An evolution model with event-based extinction^*

In a seminal paper, Bak and Sneppen [1] have proposed a simple model of evolution of species, which has inspired a wealth of studies in several areas of knowledge, including physics, biology and mathematics. In [1], species are identified with sites in a circular 1d grid, and their evolution are described through a dynamics on their fitnesses, which at each step replaces the current minimal fitness along with neighboring current fitnesses at the grid with fresh, independent fitnesses sampled from the standard uniform distribution.

In the mathematics literature, there are few rigorous results concerning the long term behavior of the Bak–Sneppen system. Noteworthy are papers of Meester and Znamenski [2, 3], which derive information on the asymptotic fitness distribution of the model.

Guiol, Machado and Schinazi [4] proposed a different, if somewhat similar dynamics, describing the evolution (in discrete time, as in [1]) of species who independently appear at each time step with probability p, and are assigned a fitness sampled from the uniform distribution on [0, 1]. Extinction occurs at each time step, also independently and with probability 1 − p, whenever there is at least one species present at the corresponding time, in which case the one with the least fitness is removed from the system. Note that the rule to 'kill the least fit' is a common feature with [1]; however, the local interactions of the Bak–Sneppen system are absent. In [4] it was proved that species fitnesses are asymptotically uniformly distributed in (f_c, 1), where f_c = (1 − p)/p, a similar result to Meester and Znamenski's.

Several extensions and generalizations of the GMS model were subsequently introduced and studied, as in Ben-Ari et al [5], Michael and Volkov [6], Bertacchi et al [7] and Grejo et al [8]. Further, Guiol et al [9] proposed a variation of the model, where the evolution is given in continuous time. See also Formentin and Swart [10] for a closely related model with a different motivation. We refer to this literature for further discussion of the models and results.

We consider here a variation of the GMS model [9] in which, as in that model, new species are born at a given rate, and at possibly another rate we have extinction of species. For each new species in the system, we associate a positive random number, chosen from a distribution F_⋆. We call this random number the fitness of the species. So far, the setting is the same as(or quite similar to the one) in [9].

Our variation is (more markedly) related with the extinction events. At the time of each such event, we have a positive threshold random variable, with distribution F_†, and all species with fitness below this threshold at that time, if any, get extinct.

We believe this is a natural variation of the GMS model, when we consider extinction of species in the natural world produced by major events such as abrupt habitat change, where conceivably each species might be affected according to its own aptitude to face the new challenge, irrespective of other species.

The random fitnesses and the random thresholds are independent of each other and of everything else in the process. We assume F_⋆ is continuous on ${\mathbb{R}}_{+}=\left[0,\infty \right)$; so, in particular, we can and will identify each species with its fitness. We also assume, for simplicity, that F_⋆ and F_† have unbounded support.

Let Π_⋆ and Π_† be independent Poisson point processes in $\mathbb{R}$ with rates λ_⋆ and λ_†, respectively. Define ${\left\{{T}_{i}\right\}}_{i\in {\mathbb{Z}}^{{\ast}}}$ as the set of birth time instants of a new species in the system, define ${\left\{{S}_{j}\right\}}_{j\in {\mathbb{Z}}^{{\ast}}}$ as the instants of time in which there is an extinction event. These sequences represent the points in the Poisson processes and are indexed in increasing order. At each time T_i ∈ Π_⋆ we assign to the newly appeared species the fitness X_i, drawn from F_⋆, and to each time S_j ∈ Π_† we associate the threshold Y_j from distribution F_†.

Given a locally finite subset A of ${\mathbb{R}}_{+}$, let η_t = {η_t(s), s ⩾ t} denote our evolution model starting from A at time t. So, at initial time $t\in \mathbb{R}$, the process has initial configuration η_t(t) = A, and at time s ∈ (t, ∞), the process has the configuration η_t(s) which is composed of all species/fitnesses either of A or that have appeared in the time interval (t, s], and that have survived the events of extinction in [t, s], i.e. species whose fitnesses are greater than the highest threshold drawn in events of extinction in [t, s] after their birth.

Notice that as long as A does not depend on t, neither does the distribution of η_t, so below we will often restrict to η₀.

Figure 1 simulates a realization of the process η_t in the time interval [t, s]. Each circle represents a species alive in the process and the gray arrow issuing from it represents its life time. Observe that the position of each species in relation to the vertical axis is given by its fitness. The three black circles represent the configuration of the process at time t. The six gray circles represent the species that were born in the time interval [t, s]; the positions that they occupy represent the respective instants of time they were born (horizontal axis) and the fitnessess that were drawn to them (vertical axis). The four withe circles represent the species that survived the extinction events and are alive at the time s. Finally, each extinction threshold is represented in figure 1 by black stars.

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1.1. Preliminary comparison to the GMS model

1.2. Records.

We derive three kinds of results. First, criteria for recurrence and transience of the empty configuration in η₀; see theorem 2.1 below; they are obtained from the analysis of a Poisson process of records. Second, we derive the existence of a limit distribution for η₀(s) as s → ∞; see theorem 2.2 below. Finally, we derive criteria for finiteness and infiniteness of the number of species present in the limit distribution; see theorem 2.3 below; curiously, the Poisson process of records of the recurrence and transience issue appears here as well, but in reverse. The behavior of $\overline{{F}_{{\dagger}}}{\circ}{\bar{{F}_{\star }}}^{-1}$ at the origin plays a determinant role in the first result, and thus, in reverse, so does that of $\bar{{F}_{\star }}{\circ}{\overline{{F}_{{\dagger}}}}^{-1}$ in the third one; as usual, for * = ⋆ and †, $\overline{{F}_{{\ast}}}=1-{F}_{{\ast}}$, and ${\overline{{F}_{{\ast}}}}^{-1}$ indicates the (right-continuous) inverse of $\overline{{F}_{{\ast}}}$, namely

$$\begin{equation*}{F}_{{\ast}}^{-1}\left(t\right)=\mathrm{inf}\left\{s:{F}_{{\ast}}\left(s\right){ >}t\right\},\quad 0{< }t{< }1.\end{equation*}$$

For the latter result we need to assume that F_† is continuous. Proofs are deferred to the last section.

Let us define the functions ${R}_{\star },{R}_{{\dagger}}:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ by

$$\begin{align*}\hfill {R}_{\star }\left(x\right)& =-\mathrm{log}\;\bar{{F}_{\star }}\left(x\right),\hfill \\ \hfill {R}_{{\dagger}}\left(x\right)& =-\mathrm{log}\;\overline{{F}_{{\dagger}}}\left(x\right).\hfill \end{align*}$$

For birth events denote by I_k the record indexes and by ${X}_{{I}_{k}}$ the record values, as follows: I₁ ≐ 1; and for k ⩾ 1

$$\begin{equation*}{I}_{k+1}\doteq \mathrm{min}\left\{i{ >}{I}_{k}:{X}_{i}{ >}{X}_{{I}_{k}}\right\}.\end{equation*}$$

Proposition 2.1. (Proposition 4.11.1, Resnick [13]). The record values ${\left({X}_{{I}_{k}}\right)}_{k{\geqslant}1}$ form a Poisson point process in ${\mathbb{R}}_{+}$ with intensity measure ∫_BR_⋆(dx), $B\in \mathcal{B}\left({\mathbb{R}}_{+}\right)$.

