The Virgin Island Model

A continuous mass population model with local competition is constructed where every emigrant colonizes an unpopulated island. The population founded by an emigrant is modeled as excursion from zero of an one-dimensional diffusion. With this excursion measure, we construct a process which we call Virgin Island Model. Furthermore, a necessary and sufficient condition for extinction of the total population is obtained for finite initial total mass.


Introduction
This paper is motivated by an open question on a system of interacting locally regulated diffusions. In (8), a sufficient condition for local extinction is established for such a system. In general, however, there is no criterion available for global extinction, that is, convergence of the total mass process to zero when started in finite total mass.
The method of proof for the local extinction result in (8) is a comparison with a mean field model (M t ) t≥0 which solves (1) dM t = κ(EM t − M t )dt + h(M t )dt + 2g(M t )dB t where (B t ) t≥0 is a standard Brownian motion and where h, g : [0, ∞) → Ê are suitable functions satisfying h(0) = 0 = g(0). This mean field model arises as the limit as N → ∞ (see Theorem 1.4 in (19) for the case h ≡ 0) of the following system of interacting locally regulated diffusions on N islands with uniform migration For this convergence, X N 0 (0), . . . , X N 0 (N − 1) may be assumed to be independent and identically distributed with the law of X N 0 (0) being independent of N . The intuition behind the comparison with the mean field model is that if there is competition (modeled through the functions h and g in (2)) among individuals and resources are everywhere the same, then the best strategy for survival of the population is to spread out in space as quickly as possible.
The results of (8) cover translation invariant initial measures and local extinction. For general h and g, not much is known about extinction of the total mass process. Let the solution (X N t ) t≥0 of (2) be started in X N 0 (i) = x½ i=0 , x ≥ 0. We prove in a forthcoming paper under suitable conditions on the parameters that the total mass |X N t | := N i=1 X N t (i) converges as N → ∞. In addition, we show in that paper that the limiting process dominates the total mass process of the corresponding system of interacting locally regulated diffusions started in finite total mass. Consequently, a global extinction result for the limiting process would imply a global extinction result for systems of locally regulated diffusions.
In this paper we introduce and study a model which we call Virgin Island Model and which is the limiting process of (X N t ) t≥0 as N → ∞. Note that in the process (X N t ) t≥0 an emigrant moves to a given island with probability 1 N . This leads to the characteristic property of the Virgin Island Model namely every emigrant moves to an unpopulated island. Our main result is a necessary and sufficient condition (see (28) below) for global extinction for the Virgin Island Model. Moreover, this condition is fairly explicit in terms of the parameters of the model. Now we define the model. On the 0-th island evolves a diffusion Y = (Y t ) t≥0 with state space Ê ≥0 given by the strong solution of the stochastic differential equation where (B t ) t≥0 is a standard Brownian motion. This diffusion models the total mass of a population and is the diffusion limit of near-critical branching particle processes where both the offspring mean and the offspring variance are regulated by the total population. Later, we will specify conditions on a, h and g so that Y is well-defined. For now, we restrict our attention to the prototype example of a Feller branching diffusion with logistic growth in which a(y) = κy, h(y) = γy(K − y) and g(y) = βy with κ, γ, K, β > 0. Note that zero is a trap for Y , that is, Y t = 0 implies Y t+s = 0 for all s ≥ 0. Mass emigrates from the 0-th island at rate a(Y t ) dt and colonizes unpopulated islands. A new population should evolve as the process (Y t ) t≥0 . Thus, we need the law of excursions of Y from the trap zero. For this, define the set of excursions from zero by (4) U := χ ∈ C (−∞, ∞), [0, ∞) : T 0 ∈ (0, ∞], χ t = 0 ∀ t ∈ (−∞, 0] ∪ [T 0 , ∞) where T y = T y (χ) := inf{t > 0 : χ t = y} is the first hitting time of y ∈ [0, ∞). The set U is furnished with locally uniform convergence. Throughout the paper, C(S 1 , S 2 ) and D(S 1 , S 2 ) denote the set of continuous functions and the set of càdlàg functions, respectively, between two intervals S 1 , S 2 ⊂ Ê. Furthermore, define The excursion measure Q Y is a σ-finite measure on U . It has been constructed by Pitman and Yor (16) as follows: Under Q Y , the trajectories come from zero according to an entrance law and then move according to the law of Y . Further characterizations of Q Y are given in (16), too. For a discussion on the excursion theory of one-dimensional diffusions, see (18). We will give a definition of Q Y later.
Next we construct all islands which are colonized from the 0-th island and call these islands the first generation. Then we construct the second generation which is the collection of all islands Note that infinitely many islands are colonized e.g. between times s 1 and s 2 .
which have been colonized from islands of the first generation, and so on. Figure 1 illustrates the resulting tree of excursions. For the generation-wise construction, we use a method to index islands which keeps track of which island has been colonized from which island. An island is identified with a triple which indicates its mother island, the time of its colonization and the population size on the island as a function of time. For χ ∈ D, let (6) I χ 0 := ∅, 0, χ be a possible 0-th island. For each n ≥ 1 and χ ∈ D, define (7) I χ n := ι n−1 , s, ψ : ι n−1 ∈ I χ n−1 , (s, ψ) ∈ [0, ∞) × D which we will refer to as the set of all possible islands of the n-th generation with fixed 0-th island (∅, 0, χ). This notation should be read as follows. The island ι n = (ι n−1 , s, ψ) ∈ I χ n has been colonized from island ι n−1 ∈ I χ n−1 at time s and carries total mass ψ(t − s) at time t ≥ 0. Notice that there is no mass on an island before the time of its colonization. The island space is defined by (8) I := {∅} ∪ χ∈D I χ where I χ := n≥0 I χ n .
