Study of the Yoccoz-Birkeland population model

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Study of the Yoccoz-Birkeland population model

Sylvain Arlot

CNRS, ´ Ecole Normale Sup´ erieure

Universit` a degli studi di Siena, Nov, 3rd 2009

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Plan

1 The model

Voles in the Arctic Description of the model Mathematical analysis

2 Simulation study Principle

Exploring the parameter space

Study of an attractor: (0.15 ; 0.30 ; 8.25)

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Clethrionomys glaerolus Microtus epiroticus on Svalbard:

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Microtus epiroticus on Svalbard

arctic environment high fertility rate

few breeding places (permafrost)

no reproduction in winter (snow)

(N. Yoccoz, Ims & Steen 1993)

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Between-year variation (N. Yoccoz et al )

0 10 20 30 40 50 60 70 80 90 100

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Seasonal variation (N. Yoccoz et al )

0 200 400 600

800

CORE

0 50 100 150 200

250

RIDGE L

0 50 100 150

200

LOWER

0 10 20 30 40 50

60

RIDGE R

0 50 100 150 200

250

BORDER

0 20 40 60 80

100

REM

90 91 92 93 94 95 96 90 91 92 93 94 95 96 90 91 92 93 94 95 96

90 91 92 93 94 95 96 90 91 92 93 94 95 96

90 91 92 93 94 95 96

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Seasonal variation (N. Yoccoz et al )

0 50 100 150 200

250 RIDGE L

90 91 92 93 94 95 96

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Possible mechanisms

High fertility rate, strongly density-dependent, and:

available food quantity (N. Yoccoz et al 2001)

predation by arctic fox (Turchin et al 2000; Frajford 2002) environmental variability (N. Yoccoz & Ims 1999)

plasticity of the maturation age (Lambin & N. Yoccoz 2001)

Are such variations still possible if the maturation age and the

environment are constant?

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Possible mechanisms

High fertility rate, strongly density-dependent, and:

available food quantity (N. Yoccoz et al 2001)

predation by arctic fox (Turchin et al 2000; Frajford 2002) environmental variability (N. Yoccoz & Ims 1999)

plasticity of the maturation age (Lambin & N. Yoccoz 2001)

Are such variations still possible if the maturation age and the

environment are constant?

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Possible mechanisms

High fertility rate, strongly density-dependent, and:

available food quantity (N. Yoccoz et al 2001)

predation by arctic fox (Turchin et al 2000; Frajford 2002) environmental variability (N. Yoccoz & Ims 1999)

plasticity of the maturation age (Lambin & N. Yoccoz 2001)

Are such variations still possible if the maturation age and the

environment are constant?

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Possible mechanisms

High fertility rate, strongly density-dependent, and:

available food quantity (N. Yoccoz et al 2001)

predation by arctic fox (Turchin et al 2000; Frajford 2002) environmental variability (N. Yoccoz & Ims 1999)

plasticity of the maturation age (Lambin & N. Yoccoz 2001)

Are such variations still possible if the maturation age and the

environment are constant?

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Possible mechanisms

High fertility rate, strongly density-dependent, and:

available food quantity (N. Yoccoz et al 2001)

predation by arctic fox (Turchin et al 2000; Frajford 2002) environmental variability (N. Yoccoz & Ims 1999)

plasticity of the maturation age (Lambin & N. Yoccoz 2001)

Are such variations still possible if the maturation age and the

environment are constant?

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Possible mechanisms

High fertility rate, strongly density-dependent, and:

available food quantity (N. Yoccoz et al 2001)

predation by arctic fox (Turchin et al 2000; Frajford 2002) environmental variability (N. Yoccoz & Ims 1999)

plasticity of the maturation age (Lambin & N. Yoccoz 2001)

Are such variations still possible if the maturation age and the

environment are constant?

