The effect of competition on the height and length of the forest of genealogical trees of a large population

The effect of competition on the height and length of the forest of genealogical trees of a

large population

M. Ba and E. Pardoux

Colloque JPS, CIRM Marseille

April 17, 2012

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 1 / 26

Introduction

Consider a Galton–Watson branching processX^m in continuous time with m ancestors at time t= 0: fork ≥1, the process

X^m jumps

(k→k+ 1, at rate µk; k→k−1, at rate λk.

For any 0≤m<n,X^n(d)= X^m+X^n−m (branching property).

In order to model competition within the population described byX^m, we superimpose to each individual a death rate due to competition, equal at timet toγX_t^m. This gives a global death rate equal at time t toγ(X_t^m)².

Set m= [Nx], µN = 2N+θ,λN = 2N andγN =γ/N, and Z^N,x = ^X_N^[Nx]. The “total population mass process”Z^N converges weakly to the solution of the Feller SDE with logistic drift

x x x 2 p _x

Introduction

Consider a Galton–Watson branching processX^m in continuous time with m ancestors at time t= 0: fork ≥1, the process

X^m jumps

(k→k+ 1, at rate µk; k→k−1, at rate λk.

For any 0≤m<n,X^n(d)= X^m+X^n−m (branching property).

In order to model competition within the population described byX^m, we superimpose to each individual a death rate due to competition, equal at timet toγX_t^m. This gives a global death rate equal at time t toγ(X_t^m)².

Set m= [Nx], µN = 2N+θ,λN = 2N andγN =γ/N, and Z^N,x = ^X_N^[Nx]. The “total population mass process”Z^N converges weakly to the solution of the Feller SDE with logistic drift

dZ_t^x =

θZ_t^x−γ(Z_t^x)²

dt+ 2p

Z_t^xdWt, Z₀^x =x. (1)

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 2 / 26

Introduction

Consider a Galton–Watson branching processX^m in continuous time with m ancestors at time t= 0: fork ≥1, the process

X^m jumps

(k→k+ 1, at rate µk; k→k−1, at rate λk.

For any 0≤m<n,X^n(d)= X^m+X^n−m (branching property).

In order to model competition within the population described byX^m, we superimpose to each individual a death rate due to competition, equal at timet toγX_t^m. This gives a global death rate equal at time t toγ(X_t^m)².

Set m= [Nx], µN = 2N+θ,λN = 2N andγN =γ/N, and Z^N,x = ^X_N^[Nx]. The “total population mass process”Z^N converges weakly to the solution of the Feller SDE with logistic drift

x x x 2 p _x

Introduction

This model has been studied by Lambert (2005) and

Pardoux-Wakolbinger (2011). The forest of those m trees is also finite a.s. . and for anyx la solution Z^x of equation (1) goes extint in finite timea.s..

On can define the height and the length of the discrete forest of genealogical trees

H^m= inf{t >0, X_t^m = 0}, L^m = Z H^m

0

X_t^mdt, for m≥1,

as well as the height and the lenght of the continuous “forest of genealogical trees”

T^x = inf{t >0, Z_t^x = 0}, S^x = Z T^x

0

Z_t^xdt .

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 3 / 26

Introduction

This model has been studied by Lambert (2005) and

Pardoux-Wakolbinger (2011). The forest of those m trees is also finite a.s. . and for anyx la solution Z^x of equation (1) goes extint in finite timea.s..

On can define the height and the length of the discrete forest of genealogical trees

H^m= inf{t >0, X_t^m = 0}, L^m = Z H^m

0

X_t^mdt, for m≥1, as well as the height and the lenght of the continuous “forest of genealogical trees”

T^x = inf{t >0, Z_t^x = 0}, S^x = Z T^x

0

Z_t^xdt

Branching process with polynomial competition

We generalize the above models, replacing the death rate γ(X_t^m)² by γ(X_t^m)^α,α >0.

Set m= [Nx], µ_N = 2N+θ,λ_N = 2N andγ_N =γ/N^α−1, and Z^N,x := ^X[Nx]_N . We show that the process Z^N converges weakly to to a Feller SDE with a negative polynomial drift.

dZ_t^x = [θZ_t^x−γ(Z_t^x)^α]dt+ 2p

Z_t^xdWt, Z₀^x =x. (2)

We study the height and the length of the genealogical tree, as an (increasing) function of the initial population in the discrete and the continuous model.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 4 / 26

Branching process with polynomial competition

We generalize the above models, replacing the death rate γ(X_t^m)² by γ(X_t^m)^α,α >0.