The Virgin Island Model is defined recursively generation by generation. The 0-th generation only consists of the 0-th island (10) V (0) := ∅, 0, Y .
The set of all islands is defined by The total mass process of the Virgin Island Model is defined by Our main interest concerns the behaviour of the law L (V t ) of V t as t → ∞.
The following observation is crucial for understanding the behavior of (V t ) t≥0 as t → ∞. There is an inherent branching structure in the Virgin Island Model. Consider as new "time coordinate" the number of island generations. One offspring island together with all its offspring islands is again a Virgin Island Model but with the path (Y t ) t≥0 on the 0-th island replaced by an excursion path. Because of this branching structure, the Virgin Island Model is a multi-type Crump-Mode-Jagers branching process (see (10) under "general branching process") if we consider islands as individuals and [0, ∞) × D as type space. We recall that a single-type Crump-Mode-Jagers process is a particle process where every particle i gives birth to particles at the time points of a point process ξ i until its death at time λ i , and (λ i , ξ i ) i are independent and identically distributed. The literature on Crump-Mode-Jagers processes assumes that the number of offspring per individual is finite in every finite time interval. In the Virgin Island Model, however, every island has infinitely many offspring islands in a finite time interval because Q Y is an infinite measure.
The most interesting question about the Virgin Island Model is whether or not the process survives with positive probability as t → ∞. Generally speaking, branching particle processes survive if and only if the expected number of offspring per particle is strictly greater than one, e.g. the Crump-Mode-Jagers process survives if and only if Eξ i [0, λ i ] > 1. For the Virgin Island Model, the offspring of an island (ι, s, χ) depends on the emigration intensities a χ(t − s) dt. It is therefore not surprising that the decisive parameter for survival is the expected "sum" over those emigration intensities We denote the expression in (14) as "expected total emigration intensity" of the Virgin Island Model. The observation that (14) is the decisive parameter plus an explicit formula for (14) leads to the following main result. In Theorem 2, we will prove that the Virgin Island Model survives with strictly positive probability if and only if Note that the left-hand side of (15) is equal to ∞ 0 a(y)m(dy) where m(dy) is the speed measure of the one-dimensional diffusion (3). The method of proof for the extinction result is to study an integral equation (see Lemma 5.3) which the Laplace transform of the total mass V solves. Furthermore, we will show in Lemma 9.8 that the expression in (14) is equal to the left-hand side of (15).
Condition (15) already appeared in (8) as necessary and sufficient condition for existence of a nontrivial invariant measure for the mean field model, see Theorem 1 and Lemma 5.1 in (8). Thus, the total mass process of the Virgin Island Model dies out if and only if the mean field model (1) dies out. The following duality indicates why the same condition appears in two situations which seem to be fairly different at first view. If a(x) = κx, h(x) = γx(K − x) and g(x) = βx with κ, γ, β > 0, that is, in the case of Feller branching diffusions with logistic growth, then model (2) is dual to itself, see Theorem 3 in (8). If (X N t ) t≥0 indeed approximates the Virgin Island Model as N → ∞, then -for this choice of parameters -the total mass process (V t ) t≥0 is dual to the mean field model. This duality would directly imply that -in the case of Feller branching diffusions with logistic growth -global extinction of the Virgin Island Model is equivalent to local extinction of the mean field model.
An interesting quantity of the Virgin Island process is the area under the path of V . In Theorem 3, we prove that the expectation of this quantity is finite exactly in the subcritical situation in which case we give an expression in terms of a, h and g. In addition, in the critical case and in the supercritical case, we obtain the asymptotic behaviour of the expected area under the path of V up to time t More precisely, the order of (16) is O(t) in the critical case. For the supercritical case, let α > 0 be the Malthusian parameter defined by (17) ∞ 0 e −αu a χ u Q Y (dχ) du = 1.
It turns out that the expression in (16) grows exponentially with rate α as t → ∞.
The result of Theorem 3 in the supercritical case suggests that the event that (V t ) t≥0 grows exponentially with rate α as t → ∞ has positive probability. However, this is not always true. Theorem 7 proves that e −αt V t converges in distribution to a random variable W ≥ 0. Furthermore, this variable is not identically zero if and only if (18) ∞ 0 a χ s e −αs ds log + ∞ 0 a χ s e −αs ds Q Y (dχ) < ∞ where log + (x) := max{0, log(x)}. This (x log x)-criterion is similar to the Kesten-Stigum Theorem (see (14)) for multidimensional Galton-Watson processes. Our proof follows Doney (4) who establishes an (x log x)-criterion for Crump-Mode-Jagers processes.
Our construction introduces as new "time coordinate" the number of island generations. Readers being interested in a construction of the Virgin Island Model in the original time coordinatefor example in a relation between V t and (V s ) s<t -are referred to Dawson and Li (2003) (3). In that paper, a superprocess with dependent spatial motion and interactive immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by a Poisson point process whose intensity measure has as one component the excursion measure of the Feller branching diffusion. In a special case (see equation (1.6) in (3) with x(s, a, t) = a, q(Y s , a) = κY s (Ê) and m(da) = ½ [0,1] (a) da), this is just the Virgin Island Model with (3) replaced by a Feller branching diffusion, i.e. a(y) = κy, h(y) = 0, g(y) = βy. It would be interesting to know whether existence and uniqueness of such stochastic integral equations still hold if the excursion measure of the Feller branching diffusion is replaced by Q Y .
