Survival and coexistence for a competition model using super-Brownian motion
Joint work with Rick Durrett and Ed Perkins
Ted Cox
The competition model is the stochastic spatial Lotka-Volterra model of Neuhauser and Pacala, Ann. Probab. (1999).
Mathematics Department Syracuse University Syracuse, NY 13244
We obtain survival and coexistence conditions via a comparison with super-Brownian motion.
4th Cornell Probability Summer School June 23 – July 3, 2008
Outline Part II 1
the Lotka-Volterra model of NP(1999)
2
Lotka-Volterra ⇒ super-Brownian motion
3
survival and coexistence
The voter model
Introduced independently by Clifford and Sudbury (1973) Holley and Liggett (1975).
Part I 1
2
3
the voter model (construction, coalescing random walks, duality, martingales) super-Brownian motion (branching random walks, martingale problem) voter models ⇒ super-Brownian motion
Model of neutral competition between types.
Voter model dynamics
Graphical construction I Or,
There is a voter at each site x of Zd holding one of two possible opinions, 0 or 1. Probability kernel p(x), x ∈ Zd symmetric, irreducible, finite range, p(0) = 0, covariance matrix σ 2 I , for example 1 if |x| = 1 p(x) = 2d
Tnx,y , n ≥ 1, x, y ∈ Zd are the “voting times” At each time Tnx,y draw an arrow from y to x the voter at x adopts the opinion of the voter at y .
Each voter has an independent mean one exponential alarm clock. When the clock at x goes off, the voter there adopts the opinion of site y with probability p(y − x), and the clock is reset.
Graphical construction II
The Tnx,y are the “arrival times” of Λx,y , which are independent, rate p(y − x) Poisson processes. ξt (x) = the opinion at x at time t
Duality t
Zd
0
1
0
1
1
0
1
0
0
1
1?
1?
1?
0?
0
1
1
0
t
1
0
Duality calculation I Duality Equation
c x ) = 1) P(ξt (x) = 1) = P(ξ0 (W t X = P(Wtx = y )ξ0 (y )
P(ξt (x) = 1 ∀ x ∈ A, ξt (y ) = 0 ∀ y ∈ B) c x ) = 1 ∀ x ∈ A, ξ0 (W cty ) = 0 ∀ y ∈ B) = P(ξ0 (W t
2
=
pt (x, y )ξ0 (y )
y
c x , x ∈ Zd Coalescing random walk system W t 1
y
X
rate one random walks on Zd with jump kernel p(x) and cx = x W 0
where pt (x, y ) = e
walks are independent until they meet, at which time they coalesce and move together
−t
∞ n (n) X t p (x, y ) n=0
n!
If ξ0 (x) are iid Bernoulli(θ), then P(ξt (x) = 1) = θ.
Duality Calculation II
Graphical construction III
P(ξt (x) = 1, ξt (y ) = 0) =
cx ) P(ξ0 (W t
=
cty ) 1, ξ0 (W
= 0)
= P(ξ0 (Wtx ) = 1, ξ0 (Wty ) = 0, τ (x, y ) > t)
ξt (x) = ξ0 (x) +
= ξ0 (x) +
Z tX 0
→ 0 as t → ∞ if d = 1, 2 where τ (x, y ) = inf{s ≥ 0 : Wsx = Wsy } If ξ0 (x) are iid Bernoulli(θ), then P(ξt (x) = 1, ξt (y ) = 0) → θ(1 − θ)P(τ (x, y ) = ∞)
= ξ0 (x) +
y
Z tX 0
h ξs− (y ) − ξs− (x) p(y − x)ds
y
i +(Λx,y (ds) − p(y − x)ds)
Z tX 0
ξs− (y ) − ξs− (x) Λx,y (ds)
y
ξs (y ) − ξs (x) p(y − x) ds + Mtx
where Mtx is a martingale with square function Rt P hM x it = 0 y 1{ξs (y ) 6= ξs (x)}p(y − x) ds
“Measure-valued” voter model, |ξ0 | < ∞ Define Xt =
P
x ξt (x)δx ,
Using ξt (x) = ξ0 (x) +
Xt (φ) = X0 (φ) +
= X0 (φ) +
hM(φ)it =
0
y
0
P x
φ(x)ξt (x).