Set m= [Nx], µ_N = 2N+θ,λ_N = 2N andγ_N =γ/N^α−1, and Z^N,x := ^X[Nx]_N . We show that the process Z^N converges weakly to to a Feller SDE with a negative polynomial drift.

dZ_t^x = [θZ_t^x−γ(Z_t^x)^α]dt+ 2p

Z_t^xdWt, Z₀^x =x. (2)

We study the height and the length of the genealogical tree, as an (increasing) function of the initial population in the discrete and the continuous model.

Results

Both IE[sup_mH^m]<∞ and IE[sup_xT^x]<∞ ifα >1, while H^m → ∞as m→ ∞and T^x → ∞ as x→ ∞ a. s. ifα≤1.

Both IE[sup_mL^m]<∞ andIE[sup_xS^x]<∞ ifα >2, whileL^m → ∞ as m→ ∞andS^x → ∞ as x→ ∞ a. s. ifα≤2.

This necessitates to define in a consistent way the population processes jointly for all initial population sizes, i. e. we will need to define the two–parameter processes {X_t^m, t ≥0,m≥1}and {Z_t^x, t ≥0,x>0}.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 5 / 26

The model with asymetric effect of the competition

The description of the process (X_t^m,t≥0) is valid for one initial conditionm, but it is not sufficiently precise to describe the joint evolution of {(X_t^m,X_tⁿ), t≥0}, 1≤m<n.

Modelize the effect of the competition in a asymmetric way. The idea is that the progenyX_t^m of them “first” ancestors should not feel the competition due to the progenyX_tⁿ−X_t^m of then−m “additional”

ancestors which is present in the populationX_tⁿ.

We order the ancestors from left to right, this order being passed to their progeny.

The model with asymetric effect of the competition

The description of the process (X_t^m,t≥0) is valid for one initial conditionm, but it is not sufficiently precise to describe the joint evolution of {(X_t^m,X_tⁿ), t≥0}, 1≤m<n.

Modelize the effect of the competition in a asymmetric way. The idea is that the progenyX_t^m of them “first” ancestors should not feel the competition due to the progenyX_tⁿ−X_t^m of then−m “additional”

ancestors which is present in the populationX_tⁿ.

We order the ancestors from left to right, this order being passed to their progeny.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 6 / 26

The model with asymetric effect of the competition

The description of the process (X_t^m,t≥0) is valid for one initial conditionm, but it is not sufficiently precise to describe the joint evolution of {(X_t^m,X_tⁿ), t≥0}, 1≤m<n.

Modelize the effect of the competition in a asymmetric way. The idea is that the progenyX_t^m of them “first” ancestors should not feel the competition due to the progenyX_tⁿ−X_t^m of then−m “additional”

ancestors which is present in the populationX_tⁿ.

We order the ancestors from left to right, this order being passed to their progeny.

Order killing

The individual placed at the position i at time t dies becuse of competition at rate γ[L_i(t)^α−(L_i(t)−1)^α],L_i(t) is the number of alive individuals at timet, who are located at his left on the planar tree.

11 18

1 2 3 4 5

Figure: Bijection between binary trees and exploration processes

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 7 / 26

Discret model in the asymetric competition picture

{X_t^m, t ≥0} is a continuous timeZ+–valued Markov process, which evolves as follows. X_t^m jumps to

(k+ 1, at rateµk;

k−1, at rateλk+γ(k−1)^α. The above description specifies well the evolution of the

twoparametersprocess{X^m_t , t ≥0,m≥0}.

For α= 1,{X_t^m, m≥1}is a Markov chain for fixed t

For α6= 1,{X_t^m, m≥1}is not a Markov chain for fixed t. The conditional law of X_tⁿ⁺¹ given X_tⁿ differs from its conditional law given (X_t¹,X_t², . . . ,X_tⁿ).

However, {X_·^m, m≥0} is a Markov chain with values in the space D([0,∞);Z+), which starts from 0 atm= 0.

Discret model in the asymetric competition picture

{X_t^m, t ≥0} is a continuous timeZ+–valued Markov process, which evolves as follows. X_t^m jumps to

(k+ 1, at rateµk;

k−1, at rateλk+γ(k−1)^α. The above description specifies well the evolution of the

twoparametersprocess{X^m_t , t ≥0,m≥0}.

For α= 1,{X_t^m, m≥1}is a Markov chain for fixed t

For α6= 1,{X_t^m, m≥1}is not a Markov chain for fixed t. The conditional law of X_tⁿ⁺¹ given X_tⁿ differs from its conditional law given (X_t¹,X_t², . . . ,X_tⁿ).

However, {X_·^m, m≥0} is a Markov chain with values in the space D([0,∞);Z+), which starts from 0 atm= 0.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 8 / 26

Discret model in the asymetric competition picture

{X_t^m, t ≥0} is a continuous timeZ+–valued Markov process, which evolves as follows. X_t^m jumps to

(k+ 1, at rateµk;

k−1, at rateλk+γ(k−1)^α. The above description specifies well the evolution of the

twoparametersprocess{X^m_t , t ≥0,m≥0}.