The next result refines the previous one.

Proposition 2.2. Let ${T}_{{I}_{k}}$ be the time of the kth record and denote by ${\Delta}{T}_{{I}_{k}}\doteq {T}_{{I}_{k+1}}-{T}_{{I}_{k}}$ the interval between two consecutive records. Then, $\left\{{\left({X}_{{I}_{k}},{\Delta}{T}_{{I}_{k}}\right)}_{k{\geqslant}1}\right\}$ is a Poisson process in ${\mathbb{R}}_{+}^{2}$ with intensity measure

$$\begin{equation*}{\hat{\mu }}_{\star }\left(C\right){\doteq \iint }_{C}{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right){\mathrm{e}}^{-{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right)s}\mathrm{d}s{R}_{\star }\left(\mathrm{d}x\right),\quad C\in \mathcal{B}\left({\mathbb{R}}_{+}^{2}\right).\end{equation*}$$

Proposition 2.3. The set of points ${\left({S}_{j},{Y}_{j}\right)}_{j{\geqslant}1}$ is a Poisson process in ${\mathbb{R}}_{+}^{2}$, denoted by ${\hat{{\Pi}}}_{{\dagger}}$, with intensity measure

$$\begin{equation*}{\mu }_{{\dagger}}\left(C\right)\doteq {\lambda }_{{\dagger}}{\iint }_{C}\mathrm{d}t{F}_{{\dagger}}\left(\mathrm{d}x\right),\quad C\in \mathcal{B}\left({\mathbb{R}}_{+}^{2}\right).\end{equation*}$$

To study recurrence and transience, we will build a ladder of records using the process of births and the fitness associated to each species. Let us denote the random region above each step of the ladder by ${\left\{{D}_{k}\right\}}_{k{\geqslant}1}$, and the region above the full ladder is denoted by D, namely, for k ⩾ 1

$$\begin{equation*}{D}_{k}\doteq \left[{T}_{{I}_{k}},{T}_{{I}_{k+1}}\right){\times}\left[{X}_{{I}_{k}},\infty \right)\quad \text{and}\quad D\doteq \bigcup _{k{\geqslant}1}{D}_{k}.\end{equation*}$$

Let Λ ≐ μ_†(D), $\mathcal{M}\doteq D\cap {\hat{{\Pi}}}_{{\dagger}}$, and $M=\#\mathcal{M}$. Observe that M counts the number of extinction events in D—as noted in Sub section 1.2, these extinction marks generate empty configurations, with no species present at and immediately after their corresponding times³. Given D, M has a Poisson distribution with mean Λ (because ${\hat{{\Pi}}}_{{\dagger}}$ is a Poisson process), so

$$\begin{equation}\mathbb{E}\left[{\mathrm{e}}^{-tM}\right]=\mathbb{E}\left[\mathbb{E}\left[{\mathrm{e}}^{-tM}\vert {\Lambda}\right]\right]=\mathbb{E}\left[{\mathrm{e}}^{-\left(1-{e}^{-t}\right){\Lambda}}\right].\end{equation} \tag{ 1 }$$

Here we allow Λ = ∞, in which case M = ∞ a.s. It may be checked, from ergodicty considerations, that {Λ = ∞} is a trivial event, and, thus, so is {M = ∞}.

Remark 2.4. Define ${h}_{{\dagger}}:{\mathbb{R}}_{+}^{2}\to {\mathbb{R}}_{+}$ by ${h}_{{\dagger}}\left(x,s\right)\doteq {\lambda }_{{\dagger}}s\overline{{F}_{{\dagger}}}\left(x\right)$. From the definition of μ_† in proposition 2.3, we have

$$\begin{equation}{\Lambda}={\mu }_{{\dagger}}\left(D\right)=\sum _{k{\geqslant}1}{\mu }_{{\dagger}}\left({D}_{k}\right)=\sum _{k{\geqslant}1}{\lambda }_{{\dagger}}{\Delta}{T}_{{I}_{k}}\overline{{F}_{{\dagger}}}\left({X}_{{I}_{k}}\right)=\sum _{k{\geqslant}1}{h}_{{\dagger}}\left({X}_{{I}_{k}},{\Delta}{T}_{{I}_{k}}\right).\end{equation} \tag{ 2 }$$

Note that $\mathbb{E}\left[M\right]=\mathbb{E}\left[\mathbb{E}\left[M\vert {\Lambda}\right]\right]=\mathbb{E}\left[{\Lambda}\right]$ and we may use Campbell's formula (see proposition 2.7 in [14]) and (2) to find

$$\begin{equation}\mathbb{E}\left[{\Lambda}\right]=\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{0}^{\infty }\frac{\overline{{F}_{{\dagger}}}\left(x\right)}{\bar{{F}_{\star }}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right).\end{equation} \tag{ 3 }$$

Proposition 2.5. For t > 0

$$\begin{equation*}\mathbb{E}\left[{\mathrm{e}}^{-tM}\right]=\mathrm{exp}\left[-{\int }_{0}^{\infty }\frac{\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)+\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right)\right].\end{equation*}$$

2.1. Recurrence/transience of η₀

We start by discussing the a.s. finiteness of M. Define the functions $\phi :{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ and ψ: [0, 1] → [0, 1] by

$$\begin{equation*}\phi \left(t\right)\doteq {\int }_{0}^{\infty }\frac{\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)+\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right),\end{equation*}$$

and

$$\begin{equation}\psi \left(u\right)\doteq \overline{{F}_{{\dagger}}}{\circ}{\bar{{F}_{\star }}}^{-1}\left(u\right).\end{equation} \tag{ 4 }$$

By monotone convergence, we have that

$$\begin{equation*}\phi \left(\infty \right)\doteq \underset{t\to \infty }{\mathrm{lim}}\phi \left(t\right)={\int }_{0}^{\infty }\frac{{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)+{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right).\end{equation*}$$

Proposition 2.6. The following statements are equivalent:

• (a)  
  ϕ(∞) < ∞;
• (b)  
  $\mathbb{P}\left(M{< }\infty \right)=1$;
• (c)  
  $\mathbb{E}\left[M\right]{< }\infty$;
• (d)  
  $\frac{\psi \left(u\right)}{{u}^{2}}$ is integrable at the origin;
• (e)  
  ψ(s⁻¹) is integrable at infinity.

