La complessa dinamica del modello di Gurtin e MacCamy Mimmo Iannelli Universit ` a di Trento IASI, Roma, January 26, 2009 – p. 1/99
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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
IASI, Roma, January 26, 2009 – p. 2/99
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
IASI, Roma, January 26, 2009 – p. 3/99
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).
IASI, Roma, January 26, 2009 – p. 4/99
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).
6
age-distribution
IASI, Roma, January 26, 2009 – p. 5/99
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).
XXXXXXXXXy
mortality
IASI, Roma, January 26, 2009 – p. 6/99
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).
-
IASI, Roma, January 26, 2009 – p. 7/99
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).
6
fertility
IASI, Roma, January 26, 2009 – p. 8/99
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).
-
IASI, Roma, January 26, 2009 – p. 9/99
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).
IASI, Roma, January 26, 2009 – p. 10/99
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).
IASI, Roma, January 26, 2009 – p. 11/99
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
IASI, Roma, January 26, 2009 – p. 12/99
Structured logistic growth
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)
IASI, Roma, January 26, 2009 – p. 13/99
Structured logistic growth
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)
with
γ(a) non-decreasing
Φ(x) decreasing
Ψ(x) increasing
IASI, Roma, January 26, 2009 – p. 14/99
Structured logistic growth
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)
with
γ(a) non-decreasing
Φ(x) decreasing
Ψ(x) increasing
β0(a) and m(a) describe how crowding impacts on different ages
IASI, Roma, January 26, 2009 – p. 15/99
Structured logistic growth
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)
with
γ(a) non-decreasing
Φ(x) decreasing
Ψ(x) increasing
β0(a) and m(a) describe how crowding impacts on different ages
R0 = basic reproduction number
IASI, Roma, January 26, 2009 – p. 16/99
Structured logistic growth
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)
with
γ(a) non-decreasing
Φ(x) decreasing
Ψ(x) increasing
β0(a) and m(a) describe how crowding impacts on different ages
R0 = the number of off-springs produced during the whole life
IASI, Roma, January 26, 2009 – p. 17/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 18/99
Structured logistic growth
1 =
a†∫
0
β(a, S∗)Π(a, S∗)da
IASI, Roma, January 26, 2009 – p. 19/99
Structured logistic growth
1 = R0Φ(S∗)
a†∫
0
β0(a)e−∫
a
0µ0(σ)dσe−Ψ(S∗)
∫a
0m(σ)dσda
IASI, Roma, January 26, 2009 – p. 20/99
Structured logistic growth
1 = R0Φ(S∗)
a†∫
0
β0(a)e−∫
a
0µ0(σ)dσe−Ψ(S∗)
∫a
0m(σ)dσda
6
decreasing as a function of S∗
IASI, Roma, January 26, 2009 – p. 21/99
Structured logistic growth
1 = R0Φ(S∗)
a†∫
0
β0(a)e−∫
a
0µ0(σ)dσe−Ψ(S∗)
∫a
0m(σ)dσda
bifurcation graph
IASI, Roma, January 26, 2009 – p. 22/99
Structured logistic growth
1 = R0Φ(S∗)
a†∫
0
β0(a)e−∫
a
0µ0(σ)dσe−Ψ(S∗)
∫a
0m(σ)dσda
bifurcation graph
trivial state
IASI, Roma, January 26, 2009 – p. 23/99
Structured logistic growth
1 = R0Φ(S∗)
a†∫
0
β0(a)e−∫
a
0µ0(σ)dσe−Ψ(S∗)
∫a
0m(σ)dσda
bifurcation graph @@I non trivial state
IASI, Roma, January 26, 2009 – p. 24/99
Structured logistic growth
Stability by linearization at p∗(a)
deviation from the steady state v(a, t) = p(a, t) − p∗(a)
IASI, Roma, January 26, 2009 – p. 25/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 26/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 27/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 28/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 29/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 30/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 31/99
Structured logistic growth
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
IASI, Roma, January 26, 2009 – p. 32/99
Structured logistic growth
Characteristic equation
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.