For α= 1,{X_t^m, m≥1}is a Markov chain for fixed t

For α6= 1,{X_t^m, m≥1}is not a Markov chain for fixed t. The conditional law of X_tⁿ⁺¹ given X_tⁿ differs from its conditional law given (X_t¹,X_t², . . . ,X_tⁿ).

However, {X_·^m, m≥0} is a Markov chain with values in the space D([0,∞);Z+), which starts from 0 atm= 0.

Discription of jointly laws

For arbitrary 0≤m<n, letV_t^m,n:=X_tⁿ−X_t^m,t ≥0. Conditionally upon {X_·<sup>`</sup>, `≤m}, and given that X_t^m =x(t),t ≥0,{V_t^m,n, t≥0}is a Z+–valued time inhomogeneous Markov process starting from V₀^m,n=n−m, whose time–dependent infinitesimal generator {Q<sub>k,`</sub>(t), k, `∈Z+}is given by

Q0,`(t) = 0, ∀`≥1, and for any k ≥1, Q_k,k+1(t) =µk,

Q_k,k−1(t) =λk+γ(x(t) +k−1)^α, Q<sub>k,`</sub>(t) = 0, ∀`6∈ {k−1,k,k+ 1}.

Moreover we have

hV^m,n,X^mi= 0.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 9 / 26

The continuous Model

{Z_t^x, t≥0,x≥0}is such that for each fixed x >0,{Z_t^x,t ≥0} is continuous process, solution of the SDE (2).

For any 0<x <y,{V_t^x,y :=Z_t^y−Z_t^x, t ≥0} solves the SDE dV_t^x,y =

θV_t^x,y−γ

(Z_t^x+V_t^x,y)^α−(Z_t^x)^α dt+ 2 q

V_t^x,ydW_t⁰, V₀^x,y =y−x,

{W_t⁰, t ≥0} is independent from the Brownian motion W which drives the SDE (2) for Z_t^x.

{Z_·^x, x≥0}is a Markov process with values in C([0,∞),IR+), starting from 0 atx = 0.

Convergence result

Let {Z˜_t^N,x, t ≥0,x ≥0} be the process such that for fixedx,

{Z˜_t^N,x, t ≥0} is the linear interpolation of{Z_t^N,x, t ≥0} between its jumps.

Theorem As N → ∞,

{Z˜_t^N,x, t≥0,x≥0} ⇒ {Z_t^x, t ≥0,x≥0}

in D([0,∞);C([0,∞);IR₊)), equipped with the Skohorod topology of the space of c`adl`ag functions of x , with values in the space C([0,∞);IR+) equipped with the topology of locally uniform convergence.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 11 / 26

Height of the discrete tree

Theorem

If0< α≤1, then

sup

m≥1

H^m = +∞ a. s.

Ifα >1, then

IE

sup

m≥1

H^m

<∞.

Proof of the Theorem[Height of the discrete tree] for α > 1

Let H₁^m= inf{s ≥0;X_s^m = 1}.

Proposition

For α >1,λ= 0,∀ m≥1,IE(H₁^m) is given by IE(H₁^m) =

m

X

k=2

1 γ(k−1)^α

∞

X

n=0

µⁿ γⁿ

1

[k(k+ 1)· · ·(k+n−1)]^α−1 <∞.

Moreover we have

H^m ≤H₁^m+GH₁²+

G

X

i=1

T_i,

where G is geometric variable with parameter _λ+µ^λ andTi is exponential with mean 1/(λ+µ).

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 13 / 26

Proof of the Theorem[Height of the discrete tree] for α ≤ 1

We have X_t^α,m ≥X_t^1,m, for all m≥1, t≥0, a. s..

{X_t^m, t ≥0} is the sum of mmutually independent copies of {X_t¹, t ≥0}. The result follows from the fact that: for allt >0 IP(H¹ >t)>0 and IP(H^m<t) = (1−IP(H¹ >t))^m.

Length of the discrete tree

Time change ofX^m: A^m_t :=

Z _t

0

X_r^mdr, η_t^m = inf{s >0; A^m_s >t}. U^m :=X^m◦η^m, and S^m = inf{r >0;U_r^m= 0}. We haveL^m =S^m since S^m =RH^m

0 X_r^mdr.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 15 / 26

Length of the discrete tree

X_t^m=m+P₁ Z t

0

µX_r^mdr

−P₂ Z t

0

[λX_r^m+γ(X_r^m−1)^α]dr

,

U_t^m =m+P1(µt)−P2

Z t

0

λ+γ(U_r^m)⁻¹(U_r^m−1)^α dr

. On the interval [0,S^m),U_t^m ≥1, and consequently we have

m−P₂ Z t

0

λU_r^m+γ(U_r^m−1)^α−1 dr

≤U_t^m

≤m+P1

Z t

0

µU_r^mdr

−P2

Z t

0

hγ

2(U_r^m−1)^α−1 i

dr

.