Definition 2.7. Let ϒ = {s ⩾ 0 : η₀(s) = ∅} denote the set of times where the species configuration is empty; in other words, ϒ is the set of times where there is no species in the system. ϒ is readily seen to a.s. consist of a collection {[a_j, b_j), j ⩾ 1}, either finite or infinite, of disjoint finite intervals. Let ϒ' = {a_j, j ⩾ 1} be the collection of left endpoints of such intervals, correspondings to the times where the species configuration freshly becomes empty. We have that ${\mathcal{M}}^{\prime }\subset {{\Upsilon}}^{\prime }$, where ${\mathcal{M}}^{\prime }$ is the projection of $\mathcal{M}$ on the time axis. (However, as pointed out above, ${\mathcal{M}}^{\prime }\ne {{\Upsilon}}^{\prime }$ a.s.; see footnote on page 5.)

The process η₀ is said to be transient if ϒ is a.s. a bounded set. On the other hand, if ϒ is a.s. an unbounded set, then we say that η₀ is recurrent.

We point out that, since the set of fitness record times $\left\{{T}_{{I}_{k}},k{\geqslant}1\right\}$ is unbounded, we have that ϒ/ϒ' is bounded if and only if ${\mathcal{M}}^{\prime }$ is bounded.

Theorem 2.1. The process η₀ is transient if

$$\begin{equation*}{\int }_{0}^{\infty }\frac{\overline{{F}_{{\dagger}}}\left(x\right)}{\bar{{F}_{\star }}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right){< }\infty ,\end{equation*}$$

and recurrent otherwise.

Example 2.8. Suppose that (X_i) and (Y_j) are exponentially distributed with parameter α_⋆ and α_†, respectively. Then,

$$\begin{equation*}{\int }_{0}^{\infty }\frac{\overline{{F}_{{\dagger}}}\left(x\right)}{\bar{{F}_{\star }}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right)=\begin{cases}\frac{{\alpha }_{\star }}{{\alpha }_{{\dagger}}-{\alpha }_{\star }},\quad \hfill & \text{if}\;\;\;{\alpha }_{{\dagger}}{ >}{\alpha }_{\star }\hfill \\ \infty ,\quad \hfill & \text{if}\;\;\;{\alpha }_{{\dagger}}{\leqslant}{\alpha }_{\star }.\hfill \end{cases}\end{equation*}$$

By proposition 2.6, if α_† ⩽ α_⋆, then M = ∞ a.s., and from theorem 2.1, η₀ is recurrent. If α_† > α_⋆, then η₀ is transient. We may compute the distribution of Λ and M in this case (it will come in handy below), as follows: by 2, $\mathbb{E}\left[{\mathrm{e}}^{-t{\Lambda}}\right]=\mathbb{E}\left[{\mathrm{e}}^{-{\sum }_{k{\geqslant}1}t.{h}_{{\dagger}}\left({X}_{{I}_{k}},{\Delta}{T}_{{I}_{k}}\right)}\right]$ and using the characterisation of a Poisson process via its Laplace functional (see theorem 3.9 on page 23 of [14])

$$\begin{align*}\hfill \mathbb{E}\left[{\mathrm{e}}^{-{\sum }_{k{\geqslant}1}t.{h}_{{\dagger}}\left({X}_{{I}_{k}},{\Delta}{T}_{{I}_{k}}\right)}\right]& =\mathrm{exp}\left[-{\int }_{\left[0,+\infty \right)}\frac{t{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right)+t{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right)\right]\hfill \\ \hfill & ={\left(\frac{{\lambda }_{\star }}{{\lambda }_{\star }+t{\lambda }_{{\dagger}}}\right)}^{\frac{{\alpha }_{\star }}{{\alpha }_{{\dagger}}-{\alpha }_{\star }}}={\left(1+t{\beta }^{-1}\right)}^{-r}.\hfill \end{align*}$$

By proposition 2.5, $\mathbb{E}\left[{\mathrm{e}}^{-tM}\right]={\left(\frac{1-p}{1-p{\text{e}}^{-t}}\right)}^{r}$, where $r=\frac{{\alpha }_{\star }}{{\alpha }_{{\dagger}}-{\alpha }_{\star }},\quad \beta =\frac{{\lambda }_{\star }}{{\lambda }_{{\dagger}}}$ and $p=\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }+{\lambda }_{{\dagger}}}$. Hence, Λ follows the gamma distribution with parameters r and β, and M follows the negative binomial distribution with parameters r and p.

2.2. Existence and in/finiteness of a long time limit distribution.

We now establish the existence of a limit distribution for η₀(s) as s → ∞. Remarkably, a ladder construction based on a record process comes up here as well, entirely parallel to that of subsection 2.1, with births and extinctions swapping roles. For that to hold we need however to assume that F_† is continuous. The ladder construction then immediately yields necessary and sufficient criteria for the almost sure in/finiteness of the number of species present in the limit distribution, identical to those for transience/recurrence of η₀, except that the symbols ⋆ and † swap roles.

As a preliminary, we will use the process of extinctions and the associated thresholds to build a ladder of records, much as in subsection 2.1. We define the kth record index J_k and the record value ${Y}_{{J}_{k}}$ as follows: J₁ ≐ −1 and; for k ⩾ 1

$$\begin{equation*}{J}_{k+1}\doteq \mathrm{max}\left\{j{< }{J}_{k}:{Y}_{j}{ >}{Y}_{{J}_{k}}\right\}\end{equation*}$$

Again by proposition 4.11.1 in [13], we get that the ${\left\{{Y}_{{J}_{k}}\right\}}_{k{\geqslant}1}$ form a Poisson point process in ${\mathbb{R}}_{+}$ with intensity measure ∫_BR_†(dx), $B\in \mathcal{B}\left({\mathbb{R}}_{+}\right)$.

Denote by ${S}_{{J}_{k}}\left(0\right)$ the time of the kth record, and by ${\Delta}{S}_{{J}_{k}}\doteq {S}_{{J}_{k}}-{S}_{{J}_{k+1}}$ the time span between two consecutive records. Similarly as in subsection 2.1, we have the following results.