Models with competition have been studied by various authors. Mueller and Tribe (1994) (15) and Horridge and Tribe (2004) (7) investigate an one-dimensional SPDE analog of interacting Feller branching diffusions with logistic growth which can also be viewed as KPP equation with branching noise. Bolker and Pacala (1997) (2) propose a branching random walk in which the individual mortality rate is increased by a weighted sum of the entire population. Etheridge (2004) (6) studies two diffusion limits hereof. The "stepping stone version of the Bolker-Pacala model" is a system of interacting Feller branching diffusions with non-local logistic growth. The "superprocess version of the Bolker-Pacala model" is an analog of this in continuous space. Hutzenthaler and Wakolbinger (8), motivated by (6), investigated interacting diffusions with local competition which is an analog of the Virgin Island Model but with mass migrating on d instead of migration to unpopulated islands.
where x ∨ y denotes the maximum of x and y. In addition, c 1 ·x ≤ a(x) ≤ c 2 ·x holds for all x ≥ 0 and for some constants c 1 , c 2 ∈ (0, ∞).
The key ingredient in the construction of the Virgin Island Model is the law of excursions of (Y t ) t≥0 from the boundary zero. Note that under Assumption A2.1, zero is an absorbing boundary for (3), i.e. Y t = 0 implies Y t+s = 0 for all s ≥ 0. As zero is not a regular point, it is not possible to apply the well-established Itô excursion theory. Instead we follow Pitman and Yor (16) and obtain a σ-finite measureQ Y -to be called excursion measure -on U (defined in (4)). For this, we additionally assume that (Y t ) t≥0 hits zero in finite time with positive probability. The following assumption formulates a necessary and sufficient condition for this (see Lemma 15.6.2 in (13)). To formulate the assumption, we define Note thatS is a scale function, that is, holds for all 0 ≤ c < y < b < ∞, see Section 15.6 in (13).
Assumption A2.2. The functions a, g and h satisfy Note that if Assumption A2.2 is satisfied, then (22) holds for all x > 0. Pitman and Yor (16) construct the excursion measureQ Y in three different ways one being as follows. The set of excursions reaching level δ > 0 hasQ Y -measure 1/S(δ). Conditioned on this event an excursion follows the diffusion (Y t ) t≥0 conditioned to converge to infinity until this process reaches level δ. From this time on the excursion follows an independent unconditioned process. We carry out this construction in detail in Section 9. In addition Pitman and Yor (16) describe the excursion measure "in a preliminary way as" where the limit indicates weak convergence of finite measures on C [0, ∞), [0, ∞) away from neighbourhoods of the zero-trajectory. However, they do not give a proof. HavingQ Y identified as the limit in (23) will enable us to transfer explicit formulas for L (Y ) to explicit formulas for Q Y . We establish the existence of the limit in (23) in Theorem 1 below. For this, let the topology on C [0, ∞), [0, ∞) be given by locally uniform convergence. Furthermore, recall Y from (3), the definition of U from (4) and the definition ofS from (20 for all bounded continuous F : C [0, ∞), [0, ∞) → Ê for which there exists an ε > 0 such that For our proof of the global extinction result for the Virgin Island Model, we need the scaling functionS in (24) to behave essentially linearly in a neighbourhood of zero. More precisely, we assumeS ′ (0) to exist in (0, ∞). From definition (20) ofS it is clear that a sufficient condition for this is given by the following assumption.
It follows from dominated convergence and from the local Lipschitz continuity of a and h that Assumption A2.3 holds if 1 0 y g(y) dy is finite. In addition, we assume that the expected total emigration intensity of the Virgin Island Model is finite. Lemma 9.6 shows that, under Assumptions A2.1 and A2.2, an equivalent condition for this is given in Assumption A2.4. We mention that if Assumptions A2.1, A2.2 and A2.4 hold, then the process Y hits zero in finite time almost surely (see Lemma 9.5 and Lemma 9.6). Furthermore, we give a generic example for a, h and g namely a(y) = c 1 y, h(y) = c 2 y κ1 − c 3 y κ2 , g(y) = c 4 y κ3 with c 1 , c 2 , c 3 , c 4 > 0. The Assumptions A2.1, A2.2, A2.3 and A2.4 are all satisfied if κ 2 > κ 1 ≥ 1 and if κ 3 ∈ [1, 2). Assumption A2.2 is not met by a(y) = κy, κ > 0, h(y) = y and g(y) = y 2 because thens(y) = y κ−1 , S(y) = y κ /κ and condition (22) fails to hold. Next we formulate the main result of this paper. Theorem 2 proves a nontrivial transition from extinction to survival. For the formulation of this result, we define If (28) fails to hold, then V t converges in distribution as t → ∞ to a random variable V ∞ satisfying for all x ≥ 0 and some q > 0. In the critical case, that is, equality in (28), V t converges to zero in distribution as t → ∞. However, it turns out that the expected area under the graph of V is infinite. In addition, we obtain in Theorem 3 the asymptotic behaviour of the expected area under the graph of V up to time t as t → ∞. For this, define dz, x ≥ 0, and similarly w id := w a with a(z) = z.
