Survival of a single mutant in one dimension

We study a one dimensional two-type contact process with equal rate of propagation (and death) of the two types. We show that the progeny of a ﬁnite number of mutants has a positive probability of survival if and only at time 0 there is at most a ﬁnite number of residents on at least one side of the mutant’s “colony”.


Introduction
The aim of this paper is to study the probability that the progeny of a single mutant in an infinite population of residents will survive. We consider this problem in the framework of the one dimensional two-type contact process.
We will prove that if the mutant has no selective advantage nor disadvantage, compared with the individuals of the resident population, then, provided we are in the supercritical case (which means that a single individual's progeny may survive for ever), a single mutant with an empty half-line in front of him, and all sites behind him occupied by resident individuals, has a progeny which survives forever with positive probability, while any finite number of mutants, with infinitely many residents on both sides, have a progeny which goes extinct a. s. Note that we define the progeny at time t of a given ancestor at time 0 as the set of individuals alive at time t, who are the descendants of that ancestor at time 0.
Let us now explain what we mean by the contact process. Note that this process is often presented in the language of infection. We shall rather consider it here as a model of the spread of a population. Consider first the usual one-type contact process with birth parameter λ > 0. This process {ξ t , t ≥ 0} is a {0, 1} -valued Markov process, hence ξ t is a random mapping which to each x ∈ associates ξ t (x) ∈ {0, 1}. The statement ξ t (x) = 1 means that the site x is occupied at time t, while ξ t (x) = 0 means that site x is empty at time t. The process evolves as follows. Let x be such that ξ 0 (x) = 1. We wait a random exponential time with parameter 1+2λ. At that time, with probability 1/(1+2λ), the individual at site x dies; with probability λ/(1 + 2λ), the individual, while continuing its own life at site x, gives birth to another individual; the newborn occupies site x + 1 if it is empty, and dies instataneously otherwise; and with probability λ/(1 + 2λ), it gives birth to a newborn who occupies site x −1 if it is empty, and dies instataneously otherwise. Then the same operation repeats itself until site x becomes empty, independently of what happened so far. The same happens at any occupied site, and the exponential clocks at various sites are mutually independent. We will use the same notation ξ t to denote the random element of {0, 1} defined above, and the random subset of consisting of all sites x ∈ where ξ t (x) = 1.
The two-type contact process {η t , t ≥ 0} is a {0, 1, 2} -valued Markov process which starts from an initial condition (A, B), where A and B are two nonintersecting subsets of , A denoting the set of sites which are occupied by type 1 individuals and B the set of sites which are occupied by type 2 individuals at time t = 0. In other words, The two-type contact process with equal birth rates λ evolves exactly like the one-type process, with each individual possibly giving birth to individuals of the same type. We shall consider in section 4 the case where the birth rate of the mutants (i. e. type 2 individuals) differs from that of the residents (i. e. type 1 individuals).
The (one-type) contact process has been extensively studied and plays a central role in the theory of interacting particle systems (see [6], [7] and references therein) but there are very few papers on the two-type contact process (see [3] and [8]). For another closely related probabilistic model of competition between species, see durneu.
