Infinite Graphs in Systematic Biology, with an Application to the Species Problem

Abstract

We argue that C. Darwin and more recently W. Hennig worked at times under the simplifying assumption of an eternal biosphere. So motivated, we explicitly consider the consequences which follow mathematically from this assumption, and the infinite graphs it leads to. This assumption admits certain clusters of organisms which have some ideal theoretical properties of species, shining some light onto the species problem. We prove a dualization of a law of T. A. Knight and C. Darwin, and sketch a decomposition result involving the internodons of D. Kornet, J. Metz and H. Schellinx. A further goal of this paper is to respond to B. Sturmfels’ question, “Can biology lead to new theorems?”

Proof of Theorem 12

In this appendix we will prove Theorem 12 (hereafter we assume its hypotheses). First we review the definitions in Kornet et al. (1995). As we state them they are not equivalent; they are altered to fit the hypotheses of Theorem 12, allowing us to simplify them.⁸

Definition 5

For any $u\in G$, write ${{\mathbb{DYN}}(u)}$ for the set $\{x\in G : u({\bf PC}_{\geq u})x\}$ and write ≧ (u) for $\{x\in G : t(x)>t(u)\}.$ If $u,v\in G$ and t(u) < t(v), define

$$ u{\bf INT}v :\Leftrightarrow u({\bf PC}_{\geq u})v \wedge \forall r[\{u({\bf PC}_{\geq u})r \wedge \left( t(r)\leq t(v) \right)\} \Rightarrow ({{\mathbb{DYN}}}(u)\cap \geqq(r) = {{\mathbb{DYN}}}(r))]. $$

If t(v) < t(u) then u INT v :⇔ v INT u, and if t(v) = t(u) then u = v by our assumption that no individuals are born simultaneously, and we define u INT u.

The main theorem of Kornet et al. (1995) implies INT is an equivalence relation.

Definition 6

An internodon is an INT-equivalence class.

We could now dive directly into the proof of Theorem 12 but we prefer to factor out a sufficient condition which might be useful in the future for proving that other species notions decompose into internodons (this condition, too, will be generalized in future work).

Proposition 13

(Sufficient conditions for internodons-decomposition) (Assuming the hypotheses of Theorem 12.) Let  $S\subseteq V.$ If S is undirancestrally closed, and for each  $u\in S$ and  ${t\in {\mathbb{R}}}$ there is some  $u'\in S$ born after t with  $u({\bf PC}_{\geq u})$ u′; then S is a union of internodons.

Proof

It suffices to let $u\in S, v\in G$, and show that if u INT v then $v\in S$.

Case 1: v is born before u. Since v INT u, in particular $v({\bf PC}_{\geq v})$ u. Thus v is an undirancestor of u, putting $v\in S$ by undirancestral closure.

Case 2: v is born after u. Since u INT v, by definition of INT we have $u({\bf PC}_{\geq u})$ v and (*) for every r such that $u({\bf PC}_{\geq u})$ r and t(r) ≤ t(v), we have ${{\mathbb{DYN}}(u)\,\cap \geqq(r)={\mathbb{DYN}}(r)}$.

By the proposition’s hypothesis, there is some $u'\in S$, born after v, such that $u({\bf PC}_{\geq u})$ u′. Letting r = v in (*), we see (since $u({\bf PC}_{\geq u})$ v and t(v) ≤ t(v)) that ${{\mathbb{DYN}}(u)\cap\geqq(v)={\mathbb{DYN}}(v). }$ Since ${u'\in {\mathbb{DYN}}(u)\cap\geqq(v)}$, this shows ${u'\in{\mathbb{DYN}}(v)}$, that is, $v({\bf PC}_{\geq v})$ u′. Thus v is an undirancestor of u′, so $v\in S$ by undirancestral closure of S. $\square$

Proof of Theorem 12

Let S be an inspecies (type II). The first hypothesis of Proposition 13 holds since S is undirancestrally closed by definition. The second hypothesis of Proposition 13 is immediate by Proposition 6. $\square$

The reader will have noticed that despite imposing such onerous hypotheses on Theorem 12, we barely seem to have actually used those hypotheses. Their key use was in simplifying the definition of internodons.