for all x ≥ 0. Otherwise, the left-hand side of (32) is infinite. In the critical case, that is, equality in (28), where the right-hand side is interpreted as zero if the denominator is equal to infinity. In the supercritical case, i.e., if (28) fails to be true, let α > 0 be such that Then the order of growth of the expected area under the path of (V s ) s≥0 up to time t as t → ∞ can be read off from The following result is an analog of the Kesten-Stigum Theorem, see (14). In the supercritical case, e −αt V t converges to a random variable W as t → ∞. In addition, W is not identically zero if and only if the (x log x)-condition (18) holds. We will prove a more general version hereof in Theorem 7 below. Unfortunately, we do not know of an explicit formula in terms of a, h and g for the left-hand side of (18). Aiming at a condition which is easy to verify, we assume instead of (18) that the second moment ( ∞ 0 a(χ s ) ds) 2 Q(dχ) is finite. In Assumption A2.5, we formulate a condition which is slightly stronger than that, see Lemma 9.8 below.
Assumption A2.5. The functions a, g and h satisfy for some and then for all x > 0.

Outline
Theorem 1 will be established in Section 9. Note that Section 9 does not depend on the sections 4-8. We will prove the survival and extinction result of Theorem 2 in two steps. In the first step, we obtain a criterion for survival and extinction in terms of Q Y . More precisely, we prove that the process dies out if and only if the expression in (14) is smaller than or equal to one. In this step, we do not exploit that Q Y is the excursion measure of Y . In fact, we will prove an analog of Theorem 2 in a more general setting where Q Y is replaced by some σ-finite measure Q and where the islands are counted with random characteristics. See Section 4 below for the definitions. The analog of Theorem 2 is stated in Theorem 5, see Section 4, and will be proven in Section 7. The key equation for its proof is contained in Lemma 5.1 which formulates the branching structure in the Virgin Island Model. In the second step, we calculate an expression for (14) in terms of a, h and g. This will be done in Lemma 9.8. Theorem 2 is then a corollary of Theorem 5 and of Lemma 9.8, see Section 10. Similarly, a more general version of Theorem 3 is stated in Theorem 6, see Section 4 below. The proofs of Theorem 3 and of Theorem 6 are contained in Section 10 and Section 6, respectively. As mentioned in Section 1, a rescaled version of (V t ) t≥0 converges in the supercritical case. This convergence is stated in a more general formulation in Theorem 7, see Section 4 below. The proofs of Theorem 4 and of Theorem 7 are contained in Section 10 and in Section 8, respectively.

Virgin Island Model counted with random characteristics
In the proof of the extinction result of Theorem 2, we exploit that one offspring island together with all its offspring islands is again a Virgin Island Model but with a typical excursion instead of Y on the 0-th island. For the formulation of this branching property, we need a version of the Virgin Island Model where the population on the 0-th island is governed by Q Y . More generally, we replace the law L (Y ) of the first island by some measure ν and we replace the excursion measure Q Y by some measure Q. Given two σ-finite measures ν and Q on the Borel-σ-algebra of D, we define the Virgin Island Model with initial island measure ν and excursion measure Q as follows.
Define the random sets of islands V (n),ν,Q , n ≥ 0, and V ν,Q through the definitions (9), (10), (11) and (12) with L (Y ) and Q Y replaced by ν and Q, respectively. A simple example for ν and Q is ν(dχ) = Q(dχ) = Eδ t →½t<L (dχ) where L ≥ 0 is a random variable and δ ψ is the Dirac measure on the path ψ. Then the Virgin Island Model coincides with a Crump-Mode-Jagers process in which a particle has offspring according to a rate a(1) Poisson process until its death at time L. Furthermore, our results do not only hold for the total mass process (13) but more generally when the islands are counted with random characteristics. This concept is well-known for Crump-Mode-Jagers processes, see Section 6.9 in (10). Assume that φ ι = φ ι (t) t∈Ê , ι ∈ I, are separable and nonnegative processes with the following properties. It vanishes on the negative half-axis, i.e. φ ι (t) = 0 for t < 0. Informally speaking our main assumption on φ ι is that it does not depend on the history. Formally we assume that Furthermore, we assume that the family {φ ι , Π ι : ι ∈ I χ } is independent for each χ ∈ D and (ω, t, χ) → φ (∅,0,χ) (t)(ω) is measurable. As a short notation, define φ is the total mass at time t of all islands which have been colonized in the last t 0 time units.
As in Section 2, we need an assumption which guarantees finiteness of The analog of Assumption A2.4 in the general setting is the following assumption.
Assumption A4.2. Both the expected emigration intensity of the 0-th island and of subsequent islands are finite: In Section 2, we assumed that (Y t ) t≥0 hits zero in finite time with positive probability. See Assumption A2.2 for an equivalent condition. Together with A2.4, this assumption implied almost sure convergence of (Y t ) t≥0 to zero as t → ∞. In the general setting, we need a similar but somewhat weaker assumption. More precisely, we assume that φ(t) converges to zero "in distribution" both with respect to ν and with respect to Q.
Assumption A4.3. The random processes φ χ (t) t≥0 : χ ∈ D and the measures Q and ν satisfy Having introduced the necessary assumptions, we now formulate the extinction and survival result of Theorem 2 in the general setting. In case of survival, the process converges weakly as t → ∞ to a probability measure L V φ,ν,Q ∞ with support in {0, ∞} which puts mass on the point ∞ where q > 0 is the unique strictly positive fixed-point of The assumption on ν to be a probability measure is convenient for the formulation in terms of convergence in probability. For a formulation in the case of a σ-finite measure ν, see the proof of the theorem in Section 7.