Let us now present a useful construction of the contact process, called the graphical representation, which is valid in both the one-type and the two-type cases (at least in the case of equal birth rates). The important feature of this construction is that processes corresponding to different initial conditions are coupled through it. Indeed, {ξ t , t ≥ 0} (resp. {η t , t ≥ 0}) is a fixed function of both the initial condition, and the set of Poisson point processes, which code all the randomness, which we now introduce.
Consider a collection {P x t , P x,+ t , P x,− t , t ≥ 0; x ∈ } of mutually independent Poisson point processes, such that the P x 's have intensity 1 while both the P x,+ 's and the P x,− 's have intensity λ, all defined on a probability space (Ω, , ). On the set × [0, ∞) we place a δ on the point (x, t) whenever t belongs to the Poisson process P x . On that set we also place an arrow from (x, t) to (x + 1, t) whenever t belongs to the Poisson process P x,+ and an arrow from (x, t) to (x − 1, t) whenever t belongs to the Poisson process P x,− .
The process {ξ A t , : t ≥ 0} is defined as follows. An open path in × [0, +∞) is a connected oriented path which moves along the time lines in the increasing t direction without passing through a δ symbol, and along birth arrows, in the direction of the arrow. Now To construct the two-type contact process, we call line of descendance an open path starting from an occupied site at time 0, and such that any arrow belonging to this path points to an unoccupied site.
Note that unlike open paths, lines of descendance depend on the initial configuration of the process. For A, B two disjoint subsets of , we define {η A,B t , t ≥ 0} as the {0, 1, 2} -valued process whose value at time t is given by ∃x ∈ B and a line of descendance from (x, 0) to ( y, t)} In biological terms, we think of the type 1 population as the resident population, and of the type two population as a mutant population.
We shall also need in this paper to define both the one-type and the two-type contact processes on subsets of of the form (−∞, a] (resp. [a, +∞)). These are defined as above, with the restriction that the Poisson processes (P a,+ , P y , P y,+ , P y,− , y > a) (resp. (P a,− , P y , P y,+ , P y,− , y < a)) are ignored.
Let {ξ A t , t ≥ 0} denote the one-type contact process starting from the configuration whose set of occupied sites is A. We will write ξ x t for ξ {x} t . We shall use the notation It follows from well-known results on the contact process, see e. g. Liggett [6], that there exists λ c < ∞ such that ρ > 0 whenever λ > λ c .
The aim of this paper is to prove This result has an interesting consequence concerning the one-type contact process {ξ A t , t ≥ 0}, which says that in the case where both sup(A) = +∞ and inf(A) = −∞, the progeny of any ancestor alive at time 0 dies out a. s., while in the case where |A| = ∞ but sup(A) < ∞, one and only one ancestor alive at time 0 has a progeny which survives for ever, see Corollary 3.12 below.
From the results needed to prove Theorem 1.1 we can also deduce: We conjecture that Theorem 1.2 holds for the two-type contact process on d for all d ≥ 1. In [8] it is proved that for d ≤ 2 and all initial configurations lim t→∞ (η t (x) = 1, η t ( y) = 2) = 0 for all x, y, while for d ≥ 3 the process admits invariant measures µ such that for all x = y, µ({η : η(x) = 1, η( y) = 2}) > 0. Although this last result may be seen as evidence favoring our conjecture (when d ≥ 3) it does not imply it nor is it implied by it.
The paper is organized as follows. In section 2, we recall and prove several results on the one-type contact process which are needed in further sections. In section 3, we study the case of a single or a finite number of mutants confronted with an infinite number of residents, in the case of equal birth rates. Theorems 1.1 and 1.2 are proved in subsections 3.3 and 3.4 respectively. Finally, in section 4, we conclude with some remarks on the case of unequal birth rates (i. e. when one of the two species has a selective advantage). We formulate one result and two conjectures.
In all of this paper, we assume that λ > λ c .

