la complessa dinamica del modello di gurtin e … complessa dinamica del modello di gurtin e maccamy...

La complessa dinamica

del modello di Gurtin e MacCamy

Mimmo Iannelli

Universit a di Trento

IASI, Roma, January 26, 2009 – p. 1/99

Outline of the talk

A chapter from the theory of age-structured populations :

Gurtin-McCamy model

Structured logistic growth

Juveniles-adults dynamics

Some recent results :

A numerical method for the analysis

Exploration of the models


Outline of the talk

A collaboration with :

F. Milner, Arizona University, Tempe, Mathematics Departm ent

C. Cusulin, Vienna University, Mathematics Department

S. Maset, Trieste University, Mathematics Department

D. Breda and R. Vermiglio, Udine University, Mathematics

Department

+ . . .

focused on numerical treatment of the Gurtin-McCamy model


Gurtin-MacCamy

The Gurtin-MacCamy system

∂p

∂t(a, t) +

∂p

∂a(a, t) + µ(a, S1(t), . . . , Sn(t))p(a, t) = 0,

p(0, t) =

∫ a†

0

β(a, S1(t), . . . , Sn(t))p(a, t) da,

Si(t) =

∫ a†

0

γi(a)p(a, t) da, i = 1, . . . , n,

p(a, 0) = p0(a).


Gurtin-MacCamy


∂p

∂t(a, t) +

∂p

∂a(a, t) + µ(a, S1(t), . . . , Sn(t))p(a, t) = 0,

p(0, t) =

∫ a†

0

β(a, S1(t), . . . , Sn(t))p(a, t) da,

Si(t) =

∫ a†

0

γi(a)p(a, t) da, i = 1, . . . , n,

p(a, 0) = p0(a).

6

age-distribution


Gurtin-MacCamy


∂p

∂t(a, t) +

∂p

∂a(a, t) + µ(a, S1(t), . . . , Sn(t))p(a, t) = 0,

p(0, t) =

∫ a†

0

β(a, S1(t), . . . , Sn(t))p(a, t) da,

Si(t) =

∫ a†

0

γi(a)p(a, t) da, i = 1, . . . , n,

p(a, 0) = p0(a).

XXXXXXXXXy

mortality


Gurtin-MacCamy


∂p

∂t(a, t) +

∂p

∂a(a, t) + µ(a, S1(t), . . . , Sn(t))p(a, t) = 0,

p(0, t) =

∫ a†

0

β(a, S1(t), . . . , Sn(t))p(a, t) da,

Si(t) =

∫ a†

0

γi(a)p(a, t) da, i = 1, . . . , n,

p(a, 0) = p0(a).

-


Gurtin-MacCamy


∂p

∂t(a, t) +

∂p

∂a(a, t) + µ(a, S1(t), . . . , Sn(t))p(a, t) = 0,

p(0, t) =

∫ a†

0

β(a, S1(t), . . . , Sn(t))p(a, t) da,

Si(t) =

∫ a†

0

γi(a)p(a, t) da, i = 1, . . . , n,

p(a, 0) = p0(a).

6

fertility


Gurtin-MacCamy


∂p

∂t(a, t) +

∂p

∂a(a, t) + µ(a, S1(t), . . . , Sn(t))p(a, t) = 0,

p(0, t) =

∫ a†

0

β(a, S1(t), . . . , Sn(t))p(a, t) da,

Si(t) =

∫ a†

0

γi(a)p(a, t) da, i = 1, . . . , n,

p(a, 0) = p0(a).

-


Gurtin-MacCamy


∂p

∂t(a, t) +

∂p

∂a(a, t) + µ(a, S1(t), . . . , Sn(t))p(a, t) = 0,

p(0, t) =

∫ a†

0

β(a, S1(t), . . . , Sn(t))p(a, t) da,

Si(t) =

∫ a†

0

γi(a)p(a, t) da, i = 1, . . . , n,

p(a, 0) = p0(a).


Gurtin-MacCamy

The basic ingredients

p(a, t) age-distribution of the population

Si(t) =

∫ a†

0

γi(a)p(a, t)da weighted selection of the population

β(a, S1(t), . . . , Sn(t)) fertility

µ(a, S1(t), . . . , Sn(t)) mortality



Logistic growth

one single size: S(t) =

∫ a†

0γ(a)p(a, t)da

fertility: β(a, x) = R0β0(a)Φ(x)

mortality: µ(a, x) = µ0(a) + m(a)Ψ(x)



