Competitive Exclusion and Axiomatic Set-Theory: De Morgan’s Laws, Ecological Virtual Processes, Symmetries and Frozen Diversity

Abstract

This work applies the competitive exclusion principle and the concept of potential competitors as simple axiomatic tools to generalized situations in ecology. These tools enable apparent competition and its dual counterpart to be explicitly evaluated in poorly understood ecological systems. Within this set-theory framework we explore theoretical symmetries and invariances, De Morgan’s laws, frozen evolutionary diversity and virtual processes. In particular, we find that the exclusion principle compromises the geometrical growth of the number of species. By theoretical extending this principle, we can describe interspecific depredation in the dual case. This study also briefly considers the debated situation of intraspecific competition. The ecological consequences of our findings are discussed; particularly, the use of our framework to reinterpret coupled mathematical differential equations describing certain ecological processes.

Appendices

Appendix 1: Resume for Symbols

• (A1) $\supset$ Perturbation.

• (A2) > Depredation.

• (A3) $\oplus$ Two independent species (independent processes).

• (A4) $\otimes$ Two interdependent species (interdependent processes).

• (A5) $\Rightarrow$ Temporal evolution.

• (A6) $\Leftrightarrow$ Equivalence.

• (A7) $\supset \subset$ Abbreviation for competition.

• (A8) $> <$ Mutual depredation.

• (A9) or Exclusion (some times $\vee$).

• (A10) $\sum$ Independent species: $A\oplus B\oplus C\oplus D\oplus \cdots$.

Appendix 2: Resume for Some Basic Processes and Definitions

• (B1) Exclusion:$\ \left\{ {\mathcal{N}}>\left( A\supset \subset B\right) \right\}$ $\Rightarrow \left\{ \left( {\mathcal{N}}>A\right)\;{\text{or}}\;\left( {\mathcal{N}}>B\right) \right\}$.

• (B2) Potential competitors: $\left\{ {\mathcal{S}}>\left( A\oplus B\right) \right\} \Rightarrow \left\{ {\mathcal{S}}>\left( A\supset \subset B\right) \right\}$.

• (B3) High intraspecific competition: ${\mathcal{N}}>(I\supset \subset I)\Rightarrow \left( {\mathcal{N}}>I\right)$ or $\left( {\mathcal{N}}>\phi \right)$.

• (B4) Dual process (D): $\left( >\right) \leftrightarrow \left( \supset \right)$.

• (B5) Inverse process (I): $\left( >\right) \leftrightarrow \left( <\right)$ and $\left( \supset \right) \leftrightarrow \left( \subset \right)$.

• (B6) First De Morgan’s law: $(A\oplus B)^{c}\Leftrightarrow \left( A^{c}\otimes B^{c}\right)$.

• (B7) A “sterile” species (M): $\left( {\mathcal{N}}>M\right) \Rightarrow \left( {\mathcal{N}}>\phi \right)$ then, from B1,

• (B8) competion with sterile: $\left\{ {\mathcal{N}}>\left( A\supset \subset M\right) \right\}$ $\Rightarrow \left\{ \left( {\mathcal{N}}>A\right)\; {\text{or}}\;\left( {\mathcal{N}}>\phi \right) \right\}$.

Point B1 corresponds to an axiom or premise. B2 and B3 correspond to definitions involving basic (operative) processes. Development of Sects. 5 and 6, also 8, are based on these mentioned points. Point B4–B5 are definitions related to virtual process. B6 is a consequence showed in Sect. 7 and related sections. Note that definitions B7 and B8, being tangential in this appendix, are motivated from Sterile Insect Technique (Alford 2015) to eradicate, for instance, fruit flies.