Next we state Theorem 3 in the general setting. For its formulation, define and similarly f Q with ν replaced by Q.
which is finite and strictly positive. Otherwise, the left-hand side of (47) is infinite. If the left-hand side of (43) is equal to one and if both f ν and f Q are integrable, where the right-hand side is interpreted as zero if the denominator is equal to infinity. In the supercritical case, i.e., if (43) fails to be true, let α > 0 be such that Additionally assume that f Q is continuous a.e. with respect to the Lebesgue measure, and that e −αt f ν (t) → 0 as t → ∞. Then the order of convergence of the expected total intensity up to time t can be read off from and from For the formulation of the analog of the Kesten-Stigum Theorem, denote by the right-hand side of (52) with ν replaced by Q. Furthermore, define for every path χ ∈ D. For our proof of Theorem 7, we additionally assume the following properties of Q.
Assumption A4.4. The measure Q satisfies where q > 0 is the unique strictly positive fixed-point of (45).
. Consequently, the Virgin Island process V φ,ν,Q t t≥0 conditioned on not converging to zero grows exponentially fast with rate α as t → ∞.

Branching structure
We mentioned in the introduction that there is an inherent branching structure in the Virgin Island Model. One offspring island together with all its offspring islands is again a Virgin Island Model but with a typical excursion instead of Y on the 0-th island. In Lemma 5.1, we formalize this idea. As a corollary thereof, we obtain an integral equation for the modified Laplace transform of the Virgin Island Model in Lemma 5.3 which is the key equation for our proof of the extinction result of Theorem 2. Recall the notation of Section 1 and of Section 4.
of random variables which is independent of φ χ and of Π χ such that and such that (63) and (64) with (11), we see that Summing over n ≥ 0 we obtain for t ≥ 0 This is equality (61). Independence of the family (60) follows from independence of (Π ι ) ι∈I χ and from independence of (φ ι ) ι∈I χ . It remains to prove (62). Because of assumption (38) the random characteristics φ ι only depends on the last part of ι. Therefore Summing over n ≥ 1 results in (62) and finishes the proof.
In order to increase readability, we introduce the following suggestive symbolic abbreviation One might want to read this as "expectation" with respect to a non-probability measure. However, (69) is not intended to define an operator.
The following lemma proves that the Virgin Island Model counted with random characteristics as defined in (39) is finite.
for all ν and the right-hand side of (72) is finite in the special case ν = Q.
Proof. We exploit the branching property formalized in Lemma 5.1 and apply Gronwall's inequality. Recall V (n),χ,Q from the proof of Lemma 5.1. The two equalities (66) and (68) imply for t ≤ T and for n ≥ 1 Using Assumption A4.1 induction on n ≥ 0 shows that all expressions in (73) and in (74) are finite in the case ν = Q. Summing (74) over n ≤ n 0 we obtain for t ≤ T . In the special case ν = Q Gronwall's inequality implies Summing (74) over n ≤ n 0 , inserting (76) into (74) and letting n 0 → ∞ we see that (70) follows from Assumption A4.1.
For the proof of (72), note that (75) with ν = δ χ and (70) imply In addition the two equalities (66) and (68) together with independence imply for t ≥ 0 and for n ≥ 1 In the special case ν = Q induction on n ≥ 0 together with (71) shows that all involved expressions are finite. A similar estimate as in (79) leads to In the special case ν = Q Gronwall's inequality together with (77) leads to which is finite by Assumption A4.1 and assumption (71). Inserting (80) into (79) and letting n 0 → ∞ finishes the proof.
In the following lemma, we establish an integral equation for the modified Laplace transform of the Virgin Island Model. Recall the definition of V φ,ν,Q t from (39).

Proof of Theorem 6
Recall the definition of (V φ,ν,Q t ) t≥0 from (39), f ν from (46) and the notation I from (69). We begin with the supercritical case and let α > 0 be the Malthusian parameter which is the unique solution of (49). Define for t ≥ 0. In this notation, equation (74) with ν replaced by Q reads as This is a renewal equation for e −αt m Q (t). By definition of α, e −αs µ Q (ds) is a probability measure. From Lemma 5.2 we know that m Q is bounded on finite intervals. By assumption, f Q is continuous Lebesgue-a.e. and satisfies (50). Hence, we may apply standard renewal theory (e.g. Theorem 5.2.6 of (10)) and obtain where we used the dominated convergence theorem. Next we consider the subcritical and the critical case. Define In this notation, equation (74) integrated over [0, t] reads as In the subcritical case, f Q and f ν are integrable. Theorem 5.2.9 in (10) applied to (88) with ν replaced by Q implies .
Inserting (89) results in (47). In the critical case, similar arguments lead to The last equality follows from (88) with ν replaced by Q and Corollary 5.2.14 of (10) with c := ∞ 0 f Q (s) ds, n := 0 and θ := ∞ 0 uµ Q (du). Note that the assumption θ < ∞ of this corollary is not necessary for this conclusion. Recall the definition of (V φ,ν,Q t ) t≥0 from (39) and the notation I from (69). As we pointed out in Section 2, the expected total emigration intensity of the Virgin Island Model plays an important role. The following lemma provides us with some properties of the modified Laplace transform of the total emigration intensity. These properties are crucial for our proof of Theorem 5. Denote by q the maximal fixed-point. Then we have for all z ≥ 0: In addition, dominated convergence together with Assumption A4.2 implies Hence, k is strictly concave. Thus, k has a fixed-point which is not zero if and only if k ′ (0) > 1. The implications (94) and (95) follow from the strict concavity of k.