ξs (y ) − ξs (x) p(y − x) ds + Mtx
(p − I )φ(x)ξs (x) ds + Mt (φ)
x
Z tX
Z tX 0
Z tX
Z tX 0
Xt (φ) =
If φ = 1, then Xt (1) = X0 (φ) +
Z tX 0
Xs ((p − I )1) ds + Mt (φ)
x
= X0 (φ) + Mt (1) so Xt (1) = |ξt | is a martingale.
Xs ((p − I )φ) ds + Mt (φ)
x
φ2 (x)1{ξs (y ) 6= ξs (x)}p(y − x) ds
x,y
Super-Brownian Motion Xt , t ≥ 0 A specific model: Introduced by Watanabe (1968) and Dawson (1977). Measure-valued (finite measures on Rd ) process. Diffusion limit of branching random walk systems. particles die and give birth to other particles particles move about on Zd multiple particles per site allowed Many variations . . .
particles at a given site x die at rate 1 give birth at rate 1 to a particle at site y with probability p(y − x)
ηt (x) = the number of particles at x at time t P |ηt | = x ηt (x) is a critical branching process
Our particles don’t actually walk.
Rescaled branching random walks Scale space:
√ Zd / N
√
pN (x) = p(x N),
Super-Brownian motion martingale problem √ x ∈ Zd / N
µ(φ) =
R
SBM(X0 , σ 2 , b)
Scale time:
√ ηtN = rate N branching random walk on Zd / N, kernel pN
Scale mass: mN = N, XtN
1 = mN
X √ x∈Zd / N
φ(x)µ(dx). Z
Xt (φ) = X0 (φ) + where Mt (φ) is a continuous
t
0
2
Xs ( σ2 ∆φ) ds + Mt (φ)
L2 -martingale, Z
hM(φ)it = b
ηtN (x) δx
t
0
Xs (φ2 ) ds
σ 2 = diffusion rate
Theorem Assume X0N (1) is bounded and X0N → X0 . Then X·N ⇒ X· as N → ∞, where X· is SBM(X0 , σ 2 , 2).
Voter model ξt as branching random walk? ξ(x) = 1 ⇔ particle at x ξ(x) = 0 ⇔ no particle at x
b = branching rate Xt (1) is a martingale.
Voter model ≈ BRW ⇒ SBM? Scale space:
√ Zd / N
Scale mass: mN =
p(y − x)1{ξt (y ) = 0} XtN =
particle at x dies at rate X p(y − x)1{ξt (y ) = 0} y
BRW behavior (?) provided the 1’s are relatively isolated.
√ x ∈ Zd / N
√ Scale time: ξtN = rate N voter model on Zd / N, kernel pN (
particle at x gives birth to a particle at y at rate
√ pN (x) = p(x N),
1 mN
X √
N N/ log N
if d ≥ 3 if d = 2
ξtN (x) δx
x∈Zd / N
Theorem Assume d ≥ 2, X0N (1) is bounded and X0N → X0 . Then X·N ⇒ X· as N → ∞ where X· is SBM(X0 , σ 2 , 2γe ).
Strategy of proof, d ≥ 3
Theorem Assume d ≥ 2, X0N (1) is bounded and X0N → X0 . Then X·N ⇒ X· as N → ∞ where X· is SBM(X0 , σ 2 , 2γe ).
∞ d SBM Xt is unique solution Z t of: ∀ φ ∈ C0 (R ), 2 Xt (φ) = X0 (φ) + Xs ( σ2 ∆φ) ds + Mt (φ) Z t 0 hM(φ)it = b Xs (φ2 ) ds
(P γe =
p(x)P(τ (0, x) = ∞) if d ≥ 3 2πσ 2 if d = 2 x
Reduction in branching rate compared to BRW (d ≥ 3). A low density limit theorem since |ξtN | = O(N), so # of 1’s O(N) ≈ d/2 → 0 sites/volume N
We know
(d ≥ 3)
0
The rescaled, measure-valued voter models XtN (φ) satisfy a “similar” martingale problem. We must show laws of X·N are tight all subsequential limits satisfy SBM martingale problem
But why γe ?
