Gabriela Marinoschi Institute of Mathematical Statistics and...

A stochastic population dynamics equation

Gabriela MarinoschiInstitute of Mathematical Statistics and Applied Mathematics of the

Romanian Academy, Bucharest

Gabriela Marinoschi, ISMMA () Atelier de travail, 13-14.09.2017, Bucarest 1 / 42

Problem presentation

dp(t, a, x) + (pa(t, a, x)− ∆p(t, a, x) + µS (t, a, x ;U(p))p(t, a, x))dt

= p(t, a, x)dW (t, a, x) in (0,T )× (0, a+)×O

−∇p(t, a, x) · ν = α0(t, a, x)p(t, a, x)+ k0(t, a, x) on (0,T )× (0, a+)× ∂O

p(t, 0, x) =∫ a+

0m0(a, x ;U(p))p(t, a, x)da in (0,T )×O

p(0, a, x) = p0(a, x) in (0, a+)×O

p(t, a, x) is the population density




= p(t, a, x)dW (t, a, x) in (0,T )× (0, a+)×O


p(t, 0, x) =∫ a+


p(0, a, x) = p0(a, x) in (0, a+)×O

µS (t, a, x ;U(p)) is the incidental mortality rate

U(p) =∫ a+

0

∫OU

γ(a, x)p(t, a, x)dxda




= p(t, a, x)dW (t, a, x) in (0,T )× (0, a+)×O


p(t, 0, x) =∫ a+


p(0, a, x) = p0(a, x) in (0, a+)×O

m0(a, x ;U(p)) is the fertility rate

U(p) =∫ a+

0

∫OU

γ(a, x)p(t, a, x)dxda



µS (t, a, x ; r), m0(a, x ; r) are positive bounded, local Lipschitz on R

∀R > 0, there exists Lµ(R) and Lm(R) such that, if |r | ≤ R, |r | ≤ R,

|µS (t, a, x ; r)− µS (t, a, x ; r)| ≤ Lµ(R) |r − r ||m0(a, x ; r)−m0(a, x ; r)| ≤ Lm(R) |r − r |

Existence, uniqueness and properties of the solution to thedeterministic nonlinear population dynamics modelM. Iannelli, F. Milner, Springer, 2017C. Cusulin, M. Iannelli, G. M., Nonlinear Anal. Real World Appl. 6(2005)C. Cusulin, M. Iannelli, G. M. Nonlinear Anal. Real World Appl. 8(2007)



A deterministic model cannot reproduce or explain the effects ofrandom fluctuations which come from the intrinsic stochastic natureof open systems.

Such effects may be induced by the interplay between nonlinearinteractions specific for natural systems and random fluctuationsgenerated by the the environment.

Demographic events are basically represented by statistical averagesand a pure stochastic contribution should be taken into account inthe equation.




= p(t, a, x)dW (t, a, x) in (0,T )× (0, a+)×O

Let (Ω,F ,P) be a probability space with the natural filtration Ft .Let W be a stochastic Gaussian process

W (t, x) =N

∑j=1

µj (a, x)βj (t)

βjNj=1 is an independent system of real-valued Brownian motions,

µj ∈ C 2([0, a+]×O), ∇µj · ν = 0 on (0, a+)× ∂O.




= p(t, a, x)dW (t, a, x) in (0,T )× (0, a+)×O

Let (Ω,F ,P) be a probability space with the natural filtration Ft .Let W be a stochastic Gaussian process

W (t, a, x) =N

∑j=1

µj (a, x)βj (t)

βjNj=1 is an independent system of real-valued Brownian motions,

µj ∈ C 2([0, a+]×O), ∇µj · ν = 0 on (0, a+)× ∂O.



Stochastic equations

dp(t) + B(t)p(t)dt + F (t)p(t)dt = p(t)dW (t)

p(0) = p0

If B is time independent and F is Lipschitz ⇐= G. Da Prato, J.Zabczyk, Cambridge University Press 2014, for Lipschitz nonlinearitiesfor the stochastic equations with multiplicative noise

If B is time independent, Fv = µS (U(v)) is locally Lipschitz ⇐= G.Da Prato, J. Zabczyk, 2014, Section 7.2 for locally Lipschitznonlinearities for the stochastic equations with an additive noise

If B is time dependent nonlinear monotone, demicontinuous andcoercive between two dual spaces ⇐= V. Barbu, M. Röckner, J. Eur.Math. Soc. 17 (2015)


