Inter-Tunneling Mechanism of Colliding Population Waves

Here we show a new interaction mechanism of colliding population waves. It provides a stable coexistence of two similar but different species competing for the same limiting resource during their asexual propagation in a limited homogeneous environment under constant conditions. The revealed mechanism opens new opportunities in conservation biology

Strong violation of the competitive exclusion principle

Bacteria and plants are able to form population waves as a result of their consumer behaviour and propagation. A soliton-like interpenetration of colliding population waves was assumed but not proved earlier. Here we show how and why colliding population waves of trophically identical but fitness different species can interpenetrate through each other without delay. We have hypothesized and revealed here that the last mechanism provides a stable coexistence of two, three and four species, competing for the same limiting resource in the small homogeneous habitat under constant conditions and without any fitness trade-offs. We have explained the mystery of biodiversity mechanistically because (i) our models are bottom-up mechanistic, (ii) the revealed interpenetration mechanism provides strong violation of the competitive exclusion principle and (iii) we have shown that the increase in the number of competing species increases the number of cases of coexistence. Thus the principled assumptions of fitness neutrality (equivalence), competitive trade-offs and competitive niches are redundant for fundamental explanation of species richness

Strong and weak competitors can coexist in the same niche

The competitive exclusion principle postulates that two trophically identical but fitness different species can not stably coexist in the same niche. However, this principle contradicts the observed nature's species richness. This fact is known as the biodiversity paradox. Here, using a simple cellular automaton model, we mechanistically show how two trophically identical, but fitness different species may stably coexist in the same niche. As environment is stable and any trade-offs are absent in this model, it strongly violates the competitive exclusion principle

A unified mechanistic model of niche, neutrality and violation of the competitive exclusion principle

The origin of species richness is one of the most widely discussed questions in ecology. The absence of unified mechanistic model of competition makes difficult our deep understanding of this subject. Here we show such a two-species competition model that unifies (i) a mechanistic niche model, (ii) a mechanistic neutral (null) model and (iii) a mechanistic violation of the competitive exclusion principle. Our model is an individual-based cellular automaton. We demonstrate how two trophically identical and aggressively propagating species can stably coexist in one stable homogeneous habitat without any trade-offs in spite of their 10% difference in fitness. Competitive exclusion occurs if the fitness difference is significant (approximately more than 30%). If the species have one and the same fitness they stably coexist and have similar numbers. We conclude that this model shows diffusion-like and half-soliton-like mechanisms of interactions of colliding population waves. The revealed mechanisms eliminate the existing contradictions between ideas of niche, neutrality and cases of violation of the competitive exclusion principle

Mechanistic mechanisms of competition and biodiversity

The nature of competition and biodiversity are open basic questions since Darwin. To investigate mechanisms of interspecific competition and their contribution in biodiversity as closely as possible we offer a white-box modelling method based on physically interpreted ecological axioms. These models are implemented as deterministic individual-based cellular automata and able to give a direct physico-mechanistic insight into studied phenomena. Competition of two trophically identical but fitness different species, competing for one limiting resource in one stable uniform habitat (which is closed for immigration, emigration, predation, herbivory and parasitism) has been investigated in conditions, which are the most unfavourable for their coexistence. The species are per capita identical in fecundity, ontogeny, regeneration features of microhabitats, and in habitat requirements. We have modelled following 8 mechanistic mechanisms of interspecific competition: 
1.	A case of the competitive exclusion when competing species differ only in fitness. 
2.	Coexistence based on periodic dominance changeovers as a consequence of environmental changes. Competing species differ only in fitness. 
3.	A strong violation of the competitive exclusion principle due to the lowered fecundity of both competitors. Competing species differ only in fitness.
4.	Coexistence based on the competition–colonisation trade-off when greater fitness is compensated by r-strategy.
5.	A competition–colonisation trade-off based on differences in ontogeny. 
6.	Competitive exclusion when recessive species drives out the dominant one having four times greater fecundity than the dominant one in stable environment (the greater fitness cannot compensate r-strategy). 
7.	An inverted competitive exclusion when recessive species drives out the dominant one by strategy of anticipatory deprivation of resources for competitor’s offsprings propagation. Recessive species drives out the dominant one in stable environment and both competing species have identical fecundity (tripod neighbourhood). Paradoxically, but the greater fitness cannot save the dominant species when the all other parameters of the species are equal.
8.	Both competing species die because the regeneration of a limiting environmental resource takes too much time and they cannot propagate. 
The revealed mechanisms of competition can be useful not only in conservation biology, but also in economics and politics. Additionally, we speculate that the simplest way to maintain biodiversity is a controlled reduction of human fertility as the decrease in biodiversity occurs largely due to humankind overloading of biosphere resources. 
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Two-loop self-energy master integrals on shell

Analytic results for the complete set of two-loop self-energy master integrals on shell with one mass are calculated.Comment: 14 pages, LaTeX, one eps-figure; in v5. misprints in the Eq.(2) and Table II correcte

Recursion relations for two-loop self-energy diagrams on-shell

A set of recurrence relations for on-shell two-loop self-energy diagrams with one mass is presented, which allows to reduce the diagrams with arbitrary indices (powers of scalar propagators) to a set of the master integrals. The SHELL2 package is used for the calculation of special types of diagrams. A method of calculation of higher order \epsilon-expansion of master integrals is demonstrated.Comment: Talk given by J.Fleischer at 6th International Workshop on Software Engineering, Artificial Intelligence, Neural Nets, Genetic Algorithms, Symbolic Algebra, Automatic Calculation (AIHENP 99), Heraklion, Crete, Greece, 12-16 April, 1999; 8 pages, LaTeX, 3 eps-figure