Proposition 2.9. The points ${\left({Y}_{{J}_{k}},{\Delta}{S}_{{J}_{k}}\right)}_{k{\geqslant}1}$ form a Poisson point process in ${\mathbb{R}}_{+}^{2}$ with intensity

$$\begin{equation*}{\hat{\mu }}_{{\dagger}}\left(C\right){\doteq \iint }_{C}{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right){\mathrm{e}}^{-{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)s}\mathrm{d}s{R}_{{\dagger}}\left(\mathrm{d}x\right),\quad C\in \mathcal{B}\left({\mathbb{R}}_{+}^{2}\right).\end{equation*}$$

Proposition 2.10. The points ${\left\{\left({T}_{-i},{X}_{-i}\right)\right\}}_{i{\geqslant}1}$ form a Poisson point process in ${\mathbb{R}}_{-}{\times}{\mathbb{R}}_{+}$, ${\mathbb{R}}_{-}=\left(-\infty ,0\right]$, denoted by ${\hat{{\Pi}}}_{\star }$, with intensity

$$\begin{equation*}{\mu }_{\star }\left({C}^{\prime }\right)\doteq {\lambda }_{\star }{\iint }_{{C}^{\prime }}\mathrm{d}t{F}_{\star }\left(\mathrm{d}y\right),\quad {C}^{\prime }\in \mathcal{B}\left({\mathbb{R}}_{-}{\times}{\mathbb{R}}_{+}\right).\end{equation*}$$

We thus have our ladder of thresholds; denote the region above each step by

$$\begin{equation*}{E}_{0}\doteq \left[{S}_{{J}_{1}},0\right){\times}{\mathbb{R}}_{+};\end{equation*}$$

$$\begin{equation*}{E}_{k}\doteq \left({S}_{{J}_{k+1}},{S}_{{J}_{k}}\right]{\times}\left({Y}_{{J}_{k}},\infty \right),k{\geqslant}1;\quad E\doteq \bigcup _{k{\geqslant}1}{E}_{k}.\end{equation*}$$

We state our existence result.

Theorem 2.2. η₀(t) converges in distribution to $\hat{\eta }$ as t → ∞, where

$$\begin{equation*}\hat{\eta }\doteq \left\{{X}_{i}:{T}_{i}\in \left({S}_{{J}_{1}},0\right]\right\}\cup \left(\bigcup _{k{\geqslant}1}\left\{{X}_{i}{ >}{Y}_{{J}_{k}}:{T}_{i}\in \left({S}_{{J}_{k+1}},{S}_{{J}_{k}}\right]\right\}\right).\end{equation*}$$

Remark 2.11. The topology for weak convergence is the usual one in the context of point processes. Our proof indeed makes use of a coupling to a sequence of processes for which the convergence is a strong one, and follows by monotonicity.

Next, we address the issue of finiteness of the number of species in $\hat{\eta }$. For each k ⩾ 0, let

$$\begin{equation*}{{\Sigma}}_{k}\doteq {\mu }_{\star }\left({E}_{k}\right);\quad {N}_{k}\doteq \#\left\{{E}_{k}\cap {\hat{{\Pi}}}_{\star }\right\}.\end{equation*}$$

Let also Σ ≐ μ_⋆(E); $N\doteq \#\left\{E\cap {\hat{{\Pi}}}_{\star }\right\}$. We may use Campbell's formula to find

$$\begin{equation*}\mathbb{E}\left[N\right]=\frac{{\lambda }_{\star }}{{\lambda }_{{\dagger}}}{\int }_{0}^{\infty }\frac{\bar{{F}_{\star }}\left(x\right)}{\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{{\dagger}}\left(\mathrm{d}x\right).\end{equation*}$$

Note that N is the number of birth events above the threshold ladder. Also, note that E has a parallel structure to that of the D ladder of subsection 2.1; the random variables Σ and N are parallel to Λ and M in that same subsection. Thus, we get parallel results, once we exchange the roles of $\left({\lambda }_{\star },{\overline{F}}_{\star }\right)$ and $\left({\lambda }_{{\dagger}},{\overline{F}}_{{\dagger}}\right)$.

From theorem 2.2, the number of species present in the limit distribution $\hat{\eta }$, denoted by $\#\hat{\eta }$, is

$$\begin{equation}\#\hat{\eta }=\sum _{k{\geqslant}0}\#\left\{{E}_{k}\cap {\hat{{\Pi}}}_{\star }\right\}=\sum _{k{\geqslant}0}{N}_{k}={N}_{0}+N.\end{equation} \tag{ 5 }$$

It is enough to consider the finiteness of N. From the parallel situation of subsection 2.1, we get the following results. For t > 0

$$\begin{align*}\hfill & \quad \quad \quad \mathbb{E}\left[{\mathrm{e}}^{-tN}\right]=\mathbb{E}\left[{\mathrm{e}}^{-\left(1-{e}^{-t}\right){\Sigma}}\right]\hfill & \hfill \\ \hfill & =\mathrm{exp}\left[-{\int }_{0}^{\infty }\frac{\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)}{{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)+\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)}{R}_{{\dagger}}\left(\mathrm{d}x\right)\right].\hfill & \hfill \end{align*}$$

Setting

$$\begin{equation*}\overline{\phi }\left(t\right)\doteq {\int }_{0}^{\infty }\frac{\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)}{{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)+\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)}{R}_{{\dagger}}\left(\mathrm{d}x\right),\quad t{ >}0,\end{equation*}$$

and

$$\begin{equation*}\overline{\psi }\left(u\right)\doteq \bar{{F}_{\star }}{\circ}{\overline{{F}_{{\dagger}}}}^{-1}\left(u\right),\quad u{ >}0,\end{equation*}$$

we have

$$\begin{equation*}\overline{\phi }\left(\infty \right)\doteq \underset{t\to \infty }{\mathrm{lim}}\overline{\phi }\left(t\right)={\int }_{0}^{\infty }\frac{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)}{{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)+{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)}{R}_{{\dagger}}\left(\mathrm{d}x\right).\end{equation*}$$

Proposition 2.12. The following statements are equivalent:

• (a)  
  $\overline{\phi }\left(\infty \right){< }\infty$;
• (b)  
  $\mathbb{P}\left(N{< }\infty \right)=1$;
• (c)  
  $\mathbb{E}\left[N\right]{< }\infty$.
• (d)  
  $\frac{\overline{\psi }\left(u\right)}{{u}^{2}}$ is integrable at the origin;
• (e)  
  $\overline{\psi }\left({s}^{-1}\right)$ is integrable at infinity.

Theorem 2.3. The number of species in the limit distribution, $\#\hat{\eta }$, is finite if

$$\begin{equation*}{\int }_{0}^{\infty }\frac{\bar{{F}_{\star }}\left(x\right)}{\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{{\dagger}}\left(\mathrm{d}x\right){< }\infty ,\end{equation*}$$

and infinite otherwise.