IASI, Roma, January 26, 2009 – p. 33/99
Structured logistic growth
1 R0
-
6
rstable
IASI, Roma, January 26, 2009 – p. 34/99
Structured logistic growth
1 R0
-
6
rstable unstable
IASI, Roma, January 26, 2009 – p. 35/99
Structured logistic growth
1 R0
-
6
rstable unstable
&%'$
IASI, Roma, January 26, 2009 – p. 36/99
Structured logistic growth
1 R0
-
6
rstable unstable
&%'$
IASI, Roma, January 26, 2009 – p. 37/99
Structured logistic growth
1 R0
-
6
rstable unstable
IASI, Roma, January 26, 2009 – p. 38/99
Structured logistic growth
1 R0
-
6
rstable unstabler
IASI, Roma, January 26, 2009 – p. 39/99
Structured logistic growth
1 R0
-
6
rstable unstabler )
'
&
$
%bifurcation point:two complex conjugate roots crossthe imaginary axis and a periodicsolution arises by Hopf bifurcation
IASI, Roma, January 26, 2009 – p. 40/99
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
IASI, Roma, January 26, 2009 – p. 41/99
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
a∗ is the maturation age
IASI, Roma, January 26, 2009 – p. 42/99
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
separated niches
Allee effect
cannibalism
IASI, Roma, January 26, 2009 – p. 43/99
Juveniles-adult dynamics
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
IASI, Roma, January 26, 2009 – p. 44/99
Juveniles-adult dynamics
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
IASI, Roma, January 26, 2009 – p. 45/99
Juveniles-adult dynamics
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
IASI, Roma, January 26, 2009 – p. 46/99
Juveniles-adult dynamics
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
IASI, Roma, January 26, 2009 – p. 47/99
Juveniles-adult dynamics
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
IASI, Roma, January 26, 2009 – p. 48/99
Juveniles-adult dynamics
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
IASI, Roma, January 26, 2009 – p. 49/99
Juveniles-adult dynamics
The case of two different ecological niches
R0
J
R0,2
R0,1
1
IASI, Roma, January 26, 2009 – p. 50/99
Juveniles-adult dynamics
The Allee effect
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+
− [θ1µ0(a) + θ2m1J ] χ[0,a∗](a)α(A)
A positive effect (a decrease of mortality) on
juveniles, due to adults presence
IASI, Roma, January 26, 2009 – p. 51/99
Juveniles-adult dynamics
The Allee effect
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+
−[θ1µ0(a) + θ2m1J ] χ[0,a∗](a)α(A)
A positive effect (a decrease of mortality) on
juveniles, due to adults presence
IASI, Roma, January 26, 2009 – p. 52/99
Juveniles-adult dynamics
The Allee effect
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+
− [θ1µ0(a) + θ2m1J ] χ[0,a∗](a)α(A)
A positive effect (a decrease of mortality) on
juveniles, due to adults presence
IASI, Roma, January 26, 2009 – p. 53/99
Juveniles-adult dynamics
The Allee effect
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+
− [θ1µ0(a) + θ2m1J ] χ[0,a∗](a)α(A)
A positive effect (a decrease of mortality) on
juveniles, due to adults presence
IASI, Roma, January 26, 2009 – p. 54/99
Juveniles-adult dynamics
The Allee effect
IASI, Roma, January 26, 2009 – p. 55/99
Juveniles-adult dynamics
The Allee effect
IASI, Roma, January 26, 2009 – p. 56/99
Juveniles-adult dynamics
Cannibalism (of adults on juveniles)
β(a, J,A) = R0bχ[a∗,a†](a)e−(b1J+b2A)
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
IASI, Roma, January 26, 2009 – p. 57/99
Juveniles-adult dynamics
Cannibalism (of adults on juveniles)
β(a, J,A) = R0bχ[a∗,a†](a)e−(b1J+b2A)
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
IASI, Roma, January 26, 2009 – p. 58/99
Juveniles-adult dynamics
Cannibalism (of adults on juveniles)
R0
J
R0,2
R0,1
1
IASI, Roma, January 26, 2009 – p. 59/99
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
β(a, S∗)v(a, t)da+
+
a†∫
0
p∗(σ)∂β
∂S(σ, S∗)dσ
a†∫
0
γ(a)v(a, t)da
IASI, Roma, January 26, 2009 – p. 60/99
A numerical method for stability analysis
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
IASI, Roma, January 26, 2009 – p. 61/99
A numerical method for stability analysis
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
IASI, Roma, January 26, 2009 – p. 62/99
A numerical method for stability analysis
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
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(a, 0) = v0(a)
IASI, Roma, January 26, 2009 – p. 63/99
A numerical method for stability analysis
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
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(a, 0) = v0(a)
IASI, Roma, January 26, 2009 – p. 64/99
A numerical method for stability analysis
Reformulation as an abstract Cauchy problem
∂v
∂t(a, t) +
∂v
∂a(a, t) + (Hv(·, t))(a) = 0
v(0, t) = K0v(·, t)v(a, 0) = v0(a)
IASI, Roma, January 26, 2009 – p. 65/99
A numerical method for stability analysis
Reformulation as an abstract Cauchy problem
∂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.