Some results on the one-type contact process
Let − be the set of integers smaller than or equal to 0 and let + be the set of integers greater than or equal to 0.
For a proof of these results the reader is referred to Theorems VI.2.19 and VI.2.24 in [6].
Let R t = sup s≤t r s .
PROOF: Let τ a = inf s : r s ≥ a . Then By the strong Markov property this is bounded below by which by symmetry is at least To do so fix > 0 and let c = 2 ρ . Then write where we have used Lemma 2.1 for the first inequality, and the L 1 convergence of r t /t for the equality. Hence lim sup Since is arbitrary the lemma is proved.
Although the following lemma is well known, we did not find it in previous publications and we include it here for the sake of completness. PROOF: Let 0 < ε < v . Then write: where the last inequality is due to the fact that ξ 0 n (x) = ξ n (x) for any x ∈ [ n , r n ]. We now show that the sum on n of each of the three terms of the right hand side above converges: For the first of these terms, the convegence is a consequece of the fact that for any n the distribution of ξ n is stochastically above the upper invariant measure of the contact process and of Theorem 1 of [5] .For the third term the convergence follows from Corollary 3.22 in Chapter VI of [6]. For the second term it follows by that same corollary applied to n and our choice of ε. We have thus proved that This, the Markov property and Theorem 3.29 in Chapter VI of [6] imply that Since ε is arbitrary and r n ≤ r n we get: (lim n r n n = v(λ)|τ 0 = ∞) = 1 and the lemma follows from the fact that sup 0≤s≤t≤1 r n+t − r n+s is bounded above by a Poisson r.v. of parameter λ.
It now follows: PROOF: There exists a strictly increasing sequence {x k , k ≥ 1} ⊂ A such that there is an infinite open path starting from each x k . Now for each n ≥ 0 and some R ∈ define It follows from the last Lemma that for R large enough, (A n ) = (A 0 ) > 0. From now on such an R is fixed. From the ergodic theorem, hence a. s. infinitely many A n occur. So almost surely, one A n with n ≥ R occurs. Now choose k large enough such that x k ≥ n. Clearly there exists an infinite open path starting from (x k , 0) which lies on the right of the line {(v t, t), t ≥ 0}.
Although our next result is well-known, we could not find it explicitly stated in the litterature. It follows immediately from our previous Corollary and the fact that v(λ) > 0 whenever λ > λ c . Let µ + denote the upper invariant measure for the contact process on . This is defined as follows. Denote by {χ t , t ≥ 0} the one-type contact process on . This process takes its values in {0, 1} . In accordance with the above conventions, for A ⊂ , we write χ A t for the contact process on starting with the initial condition Then µ + is the weak limit, as t → ∞, of the law of For the proof of this result, we will need the following PROOF: We first exploit the well-known self-duality of the contact process. Since there is a one to one correspondance between the open paths from some ( y, 0), y ∈ , to some (x, t), x ∈ (0, n] and the open paths from some (x, 0), x ∈ (0, n] to some ( y, t), y ∈ obtained by reversing the directions of the arrows, Letting t → ∞ in the above identity yields The last right hand side is the probability that there is an infinite open path starting from some Part b) follows from part a) and in view of Lemma 2.7, to prove part a) it suffices to show that for the contact process on there exist constants K, c > 0 such that It follows from Corollary VI.3.22 in [6] that for some K 1 , c > 0 we have: and that Therefore, using (2.1) we get for some constant K 3 . We have shown in particular that It follows from (2.2) that the second term of the right hand side decays exponentially in n. Hence, the lemma will be proved if we show that the first term also decays exponentially in n. To do so, let σ = inf{t : r t ≤ −n} and let Y n be a Poisson random variable of parameter 2λβ n. It now follows from the Strong Markov property applied at the stopping time σ that: Since, given our choice of β, lim n→∞ (Y n ≤ (1 + vβ)n) = 1 and by (2.1), (r 2β n ≤ vβ n) decays exponentially in n, the same happens to (σ ≤ β n) = (inf 0≤t≤β n r t ≤ −n).
Let T −1 be the operator on the set of probability measures on {0, 1} defined by The natural partial order on {0, 1} induces a partial order on the set of probability measures on {0, 1} which we denote by ≤. Recalling that µ + is the upper invariant measure for the contact process on , we have PROOF: Consider the contact process {χ t , t ≥ 0} this time on ∪{0}, starting again from χ 0 ≡ 1. Let now {χ t , t ≥ 0} denote the same process, with the same initial condition and the same realization of the graphical representation, except that we delete all arrows between states 0 and 1. The restriction to of the asymptotic (as t → ∞) law of χ t coincides with µ + , while the same law associated with χ t coincides with T −1 (µ + ).The result follows from the fact that for all t > 0, x ≥ 1, Our next proposition is taken from [1] (See Theorem 2 in that reference). Although there the result is stated and proved for the contact process on , their proof also holds for the contact process on .
PROOF: Consider the contact process {χ t , t ≥ 0} on , starting from χ 0 ≡ 1. We deduce from Proposition 2.9: It then follows from Lemma 2.11 below that It remains to let t → ∞.
Lemma 2.11. Let {χ t , t ≥ 0} denote the contact process on , starting from any deterministic initial condition. For any t > 0, the law of χ t has positive correlations.
PROOF: For the contact process on [1, · · · , n], the result follows from Theorem 2.14 on page 80 of Liggett [6]. Our result then follows by letting n → ∞.
Note that Lemma 2.11 applies as well to the contact process {ξ t , t ≥ 0} on .
Let , P x,− t , x ∈ } as explained above. Note that on the event S x, y the process starting from {x} survives but this does not mean that if we start from {x, y} the progeny of (say) x lives forever. We now show that (recall the definition of ρ in (1.1)) Lemma 2.12. For all x, y ∈ (S x, y ) ≥ ρ 2 .
PROOF: Denoting by µ the upper invariant measure of the contact process {ξ t , t ≥ 0} on , i. e. µ is the limit as t → ∞ of the law of ξ t , we have by the same duality argument already used in the proof of Lemma 2.7 the identities Letting t → ∞ in the result of Lemma 2.11 applied to the contact process on implies that µ has positive correlations. Hence The result follows from this inequality and the three above identities.
We now fix some λ > λ c and let v = v(λ). We pick From now on t 0 will be a large enough multiple of 2 v so that the following holds : where Let us define new processes. For any z ∈ , we write where as usual the sup (resp. the inf) over an empty set is −∞ (resp. +∞). From now on, t 0 will be a large enough multiple of 2 v such that both the inequality (2.3) and the conclusion of Lemma 2.13 hold.