Example 2.13. Let (X_i) and (Y_j) be as in example 2.8. We have then

$$\begin{equation*}{\int }_{0}^{\infty }\frac{\bar{{F}_{\star }}\left(x\right)}{\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{{\dagger}}\left(\mathrm{d}x\right)=\begin{cases}\frac{{\alpha }_{{\dagger}}}{{\alpha }_{\star }-{\alpha }_{{\dagger}}},\quad \hfill & \text{if}\;\;\;{\alpha }_{\star }{ >}{\alpha }_{{\dagger}}\hfill \\ \infty ,\quad \hfill & \text{if}\;\;{\alpha }_{\star }{\leqslant}{\alpha }_{{\dagger}}.\hfill \end{cases}\end{equation*}$$

By the theorem 2.3, if α_⋆ ⩽ α_†, then $\#\hat{\eta }=\infty$ a.s. If α_⋆ > α_†, then $\#\hat{\eta }{< }\infty$ a.s.; let us find its distribution. We can show that, as in example 2.8, N follows the negative binomial distribution with parameters $\frac{{\alpha }_{{\dagger}}}{{\alpha }_{\star }-{\alpha }_{{\dagger}}}$ and $\frac{{\lambda }_{\star }}{{\lambda }_{\star }+{\lambda }_{{\dagger}}}$. From the joint distribution of (N₀, Σ₀), discussed in the proof of theorem 2.3 below, we have that N₀ follows the negative binomial distribution with parameters 1 and $\frac{{\lambda }_{\star }}{{\lambda }_{\star }+{\lambda }_{{\dagger}}}$. It may be checked that N₀ and N are independent, and we find from (5) that $\#\hat{\eta }$ follows the negative binomial distribution with parameters $\frac{{\alpha }_{\star }}{{\alpha }_{\star }-{\alpha }_{{\dagger}}}$ and $\frac{{\lambda }_{\star }}{{\lambda }_{\star }+{\lambda }_{{\dagger}}}$.

Remark 2.14. We might label the case of ∞ in both criteria in theorems 2.1 and 2.3 as null recurrent. This is of course the case when F_⋆ = F_†, which then becomes a natural candidate for comparison with the GMS model below the critical point, which also is recurrent and has infinitely many species in its limiting distribution, as anticipated in subsection 1.1, and is thus null recurrent in the same way. However, as we briefly discuss below—see remark 3.1—, in this case the models are different in another fundamental level, namely the behavior of N^f, the number of species in the limiting distribution with fitness below f, as f → ∞ (in our setting; in the setting of [9], the limit should be taken as f ↑ f_c). While for the GMS model the suitably rescaled object converges in distribution as the scale parameter vanishes to a nontrivial distribution—see theorem 1.2 of [10]—, we have a law of large numbers type of convergence to a constant in our model. In the terms proposed in the latter reference, this puts the two models in different universality classes.

Our proofs of propositions 2.2 and 2.3 use definition 5.3 (K-marking process), as well as theorem 5.6 (Marking theorem) in [14].

Proof of Proposition 2.2. Conditional on $\left\{{\left({X}_{{I}_{k}}\right)}_{k{\geqslant}1}={\left({x}_{k}\right)}_{k{\geqslant}1}\right\}$, the random variables ${\Delta}{T}_{{I}_{k}}$ are independent of each other and follow the exponential distribution of rate ${\lambda }_{\star }\bar{{F}_{\star }}\left({x}_{k}\right)$. Denote by K_⋆(x, B) the following probability kernel: for x ⩾ 0, and B = (s₁, s₂] ⊂ [0, ∞), let ${K}_{\star }\left(x,\left({s}_{1},{s}_{2}\right]\right)\doteq {\mathrm{e}}^{-{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right){s}_{1}}-{\mathrm{e}}^{-{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right){s}_{2}}$, and extend the definition for Borelians B in the usual way.

The points ${\left({X}_{{I}_{k}},{\Delta}{T}_{{I}_{k}}\right)}_{k{\geqslant}1}$ form a K_⋆-marking of the Poisson process of proposition 2.1. The result follows by the Marking Theorem (theorem 5.6 of [14]).□

Proof of Proposition 2.3. The random variables ${\left({Y}_{j}\right)}_{j{\geqslant}1}$ are independent of each other and independent of ${\left({S}_{j}\right)}_{j{\geqslant}1}$. Then, the points ${\left({S}_{j},{Y}_{j}\right)}_{j{\geqslant}1}$ form a ${\mathbb{Q}}_{{\dagger}}$-marking independent of ${\left({S}_{j}\right)}_{j{\geqslant}1}$, where ${\mathbb{Q}}_{{\dagger}}$ is the measure induced by Y. The result follows again by the Marking theorem. □

Proof of Proposition 2.5. From equations (1) and (2)

$$\begin{align}\hfill & \mathbb{E}\left[{\mathrm{e}}^{-tM}\right]=\mathbb{E}\left[{\mathrm{e}}^{-\left(1-{e}^{-t}\right){\Lambda}}\right]=\mathbb{E}\left[{\mathrm{e}}^{-\sum _{k{\geqslant}1}\left(1-{e}^{-t}\right){h}_{{\dagger}}\left({X}_{{I}_{k}},{\Delta}{T}_{{I}_{k}}\right)}\right]\hfill \\ \hfill & =\mathrm{exp}\left[{-\iint }_{{\left[\left.0,\infty \right)\right.}^{2}}\left(1-{\mathrm{e}}^{-\left(1-{e}^{-t}\right){h}_{{\dagger}}\left(x,s\right)}\right){\hat{\mu }}_{\star }\left(\mathrm{d}x,\mathrm{d}s\right)\right]\hfill \end{align} \tag{ 6 }$$

$$\begin{equation}=\mathrm{exp}\left[{-\iint }_{{\left[\left.0,\infty \right)\right.}^{2}}\left(1-{\mathrm{e}}^{-\left(1-{e}^{-t}\right){\lambda }_{{\dagger}}s\overline{{F}_{{\dagger}}}\left(x\right)}\right){\lambda }_{\star }\bar{{F}_{\star }}\left(x\right){\mathrm{e}}^{-{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right)s}\mathrm{d}s{R}_{\star }\left(\mathrm{d}x\right)\right].\end{equation} \tag{ 7 }$$

In (6) we used the characterisation of a Poisson process via its Laplace functional (see theorem 3.9 on page 23 of [14]), and (7) follows by proposition 2.2.