The method of proof (cf. Section 6.5 in (10)) of the extinction result for a Crump-Mode-Jagers process (J t ) t≥0 is to study an equation for (Ee −λJt ) t≥0,λ≥0 . The Laplace transform (Ee −λJt ) λ≥0 converges monotonically to P(J t = 0) as λ → ∞, t ≥ 0. Furthermore, P(J t = 0) = P(∃s ≤ t : J s = 0) converges monotonically to the extinction probability P(∃s ≥ 0 : J s = 0) as t → ∞. Taking monotone limits in the equation for (Ee −λJt ) t≥0,λ≥0 results in an equation for the extinction probability. In our situation, there is an equation for the modified Laplace transform (L t (λ)) t>0,λ>0 as defined in (98) below. However, the monotone limit of L t (λ) as λ → ∞ might be infinite. Thus, it is not clear how to transfer the above method of proof. The following proof of Theorem 2 directly establishes the convergence of the modified Laplace transform.
Proof of Theorem 5. Recall q from Lemma 7.1. In the first step, we will prove for all λ > 0. Set L t (0) := 0. It follows from Lemma 5.2 that (L t ) t≤T is bounded for every finite T . Lemma 5.3 with ν replaced by Q provides us with the fundamental equation Based on (99), the idea for the proof of (98) is as follows. The term λφ χ (t) vanishes as t → ∞. If L t converges to some limit, then the limit has to be a fixed-point of the function By Lemma 7.1, this function is (typically strictly) concave. Therefore, it has exactly one attracting fixed-point. Furthermore, this fact forces L t to converge as t → ∞.
We will need finiteness of L ∞ := lim sup t→∞ L t . Looking for a contradiction, we assume L ∞ = ∞. Then there exists a sequence (t n ) n∈AE with t n → ∞ such that L tn ≤ sup t≤tn L t ≤ L tn +1. We estimate The last summand converges to zero by Assumption A4.3 and is therefore bounded by some constant c. Inequality (101) leads to the contradiction The last equation is a consequence of (96) and the assumption L ∞ = ∞. Next we prove L ∞ ≤ q using boundedness of (L t ) t≥0 . Let (t n ) n∈AE be a sequence such that lim n→∞ L tn = L ∞ < ∞. Then a calculation as in (101) The last summand is equal to zero by Assumption A4.3. The first summand on the right-hand side of (103) is dominated by which is finite by boundedness of (L t ) t≥0 and by Assumption A4.2. Applying dominated convergence, we conclude that L ∞ is bounded by Thus, Lemma 7.1 implies lim sup t→∞ L t ≤ q. Assume q > 0 and suppose that m := lim inf t→∞ L t = 0. Let (t n ) n∈AE be such that 0 < L tn ≥ inf 1≤t≤tn L t ≥ cL tn → 0 as n → ∞ and t n + 1 ≤ t n+1 → ∞. By Lemma 7.1, there is an n 0 and a c < 1 such that c tn 0 0 a χ s dsQ(dχ) > 1. We estimate In order to prove m ≥ q, let (t n ) n∈AE be such that lim n→∞ L tn = m > 0. An estimate as above together with dominated convergence yields Therefore, Lemma 7.1 implies lim inf t→∞ L t = m ≥ q, which yields (98). Finally, we finish the proof of Theorem 5. Applying Lemma 5.3, we see that The first summand on the right-hand side of (109) converges to zero as t → ∞ by Assumption A4.3. By the first step (98), L t → q as t → ∞. Hence, by the dominated convergence theorem and Assumption A4.2, the left-hand side of (109) converges to zero as t → ∞. As ν is a probability measure by assumption, we conclude This implies Theorem 5 as the Laplace transform is convergence determining, see e.g. Lemma 2.1 in (5).

The supercritical Virgin Island Model. Proof of Theorem 7
Our proof of Theorem 7 follows the proof of Doney (1972) (4) for supercritical Crump-Mode-Jagers processes. Some changes are necessary because the recursive equation (99) differs from the respective recursive equation in (4). Parts of our proof are analogous to the proof in (4) which we nevertheless include here for the reason of completeness. Lemma 8.9 and Lemma 8.10 below contain the essential part of the proof of Theorem 7. For these two lemmas, we will need auxiliary lemmas which we now provide.
We assume throughout this section that a solution α ∈ Ê of equation (34) exists. Note that this is implied by A4.2 and Q ∞ 0 a(χ s ) ds > 0 > 0. Recall the definition of µ Q from (82).
Lemma 8.1. The operator H α is contracting in the sense that for all ψ 1 , ψ 2 ∈ D.
Proof. The lemma follows immediately from |e −x − e −y | ≤ |x − y| and from the definition (82) of µ Q .

Lemma 8.2. The operator H α is nondecreasing in the sense that
Proof. The lemma follows from 1 − e −cx being increasing in x for every c > 0.
Proof. The assertion follows from the basic fact that f is nondecreasing.
The following lemma, due to Athreya (1), translates the (x log x)-condition (58) into an integrability condition on η . For completeness, we include its proof.
It is a basic fact that

The limiting equation
In the following two lemmas, we consider uniqueness and existence of a function Ψ which satisfies: where q ≥ 0 is as in Lemma 7.1. Notice that the zero function does not satisfy (120)(c). First, we prove uniqueness.
for every λ ≥ 0 where Ψ is the unique solution of (120).