The two-type contact process with equal birth rates
Let η t denote the contact process with two types.
t , t ≥ 0} now denotes the contact process where at time zero A is the set of sites occupied by individuals of type 1, and B is the set of sites occupied by individuals of type 2. The dynamics is the same as before, using the same construction with the same collection of Poisson processes, except that now an individual of type α ∈ {1, 2} located at site z gives birth at time t to an individual of the same type at site z + 1 (resp. at site z − 1), if t is a point of the Poisson process P x,+ (resp. P x,− ) and the site z + 1 (resp. z − 1) is not occupied at time t.
In other words, the 1's and 2's together evolve like a one-type contact process, and all descendants of a 1 (resp. of a 2) are 1's (resp. 2's).

A single mutant in front of an infinite number of residents may survive
In this subsection, we consider the process {η A,B t , t ≥ 0} only in the case where A < B, meaning that all points in A are located on the left of each point of B. In other words, the initial configurations belong to the set: Given the nearest neighbor character of our process, whenever it starts in , it remains in with probability 1.

PROOF:
By Lemma 2.13 and symmetry arguments we have: Consequently for the initial configuration {0}{v t 0 } the first of these paths is always occupied by a type 1 particle. Therefore, adding to the initial configuration extra 1-type particles to the left of the origin does not alter the process to the right of that open path. Hence We will also need the following Lemma: converges to 0.
PROOF: It suffices to show that ], hence we only need to prove that for the one type contact process converges to 0 as t 0 goes to infinity. But on the event C(t 0 , ) the set of occupied points in the is the same whether the initial condition of the process is or {v t 0 } . Since starting from we have more occupied points than under the upper invariant measure the result follows from the fact that under the upper invariant measure the probability of having an empty interval of length n tends to 0 as n tends to infinity.
From now on we shall use { a ζ t , t ≥ 0} to denote the two-type contact process on (−∞, a]. Now we can prove: PROOF: In this proof we will consider the two type contact process on both and (−∞, 3 2 v t 0 ]. These two processes are constructed on the same probability space with the same Poisson processes. For the second of these pocesses The existence of the first of these paths has a probability which converges to 0 as t 0 goes to infinity by Lemma 2.2.
By reversing the arrows and using symmetry and again Lemma 2.2, we see that the same happens to the second path.
Hence, if we define converges to 0 as t 0 goes to infinity . The result follows from Corollary 3.1 and the following claim: starting both η t and 3v t 0 To justify this claim note first that on the event G there exists a rightmost open path from (v t 0 , 0) , which remains to the left of the line x = We now introduce the following partial order on {0, 1, 2} : (3.1) Intuitively means "more 1's" and "fewer 2's".
Remark 3.5. The reader might think that a more natural definition of the inequality in distribution would be to say that µ 1 µ 2 whenever µ 1 ( f ) ≥ µ 2 ( f ) for all f : {0, 1, 2} → which are increasing in the sense that η 1 η 2 implies f (η 1 ) ≥ f (η 2 ). Theorem II.2.4 in [6] says that for the standard partial order on {0, 1} the two definitions are equivalent. It is clear that this theorem can be extended to our partial order, but we shall not need this result here.
Note that η 1 η 2 implies br(η 1 ) ≥ br(η 2 ) and that if γ ζ, the coupling between the contact processes starting form different initial conditions deduced from the graphical representation produces the property In the sequel for any probability measure µ on {0, 1, 2} and any i ∈ , T i (µ) will denote the measure µ translated by i. That is the measure such that for all n ∈ , all x 1 < x 2 < · · · < x n and all possible values of a 1 , . . . , a n we have: Moreover, if µ is a measure on A [n,∞) where A is any non-empty subset of {0, 1, 2}, then T i (µ) will be the measure on A [n+i,∞) satisfying (*).