Integrating the above expression in s, we have

$$\begin{align}\hfill \mathbb{E}\left[{\mathrm{e}}^{-tM}\right]& =\mathrm{exp}\left[-{\int }_{0}^{\infty }\left(1-\frac{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)}{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)+\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}\right){R}_{\star }\left(\mathrm{d}x\right)\right]\hfill \\ \hfill & =\mathrm{exp}\left[-{\int }_{0}^{\infty }\frac{\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)+\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right)\right].\hfill \end{align}$$

□

Proof of Proposition 2.6. To prove (a) ⇒ (b) note that

$$\begin{equation}0{\leqslant}\frac{\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right)+{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{\leqslant}\frac{\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right)+\left(1-{\mathrm{e}}^{-t}\right){\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{\leqslant}\frac{{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\overline{{F}_{\star }}\left(x\right)+{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}.\end{equation} \tag{ 8 }$$

Integrating (8) against R_⋆(dx) we get,

$$\begin{equation}\left(1-{\mathrm{e}}^{-t}\right)\phi \left(\infty \right){\leqslant}\phi \left(t\right){\leqslant}\phi \left(\infty \right).\end{equation} \tag{ 9 }$$

If ϕ(∞) < ∞, by monotone convergence theorem, lim_t↓0 ϕ(t) = 0. Then,

$$\begin{equation*}\underset{t{\downarrow}0}{\mathrm{lim}}\;\mathbb{E}\left[{\mathrm{e}}^{-tM}\right]=\underset{t{\downarrow}0}{\mathrm{lim}}\;\mathrm{exp}\left[-\phi \left(t\right)\right]=\mathrm{exp}\left[-\underset{t{\downarrow}0}{\mathrm{lim}}\;\phi \left(t\right)\right]=1\end{equation*}$$

and, on the other hand,

$$\begin{align*}\hfill \underset{t{\downarrow}0}{\mathrm{lim}}\;\mathbb{E}\left[{\mathrm{e}}^{-tM}\right]& =\underset{t{\downarrow}0}{\mathrm{lim}}\sum _{m{\geqslant}0}{\mathrm{e}}^{-tm}\mathbb{P}\left(M=m\right)+0\cdot \mathbb{P}\left(M=\infty \right)\hfill \\ \hfill & =\sum _{m{\geqslant}0}\mathbb{P}\left(M=m\right)=\mathbb{P}\left(M{< }\infty \right)\hfill \end{align*}$$

To prove (b) ⇒ (a) suppose ϕ(∞) = ∞. By (9), ϕ(t) = +∞ for each t > 0. Then,

$$\begin{align*}\hfill & \mathbb{E}\left[{\mathrm{e}}^{-tM}\right]=\mathrm{exp}\left[-\phi \left(t\right)\right]=0\quad \text{for} \text{each}\;\;t{ >}0\hfill \\ \hfill {\Rightarrow}\;& \mathbb{P}\left({\mathrm{e}}^{-tM}=0\right)=1\quad \text{for} \text{each}\;\;t{ >}0\hfill \\ \hfill {\Rightarrow}\;& \mathbb{P}\left(M=\infty \right)=1.\hfill \end{align*}$$

By contradiction we get (b) ⇒ (a).

Since trivially (c) ⇒ (b), we have from the above that (c) ⇒ (a).

To show that (a) ⇒ (c), we will change variables. Let $y={R}_{\star }\left(x\right)=-\mathrm{log}\;\bar{{F}_{\star }}\left(x\right)$, so $x={\bar{{F}_{\star }}}^{-1}\left({\mathrm{e}}^{-y}\right)$. From this and (4), $\overline{{F}_{{\dagger}}}\left(x\right)=\overline{{F}_{{\dagger}}}{\circ}{\bar{{F}_{\star }}}^{-1}\left({\mathrm{e}}^{-y}\right)=\psi \left({\mathrm{e}}^{-y}\right)$. Making u = e^−y, we find that

$$\begin{equation*}\begin{aligned}\hfill \phi \left(\infty \right)& ={\int }_{0}^{\infty }\frac{{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{{\lambda }_{\star }\bar{{F}_{\star }}\left(x\right)+{\lambda }_{{\dagger}}\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right)\hfill \\ \hfill & ={\int }_{0}^{\infty }\frac{{\lambda }_{{\dagger}}\psi \left({\mathrm{e}}^{-y}\right)}{{\lambda }_{\star }{\mathrm{e}}^{-y}+{\lambda }_{{\dagger}}\psi \left({\mathrm{e}}^{-y}\right)}\mathrm{d}y={\int }_{0}^{1}\frac{{\lambda }_{{\dagger}}\psi \left(u\right)}{{\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(u\right)}\frac{\mathrm{d}u}{u},\hfill \end{aligned}\end{equation*}$$

where the second equality is quite clear if F_⋆ is strictly increasing, but holds in general (see e.g. the [15] for a discussion and justification).

If ϕ(∞) < ∞, then dominated convergence yields

$$\begin{equation*}\underset{\varepsilon {\downarrow}0}{\mathrm{lim}}{\int }_{\varepsilon }^{2\varepsilon }\frac{{\lambda }_{{\dagger}}\psi \left(u\right)}{\left({\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(u\right)\right)u}\mathrm{d}u=\underset{\varepsilon {\downarrow}0}{\mathrm{lim}}{\int }_{0}^{1}\frac{{\lambda }_{{\dagger}}\psi \left(u\right)}{\left({\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(u\right)\right)u}{I}_{\left(\varepsilon ,2\varepsilon \right)}\left(u\right)\mathrm{d}u=0.\end{equation*}$$

Since

$$\begin{equation*}0{\leqslant}{\int }_{\varepsilon }^{2\varepsilon }\frac{{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}{\left({\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)\right)u}\mathrm{d}u{\leqslant}{\int }_{\varepsilon }^{2\varepsilon }\frac{{\lambda }_{{\dagger}}\psi \left(u\right)}{\left({\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(u\right)\right)u}\mathrm{d}u,\end{equation*}$$

we get that

$$\begin{equation}\underset{\varepsilon {\downarrow}0}{\mathrm{lim}}{\int }_{\varepsilon }^{2\varepsilon }\frac{{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}{\left({\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)\right)u}\mathrm{d}u=0.\end{equation} \tag{ 10 }$$

Solving the integral,

$$\begin{equation*}{\int }_{\varepsilon }^{2\varepsilon }\frac{{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}{\left({\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)\right)u}\mathrm{d}u=\mathrm{log}\left(1+\frac{{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}{{\lambda }_{\star }2\varepsilon +{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}\right),\end{equation*}$$

and from (10)

$$\begin{equation*}\underset{\varepsilon {\downarrow}0}{\mathrm{lim}}\;\mathrm{log}\left(1+\frac{{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}{{\lambda }_{\star }2\varepsilon +{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}\right)=0{\Rightarrow}\underset{\varepsilon {\downarrow}0}{\mathrm{lim}}\frac{{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}{{\lambda }_{\star }2\varepsilon +{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}=0{\Rightarrow}\underset{\varepsilon {\downarrow}0}{\mathrm{lim}}\frac{{\lambda }_{{\dagger}}\psi \left(\varepsilon \right)}{{\lambda }_{\star }\varepsilon }=0.\end{equation*}$$