Hypotheses

k0, and p0 are random functions

p0(·, a, x) is measurable with respect to F0, a.a. (a, x)

k0(·, t, a, x) is Ft -adapted, a.a. (t, a, x)


Notation

H = L2(O), V = H1(O), V ′ = (H1(O))′

H = L2(0, a+;H), V = L2(0, a+;V ), V ′ = L2(0, a+;V ′)


DefinitionA solution p is an Ft -adapted process,

p ∈ C ([0,T ];H) ∩ L2(0,T ;V) ∩ C ([0, a+]; L2(0,T ;H)), P-a.s.,

(p(t),ψ)H

+∫ t

0

∫Op(τ, a+, x)ψ(a+, x)dxdτ −

∫ t

0

∫ a+

0

∫Opψadxdadτ

−∫ t

0

∫ a+

0

∫Om0(a, x ;U(p))pψ(0, x)dxdadτ

+∫ t

0

∫ a+

0

∫O(∇p · ∇ψ+ µS (τ, a, x ;U(p))pψ)dxdadτ

+∫ t

0

∫ a+

0

∫O(α0p + k0)ψdσdadτ

= (p0,ψ)H +∫ t

0(p(τ)dW (τ),ψ)H , for all ψ ∈ V , with ψa ∈ V ′.


Presentation outline

1 Rescalling transformation (V. Barbu, M. Röckner, 2015)

p(t, a, x) = eW (t ,a,x )y(t, a, x)

system in y and intermediate results

2 Proof that the random system in y has a unique solution for allω ∈ Ω

3 Proof that the stochastic system in p has a unique solution.




p(t, a, x) = eW (t ,a,x )y(t, a, x)

system in y and intermediate results2 Proof that the random system in y has a unique solution for all

ω ∈ Ω

3 Proof that the stochastic system in p has a unique solution.




p(t, a, x) = eW (t ,a,x )y(t, a, x)

system in y and intermediate results2 Proof that the random system in y has a unique solution for all

ω ∈ Ω3 Proof that the stochastic system in p has a unique solution.


1. Rescalling transformation

yt + ya − ∆y + g1y + g2 · ∇y + µS (t, a, x ;U(eW y))y = 0 (y)

in (0,T )× (0, a+)×O

−∇y · ν = α(t, a, x)y + k(t, a, x) in (0,T )× (0, a+)× ∂O

y(t, 0, x) =∫ a+

0m(t, a, x ;U(eW y))y(t, a, x)da in (0,T )×O

y(0, a, x) = y0(a, x) = p0(a, x) in (0, a+)×Owhere

g1 = Wa − ∆W − |∇W |2 + µ, g2 = −2∇W , µ =12

N

∑j=1

µ2j (a, x)

α = α0, k = k0e−W , m(t, a, x) = m0(a, x ;U(eW y))eW (t ,a,x )−W (t ,0,x )


1. Intermediate results

Yt + Ya − ∆Y + f1Y + f2 · ∇Y + E1(t, a, x ;Y ) = f (Y )

in (0,T )× (0, a+)×O

−∇Y · ν = YfΓ + f 0Γ in (0,T )× (0, a+)× ∂O

Y (t, 0, x) =∫ a+

0E2(t, a, x ;Y )da in (0,T )×O

Y (0, a, x) = Y0(a, x) in (0, a+)×O

Ei (t, a, x ; ·) : (0,T )× (0, a+)×O ×H → H Lipschitz on H.



LemmaSystem (Y ) has a unique solution

Y ∈ C ([0,T ];H) ∩ C ([0, a+]; L2(0,T ;H)) ∩ L2(0,T ;V).



Yt + Ya − ∆Y + f1Y + f2 · ∇Y = f (Y )

in (0,T )× (0, a+)×O

−∇Y · ν = YfΓ in (0,T )× (0, a+)× ∂O

Y (t, 0, x) = 0 in (0,T )×OY (0, a, x) = Y0(a, x) in (0, a+)×O



Proof.Introduce A(t) : D(A(t)) ⊂ H → H

D(A(t)) = v ∈ V ; va ∈ V ′, v(0, x) = 0, A(t)v ∈ H

〈A(t)v ,ψ〉V ′,V =∫ a+

0〈va(a),ψ(a)〉V ′,V da+

∫ a+

0

∫O∇v · ∇ψdxda

+∫ a+

0

∫∂OvfΓ(t, a, x)ψdσda

+∫ a+

0

∫O(f1v +∇v · f2)ψdxdadt, for all ψ ∈ V .