Logistic growth


∫ a†

0γ(a)p(a, t)da



with

γ(a) non-decreasing

Φ(x) decreasing

Ψ(x) increasing



Logistic growth


∫ a†

0γ(a)p(a, t)da



with


Φ(x) decreasing

Ψ(x) increasing

β0(a) and m(a) describe how crowding impacts on different ages



Logistic growth


∫ a†

0γ(a)p(a, t)da



with


Φ(x) decreasing

Ψ(x) increasing


R0 = basic reproduction number



Logistic growth


∫ a†

0γ(a)p(a, t)da



with


Φ(x) decreasing

Ψ(x) increasing


R0 = the number of off-springs produced during the whole life



The search for a stationary state p∗(a)

∂p∗

∂a(a) + µ(a, S∗)p∗(a) = 0

=⇒ p∗(a) = Π(a, S∗)p∗(0), Π(a, S) = e−

a∫

0

µ(σ,S)dσ

1 =

a†∫

0

β(a, S∗)Π(a, S∗)da, p∗(0) =S∗

a†∫

0

γi(a)Π(a, S∗)da



1 =

a†∫

0

β(a, S∗)Π(a, S∗)da



1 = R0Φ(S∗)

a†∫

0

β0(a)e−∫

a

0µ0(σ)dσe−Ψ(S∗)

∫a

0m(σ)dσda



1 = R0Φ(S∗)

a†∫

0

β0(a)e−∫

a


∫a

0m(σ)dσda

6

decreasing as a function of S∗



1 = R0Φ(S∗)

a†∫

0

β0(a)e−∫

a


∫a

0m(σ)dσda

bifurcation graph



1 = R0Φ(S∗)

a†∫

0

β0(a)e−∫

a


∫a

0m(σ)dσda

bifurcation graph

trivial state



1 = R0Φ(S∗)

a†∫

0

β0(a)e−∫

a


∫a

0m(σ)dσda

bifurcation graph @@I non trivial state



Stability by linearization at p∗(a)

deviation from the steady state v(a, t) = p(a, t) − p∗(a)





∂v

∂t(a, t) +

∂v

∂a(a, t) + µ(a, S∗)v(a, t)+

+p∗(a)∂µ

∂S(a, S∗)

a†∫

0

γ(a)v(a, t)da = 0

v(0, t) =

a†∫

0

β(a, S∗)v(a, t)da+

+

a†∫

0

p∗(σ)∂β

∂S(σ, S∗)dσ

a†∫

0

γ(a)v(a, t)da





∂v

∂t(a, t) +

∂v

∂a(a, t) + µ(a, S∗)v(a, t)+

+p∗(a)∂µ

∂S(a, S∗)

a†∫

0

γ(a)v(a, t)da = 0

v(0, t) =

a†∫

0


+

a†∫

0

p∗(σ)∂β

∂S(σ, S∗)dσ

a†∫

0

γ(a)v(a, t)da



Characteristic equation

det

∣∣∣∣∣∣∣

1 − K00(λ) −b∗ − K01(λ)

−K10(λ) 1 − K11(λ)

∣∣∣∣∣∣∣= 0

K00(t) = β(t, S∗)Π(t, S∗) K10(t) = γ(t)Π(t, S∗)

K01(t) = −p∗(0)

∫a†

0

∂µ

∂S(σ, S∗)K00(t + σ)dσ

K11(t) = −p∗(0)

∫a†

0

∂µ

∂S(σ, S∗)K10(t + σ)dσ

b∗ = p∗0(0)

∫a†

0

∂β

∂S(σ, S∗)Π(σ, S∗)dσ




det

∣∣∣∣∣∣∣

1 − K00(λ) −b∗ − K01(λ)

−K10(λ) 1 − K11(λ)

∣∣∣∣∣∣∣= 0


K01(t) = −p∗(0)

∫a†

0

∂µ

∂S(σ, S∗)K00(t + σ)dσ

K11(t) = −p∗(0)

∫a†

0

∂µ

∂S(σ, S∗)K10(t + σ)dσ

b∗ = p∗0(0)

∫a†

0

∂β





det

∣∣∣∣∣∣∣

1 − K00(λ) −b∗ − K01(λ)

−K10(λ) 1 − K11(λ)

∣∣∣∣∣∣∣= 0

If all characteristic roots have negative real partthen the steady state p∗(a) is stable.If at least one of the characteristic roots has apositive real part then the state is unstable.



