/nm)Nd + Nm). (Eq. 8)

From this equation one immediately sees that: If nd > nm, then always N'm < Nm, and the numbers of the monoikous form gradually decline to zero. If the reverse nd < nm is true, then it increases to Nm = N*, the biological limit (The mathematical possibility of it increasing to Nm = N* + R* is biologically meaningless). This produces the following equilibria:

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when nd > nm : Ňm = 0, Ňd =N*+R*,when nd = nm : Ňm and Ňd , any value according to condition (2) (Eq. 9)when nd < nm : Ňm = N*, Ňd =R*.

In this latter type, the dioikous form "gives up" with almost an altruistic effect the advantage it has in the new habitat and so the competition drops to the level of selection. Now the decline in the disadvantaged (here monoikous) species which becomes restricted to its own habitat [need to work on this sentence]. - In Fig. 2c the monoicous species wins (nm/nd =2) thus Ňm= 4, Ňd = 2.

For some cases in which the existence places are not actively chosen an additional consideration is relevant. One thinks somewhat of two related grass types, one of which can germinate and survive on poor (R*) and good (N*) soil, while the other monoikous one can only do so on the latter. The seeds will be blown onto both locations by the wind. On N* the same proportion of both seeds grow, but the monoikous species dies on R*-soil so that these remaining existence places are now free and usable by the dioikous species. At equilibrium, all the R* existence places will be occupied by the dioikous species, and correspondingly the N*-places will be divided according to the proportion of the two types of seeds falling there. So this scenario leads to - cf. equation (3) and the following - to the "unregulated type", which thus here, where there is passive occupation of existence places, so there is barely a difference but it can occur in cases (as most) where the seed number is particularly high. Other somewhat modified examples can be suggested in the case of parasites whose larvae feed on various animals (tapeworms), but which can only develop further in particular circumstances. (Necessary conditions are that there is a limit to the number of parasites that can live on a host. Further the mutual impact of the parasite and the host on each other's numbers would also play a role).

p. 529A modification of this type with passive occupation of existence places would be if the

monoikous grass germinated on the bad soil (R*) but did not reach flowering (reproduction). It would then actually occupy some of the R*-places and these would remain unavailable. For the monoikous form there would be little difference to the above, but for the dioikous form a proportion of the existence places would be lost. Thereby its advantage over the monoikous form would be reduced.

The overall result is presented in tabular form next to Fig. 3. As the strongly regulated type represents an unrealistic limiting case and also because there are no infinite population numbers, we are only interested in the section inside the marked border. Moreover when the selectively superior form normally eliminates the inferior, then it will be evn more the case if the superior form is dioikous and advantaged thus in other ways. In agreement with the forgoing result in Section V we can note the important result: A mutation which makes a new niche in the habitat usable and so increases the number of existence places will spread even if the mutants in the original individuals are strongly inferior (assuming, that the mutants can survive at all and are therefore present). A small ecological advantage can therefore compensate for an arbitrarily large selective disadvantage. Moreover, the mutants will not only occupy the new existence places, but always increase their numbers at the expense of the non-mutant individuals whose numbers will decrease to a stable equilibrium between mutants

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and non-mutants. The reduction in the number of individuals of the monoikous species will depend above all on the size (R*) of the newly colonized niche. The pair of equations (6) can serve as a basis for the numerical value of the equilibrium, as this type of realistic competition may be the most common. If the dioikous mutant has an equally high fitness as the original individuals then the latter will be eliminated, but this can (with certain types of competition) also happen if the mutants show a lower fitness than the non-mutants.

The general meaning of these result will be expanded on later. Here a brief addition is presented for completeness.

Up to now there has been discussion sometimes of species, of races or of mutants. The start of competition between two species which cannot be crossed can be thought of as an immigration of the dioikous type into the region of the monoikous one. For mutational origin of the dioikous type, the genetic difference monoikous-dioikous will as a rule

p. 530be Mendelian. A simple consideration shows that, regardless of the particular type of inheritance, the results obtained so far still hold. Consider for example the weakly regulated type, where the dioikous type is dominant (alleles d and m). Then the individuals of the two races when all the existence places are filled are,

Ňm = m2(N*+R*), Ňd = (d2 + 2dm)(N*+R*) (Eq. 10)

According to the law of constancy of genotype classes this relationship will from now stay the same in the absence of selection. That this assumption is met when Nm and Nd meet the equilibrium conditions (6) follows from (6) and (10) such that m2 = Ňm/(N*+R*) and d=1-m.

