Quasi-stationary distributions and diffusion models in population - PowerPoint PPT Presentation

Quasi-stationary distributions and diffusion models in population dynamics P. Cattiaux, P. Collet, A. Lambert, S. Martinez, S. Méléard, J. San Martin, Ecole Polytechnique, Universidad de Chile, Université Paris 6, Université Toulouse III Journées de Probabilités, 11 septembre 2007

Quasi-stationary Distributions The aim : To study the asymptotic behavior of the size ( Z t ) t of some isolated biological population. ◮ No immigration ◮ Competition for limited resources implies extinction after some finite time T 0 . ◮ Z t ≥ 0 , ∀ t and 0 is an absorbing point. ◮ The population size fluctuates for large amounts of time before extinction : captured by the notion of quasi-stationarity. References : Pollett, Seneta ;Vere-Jones, Van Doorn, Ferrari ;Kesten ; Martinez ;Picco, Collet ;Martinez ;San Martin, Gosselin, Steinsaltz ;Evans, Lambert

Quasi-stationary distribution Definition : ν quasi-stationary distribution (QSD) if ∀ A ⊂ ( 0 , + ∞ ) , ∀ t , P ν ( Z t ∈ A | T 0 > t ) = ν ( A ) . Example : If ∃ µ probability measure on R ∗ + such that t → + ∞ P x ( Z t ∈ A | T 0 > t ) = µ ( A ) , lim x fixed population size , then µ is a QSD. µ is called Yaglom limit. Definition : Q -process : law of the process conditioned to be never extinct. ∀ B s ∈ F s , Q x ( B s ) = lim t → + ∞ P x ( Z ∈ B s | T 0 > t ) . Question : Long time behavior of this process ?

Population Dynamics Our aim : To study QSD and Q -processes for population dynamics obtained as scaling limits of generalized birth-death processes. ◮ Birth-death process ( Z N t ) t with absorbing state 0. ◮ renormalized by weights 1 N : values in 1 N N . ◮ birth rates b N ( z ) , b N ( 0 ) = 0. ◮ death rates d N ( z ) , d N ( 0 ) = 0.

Convergence Theorem Assumptions : ◮ ∃ γ ≥ 0 and a smooth funtion h with h ( 0 ) = 0, such that b N ( z ) − d N ( z ) → h ( z ) ; b N ( z ) + d N ( z ) → γ z . N N 2 ◮ The function h is the limiting growth rate. ◮ γ is a demographic parameter describing the ecological timescale. ◮ ( Z N 0 ) N converges as N → ∞ : population size of order N . ◮ Then, (Lipow or Joffe-Métivier), ( Z N t , t ≥ 0 ) ⇒ ( Z t , t ≥ 0 ) .

The limiting process ◮ If γ = 0, dynamical system dZ t = h ( Z t ) dt . ◮ 0 (unstable) equilibrium ( h ( 0 ) = 0), ◮ Existence of a non-trivial stable equilibrium. ◮ If γ > 0, � dZ t = γ Z t dB t + h ( Z t ) dt . ◮ Acceleration of ecological process ⇒ noise (demographic stochasticity). h ( z ) ◮ The function is the mean individual growth rate. z ◮ If the growth function h ≡ 0, Z is a Feller diffusion. ◮ The process Z is called a generalized Feller diffusion.

Examples ◮ h ( z ) = rz : continuous state branching process. ◮ h ( z ) = rz − cz 2 : logistic branching process (Lambert). � � ◮ h ( z ) = ( rz − cz 2 ) z K 0 − 1 : Allee effect, the individual growth rate h ( z ) increases, then decreases. (Cooperation, z then competition). Continuous state branching process (Lambert) : ◮ subcritical case r < 0 : infinite number of QSD ◮ critical case r = 0 : no QSD ◮ supercritical case r > 0 : no sense but L ( Z | extinction ) ∼ CSBP ( − r ) .