IASI, Roma, January 26, 2009 – p. 66/99
A numerical method for stability analysis
Reformulation as an abstract Cauchy problem
∂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)
IASI, Roma, January 26, 2009 – p. 67/99
A numerical method for stability analysis
Reformulation as an abstract Cauchy problem
∂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(·)
IASI, Roma, January 26, 2009 – p. 68/99
A numerical method for stability analysis
Reformulation as an abstract Cauchy problem
∂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
IASI, Roma, January 26, 2009 – p. 69/99
A numerical method for stability analysis
Discrete approximation of the generator
IASI, Roma, January 26, 2009 – p. 70/99
Recent results: a numerical method for stability analysis
Discrete approximation of the generator
[0, a†] ΩN =θi =
a†
2 cos(
N−iN π
)+
a†
2 : i = 0, . . . , N
IASI, Roma, January 26, 2009 – p. 71/99
A numerical method for stability analysis
Discrete approximation of the generator
[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
IASI, Roma, January 26, 2009 – p. 72/99
A numerical method for stability analysis
Discrete approximation of the generator
[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
IASI, Roma, January 26, 2009 – p. 73/99
A numerical method for stability analysis
Discrete approximation of the generator
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
IASI, Roma, January 26, 2009 – p. 74/99
Exploration of juveniles-adults dinamics
Back to adults-juveniles competition:the case of separate niches
R0
J
R0,2
R0,1
1
IASI, Roma, January 26, 2009 – p. 75/99
Exploration of juveniles-adults dinamics
Separate niches: exploring the bifurcation graph
R0
J
R0,2
R0,1
1
IASI, Roma, January 26, 2009 – p. 76/99
Exploration of juveniles-adults dinamics
Separate niches: exploring the bifurcation graph
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
IASI, Roma, January 26, 2009 – p. 77/99
Exploration of juveniles-adults dinamics
Separate niches: exploring the bifurcation graph
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
IASI, Roma, January 26, 2009 – p. 78/99
Exploration of juveniles-adults dinamics
Separate niches: exploring the bifurcation graph
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
IASI, Roma, January 26, 2009 – p. 79/99
Exploration of juveniles-adults dinamics
Separate niches: exploring the bifurcation graph
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
IASI, Roma, January 26, 2009 – p. 80/99
Exploration of juveniles-adults dinamics
Separate niches: exploring the bifurcation graph
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
PPi mbifurcationB munstableC
mbifurcationDPPPPi
IASI, Roma, January 26, 2009 – p. 81/99
Exploration of juveniles-adults dinamics
Separate niches: exploring the bifurcation graph
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
PPi mbifurcationB munstableC
mbifurcationDPPPPi
munstabletwo complex roots
EAAAAAU
IASI, Roma, January 26, 2009 – p. 82/99
Exploration of juveniles-adults dinamics
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
IASI, Roma, January 26, 2009 – p. 83/99
Exploration of juveniles-adults dinamics
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
IASI, Roma, January 26, 2009 – p. 84/99
Exploration of juveniles-adults dinamics
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
IASI, Roma, January 26, 2009 – p. 85/99
Exploration of juveniles-adults dinamics
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
IASI, Roma, January 26, 2009 – p. 86/99
Exploration of juveniles-adults dinamics
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
IASI, Roma, January 26, 2009 – p. 87/99
Exploration of juveniles-adults dinamics
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
IASI, Roma, January 26, 2009 – p. 88/99
Exploration of juveniles-adults dinamics
Orbits by numerical computation of the solution
IASI, Roma, January 26, 2009 – p. 89/99
Exploration of juveniles-adults dinamics
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
IASI, Roma, January 26, 2009 – p. 90/99
Future work
IASI, Roma, January 26, 2009 – p. 91/99
Future work
systematic use of the numerical method for a complete
analysis of some specific population models
IASI, Roma, January 26, 2009 – p. 92/99
Future work
systematic use of the numerical method for a complete
analysis of some specific population models
extension of the method to age structured models with
diffusion
IASI, Roma, January 26, 2009 – p. 93/99
Future work
systematic use of the numerical method for a complete
analysis of some specific population models
extension of the method to age structured models with
diffusion
extension to epidemic models
IASI, Roma, January 26, 2009 – p. 94/99
Future work
systematic use of the numerical method for a complete
analysis of some specific population models
extension of the method to age structured models with
diffusion
extension to epidemic models
building of a (friendly enough) simulation system includin g
IASI, Roma, January 26, 2009 – p. 95/99
Future work
systematic use of the numerical method for a complete
analysis of some specific population models
extension of the method to age structured models with
diffusion
extension to epidemic models
building of a (friendly enough) simulation system includin g
numerical methods for the computation of the solution
IASI, Roma, January 26, 2009 – p. 96/99
Future work
systematic use of the numerical method for a complete
analysis of some specific population models
extension of the method to age structured models with
diffusion
extension to epidemic models
building of a (friendly enough) simulation system includin g
numerical methods for the computation of the solution
computation of steady states
IASI, Roma, January 26, 2009 – p. 97/99
Future work
systematic use of the numerical method for a complete
analysis of some specific population models
extension of the method to age structured models with
diffusion
extension to epidemic models
building of a (friendly enough) simulation system includin g
numerical methods for the computation of the solution
computation of steady states
stability analysis via numerical computation of
characteristic roots
IASI, Roma, January 26, 2009 – p. 98/99
THANK YOU FORYOUR ATTENTION
IASI, Roma, January 26, 2009 – p. 99/99
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