VII. The stability of the equilibriumFor later considerations, the stability of the equilibria obtained in the previous section are

of particular interest.let us say that at the outset the living spaces are all occupied (Nd+Nm=N*+R*), which means that the moment the population numbers (Nd , Nm) deviate from the equilibrium values (Ňm , Ňd ), for example by chance deviations deviate in one direction or another, then in the next period these values will tend to go back to the equilibria, similarly to a stone which has been pushed away from a hollow will tend to roll back to its deepest point. Unstable equilibriua, comparable with a rock on a spire or saddle, which would roll to a new point on the slightest disturbance are biologically unimportant (e.g. Nd = Nm = 0), and neutral equilibria, equivalent to a rock lying on a flat surface such that after every accidental shift it stays in the new location, are biologically rare (cf. equation 9 nm=nd) and for other reasons of little practical importance.

It is easy to show in particular cases that one is dealing with stable equilibria. Thus for the weakly regulated type from (4) and (%) and using the numbers in Fig. 2 (N*=4, R*=2, nm=6, nd=3) that the numbers of Nm change each generation as N'm = 6 Nm:(3+Nm). If one begins with Nm=4 then the N'm-series converging to the equilibrium 3 follows 4.00, 3.43, 3.20, 3.09, 3.05, etc. and similarly if one begins with Nm=2 it follows the series 2.00, 2.40. 2.67, 2.82, 2.91, etc. The same is true for the two other and for all

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the in between types, where it must be the case that the dioikous species cannot be displaced from its 'reserve' R*.

In the following it can only be briefly shown how the stability of the equilibrium can be shown mathematically, above all to show the similarity to the findings of VOLTERRA's Theory of biological communities.

p. 531The numbers of individuals in a species with a progeny number of n and a mortality rate v

increases from its value N0 in the 0-th generation to the r-th generation according to

Nr = N0[n(1-v)]r (Eq. 11)

After getting to "constant increase" this can be written as

dN/dt = ε N, (Eq.12)

then the growth rate ε is related to the biological constants n, v, and the generation time, in the relationship8 (Footnote 8 Then following (8), till the population saturates [? ver-s-facht], x = log s: log [n(1-v)] generations, thus the time is xT. After (12) it leads to the same effect of time (loge s)/ε. That the rwo are identical follows from (13).)

ε = (1/T) loge[n(1-v)] (Eq. 13)

Now we consider two species that differ in their reproductive rates (n1, n2) and their mortality rates (v1, v2). Let these population numbers be N0

1 and N02. The for the next generation it follows that

N1:N2 = n1(1-v1) N01 : n2(1-v2)N0

2 (Eq. 14)

If we take the log of (14) and divide through by the generation time (for which we can write t+T-t), it follows

log N1 - log N01/((t + Y) - t) = log N2 - log N0

2/((t + Y) - t) + (log n1(1-v1) - log n2(1-v2))/T (15)[typo in original equation]

If we write ∆t instead of T, so there is a difference quotient left and right of the equals sign. By transformation to instantaneous growth ( ∆t → dt), that is to the differential quotient, then after consideration of (13), (15) becomes

(1/N1) (dN1/dt) = (1/N2) (dN2/dt) + ε1 - ε2 (16)

Besides this the population numbers must satisfy the following, which is formed and based on complete analogy with (1), and leads to the total population (N1+N2)

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d (N1 + N2) = c ( ε1 N1 + ε2 N2 ).dt.(N* - N1-N2). (17)

Here furthermore c is equal to 1/N*. From (16) and (17) one obtains the final formula:

dN1/dt=(N1/N*) ε1 (N* - N1 - (ε2/ε1 )N2), dN2/dt=(N2/N*) ε2 (N* - (ε1/ε2 ) N1 - N2) (18)

Each of these is formed by analogy with (1b), except that in the parentheses, thus "the probability that a new existence place will be captured", the alternative species is weighted with a "weight" (ε2/ε1 , ε1/ε2 ) that carries the calculation of the differences in the reproductive and mortality rates.

This somewhat roundabout way has been chosen to show that in the relations expressed in the VERHULST equation can also be applied to nutrient or other competition and then the system (pair) of equations lead, such as (18) which is completely analogous to VERHULST's and in which the biological meaning of each of the constants presented is known. The equilibrium is obtained by setting dN1/dt= dN2/dt=0. For (18) 3 equilibria result: 1. N1 = 0, N2 = N*; 2. N2 =0, N1= N*; 3. N1 = N2 = 0.

p. 532Of these the only stable one s are 9 (Footnote9: The calculations from physics to determine whether the equilibrium is stable , etc. has been most clearly presented for the purpose of biologists by LOTKA (1923).) where the species with the smaller ε dies out and thus the species with the larger ε takes over all the existence places; the species with the higher fitness or reproductive rate eliminates the other species. Only for the very unlikely case of ε1= ε2 can both species leav in peaceful coexistence and the equilibrium would be neutral.