1. (Unscaled) voter model: Z tX Xt (φ) = X0 (φ) + Xs ((p − I )φ) ds + Mt (φ) 0
2. Rescaled voter models: XtN (φ)
=
X0N (φ)
Since N(pN − I )φ ≈ XtN (φ)
≈
Z
+
x
t
0
0
XsN (N(pN
− I )φ) ds +
MtN (φ)
σ2 2 ∆φ,
X0N (φ)
Z +
0
t
1. (Uunscaled) voter model: Z tX hM(φ)it = φ2 (x)p(y − x)1{ξs (y ) 6= ξs (x)} ds
2 XsN ( σ2 ∆φ) ds
+
MtN (φ)
x,y
2. Rescaled voter models: Z 1 tX 2 N hM (φ)it = φ (x)pN (y − x)1{ξsN (y ) 6= ξsN (x)} ds N 0 x,y Z t ≈ 2γe XsN (φ2 ) ds?? γe =
0
P x
p(x)P(τ (0, x) = ∞)
E hM N (φ)it ≈ 2γe E
Rt 0
XsN (φ2 )ds? εN → 0 and NεN → ∞
1 E hM (φ)it = N N
2 ≈ N
Z tX 0
2
x)E 1{ξsN (y )
2
x)E ξsN (x)1{ξsN (y )
φ (x)pN (y −
x,y
Z tX 0
φ (x)pN (y −
x,y
6=
ξsN (x)} ds = 0} ds
Z 2 tX 2 φ (x)pN (y − x)E ξsN (x)P(τ (x, y ) = ∞) ds?? ≈ N 0 x,y Z 2γe t X 2 = φ (x)E ξsN (x) ds N 0 x Z t = 2γe XsN (φ2 ) ds 0
P(τ N (x, y ) > εN ) ≈ P(τ (x, y ) = ∞) x ≈ x 0 and y ≈ y 0 but P(τ (x 0 , y 0 ) < ∞) ≈ 0
P(ξsN (x) = 1, ξsN (y ) = 0) / ξ0N , τN (x, y ) > εN ) = P(Wsx ∈ ξ0N , Wsy ∈ 0
≈ P(τN (x, y ) > εN ) · P(Wsx ∈ ξ0N ) · P(Wsy ∈ / ξ0N ) 0
≈ P(τ (x, y ) = ∞) · P(Wsx ∈ ξ0N ) · 1 = P(τ (x, y ) = ∞) · P(ξsN (x) = 1)
Rescaled branching random walks II
√ √ √ Scale space: Zd / N, pN (x) = p(x N), x ∈ Zd / N
σ2 Xt (φ) = X0 (φ) + 2
Scale time: a particle at x dies at rate N+d gives birth at rate (N+b)pN (y − x) to a particle at y Scale mass: mN = N, XtN =
1 mN
Super-Brownian Motion SBM(X0 , σ 2 , b, g )
X √ x∈Zd / N
Z 0
t
Z Xs (∆φ) ds + Mt (φ) + g
where Mt (φ) is a continuous L2 -mg with hM(φ)it = b σ 2 = diffusion rate
ηtN (x) δx
Theorem Assume X0N (1) is bounded and X0N → X0 . Then X·N ⇒ X· as N → ∞, where X· is SBM(X0 , σ 2 , 2, g ), where g = b − d.
0
b = branching rate g = growth rate g > 0 implies P( lim Xt (1) = ∞) > 0. t→∞
t
Xs (φ) ds
Rt 0
Xs (φ2 ) ds
The NP stochastic spatial Lotka-Volterra model
Neuhauser and Pacala (1999), a model for competition between two species each site of Zd occupied by a single individual
Ingredients nonnegative mortality constants α0 , α1 symmetric, irreducible probability kernel p(x) on Zd , p(0) = 0, covariance matrix σ 2 I . nearest neighbor case: p(x) =
types are 0 and 1
1 if |x| = 1, x ∈ Zd 2d
local frequencies of type i for site x in configuration ξ, X p(y − x)1{ξ(y ) = i} i = 0, 1 fi (x, ξ) =
individuals die and get replaced
y
ξt (x) = the type of individual at site x at time t
Fecundity parameter λ = 1 (for now).