In this section, we define the excursion measureQ Y and prove the convergence result of Theorem 1. We follow Pitman and Yor (16) in the construction of the excursion measure. Under Assumptions A2.1 and A2.2, zero is an absorbing point for Y . Thus, we cannot simply start in zero and wait until the process returns to zero. Informally speaking, we instead condition the process to converge to infinity. One way to achieve this is by Doob's h-transformation. Note that S (Y t∧Tε ) t≥0 is a bounded martingale for every ε > 0, see Section V.28 in (17). In particular, The sequence of processes (Y ↑,ε t ) t≥0 , ε > 0 is consistent in the sense that for all 0 ≤ y ≤ ε and δ > 0. Therefore, we may define a process Y ↑ = (Y ↑ t ) 0≤t≤T∞ which coincides with (Y ↑,ε t ) t≥0 until time T ε for every ε > 0. Note that the ↑-diffusion possibly explodes in finite time.
The following important observation of Williams has been quoted by Pitman and Yor (16). Because we assume that zero is an exit boundary for (Y t ) t≥0 , zero is an entrance boundary but not an exit boundary for the ↑-diffusion. More precisely, the ↑-diffusion started at its entrance boundary zero and run up to the last time it hits a level y > 0 is described by Theorem 2.5 of Williams (20) as the time reversal back from T 0 of the ↓-diffusion started at y, where the ↓-diffusion is the process (Y t ) t≥0 conditioned on T 0 < ∞. Hence, the process Y ↑ t t≥0 may be started in zero but takes strictly positive values at positive times.
Pitman and Yor (16) define the excursion measureQ Y as follows. Under that is, conditional on "excursions reach level ε", an excursion follows the ↑-diffusion until time T ε and then follows the dynamics of (Y t ) t≥0 . In addition,Q Y T ε < T 0 = 1 S(ε) . With this in mind, define a processŶ ε := Ŷ ε t t≥0 which satisfies for y ≥ 0. In addition, (Ŷ ε t , t ≤ T ε ) and (Ŷ ε t , t ≥ T ε ) are independent. Define the excursion measureQ Y on U by This is well-defined if (167) holds for all ε, δ > 0. The critical part here is the path between T ε and T ε+δ . Therefore, (167) follows from The first equality is equation (21) with c = 0, y = ε and b = ε + δ. The last equality is the strong Markov property of Y ↑,ε+δ . The last but one equality is the following lemma.
Lemma 9.1. Assume A2.1 and A2.2. Let 0 < y < ε. Then Proof. We begin with the proof of independence of (Ŷ ε t , t ≤ T ε ) and of (Ŷ ε t , t ≥ T ε ). Let F and G be two bounded continuous functions on the path space. Denote by F Tε the σ-algebra generated by (Y t ) t≤Tε . Then The last equality is the strong Markov property of Y . Choosing F ≡ 1 in (170) proves that the left-hand side of (169) satisfies (165). In addition, equation (170) proves the desired independence. For the proof of we repeatedly apply the semigroup (161) of (Y ↑,ε t ) t≥0 to obtain for bounded, continuous functions f 1 , ..., f n and time points 0 ≤ t 1 < ... < t n . By equation (21) with c = 0, P y -almost surely where F tn∧Tε is the σ-algebra generated by (Y s ) s≤tn∧Tε . Insert this identity in the right-hand side of (172) to obtain This proves (171) because finite-dimensional distributions determine the law of a process. Now we prove convergence to the excursion measureQ Y .
Proof of Theorem 1. Let F : C [0, ∞), [0, ∞) → Ê be a bounded continuous function for which there exists an ε > 0 such that F (χ)½ T0<Tε = 0 for every path χ. Let 0 < y < ε. By Lemma 9.1, we obtain The last equality is the strong Markov property of the ↑-diffusion. The random time T y converges to zero almost surely as y → 0. Another observation we need is that every continuous path (χ t ) t≥0 is uniformly continuous on any compact set [0, T ]. Hence, the sequence of paths (χ Ty +t ) t≥0 , y > 0 converges locally uniformly to the path χ t t≥0 almost surely as y → 0. Therefore, the dominated convergence theorem implies Putting (175) and (176) together, we arrive at which proves the theorem.
We will employ Lemma 9.1 to calculate explicit expressions for some functionals ofQ Y . For example, we will prove in Lemma 9.8 together with Lemma 9.6 that Furthermore, let the continuous function ψ : [0, ∞) → Ê be nonnegative and nondecreasing. Then for every b ≤ ∞ and m ∈ AE ≥0 .
Proof. W.l.o.g. assume m ≥ 1. Let ε > 0 be such that ε < inf supp f and let y < ε. Using Lemma 9.1, we see that the left-hand side of (179) is equal to The second equality is the strong Markov property of Y ↑,ε and the change of variable s → s − T y . For the convergence, we applied the monotone convergence theorem.
The explicit formula on the right-hand side of (178) originates in the explicit formula (180) below, which we recall from the literature.
Let (Ỹ t ) t≥0 be a Markov process with càdlàg sample paths and state space E which is a Polish space. For an open set O ⊂ E, denote by τ the first exit time of (Ỹ t ) t≥0 from the set O. Notice that τ is a stopping time. For m ∈ AE 0 , define for a given function f ∈ C O, [0, ∞) . In the following lemma, we derive an expression for w 2 for which Lemma 9.3 is applicable.
for all y ∈ E.