Thus for the previous case where the species only differ in their fitness or reproductive rate, no equilibrium is possible in which both species coexist permanently, and depends mathematically on the fact that when the parts in brackets in (18) are set eequal to each other no solution is possible and this further indicates that the coefficient of N1 in the first equation is the same as that in the second. If we allows arbitrary coefficients:

dN1/dt=a1 N1 + a11 N21 + a12 N1 N2 (19)

dN2/dt=a1 N2 + a21 N1 N2 + a22 N22 ,

so it is possible with certain magnitudes of these coefficients for both species to live together at the same time ( a11/a12 > a1/a2> a11/a21) and this equilibrium would be stable. Now it was shown here that for two species that differ only in their fitness or reproductive rate, but which consume the same nutrients and other resources of the environment in the same way, the one with the smallest ε dies out. The same is true if in two species one eats more, for example is bigger. Mathematically it could be that some individuals of the first species equal [gleichsetzen?] one of the second. On the other hand the above consideration of equal consumption of resources could be relaxed such that they do not need to rely on them, that they are present , and thus in part in excess and remaining unused. However, if one is dealing with differences in consumption of a scare factor of the resources present then the number of existence places for each species differs and instead equations of the form (19) , which allow stable equilibrium, then with complicated types of competition equations, besides the ones enumerated in (19), there would be

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additional parameters. It would be excessive to present the mathematics. It can be finally noted that the general equations (19) actually also can be relevant for species which consume exactly the same resources, but which do not stand in a predator-prey relationship to each other, in the case where their reproduction is dependent on given conditions, but where these latter nearly always have a very " "unbiological" character. Thus for example two species would coexist when the one with the larger εin th case where its population density exceeded a certain value, its competitiveness would be reduced, if it thus altruistically left the disadvantaged species some of the existence places.

VIII. Discussion, inferences, and implications

The information developed in the previous section is in no way new; however, apparently no one has made comprehensive inferences from them.

1. Biomathematical literature. VOLTERRA only dealt with the exact case in which two competing species strive for the same resources. Indeed he also presented our formula (19) in 1926, but did not emphasize the biological meaning of the six constants in the equation. LOTKA (1932) similarly only dealt with the consumption of the same nutrient

p. 533by both species, but mentioned briefly that an extension od this "narrow and unnatural restriction" must lead to an equilibrium in which neither species would die out. D'ANCONA (1939, p. 61) only made brief reference to LOTKA. KOSTIZIN (1937) considered both our pair of equations (19) from a purely mathematical standpoint without consideration of its biological meaning.

GAUSE (1932 ff.) finally gave the general formula (19) in a somewhat simpler form (here, our symbols are used):

dN1/dt = ε1 N1 [N1* -(N1 + αN2)/N1] (20) dN2/dt = ε2 N2 [N2* -(N2 + βN1)/N2] [Typo in subscript in original]

Clearly (20) is derived from (18), because on the basis of the broadened assumptions, both species do not consume the same nutrients, and thus in their common habitat they have a different number of existence places, the total N* in (18) that must be indicated by N1* and N2*.The constants α and β were empirically determined by GAUSE, e.g. (1933) for two yeast races on the basis of the population growth as α = 1.25 and β=0.850, and on the basis of the quantity of alchohol produced as α = 1.25 and β=0.80. Thereby GAUSE evidently missed that in the absence of disturbing conditions according to (18) that it must be that α . β = 1, which is the case for the above example (1.25 x 0.80 =1), and likewise that α and β are not new constants but represented only the relationship of the two reproductive coefficients (ε1/ε2 ; ε2/ε1).

2. Further investigations of GAUSE (1932 f, summary 1934, 1935 f.). In the experimental findings of this later work on competition between yeast and Paramecium species the relationship α . β = 1 no longer held., because at least three other things were predominant: a) it could be one species

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possesses a real selection advantage over the other, as for example P. caudatum over P. aurelia. Here thus both species achieve an equilibrium where the selectively disadvantaged species is eliminated; b) it could be one of the two competing species is bigger than the other so that the majority of these are the latter, for example P. caudatum > P. aurelia; c) the could additionally be differences that are hard to detect, e.g. damage through production of autoexcretions (such as alchohol in yeast).

3. Inferences. If two coexisting species have separate special nutrient/ food sources (R*1, R*2), different reproductive rates (ε1, ε2), different nutrient consumption per individual/time period (A1, A2), then the population equations would have at least eight constants, and even here some others would still be left out. Among these f1,2 can be included in ε1,2. Moreover there is - as can only be interpreted here on a spatial basis - at least for lower organisms a connection between A1,2 and ε1,2. Because , namely, in microorganisms energy consumption follows laws of surface area (LUDWIG 1928a) and the output for the exchange of materials necessary for existence and other than for transport (movement) are minimal, a linearly 10 times larger animal consumes 100 times as much energy and cannot use this

p. 534 amount for building its body, thus for growth and reproduction. That it must increase its size 1000 times, so it will reproduce 10 times as slowly as the smaller one and as a result will be eliminated by it (cf. Bacillus-Paramecium, LUDWIG 1928a). Also the selective disadvantage of the larger P. caudatum against P. aurelia (GAUSE) is a case in point. Thus for the microbial kingdom, the A1, 2 could be thought of as included in ε1,2.