More generally, if there exists h ∞ = lim z →∞ h ( z ) ∈ [ −∞ , + ∞ ] , then h ∞ determines the long time behavior of the process. 3 cases : ◮ If h ∞ = −∞ , subcritical case, process a.s. absorbed at 0 in finite time. Existence of a Yaglom limit. ◮ If h ∞ ∈ ( −∞ , + ∞ ) , critical case. Nothing known concerning QSD. Proposition : If lim z → + ∞ h ( z ) = + ∞ , then the ◮ generalized Feller diffusion Z conditioned on eventual extinction satisfies � u ′ ( Y t ) � � dY t = γ Y t dB t + h ( Y t ) + γ Y t dt , u ( Y t ) where u ( y ) = P y ( lim t Z t = 0 ) and h ( y ) + γ y u ′ ( y ) u ( y ) ∼ y →∞ − h ( y ) .

Existence and uniqueness of a QSD in the subcritical case Theorem : Assume h ( z ) lim √ z = −∞ (strong competition in large population) , z →∞ zh ′ ( z ) lim h ( z ) 2 = 0 (technical assumption, fulfilled for most z →∞ classical biological models) .

Then there exists a probability measure ν such that ◮ For each initial law with bounded support (in particular for each Dirac measure), L ( Z t | T 0 > t ) ⇒ ν, exponentially fast . (1) ⇒ existence of a QSD (Yaglom limit) . ◮ The Q -process is well defined and converges, when t → + ∞ , to a measure absolutely continuous w.r.t. ν . ◮ If � ∞ 1 − h ( z ) dz < ∞ , 1 then Z comes down from infinity and (1) holds for all initial law : ⇒ uniqueness of the QSD .

The associated Kolmogorov equation � Z t We introduce X t = 2 γ . Then � γ X 2 1 2 � t dX t = dB t − dt + h dt . 2 X t 4 γ X t Example : h ( z ) = rz − cz 2 ,then � rX t 2 − c γ X 3 � 1 t dX t = dB t − dt + dt . 2 X t 8 The diffusion has the form : dX t = dB t − q ( X t ) dt , 1 where q ( x ) ∼ x → 0 2 x and q ( x ) �→ x → + ∞ + ∞ .

Study of QSD for the diffusion dX t = dB t − q ( X t ) dt . Mandl (1961), Collet, Martinez, San Martin (1995) and Evans, Steinsaltz (2007) under Mandl’s conditions : q is C 1 up to 0 and doesn’t grow to fast to infinity at ∞ . Here ◮ q ∈ C 1 (] 0 , + ∞ [) . ◮ Define T y : first time the process hits y . ◮ Explosion time : τ = T 0 ∧ T ∞ . Assumption (H1) : ∀ x , P x ( τ = T 0 < + ∞ ) = 1 . Remark : (H1) satisfied if X comes as previously from a generalized Feller diffusion.

Reference measure : � x For Q ( x ) = 1 2 q ( u ) du , let us define on ( 0 , + ∞ ) the measure µ ( dx ) = e − Q ( x ) dx . Remark : µ is not necessarily bounded : Q ( x ) ∼ x → 0 ln x if q ( x ) ∼ x → 0 − 1 2 x . Nevertheless L 2 ( µ ) is the natural space to work with. Theorem : (By Girsanov’s theorem) � ∞ P t g ( x ) = E ( g ( X t ) 1 t < T 0 ) = g ( y ) r ( t , x , y ) µ ( dy ) 0 and if ∃ C > 0 such that ∀ y > 0, q 2 ( y ) − q ′ ( y ) ≥ − C , then r ∈ L 2 ( µ ) and � ∞ 1 r 2 ( t , x , y ) µ ( dy ) ≤ e Ct e Q ( x ) . √ 2 π t 0