Dynamics for LV process ξt
LV rate function
At site x at time t with fi = fi (x, ξt− ), 0→1
at rate
f1 · (f0 + α0 f1 )
1→0
at rate
f0 · (f1 + α1 f0 )
Interpretation: if ξt− (x) = 0, then death at rate (f0 + α0 f1 ) replacement by an individual of type 1 with probability f1
P ξt+h (x) 6= ξt (x) | Ft = c(x, ξt )h + o(h) as h ↓ 0
c(x, ξ) =
f1 · (f0 + α0 f1 )
if ξ(x) = 0
f0 · (f1 + α1 f0 )
if ξ(x) = 1
fi = fi (x, ξ) =
P y
p(x, y )1{ξ(y ) = i}
i = 0, 1
Graphical construction: Yes
Death incorporates interspecific competition rate:
αi
intraspecific competition rate:
1
Explicit calculations: difficult because no simple dual process, except . . .
(α0 , α1 ) = (1, 1) is the voter model fi = fi (x, ξ) =
P
y p(x, y )1{ξ(y ) = i}
The LV rate function is ( f1 · (f0 + α0 f1 ) c(x, ξ) = f0 · (f1 + α1 f0 )
i = 0, 1
if ξ(x) = 0 if ξ(x) = 1
Set α0 = α1 = 1, use f0 + f1 = 1, get voter model rate function ( f1 if ξ(x) = 0 c(x, ξ) = f0 if ξ(x) = 1 Our focus: The behavior of the LV process for (α0 , α1 ) ≈ (1, 1)
Answers for the voter model (α0 , α1 ) = (1, 1)
Survival and coexistence Let |ξ| =
X
ξ(x), the number of 1’s in ξ.
x
Survival (of 1’s): For |ξ0 | = 1, P(|ξt | > 0 ∀ t ≥ 0) > 0 Coexistence: ∃ stationary distribution µ with µ infinitely many individuals of each type = 1 Goal: Given p(x), determine what parameter values α0 , α1 correspond to survival and/or coexistence.
LV = voter model + perturbation? 0 → 1 at rate f1 (f0 + α0 f1 +f1 − f1 ) = f1 (1 + (α0 − 1)f1 ) = f1 + (α0 − 1)f12
For all d ≥ 1: no survival |ξt | is a martingale. For d ≤ 2: no coexistence.
So the LV rate function is ( f1 + (α0 − 1)f12 c(x, ξ) = f0 + (α1 − 1)f02
if ξ(x) = 0 if ξ(x) = 1
For d ≥ 3: coexistence . For (α0 , α1 ) near (1, 1), LV = voter model + small perturbation
Strategy for proving survival/coexistence
Prove analogue of VM ⇒ SBM for LV process for (α0 , α1 ) ≈ (1, 1) and identify limiting parameters.
Scaled measure-valued L-V processes XtN , N = 1, 2, . . . Scale space/time/mass as before. √ space Zd / N ( N if d ≥ 3 mass mN = N/ log N if d = 2 time
When g > 0 (so limiting SBM survives) argue that the approximating LV processes must survive. If both the 1’s and 0’s survive there should be coexistence.
Put αi = 1 + 0→1
at rate
1→0
at rate
θi N , ξ has rates N t N f1 + (α0N − 1)f12 = Nf1 + θ0 f12 N f0 + (α1N − 1)f02 = Nf0 + θ1 f02
Measure-valued LV process
XtN =
1 mN
X √ x∈Zd / N
ξtN (x) δx
Convergence Theorem (CP) Assume d ≥ 2 , X0N (1) is bounded, αiN = 1 + Then
θi and X0N → X0 . mN
X·N ⇒ X·
where X· is SBM(X0 , 2γe , σ 2 /2, g ), with ( γ0 θ0 − γ1 θ1 if d ≥ 3 g= γ ∗ (θ0 − θ1 ) if d = 2
p(e)p(e 0 )P τ (0, e) = τ (0, e 0 ) = ∞ X γ0 = p(e)p(e 0 )P τ (0, e) = τ (0, e 0 ) = ∞, τ (e, e 0 ) < ∞
γ1 =
X
γ0 < γ1 ∗
γ = 2πσ
2
Z 0
∞
X
p(e)p(e 0 )a(y − x)
x,y ,e,e 0
P τ (0, e) ∧ τ (0, e 0 ) > τ (e, e 0 ) ∈ ds, Ws0 = x, Wse = y γ ∗ , γ0 , γ1 are coalescing random walk quantities
d ≥ 3 survival parameter values?