Proof.Introduce the Cauchy problem

dYdt(t) + A(t)Y (t) = f (t), a.e. t ∈ (0,T ), (CY )

Y (0) = Y0

(i) D(A(t)) is independent of t (D(A(t)) = H)(ii) A(t) is quasi monotone on H for all t ∈ [0,T ](iii) For each u ∈ H, t → Jλ(t)u is Lipschitz

Let f ∈ W 1,1(0,T ;H), Y0 ∈ D(A(t)) =⇒ (CY ) has a unique strongsolution

Y ∈ W 1,∞(0,T ;H) ∩ C ([0,T ];H) ∩ L∞(0,T ;D(A(t)).



Yt + Ya − ∆Y + f1Y + f2 · ∇Y = f (Y )

in (0,T )× (0, a+)×O

−∇Y · ν = YfΓ + f 0Γ in (0,T )× (0, a+)× ∂O

Y (t, 0, x) = F (t, x) ∈ L2(0,T ;H) in (0,T )×O

Y (0, a, x) = Y0(a, x) in (0, a+)×O



Proof.Let f ∈ L2(0,T ;H), Y0 ∈ H .Define FΓ(t) ∈ V ′ by

〈FΓ(t),ψ〉V ′,V = −∫ a+

0

∫∂Of 0Γ (t, s, σ)ψ(a, σ)dσda,

for all ψ ∈ V , a.e. t ∈ (0,T ).



Proof.Regularize all functions.

f n ∈ W 1,1(0,T ;H), f n → f strongly in L2(0,T ;H)F nΓ ∈ W 1,1(0,T ;H), F nΓ → FΓ strongly in L2(0,T ;V ′)Y n0 ∈ D(A(t)), Y n0 → Y0 strongly in HF n ∈ C ([0,T ]×O), F n → F strongly in L2(0,T ;H)



Proof.

Homogenize the b.c. at a = 0 by setting Y n = Y n − F n

dY n

dt(t) + A(t)Y n(t) = f n(t), a.e. t ∈ (0,T ), (CY )

Y (0) = Y0

Y nn is Cauchy in C ([0,T ];H) ∩ L2(0,T ;V) ∩ C ([0, a+]; L2(0,T ;H))Y n → Y strongly as n→ ∞



Yt + Ya − ∆Y + f1Y + f2 · ∇Y + E1(t, a, x ;Y ) = f (Y )

in (0,T )× (0, a+)×O

−∇Y · ν = YfΓ + f 0Γ in (0,T )× (0, a+)× ∂O

Y (t, 0, x) =∫ a+

0E2(t, a, x ;Y )da in (0,T )×O

Y (0, a, x) = Y0(a, x) in (0, a+)×O



Yt + Ya − ∆Y + f1Y + f2 · ∇Y + E1(t, a, x ; ζ) = f (Y )

in (0,T )× (0, a+)×O

−∇Y · ν = YfΓ + f 0Γ in (0,T )× (0, a+)× ∂O

Y (t, 0, x) =∫ a+

0E2(t, a, x ; ζ)da in (0,T )×O

Y (0, a, x) = Y0(a, x) in (0, a+)×OFixed point ζ ∈ C ([0,T ];H)


2. Main results: system (y)

TheoremFor each fixed ω ∈ Ω, system (y) has a unique solution

y ∈ C ([0,T ];H) ∩ L2(0,T ;V) ∩ C ([0, a+]; L2(0,T ;H)),

which is Ft -adapted. The solution satisfies the estimate

‖y(t)‖2H + ‖y(a)‖2L2(0,T ;H ) +

∫ t

0‖y(t)‖2V dτ

≤ Cest

(‖y0‖2H +

∫ t

0‖k(τ)‖2L2(0,a+;L2(∂O )) dτ

)Cest = c0ec1(1+‖g1‖∞+‖g2‖

2∞+a

+m2∞+µ∞)T .