As before µ + denotes the upper invariant mesure for the contact process on and µ + 2 will be the measure obtained from µ + by means of the map: F : {0, 1} → {0, 2} given by F (η)(x) = 2η(x). With a slight abuse of notation the measures µ + and µ + 2 will also be seen as measures on {0, 1, 2} and a similar abuse of notation will be applied to the translates of these measures.
We start the process {η t , t ≥ 0} from the initial distribution µ determined by is the point mass on the configuration In the sequel η 0 will denote a random initial configuration distributed according to µ. In other words, we assume that η 0 = η 0 .
We now proceed as follows. We partition the probability space into a countable number of events: H, J 0 , J 1 , . . . and let the process run on a time interval of length t 0 . Then we show that the distribution of η t 0 conditioned on any event of the partition is than a convex combination of translations ofμ. Hence the unconditioned distribution of η t 0 is also such a convex combination. Then we replace η t 0 by a random configuration η 1 whose distribution is this convex combination and let the process run on another time interval of length t 0 and so on. Our partition of the probability space is given by : where Q t 0 = max{R t 0 , v t 0 } (recall that R t = sup s≤t r s ).
Since the initial distribution considered here is than the initial distribution of Corollary 3.3, we have We also claim that conditioned on H, the distribution of 3v t 0 2 ξ t 0 is ≥ T 3v t 0 2 µ + 2 (this follows from Lemma 2.8 and the fact that the process 3v t 0 2 ξ t is independent of H).
Therefore, the distribution of η t 0 conditioned on H is ν where ν is determined by: 1. The projection of ν on {0, 1, 2} (−∞, 3v t 0 2 ] is the point mass on the configuration It follows from Lemma 2.10 (applied to µ A similar argument shows that the conditional distribution of η t 0 given J m is where Y is distributed as above.
It follows from the above arguments that η t 0 µ 1 in distribution, where We can now state: Proposition 3.6. If t 0 is large enough, there exists a positive integer valued random variable Z(t 0 ) such that has an exponentially decaying tail.

PROOF:
Part a) follows from (3.4) and part b) follows from part a) of Lemma 2.6 and the fact that R t 0 is bounded by a Poisson random variable of parameter λt 0 . To prove part c) write Hence it follows from Lemma 2.2 that lim sup and the result follows from (3.3).
We can now prove: Let α t denote the number of descendants at time t of the unique initial type 2 individual (hence α t denotes also the number of type 2 individuals at time t). On the event that the lineage of the unique type 2 individual survives for ever we have α t → ∞ as t → ∞ a. s. Hence if that event has positive probability, (α t ) → ∞ as t → ∞. Consequently for any δ > 0, Denote by r t (x) the supremum of the set of sites occupied by the descendants of the individual (x, 0). Clearly, whatever the initial configuration is where as above From the result recalled at the beginning of section 2, there exists T such that Recall that in our initial configuration all sites are occupied (who occupies each site is irrelevant to contradict the fact that T δ < ∞, which we now do).
For n odd, let Z t (n) be the number of sites which at time t are in a line of descendance starting at time 0 in the interval [−(n − 1)/2, . . . , (n − 1)/2]. Now by stationarity whenever t ≥ T δ , On the other hand, if t ≥ T , by symmetry, Choosing n > 2t(v + 1)/δ, the last two inequalities yield a contradiction.
In order to deduce Theorem 3.9 from Proposition 3.10, we shall need the following Lemma where, as above, the η t 's for various initial conditions are defined with the same unique graphical representation.
From Corollary 3.8, symmetry and translation invariance, On the set {(x, t) : x ∈ , t ≥ 0} the Poisson processes used in the construction are n-fold mixing with respect to translations on for any n ∈ . Since x n+1 ≥ x n + 1, this implies that for all k ≥ 1 Consequently ∩ n≥0 C c n ≤ (1 − γ) k for all k ≥ 1. This shows that ∪ n≥0 C n = 1.
The result for the D m 's is proved similarly. From the last Lemma we know that (∪ n,m E n,m ) = 1. Hence, it suffices to show that for all n, m ∈ , we have: But on the event E n,m the evolution of "2"'s is not altered by adding "1"'s to the left of y m or to the right of x n . Therefore the result follows from Proposition 3.10.