Length of the discrete tree

X_t^m=m+P₁ Z t

0

µX_r^mdr

−P₂ Z t

0

[λX_r^m+γ(X_r^m−1)^α]dr

,

U_t^m =m+P1(µt)−P2

Z t

0

λ+γ(U_r^m)⁻¹(U_r^m−1)^α dr

.

On the interval [0,S^m),U_t^m ≥1, and consequently we have m−P₂

Z t

0

λU_r^m+γ(U_r^m−1)^α−1 dr

≤U_t^m

≤m+P1

Z t

0

µU_r^mdr

−P2

Z t

0

hγ

2(U_r^m−1)^α−1 i

dr

.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 16 / 26

Length of the discrete tree

Theorem Ifα≤2, then

sup

m≥0

L^m=∞ a. s..

Ifα >2, then

IE

sup

m≥0

L^m

<∞.

Height of the continuous tree

Consider again {Z_t^x, t≥0}solution of (2). We have Theorem

If0< α <1,0<IP(T^x =∞)<1if θ >0, while T^x <∞ a. s. if θ= 0.

Ifα= 1, T^x <∞a. s. if γ ≥θ, while 0<IP(T^x =∞)<1ifγ < θ.

Ifα >1, T^x <∞ a. s.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 18 / 26

Height of the continuous tree

Theorem

It α≤1, then T^x → ∞a. s., as x → ∞.

Ifα >1, thenIE[sup_x>0T^x]<∞.

Proof of of Theorem 5 for α > 1

We first need to establish some preliminary results on SDEs with infinite initial condition. Let f :IR+→IR be locally Lipschitz and such that

x→∞lim

|f(x)|

x^α = 0. (3)

Theorem

Let α >1,γ >0and f satisfy the assumption (3). Then there exists a minimal X ∈C((0,+∞);IR) which solves

(dX_t = [f(X_t)−γ(X_t)^α]1_{Xt≥0}dt+dW_t;

Xt → ∞, as t →0. (4)

Moreover, if T₀ := inf{t>0, X_t = 0}, thenIE[T₀]<∞.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 20 / 26

Proof of the Theorem for α > 1

The processY_t^x :=p

Z_t^x solves the SDE dY_t^x =

θ

2Y_t^x −γ

2(Y_t^x)^2α−1− 1 8Y_t^x

dt+dWt, Y₀^x =√ x.

By a well–known comparison theorem, Y_t^x ≤U_t^x, whereU_t^x solves dU_t^x =

θ

2U_t^x−γ

2(U_t^x)^2α−1

dt+dW_t, U₀^x =√ x.

The result follows from the previous Theorem.

Proof of the Theorem for α < 1

The result is equivalent to the fact that the time to reach 1, starting from x >1, goes to∞ asx → ∞. But whenZ_t^x ≥1, a comparison of SDEs for various values of α shows that it suffices to consider the caseα= 1. But in that case,Tⁿ is the maximum of the extinction times ofn mutually independent copies of Z_t¹, hence the result.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 22 / 26

Length of the continuous tree

Theorem

Ifα≤2, then S^x → ∞ a. s. as x → ∞.

Ifα >2, thenIE[sup_x>0S^x]<∞.

Proof of the Theorem for α > 2

Time change ofZ^x: At =

Z t

0

Z_s^xds, η(t) = inf{s >0, As >t}.t≥0, and U_t^x =Z^x◦η(t) The processU^x solves the SDE

dU_t^x =

θ−γ(U_t^x)^α−1

dt+ 2dWt, U₀^x =x. (5) Let τ^x := inf{t >0, U_t^x = 0}. It follows from the above that

η(τ^x) =T^x, henceS^x =τ^x. The result follows for α >2.

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 24 / 26

Proof of the Theorem for α ≤ 2

It suffice to consider the case α= 2. In that case, we have U_t^x =e^−γtx+

Z t

0

e^−γ(t−s)[θds+ 2dWs], hence

S^x = inf

t >0, Z t

0

e^γs(θds+ 2dWs)≤ −x

, which clearly goes to infinity, as x→ ∞.

Thanks!

M. Ba and E. Pardoux (Colloque JPS, CIRM Marseille)Marseille, 17Avril 2012 April 17, 2012 26 / 26