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:
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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)).
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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.
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.
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Or 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
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 fraction
c) 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 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.
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(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, 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. Ecomutations
In 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