Proof of Theorem 1.1
The only if part follows from Theorem 3.9. Let us prove the if part. We consider the case where |A ∩ B + | < ∞. The other case is treated similarly.
We let Hence from the strong Markov property it remains to show that whenever A ∩ B + = , A,B (the type 2 population survives for ever) > 0.
This last statement follows from translation invariance, (3.2) and Corollary 3.8.

Proof of Theorem 1.2
By the Markov property and symmetry it suffices to show that the theorem holds for some A and B.
To prove this, let (x n ) n≥0 and ( y m ) m≥0 be as in the statement of Lemma 3.11 and let C n and D m be as in the proof of that lemma. It follows from that same lemma that there exist n and m such that (C n ∩ D m ) > 0. This implies that { y m }{x n } (∀ t > 0 ∃ x, y : η t (x) = 1, η t ( y) = 2) > 0.
Hence, the theorem holds when A = {x n } and B = { y m }.

Corollary for the one-type contact process
The following is an immediate consequence of the above results. 2. if |A| = +∞ but sup A < ∞, then exactly one individual has a progeny which survives for ever.
PROOF: The first statement is a consequence of Theorem 3.9. For the second statement first note that it follows from (3.2) and Corollary 3.8 that for any initial condition having a rightmost individual, the probability that this individual has a progeny which survives forever is bounded below by γ > 0. We then define an increasing sequence of stopping times: τ 1 is the smallest time at which the progeny of the rightmost initial individual dies out, τ 2 is the smallest time at which the progeny of the rightmost individual at time τ 1 dies out and so on. It then follows from a repeated application of the Strong Markov Property that (τ n < ∞) ≤ γ n . Hence, with probability 1 for some k, τ k = ∞ which implies that at least one individual has a progeny wchich survives forever. Suppose now that two individuals, say x < y, have a progeny which survives for ever with positive probability. Adding infinitely many individuals at time t = 0 on the right of y cannot possibly modify the fate of the progeny of x. This would mean that the progeny of x would survive for ever with positive probability, in the presence of infinitely many individuals at time t = 0 on both of its sides. This contradicts Theorem 3.9.

Remarks about the case of unequal birth rates
Assume that the type 1 individuals have the birth rate µ, and type 2 individuals have the birth rate λ.
It is not hard to deduce from our argument that for µ > λ c there exists > 0 such that the conclusion of 1.1 remains true if µ − < λ < µ. However, we conjecture that this is not the case for all values of λ in the interval (λ c , µ). Consider now the right contact process, where each individual gives birth to offsprings on its right at rate λ, and does not give birth to any offspring on its left. Let now λ cc denote the critical value of the parameter λ, such that whenever λ > λ cc , the one-type right contact process starting from {0} has a positive probability of survival. Going back to our two-types contact process, whenever λ > λ cc , whatever the value of µ may be, the progeny of a single type 2 individual with a finite number of type 1 individuals on its right at time 0, has a positive probability of survival.
In the other direction, we conjecture that if the rates favor type 2 individuals (i.e. λ > µ > λ c ) then a unique type 2 individual has a positive probability of having descendants at all times even when all remaining sites are occupied at time 0 by type 1 individuals.