1 R0

-

6

rstable



1 R0

-

6

rstable unstable



1 R0

-

6

rstable unstable

&%'$



1 R0

-

6

rstable unstable



1 R0

-

6

rstable unstabler



1 R0

-

6

rstable unstabler )

'

&

$

%bifurcation point:two complex conjugate roots crossthe imaginary axis and a periodicsolution arises by Hopf bifurcation


Juveniles-adult dynamics

The example of juveniles-adults dynamics

two selected groups

J(t) =

∫ a∗

0

p(a, t) da, juveniles

A(t) =

∫ a†

a∗

p(a, t) da, adults




two selected groups

J(t) =

∫ a∗

0


A(t) =

∫ a†

a∗

p(a, t) da, adults

a∗ is the maturation age




two selected groups

J(t) =

∫ a∗

0


A(t) =

∫ a†

a∗

p(a, t) da, adults

separated niches

Allee effect

cannibalism



The case of two different ecological niches

β(a, J,A) = R0bχ[a∗,a†](a)e−(b1J+b2A)

µ(a, J,A) = µ0(a) + m1χ[0,a∗](a)J + m2χ[a∗,a†](a)A




R0

J

R0,2

R0,1

1



The Allee effect


µ(a, J,A) = µ0(a) + m1χ[0,a∗](a)J + m2χ[a∗,a†](a)A+

− [θ1µ0(a) + θ2m1J ] χ[0,a∗](a)α(A)

A positive effect (a decrease of mortality) on

juveniles, due to adults presence



The Allee effect



−[θ1µ0(a) + θ2m1J ] χ[0,a∗](a)α(A)





The Allee effect



− [θ1µ0(a) + θ2m1J ] χ[0,a∗](a)α(A)





The Allee effect



Cannibalism (of adults on juveniles)


µ(a, J,A) = µ0(a) + m1χ[0,a∗](a)A

1 + θJ

A negative effect (increase of mortality) on

juveniles, due to predation by adults, regulated by a

functional response of Holling type





µ(a, J,A) = µ0(a) + m1χ[0,a∗](a)A

1 + θJ

A negative effect (increase of mortality) on

juveniles, due to predation by adults, regulated by a

functional response of Holling type




R0

J

R0,2

R0,1

1


A numerical method for stability analysis

The starting point: linearization at a steady state p∗(a)

∂v

∂t(a, t) +

∂v

∂a(a, t) + µ(a, S∗)v(a, t)+

+p∗(a)∂µ

∂S(a, S∗)

a†∫

0

γ(a)v(a, t)da = 0

v(0, t) =

a†∫

0


+

a†∫

0

p∗(σ)∂β

∂S(σ, S∗)dσ

a†∫

0

γ(a)v(a, t)da



The starting point: the resulting characteristic equation

det

∣∣∣∣∣∣∣∣∣

1 − K00(λ) −b∗ − K01(λ)

−K10(λ) 1 − K11(λ)

∣∣∣∣∣∣∣∣∣

= 0

The goal: to approximate the roots



The starting point: the resulting characteristic equation

det

∣∣∣∣∣∣∣∣∣

1 − K00(λ) −b∗ − K01(λ)

−K10(λ) 1 − K11(λ)

∣∣∣∣∣∣∣∣∣

= 0

The goal: to approximate the roots

reformulation of the linearization as an abstract Cauchy pr oblem

discrete approximation of the generator

computation of the spectrum of the approximated generator



Reformulation as an abstract Cauchy problem

∂v

∂t(a, t) +

∂v

∂a(a, t) + µ(a, S∗)v(a, t)+

+p∗(a)∂µ

∂S(a, S∗)

a†∫

0

γ(a)v(a, t)da = 0

v(0, t) =

a†∫

0


+

a†∫

0

p∗(σ)∂β

∂S(σ, S∗)dσ

a†∫

0

γ(a)v(a, t)da

v(a, 0) = v0(a)




∂v

∂t(a, t) +

∂v

∂a(a, t) + µ(a, S∗)v(a, t)+

+p∗(a)∂µ

∂S(a, S∗)

a†∫

0

γ(a)v(a, t)da = 0

v(0, t) =

a†∫

0


+

a†∫

0

p∗(σ)∂β

∂S(σ, S∗)dσ

a†∫

0

γ(a)v(a, t)da

v(a, 0) = v0(a)




∂v

∂t(a, t) +

∂v

∂a(a, t) + (Hv(·, t))(a) = 0

v(0, t) = K0v(·, t)v(a, 0) = v0(a)




∂v

∂t(a, t) +

∂v

∂a(a, t) + (Hv(·, t))(a) = 0

v(0, t) = K0v(·, t)v(a, 0) = v0(a)

d

dtu(t) = Au(t), t ≥ 0,

u(0) = u0 ∈ X.




∂v

∂t(a, t) +

∂v

∂a(a, t) + (Hv(·, t))(a) = 0

v(0, t) = K0v(·, t)v(a, 0) = v0(a)

d

dtu(t) = Au(t), t ≥ 0,

u(0) = u0 ∈ X.