We may thus find δ ∈ (0, 1) such that λ_†ψ(u) ⩽ λ_⋆u for each u ∈ (0, δ). Thus,

$$\begin{equation*}{\int }_{0}^{\delta }\frac{{\lambda }_{{\dagger}}\psi \left(u\right)}{{\lambda }_{\star }u}\frac{\mathrm{d}u}{u}{\leqslant}2{\int }_{0}^{\delta }\frac{{\lambda }_{{\dagger}}\psi \left(u\right)}{{\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(u\right)}\frac{\mathrm{d}u}{u}{\leqslant}2{\int }_{0}^{1}\frac{{\lambda }_{{\dagger}}\psi \left(u\right)}{{\lambda }_{\star }u+{\lambda }_{{\dagger}}\psi \left(u\right)}\frac{\mathrm{d}u}{u}{< }\infty .\end{equation*}$$

Thus, since also ψ(u) ⩽ 1, we have

$$\begin{equation*}\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{0}^{1}\frac{\psi \left(u\right)}{{u}^{2}}\mathrm{d}u=\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{0}^{\delta }\frac{\psi \left(u\right)}{{u}^{2}}\mathrm{d}u+\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{\delta }^{1}\frac{\psi \left(u\right)}{{u}^{2}}\mathrm{d}u{< }\infty .\end{equation*}$$

Rewriting (3) in terms of ψ, we have

$$\begin{equation}\mathbb{E}\left[M\right]=\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{0}^{\infty }\frac{\overline{{F}_{{\dagger}}}\left(x\right)}{\bar{{F}_{\star }}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right)=\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{0}^{\infty }\frac{\psi \left({\mathrm{e}}^{-y}\right)}{{\mathrm{e}}^{-y}}\mathrm{d}y=\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{0}^{1}\frac{\psi \left(u\right)}{{u}^{2}}\mathrm{d}u,\end{equation} \tag{ 11 }$$

and the finiteness of the latter expression establishes that (a) ⇒ (c).

(c) ⇔ (d) follows readily from (11). To prove (d) ⇔ (e) make the following substitution

$$\begin{equation*}s={u}^{-1}{\Rightarrow}\frac{\mathrm{d}s}{\mathrm{d}u}=-{u}^{-2}{\Rightarrow}\mathrm{d}s=-\frac{\mathrm{d}u}{{u}^{2}}\end{equation*}$$

and we can rewrite $\mathbb{E}\left[M\right]$ as

$$\begin{equation*}\mathbb{E}\left[M\right]=-\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{\infty }^{1}\psi \left({s}^{-1}\right)\mathrm{d}s=\frac{{\lambda }_{{\dagger}}}{{\lambda }_{\star }}{\int }_{1}^{\infty }\psi \left({s}^{-1}\right)\mathrm{d}s.\end{equation*}$$

Note that

$$\begin{equation*}{\int }_{1}^{\delta }\psi \left({s}^{-1}\right)\mathrm{d}s{< }\infty \quad \text{for} \text{each}\;\;\delta \in {\mathbb{R}}^{+};\end{equation*}$$

as a consequence

$$\begin{equation*}\mathbb{E}\left[M\right]{< }\infty {\Leftrightarrow}{\int }_{\delta }^{\infty }\psi \left({s}^{-1}\right)\mathrm{d}s{< }\infty \quad \text{for} \text{some}\;\;\delta \in {\mathbb{R}}^{+}.\end{equation*}$$

□

Proof of Theorem 2.1. By proposition 2.6, if ${\int }_{0}^{\infty }\frac{\overline{{F}_{{\dagger}}}\left(x\right)}{\overline{{F}_{\star }}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right)=\infty$, then M = ∞ a.s. Since ${\mathcal{M}}_{k}\doteq {D}_{k}\cap {\hat{{\Pi}}}_{{\dagger}}$ is a.s. finite for every k ⩾ 1, we have that $\mathcal{M}$ is a.s. unbounded, and thus so is ϒ' and ϒ.

If ${\int }_{0}^{\infty }\frac{\overline{{F}_{{\dagger}}}\left(x\right)}{\overline{{F}_{\star }}\left(x\right)}{R}_{\star }\left(\mathrm{d}x\right){< }\infty$, then by proposition 2.6, we have that a.s. M < ∞, and thus $\mathcal{M}$ is bounded, and thus so is ϒ' and ϒ, as follows from what has been pointed out at the end of definition 2.7 above. □

Proof of Theorem 2.2. By the homogeneity of the model, the process η_t = {η_t(t + s) : s ⩾ 0} has the same distribution for each $t\in \mathbb{R}$. In particular,

$$\begin{equation}\mathbb{P}\left({\eta }_{0}\left(t\right)\in \cdot \vert {\eta }_{0}\left(0\right)=A\right)=\mathbb{P}\left({\eta }_{-t}\left(0\right)\in \cdot \vert {\eta }_{-t}\left(-t\right)=A\right),\end{equation} \tag{ 12 }$$

where A is a locally finite subset of [0, ∞). Let ${\left({t}_{l}\right)}_{l{\geqslant}1}$ be an increasing sequence in ${\mathbb{R}}^{+}$ such that t_l → ∞ as l → ∞ and A as above; take a sequence of processes ${\left({\eta }_{-{t}_{l}}\right)}_{l{\geqslant}1}$ such that ${\eta }_{-{t}_{l}}\left(-{t}_{l}\right)=A$, l ⩾ 1. Define the sequence ${\left({\eta }_{l}^{\prime }\right)}_{l{\geqslant}1}$ by

$$\begin{equation*}{\eta }_{l}^{\prime }\doteq \begin{cases}\left\{{X}_{i}:{T}_{i}\in \left(-{t}_{l},0\right]\right\},\quad \quad \quad \quad \quad \quad \quad \text{if}\quad {{\Pi}}_{{\dagger}}\cap \left(-{t}_{l},0\right)=\varnothing ;\quad \hfill \\ \left\{{X}_{i}:{T}_{i}\in \left({S}_{{J}_{1}},0\right]\right\}\cup \left({\cup }_{k=1}^{{\hat{k}}_{l}-1}\left\{{X}_{i}{ >}{Y}_{{J}_{k}}:{T}_{i}\in \left({S}_{{J}_{k+1}},{S}_{{J}_{k}}\right]\right\}\right)\quad \hfill \\ \quad \quad \cup \left\{{X}_{i}{ >}{Y}_{{J}_{{\hat{k}}_{l}}}:{T}_{i}\in \left(-{t}_{l},{S}_{{J}_{{\hat{k}}_{l}}}\right]\right\},\quad \quad \quad \text{otherwise},\quad \hfill \end{cases}\end{equation*}$$

where ${\hat{k}}_{l}\doteq \mathrm{max}\left\{k:{S}_{{J}_{k}}{ >}-{t}_{l}\right\}$, and the union ${\cup }_{k=1}^{{\hat{k}}_{l}-1}$ above is empty if ${\hat{k}}_{l}=1$. Note that ${\left({\hat{k}}_{l}\right)}_{l{\geqslant}1}$ is an increasing sequence with ${\hat{k}}_{l}\to \infty$ as l → ∞. Thus, the sequence ${\left({\eta }_{l}^{\prime }\right)}_{l{\geqslant}1}$ is increasing and ${\mathrm{lim}}_{l\to \infty }\;{\eta }_{l}^{\prime }={\bigcup }_{l{\geqslant}1}{\eta }_{l}^{\prime }=\hat{\eta }$, with

$$\begin{equation*}\hat{\eta }\doteq \left\{{X}_{i}:{T}_{i}\in \left({S}_{{J}_{1}},0\right]\right\}\cup \left(\bigcup _{k{\geqslant}1}\left\{{X}_{i}{ >}{Y}_{{J}_{k}}:{T}_{i}\in \left({S}_{{J}_{k+1}},{S}_{{J}_{k}}\right]\right\}\right).\end{equation*}$$