4. Further considerations. Initially ecomutations do not necessarily need an increase in the number of existence places from the preexisting N* by R* (as in Fig. 4a), but only a shift in the existence regions (Fig. 4b): for the original race the existence places are indicated by the heavily outlined ovals, for the new one by the thinly outlined ovals so that in total there are three goups: R*1 , only occupiable by the original race; R*2 only by the mutants; N*, by both. Then we modify the results obtained up to now, that neither of the two races can extinguish the other , regardless of whether the older or newer has a greater fitness, because each possesses its own reserve R*1 or R*2. In the table next to Fig. 3 the possibility of the extinction of the original race (there the "monoikous") is removed. The equilibrium values are calculable analogously to the above, and depend on the relationship n1/n2 as ell as the values of R*1 ,R*2, and N*. The equilibria are again stable.

This also holds for the case of three or four species, about which we cannot go into any details (cf. Fig. 4c,d). Should a new species find a small niche in the a habitat occupied by n species, so it can persist there regardless of whatever selective disadvantages, and indeed increase its population size the more euryoikous it is (Fig. 4e, existence region in dashes, niche black). If in a habitat there are practically no more niches free, then this can be termed fully used up. Nevertheless, on probabilistic grounds a new "plyoikous" species can still annidate if from among the legitimate competitors it steals a small proportion of the "scarcity factors" (Fig. 4g). Anyway, even the simplest case with 3 species, where one can only use R1, the other only R2,

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and the third both, leads to mathematically involved equations which have to be presented elsewhere because of the large number of constants (N1,2,3; R1,2 , [sic] n1,2,3; ε1,2,3).

IX. Annidation (Einnischung) as a fifth evolutionary factor.Selection theory (see Ch. 1) has long considered that a new mutant is advantaged or

disadvantaged relative to the original species that it or the other will eventually be eliminated. In this regard, we have shown above that there have to be sharp differences according to whether a mutation brings a) a real selective advantage or b) an ecological advantage or

p. 535c) both. It remains certain that ecomutations must occur even if up to now precise studies are absent. This Einnischung or Annidation10 [10Footnote: annidare (ital: annidarsi) = nest in; innidare is less usable] is a fifth, and up to now neglected, evolutionary factor. It concerns new forms (=additional) of use of a common habitat, and implies a deviation from the premises on which the HARDY-WEINBERG law is based but which till now have not been explicitly stated.

p. 536Many antecedents for our conclusions have been present for a long time: the idea of preadaptation going back to CUENOT, the niche concept of ELTON; the conclusions of GOLDSCHMIDT (1940,1948) and others on "ecotypes" and "ecospecies", etc. The essentials of our findings are:

1. by ecomutations, diversification into new races and species can occur in the same place and without spatial separation;

2. very closely related species (e.g. insects: cereal click beetles, leaf hoppers of the meadows, etc.) can coexist permanently together in a very small living space; that

3. the average population extent of the co-existing species must be different; that4. the annidation of a new mutant, race or species can occur much more quickly than in

the case of selection because the new type of individual in its so-called new niche finds an unoccupied space and does not, as in the case of selection, have to gradually displace a prior resident; that

5. an ecomutation can be severely disadvantaged relative to its "competitors", because it is not displacable from its niche; and thereby

6. perhaps the origin of many selectively worthless to "deleterious" (atelic or dystelic) characters is plausible. In a special niche it is possible, for example in a fly species where long eye stalks bring an advantage, for this stalk to gradually increase to extremes, while within the niche competition wins, - while elsewhere large material savings are present; and finally

7. for extinction there may be no strong systematic clues: if the habitat of a new mutationally arisen or immigrant species also includes the niche of a species that has been specialized there for a long time, then this can be quickly totally eliminated.

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I have (1948) already briefly applied these thoughts to examples. LACK (1945), out of purely empirical grounds, the Galapagos-Geospizids came to similar results. Also human culture today, not to say also in the future, finds itself in a special niche out of which it will be hard to force and which can therefore lead to "luxurious" characteristics.

I thank Dr. HASSENSTEIN (Kaiser Wilhelm Institut, Wilhelmshaven) for the stimulation of the suggestion of the three competition types named in section VI. This, as well as on the selective and ecological value of body size, have been considered in other places, similarly with a rigorous biomathematical treatment of the expected course of population change under competition.

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