αiN = 1 +
d ≥ 3 Lotka-Volterra survival
θi mN
α1
6
slope m0
Limiting SBM growth rate is g = γ0 θ0 − γ1 θ1 1
Let m0 = γ0 /γ1 < 1, g > 0 iff θ1 < m0 θ0
........................ .................... ................. .............. ............ ......... ........ ....... ...... ..... ..... . . . . ... ... ... ... ... .. ..
slope p∗
Survival
Replacing θi with αi − 1 suggests survival for (α0 , α1 ) near (1, 1) below the line α1 − 1 < m0 (α0 − 1)
slope 1/p∗ α0
1 γ0 0. given ε > 0 can choose large K , L, M, T such that
Based on Bramson/Durrett method.
e0 (I0 ) ≥ M ⇒ P(X eT (I−1 ) ∧ X eT (I1 ) ≥ M) > 1 − ε X
On large scales, can compare LV with supercritical oriented percolation. Difficulties with method involve our inability to compute second moments.
I−1 ˜I0
I1
I0
I0 = [−L, L] × R d−1 , Im = I0 + 2me1 eI0 = K I0 et is Xt restricted to eI × [0, T ]. X
2. Lotka-Volterra survival (finite time)
3. Lotka-Volterra survival (infinite time)
Lotka-Volterra models XtN with parameters αiN and limiting growth rates gN → g = γ0 θ0 − γ1 θ1 > 0 for all large N, e N (I N ) ≥ M ⇒ P(X e N (I N ) ∧ X e N (I N ) ≥ M) > 1 − ε X 0 0 NT −1 NT 1 N I−1
˜I N 0
I1N
I0N
√ √ I0N = √NI0 , ImN = I0N + 2m Ne1 eI N = NeI0 0 e N is X N restricted to eI N × [0, NT ]. X t
t
NT 0
T 0
The survival curve, d ≥ 3
Extinction?
Define the survival curve α1 = h(α0 ) by h(α0 ) = sup{α1 : survival for (α0 , α1 )}
Is the slope m0 correct? Is h0 (1) = m0 ? Need to supplement survival result with extinction result.
Assume (α0 , α1 ) ≈ (1, 1), above our line, so g < 0. α1
The SBM approximation g < 0 does not seem to be enough to obtain extinction, ξt = 0 eventually.
6
Need to use hydrodynamical approach, with fast voter instead of fast stirring.
h(α0)
... .... .... .... .... ..... ..... ...... ..... ...... ...... ...... ...... ...... . . . . . . ..... ....... ....... ....... ......... ......... ......... ........ ......... ......... ........ ........ ....... ....... ....... ....... . . . . . ... ....... ...... ...... ..... ..... .... ...
Extinction
1
Survival
α0
1
-
h0 (1) ?
We know h0 (1−) ≤ m0 < 1 and h0 (1+) ≥ m0
Theorem (CDP) For d ≥ 3,
h0 (1−) = m0
1
h0 (1+) = 1 1
2
3
α1
6
h(α0)
1
.. .... .... .... . . . . ... .... .... ..... . . . . . . . . . . . ............. ............. ................... Survival
1
4
5
α0
-
Rescaled voter models converge to super-Brownian motion (Cox, Durrett Perkins). Ann. Probab. 28 (2000), 185-234. Super-Brownian limits of voter model clusters (Bramson, Cox and LeGall). Ann. Probab. 29 (2001), 1001-1032. Rescaled Lotka-Volterra models converge to super-Brownian motion (Cox and Perkins). Ann. Probab. 33 (2005), 904-947. Survival and coexistence in stochastic spatial Lotka-Volterra (Cox and Perkins). Prob. Theory Rel. Fields. 139 (2007), 89-142. Renormalization of the two-dimensional Lotka-Volterra model (Cox and Perkins). Ann. Appl. Probab. 18 (2008), 747-812.