Define the sequence ${\left({\eta }_{l}^{\prime \prime }\right)}_{l{\geqslant}1}$ by

$$\begin{equation*}{\eta }_{l}^{\prime \prime }\doteq \begin{cases}A,\quad \hfill & \text{if}\quad {{\Pi}}_{{\dagger}}\cap \left(-{t}_{l},0\right)=\varnothing \hfill \\ A\cap \left({Y}_{{J}_{{\hat{k}}_{l}}},\infty \right),\quad \hfill & \text{otherwise}.\hfill \end{cases}\end{equation*}$$

By proposition 2.9, ${Y}_{{J}_{k}}\to \infty$ as k → ∞, since R_†(x) < ∞ for every x > 0 and R_†(x) → ∞ as x → ∞. Thus, ${\left({Y}_{{J}_{{\hat{k}}_{l}}}\right)}_{l{\geqslant}1}$ is an increasing sequence with ${\mathrm{lim}}_{l\to \infty }\;{Y}_{{J}_{{\hat{k}}_{l}}}=\infty$, since ${\hat{k}}_{l}\to \infty$ as l → ∞. Therefore, ${\left({\eta }_{l}^{\prime \prime }\right)}_{l{\geqslant}1}$ is a decreasing sequence and ${\mathrm{lim}}_{l\to \infty }\;{\eta }_{l}^{\prime \prime }={\bigcap }_{l{\geqslant}1}{\eta }_{l}^{\prime \prime }=\varnothing$. Because ${\eta }_{-{t}_{l}}\left(0\right)={\eta }_{l}^{\prime }\cup {\eta }_{l}^{\prime \prime }$ for each l ⩾ 1, we have that ${\mathrm{lim}}_{l\to \infty }\;{\eta }_{-{t}_{l}}\left(0\right)={\mathrm{lim}}_{l\to \infty }\;{\eta }_{l}^{\prime }\cup {\mathrm{lim}}_{l\to \infty }\;{\eta }_{l}^{\prime \prime }=\hat{\eta }$, and notice that the limit does not depend on the choice of ${\left({t}_{l}\right)}_{l{\geqslant}1}$. Thus ${\mathrm{lim}}_{t\to \infty }\;{\eta }_{-t}\left(0\right)=\hat{\eta }$ a.s., and, by (12), ${\eta }_{0}\left(t\right)\to \hat{\eta }$ in distribution as t → ∞.□

Proof of Proposition 2.12. The proof is analogous to that of proposition 2.6, using the same ideas, and changing the roles of $\left({\lambda }_{\star },{\overline{F}}_{\star }\right)$ and $\left({\lambda }_{{\dagger}},{\overline{F}}_{{\dagger}}\right)$. □

Proof of Theorem 2.3. N₀ has a Poisson distribution with parameter Σ₀. Since Σ₀ = μ_⋆(E₀) = λ_⋆(0 − S₋₁) ∼ exponential (λ_†/λ_⋆), we have that $\mathbb{E}\left[{N}_{0}\right]=\mathbb{E}\left[{{\Sigma}}_{0}\right]={\lambda }_{\star }/{\lambda }_{{\dagger}}{< }\infty$, and thus N₀ < ∞ a.s.

By (5) and proposition 2.12, if ${\int }_{0}^{\infty }\frac{\overline{{F}_{\star }}\left(x\right)}{\overline{{F}_{{\dagger}}}\left(x\right)}{R}_{{\dagger}}\left(\mathrm{d}x\right){< }\infty$, then N < ∞ a.s., and thus $\#\hat{\eta }{< }\infty$ a.s. Otherwise, N = ∞ a.s., and we have $\#\hat{\eta }=\infty$ a.s. □

Remark 3.1. As a final remark, we argue in a few brief steps that, as pointed out in remark 2.14 above, when F_⋆ = F_† = F, then N^f, the number of species in the limiting distribution with fitness below f, when properly rescaled, converges (a.s.) to a nontrivial constant. First, by the above construction of the limiting distribution of species fitnesses, it is enough to consider the intensity measure of the region bounded by the ladder of records, the x-axis and the horizontal line through (0, f). This in turn may be readily written as

$$\begin{equation}\sum _{i=1}^{\vert \mathcal{P}\cap \left[0,f\right]\vert }\left(1-\frac{\overline{F}\left(f\right)}{\overline{F}\left({X}_{i}\right)}\right){\mathcal{E}}_{i}=\sum _{i=1}^{\vert \mathcal{P}\cap \left[0,f\right]\vert }{\mathcal{E}}_{i}-\overline{F}\left(f\right)\sum _{i=1}^{\vert \mathcal{P}\cap \left[0,f\right]\vert }\frac{1}{\overline{F}\left({X}_{i}\right)}{\mathcal{E}}_{i}\end{equation} \tag{ 13 }$$

where $\mathcal{P}=\left\{{X}_{1},{X}_{2},\dots \right\}$ is a Poisson point process on ${\mathbb{R}}_{+}$ of intensity measure ∫_BR(dx), $B\in \mathcal{B}\left({\mathbb{R}}_{+}\right)$, with $R\left(x\right)=-\mathrm{log}\;\overline{F}\left(x\right)$, and ${\mathcal{E}}_{1},{\mathcal{E}}_{2},\dots$ are i.i.d. standard exponential random variables. Clearly, the first term on the right hand side of (13), when scaled by R(f), converges to 1 almost surely as f → ∞, by the law of large numbers. We leave as an exercise to check, using well known properties of Poisson point processes such as $\mathcal{P}$, that the second term, when scaled by R(f), vanishes almost surely as f → ∞; indeed, without scaling, it is stochastically bounded uniformly by a non degenerate random variable; we immediately get convergence to 0 in probability of the scaled quantity, and a closer look reveals that this can be strengthened to a.s. convergence.