−∫ T

0

∫ a+

0

∫Oyψtdxdadt −

∫ a+

0

∫Oy0ψ(0, a, x)dxda

+∫ T

0

∫Oy(t, a+, x)ψ(t, a+, x)dxdt −

∫ T

0

∫ a+

0

∫Oyψadxdadt

−∫ T

0

∫O

(∫ a+

0m(t, a, x ;U(eW y))da

)ψ(t, 0, x)dxdt

+∫ T

0

∫ a+

0

∫∂O(αyψ+ kψ)dσdadt +

∫ T

0

∫ a+

0

∫O∇y · ∇ψdxdadt

+∫ T

0

∫ a+

0

∫O

(yg1ψ+ ψg2 · ∇y + µS (t, a, x ;U(e

W y))yψ)dxdadt

= 0,

for all ψ ∈ W 1,2(0,T ;H) ∩ L2(0,T ;V), ψa ∈ L2(0,T ;V ′),ψ(T , a, x) = 0.



Proof.Step 1. Consider a mollifier ρε and define

Wε(t, a, x) =∫ T

0W (t, a, x)ρε(t − s)ds.

Wε → W strongly in C ([0,T ];C 2([0, a+]×O)) as ε→ 0.

Replace (y) by (yε)



yt + ya − ∆y + g1ε(t, a, x)y + g2ε(t, a, x) · ∇y + S1(t, a, x ; y) = 0 (yε)

−∇y · ν = αε(t, a, x)y + kε(t, a, x) in (0,T )× (0, a+)× ∂O

y(t, 0, x) =∫ a+

0S2(t, a, x ; y)da in (0,T )×O

y(0, a, x) = y0(a, x) in (0, a+)×O

S1(t, a, x ; y) = µS (t, a, x ;U(eWεy))y , S2(a, x ; y) = m(t, a, x ;U(eWεy))y

Si are local Lipschitz on H !!!



Truncation approximation: replace S1 and S2 by

SNi (t, a, x ; u) =

Si (t, a, x ; u), ‖u‖H ≤ NSi(t, a, x ; Nu

‖u‖H

), ‖u‖H > N

SNi (t, a, x ; u) are Lipschitz continuous on H

Problem (yNε ) has a unique solution∥∥∥yNε (t)∥∥∥2H ≤ c0ec1(1+‖g1‖∞+‖g2‖2∞+a

+m2∞+µ∞)T ×(‖y0‖2H + ‖k‖

2L2(0,T ;L2(0,a+;L2(∂O )))

):= R20

Set ∥∥∥yNε (t)∥∥∥H ≤ R0 < [R0] + 1 := N0

Choose N ≥ N0,∥∥yNε (t)∥∥H ≤ N0 < N, SNi (t, a, x ; y) = Si (t, a, x ; y)

=⇒ yN0ε = yε.



Truncation approximation: replace S1 and S2 by

SNi (t, a, x ; u) =

Si (t, a, x ; u), ‖u‖H ≤ NSi(t, a, x ; Nu

‖u‖H

), ‖u‖H > N

SNi (t, a, x ; u) are Lipschitz continuous on HProblem (yNε ) has a unique solution∥∥∥yNε (t)∥∥∥2H ≤ c0ec1(1+‖g1‖∞+‖g2‖

2∞+a

+m2∞+µ∞)T ×(‖y0‖2H + ‖k‖

2L2(0,T ;L2(0,a+;L2(∂O )))

):= R20

Set ∥∥∥yNε (t)∥∥∥H ≤ R0 < [R0] + 1 := N0

Choose N ≥ N0,∥∥yNε (t)∥∥H ≤ N0 < N, SNi (t, a, x ; y) = Si (t, a, x ; y)

=⇒ yN0ε = yε.



Proof.Step 2. Some estimates imply that yεε is CauchyPass to the limit as ε→ 0, and get that y is a solution.



−∫ T

0

∫ a+

0

∫Oyψtdxdadt −

∫ a+

0

∫Oy0ψ(0, a, x)dxda

+∫ T

0

∫Oy(t, a+, x)ψ(t, a+, x)dxdt −

∫ T

0

∫ a+

0

∫Oyψadxdadt

−∫ T

0

∫O

(∫ a+

0m(t, a, x ;U(eW y))da

)ψ(t, 0, x)dxdt

+∫ T

0

∫ a+

0

∫∂O(αyψ+ kψ)dσdadt +

∫ T

0

∫ a+

0

∫O∇y · ∇ψdxdadt

+∫ T

0

∫ a+

0

∫O

(yg1ψ+ ψg2 · ∇y + µS (t, a, x ;U(e

W y))yψ)dxdadt

= 0,

for all ψ ∈ W 1,2(0,T ;H) ∩ L2(0,T ;V), ψa ∈ L2(0,T ;V ′),ψ(T , a, x) = 0.