Define : � ∞ < f , g > µ = f ( y ) g ( y ) µ ( dy ) . 0 Define the symmetric form defined for f , g ∈ C ∞ c (( 0 , + ∞ )) by E ( f , g ) = < f ′ , g ′ > µ , Dirichlet forms theory (Fukushima) : ◮ P t extends to a symmetric sub-Markovian semi-group of contractions on L 2 ( µ ) . ◮ Its generator L is non-positive self-adjoint on L 2 ( µ ) with 2 f ′′ − qf ′ . c (( 0 , + ∞ )) , Lf = 1 domain D ( L ) and for f ∈ C ∞

Spectral theory Assumption (H2) : (i) ∃ C > 0 such that ∀ y > 0, q 2 ( y ) − q ′ ( y ) ≥ − C . (ii) lim y → + ∞ q 2 ( y ) − q ′ ( y ) = + ∞ . Spectral Theory in L 2 ( µ ) : Assume (H1) and (H2), then ◮ ( − L ) has a purely discontinuous spectrum 0 < λ 1 < . . . < λ n . . . , each λ k is simple. ◮ ( η k ) BON of eigenfunctions, and η 1 ( x ) > 0, ∀ x > 0. k e − λ k t < η k , f > µ η k . ◮ For f ∈ L 2 ( µ ) , P t f = L 2 � ◮ < P t f , g > µ ∼ t →∞ e − λ 1 t < η 1 , f > µ < η 1 , g > µ .

Proof : We transform the spectral theory for a Fokker-Planck operator in a spectral theory for a Schrödinger operator. For f ∈ L 2 ( dx ) , we set ˜ P t f ( x ) = e − Q / 2 P t ( f e Q / 2 ) . ˜ P t is a strongly continuous semi-group on L 2 ( dx ) with generator L = 1 2 ∆ − 1 2 ( q 2 − q ′ ) , ˜ on C ∞ c (( 0 , + ∞ )) . Rem : The potential q 2 − q ′ is not in L ∞ loc near 0. Adaptation of Berezin-Shubin.

The Yaglom limit For f ∈ L 2 ( µ ) , e − λ k t < η k , f > µ η k . � P t f = L 2 k Heuristically, e − λ 1 t < η 1 , 1 A > µ η 1 , P t 1 A ∼ t →∞ e − λ 1 t < η 1 , 1 > µ η 1 . P t 1 R ∗ ∼ t →∞ + Then if x > 0, P t 1 A ( x ) < η 1 , 1 A > µ + ( x ) �→ t →∞ . P t 1 R ∗ < η 1 , 1 > µ η 1 d µ A good candidate to be the Yaglom limit is ν 1 = <η 1 , 1 > µ , if η 1 ∈ L 1 ( µ ) .

Theorem : Assume (H1), (H2), and η 1 ∈ L 1 ( µ ) , and consider η 1 d µ ν 1 = <η 1 , 1 > µ . Then ◮ P x ( T 0 > t ) ∼ t →∞ e − λ 1 t η 1 ( x ) < η 1 , 1 > µ . ◮ ν 1 is a Yaglom limit : ∀ x > 0 , lim t →∞ P x ( X t ∈ A | T 0 > t ) = ν 1 ( A ) . ◮ ν 1 is a QSD and P ν 1 ( T 0 > t ) = e − λ 1 t . ◮ Speed of convergence : If moreover η 2 ∈ L 1 ( µ ) , t →∞ e ( λ 2 − λ 1 ) t ( P x ( X t ∈ A | T 0 > t ) − ν 1 ( A )) < + ∞ . lim

The Q -process Theorem : 1) Assume B s ∈ F s . Then for all x > 0, t → + ∞ P x ( X ∈ B s | T 0 > t ) = Q x ( B s ) . lim Q x has the transition probability densities (w.r.t. Lebesgue measure) q ( s , x , y ) = e λ 1 s η 1 ( y ) η 1 ( x ) r ( s , x , y ) e − Q ( y ) . 2) For any Borel set A , � η 2 s → + ∞ Q x ( ω s ∈ A ) = lim 1 ( y ) µ ( dy ) . A The stationary measure of the Q -process is absolutely continuous w.r.t. ν 1 .