Proof. Let y ∈ E be fixed. For the proof of (182), we apply Fubini to obtain The last equality follows from Fubini and a change of variables. The stopping time τ can be expressed as τ = F (Ỹ u ) u≥0 with a suitable path functional F . Furthermore, τ satisfies for r, s ≥ 0. Therefore, the right-hand side of (184) is equal to The last but one equality is the Markov property of (Ỹ t ) t≥0 . This proves (182). For the proof of (183), break the symmetry in the square of w 2 (y) to see that w 2 (y) is equal to This finishes the proof.
We will need that (Y t ) t≥0 dies out in finite time. The following lemma gives a condition for this. RecallS(∞) := lim y→∞S (y). Proof. On the event {Y t ≤ K}, we have that almost surely. The last inequality follows from Lemma 15.6.2 of (13) and Assumption A2.2. Therefore, Theorem 2 of Jagers (11) implies that, with probability one, either (Y t ) t≥0 hits zero in finite time or converges to infinity as t → ∞. With equation (21), we obtain This proves the assertion.
For the rest of the proof, assume that f (z)/z is bounded by c f . Let c 1 , c 2 be the constants from A2.1. Note that f (z) ≤ c f z ≤ The convergence (24) of Theorem 1 also holds for (χ s ) s≥0 → f (χ t ), t fixed, if f (y)/y is a bounded function. For this, we first estimate the moments of (Y t ) t≥0 . Lemma 9.9. Assume A2.1. Let (Y t ) t≥0 be a solution of equation (3) and let T be finite. Then, for every n ∈ AE ≥2 , there exists a constant c T such that for all y ≥ 0 and every stopping time τ .
Proof. The proof is fairly standard and uses Itô's formula and Doob's L p -inequality.
Right-continuity of the function t → f (Y ↑ t ) S(Y ↑ t ) ½ t<T b implies the last equality in (208). Now we let b → ∞ in (208) and apply monotone convergence to obtain The following estimate justifies the interchange of the limits lim The last equality follows fromS ′ (0) ∈ (0, ∞) and from Lemma 9.9. Putting (211) and (210) together, we get Note that (212) is bounded in t > 0 because of f (y) ≤ c f y ∨ y n and Lemma 9.7. We finish the proof of the first equality in (206) by proving that the limits lim ε→0 and lim y→0 on the right-hand side of (207) interchange.
The first equality is (212) with f replaced by f − f ε . The last equality follows from the dominated convergence theorem. The function f ε /S converges to f /S for every y > 0 as ε → 0. Note that Y ↑ t > 0 almost surely for t > 0. Integrability of f (Y ↑ t ) S(Y ↑ t ) ½ t<T∞ follows from finiteness of (212).
We have settled equation (178) in Lemma 9.8 (together with Lemma 9.6). A consequence of the finiteness of this equation is that lim inf t→∞ χ t dQ Y = 0. In the proof of the extinction result for the Virgin Island Model, we will need that χ t dQ Y converges to zero as t → ∞. This convergence will follow from equation (178) if [0, ∞) ∋ t → χ t dQ Y is globally upward Lipschitz continuous. We already know that this function is bounded in t by Lemma 9.10. Proof. We will prove that the function [0, ∞) ∋ t → χ n t dQ Y is globally upward Lipschitz continuous. The assertion then follows from the finiteness of (199) with f (z) replaced by z n and with m = 1. Recall τ K , c h and c g from the proof of Lemma 9.9. From (3) and Itô's lemma, we obtain for y > 0 and 0 ≤ s ≤ t E y Y n r∧τK + Y n−1 r∧τK dr wherec := n c h + (n − 1)c g . Letting K → ∞ and then y → 0, we conclude from the dominated convergence theorem, Lemma 9.9 and Lemma 9.10 that 10 Proof of Theorem 2, Theorem 3 and of Theorem 4 We will derive Theorem 2 from Theorem 5 and Theorem 3 from Theorem 6. Thus, we need to check that Assumptions A4.1, A4.2 and A4.3 with φ(t, χ) := χ t , ν := L x (Y ) and Q := Q Y hold under A2.1, A2.2, A2.3 and A2.4. Recall that Q Y =S ′ (0)Q Y ands(0) =S ′ (0)s(0). Assumption A4.1 follows from Lemma 9.9 and Lemma 9.10. Lemma 9.6 and Lemma 9.8 imply A4.2. Lemma 9.5 together with Lemma 9.6 implies that (Y t ) t≥0 hits zero in finite time almost surely. The second assumption in A4.3 is implied by Lemma 9.11 with n = 1 and Assumption A2.4. By Theorem 5, we now know that the total mass process (V t ) t≥0 dies out if and only if condition (43) is satisfied. However, by Lemma 9.8 with m = 1 and f (·) = a(·), condition (43) is equivalent to condition (28). This proves Theorem 2 For an application of Theorem 6, note that f ν and f Q are integrable by Lemma 9.6 and Lemma 9.8, respectively. In addition, Lemma 9.6 and Lemma 9.8 show that Similar equations hold for w ′ id (0) and w a (x). Moreover, the denominators in (33) and (48) are equal by Lemma 9.8,equation (200), together with Lemma 9.6. Therefore, equations (32) and (33) follow from equations (47)  and Lemma 9.11 with n = 1 together with Assumption A2.4. Furthermore, Lemma 9.10 together with Lemma 9.7 and the dominated convergence theorem implies continuity of f Q . Therefore, Theorem 6 implies (51) which together with (52) reads as (35).