'

&

$

%

L1

([0, a†], R)




∂v

∂t(a, t) +

∂v

∂a(a, t) + (Hv(·, t))(a) = 0

v(0, t) = K0v(·, t)v(a, 0) = v0(a)

d

dtu(t) = Au(t), t ≥ 0,

u(0) = u0 ∈ X.

'

&

$

%9

)

u(t) ≡ v(·, t)u0 ≡ v0(·)




∂v

∂t(a, t) +

∂v

∂a(a, t) + (Hv(·, t))(a) = 0

v(0, t) = K0v(·, t)v(a, 0) = v0(a)

d

dtu(t) = Au(t), t ≥ 0,

u(0) = u0 ∈ X.

'

&

$

%

Aϕ = −ϕ′ −Hϕ

D (A) = ϕ ∈ X | ϕ′ ∈ X, ϕ(0) = K0ϕ



Discrete approximation of the generator


Recent results: a numerical method for stability analysis


[0, a†] ΩN =θi =

a†

2 cos(

N−iN π

)+

a†

2 : i = 0, . . . , N




[0, a†] ΩN =θi =

a†

2 cos(

N−iN π

)+

a†

2 : i = 0, . . . , N

ϕ ∈ X y ∈ XN∼= CN

set yi = ϕ(θi), i = 1, . . . , N




[0, a†] ΩN =θi =

a†

2 cos(

N−iN π

)+

a†

2 : i = 0, . . . , N

ϕ ∈ X y ∈ XN∼= CN

set yi = ϕ(θi), i = 1, . . . , N

A AN : XN → XN ,

build ϕN an interpolating polynomial through yi such that

ϕN (0) = K0ϕN

compute zi = −ϕ′N (θi) − (HϕN ) (θi), i = 1, . . . , N

set (ANy)i = zi




the eigenvalues of AN approximate the eigenvalues of A

If λ is an eigenvalue of A with multiplicity ν, then for N sufficiently large,

AN has exactly ν eigenvalues λi, i = 1, . . . , ν, such that

max1≤i≤ν

|λ − λi| ≤(

C2

C3

)1/ν(

εN +1√N

(C1

N

)N)1/ν


Exploration of juveniles-adults dinamics

Back to adults-juveniles competition:the case of separate niches

R0

J

R0,2

R0,1

1



Separate niches: exploring the bifurcation graph

R0

J

R0,2

R0,1

1




R0

J

R0,2

R0,1

1

−8 −6 −4 −2 0 2−40

−30

−20

−10

0

10

20

30

40

ℜ (λ)ℑ

(λ)

K

mAA

PPi




R0

J

R0,2

R0,1

1

−8 −6 −4 −2 0 2−40

−30

−20

−10

0

10

20

30

40

ℜ (λ)ℑ

(λ)

K

mstableA

A

PPi




R0

J

R0,2

R0,1

1

−8 −6 −4 −2 0 2−40

−30

−20

−10

0

10

20

30

40

ℜ (λ)ℑ

(λ)

J

mstableA

B

PPi mbifurcationB




R0

J

R0,2

R0,1

1

−8 −6 −4 −2 0 2−40

−30

−20

−10

0

10

20

30

40

ℜ (λ)ℑ

(λ)

I

mstableA

C

PPi mbifurcationB munstableC




R0

J

R0,2

R0,1

1

−8 −6 −4 −2 0 2−40

−30

−20

−10

0

10

20

30

40

ℜ (λ)ℑ

(λ)

H

mstableA

D


mbifurcationDPPPPi




R0

J

R0,2

R0,1

1

−8 −6 −4 −2 0 2−40

−30

−20

−10

0

10

20

30

40

ℜ (λ)ℑ

(λ)

E

mstableA

E


mbifurcationDPPPPi

munstabletwo complex roots

EAAAAAU



A complete pattern

−7 −6 −5 −4 −3 −2 −1 0 1−1

0

1

2

3

4

5

ℜ (λ)

ℑ(λ

)

R0

J

R0,2

R0,11



Orbits by numerical computation of the solution



Orbits by numerical computation of the solution

0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

J

A

R0=340

R0=300

R0=200R

0=100R

0=50

R0=30

R0=75R

0=35

R0=24

R0=411

R0=150


Future work


Future work

systematic use of the numerical method for a complete

analysis of some specific population models


Future work



extension of the method to age structured models with

diffusion


Future work




diffusion

extension to epidemic models


Future work




diffusion


building of a (friendly enough) simulation system includin g


Future work




diffusion



numerical methods for the computation of the solution


Future work




diffusion




computation of steady states


Future work




diffusion




computation of steady states

stability analysis via numerical computation of

characteristic roots


THANK YOU FORYOUR ATTENTION


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