Define X = u ∈ V ; ua ∈ V ′, A(t) : V ∩ C([0, a+];H)→ X ′

⟨A(t)v ,ψ0

⟩X ′,X

=∫Ov(a+, x)ψ0(a

+, x)dx −∫ a+

0

∫Ov(ψ0)adxda

−∫O

(∫ a+

0m(t, a, x ;U(v))vda

)ψ0(0, x)dx

+∫ a+

0

∫O

µS (t, a, x ;U(v))vψ0dxda

+∫ a+

0

∫O(∇v · ∇ψ0 + vg1ψ0 + ψ0g2 · ∇v)dxda

+∫ a+

0

∫∂O(αv + k)ψ0dσda



dydt+ A(t)y = 0, in D′(0,T ;X ′)

∥∥∥∥dydt (ϕ)∥∥∥∥X ′≤ C ‖ϕ‖L2(0,T )

dydt∈ L2(0,T ;X ′).


3. Main results: system (p)

TheoremUnder the initial assumptions the stochastic problem (p) has a uniquesolution Ft -adapted

p ∈ C ([0,T ]; L2(H)) ∩ L2(0,T ;V)) ∩ C ([0, a+]; L2(0,T ;H)).



Proof.

dydt(t) + A(t)y(t) = 0, a.e. t ∈ (0,T ),

y(0) = y0

Consider a mollifier ρε and define

yε(t) = (y ∗ ρε)(t) =∫ T

0y(t − s)ρε(s)ds

Obviously, yε → y strongly inC ([0,T ];H) ∩ C ([0, a+]; L2(0,T ;H)) ∩ L2(0,T ;V).



Proof.dyε

dt(t) + (ρε ∗ A(t)y)(t) = 0,

eWdyε

dt(t) + eW (ρε ∗ A(t)y)(t) = 0

Denote pε := eW yε,

pε → eW y := p strongly inC ([0,T ];H) ∩ C ([0, a+]; L2(0,T ;H)) ∩ L2(0,T ;V)

pε(t)− pε(0)−∫ t

0µpεdτ −

∫ t

0pεdW (τ)

+∫ t

0eW (τ)(ρε ∗ A(τ)y)(τ)dτ = 0



Proof.Then, test at ψ0 ∈ X , and pass to the limit as ε→ 0

(p(t),ψ0)H − (p(0),ψ0)H −∫ t

0(µp(τ),ψ0)H dτ

−N

∑k=1

∫ t

0

∫ a+

0

∫O

µkψ0p(τ)dxdadβk (τ)

+∫ t

0

⟨A(τ)y(τ), eW (τ)ψ0

⟩X ′,X

dτ = 0.



(p(t),ψ)H

+∫ t

0

∫Op(τ, a+, x)ψ(a+, x)dxdτ −

∫ t

0

∫ a+

0

∫Opψadxdadτ

−∫ t

0

∫ a+

0

∫Om0(a, x ;U(p))pψ(0, x)dxdadτ

+∫ t

0

∫ a+

0

∫O(∇p · ∇ψ+ µS (τ, a, x ;U(p))pψ)dxdadτ

+∫ t

0

∫ a+

0

∫O(α0p + k0)ψdσdadτ

= (p0,ψ)H +∫ t

0

∫ a+

0

∫OpψdxdadW (τ), for all ψ ∈ V , with ψa ∈ V ′.



Proof.

p is Ft -adapted because it is a limit of Ft -adapted processes.


References

V. Barbu, M. Röckner, Stochastic variational inequalities and applications tothe total variation flow perturbed by linear multiplicative noise, Arch.Rational Mech. Anal. 209 (2013), 797—834.

V. Barbu, M. Röckner, An operatorial approach to stochastic partialdifferential equations driven by linear multiplicative noise, J. Eur. Math. Soc.17 (2015), 1789-1815.

C. Cusulin, M. Iannelli, G. Marinoschi, Convergence in a multi-layerpopulation model with age-structure, Nonlinear Anal. Real World Appl. 8(2007), 887-902.

G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions,Second Edition, Series: Encyclopedia of Mathematics and its Applications152, Cambridge University Press, 2014.

C. Prévôt, M. Röckner, A Concise Course on Stochastic Partial DifferentialEquations, Springer LN in Math. 1905, Berlin, 2007.


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