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The model (J.-C. Yoccoz & Birkeland 1998)

N(t ) = Z A

A

S(a)m ρ (t − a)N(t − a)m(N(t − a))da (female and mature population; A 0 : maturation age)

Fertility: m(N) =

( m 0 if N ≤ 1 m ₀ N ^−γ otherwise Seasonal parameter: m _ρ (t) =

( 0 if 0 ≤ t ≤ ρ mod. 1 1 if ρ ≤ t ≤ 1

Survival rate: S(a) = 1 − a

A ₁

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The model (J.-C. Yoccoz & Birkeland 1998)

N(t ) = Z A

A

S(a)m ρ (t − a)N(t − a)m(N(t − a))da (female and mature population; A 0 : maturation age)

Fertility: m(N) =

( m 0 if N ≤ 1 m ₀ N ^−γ otherwise Seasonal parameter: m _ρ (t) =

( 0 if 0 ≤ t ≤ ρ mod. 1 1 if ρ ≤ t ≤ 1

Survival rate: S(a) = 1 − a

A ₁

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The model (J.-C. Yoccoz & Birkeland 1998)

N(t ) = Z A

A

S(a)m ρ (t − a)N(t − a)m(N(t − a))da (female and mature population; A 0 : maturation age)

Fertility: m(N) =

( m 0 if N ≤ 1 m ₀ N ^−γ otherwise Seasonal parameter: m _ρ (t) =

( 0 if 0 ≤ t ≤ ρ mod. 1 1 if ρ ≤ t ≤ 1

Survival rate: S(a) = 1 − a

A ₁

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Parameters (N. Yoccoz, Ims, Steen 1993)

fertility m ₀ ≈ 54 .

maturation age A 0 ∈ [0.14 ; 0.20] , minimum = 0.10 . maximal age A 1 ≈ 2 .

winter length ρ ∈ [0.35 ; 0.45] (temperate climate) ou ρ ≈ 0.7 (arctic climate).

winter survival rate ≈ 0.1 summer survival rate ≈ 0.85 . density-dependence γ

spring length ε .

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Parameters (N. Yoccoz, Ims, Steen 1993)

fertility m ₀ ≈ 54 .

maturation age A 0 ∈ [0.14 ; 0.20] , minimum = 0.10 . maximal age A 1 ≈ 2 .

winter length ρ ∈ [0.35 ; 0.45] (temperate climate) ou ρ ≈ 0.7 (arctic climate).

winter survival rate ≈ 0.1 summer survival rate ≈ 0.85 .

density-dependence γ

spring length ε .

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Parameters (N. Yoccoz, Ims, Steen 1993)

fertility m ₀ ≈ 54 .

maturation age A 0 ∈ [0.14 ; 0.20] , minimum = 0.10 . maximal age A 1 ≈ 2 .

winter length ρ ∈ [0.35 ; 0.45] (temperate climate) ou ρ ≈ 0.7 (arctic climate).

winter survival rate ≈ 0.1 summer survival rate ≈ 0.85 . density-dependence γ

spring length ε .

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Mathematical analysis: dynamical system

∀t ₀ ∈ [0, 1), Y _t

is the set of continuous functions N:

[−A ₁ , 0] 7→ [0, +∞) such that

N(0) = Z A

A

S(a)m _ρ (t ₀ − a)N(−a)m(N(−a))da

Phase space:

Y ^] = {(t, N) / t ∈ R / Z , N ∈ Y _t } Semi-group (T ^s ) s≥0 : T ^s (t, N) = (t + s mod. 1, N _t ^s ) N

(−a) =

( N(s − a) if 0 ≤ s ≤ a ≤ A

R

A₀

S (b)N(s − a − b)m(N(s − a − b))m

(t + s − a − b)db otherwise

⇒ well-defined for 0 ≤ s ≤ A ₀ ,

⇒ extended to s ≥ 0 by the semi-group property.

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Mathematical analysis: dynamical system

∀t ₀ ∈ [0, 1), Y _t

is the set of continuous functions N:

[−A ₁ , 0] 7→ [0, +∞) such that N(0) =

Z A

A

S(a)m _ρ (t ₀ − a)N(−a)m(N(−a))da Phase space:

Y ^] = {(t, N) / t ∈ R / Z , N ∈ Y _t }

Semi-group (T ^s ) s≥0 : T ^s (t, N) = (t + s mod. 1, N _t ^s ) N

(−a) =

( N(s − a) if 0 ≤ s ≤ a ≤ A

R

A₀

S (b)N(s − a − b)m(N(s − a − b))m

(t + s − a − b)db otherwise

⇒ well-defined for 0 ≤ s ≤ A ₀ ,

⇒ extended to s ≥ 0 by the semi-group property.

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Mathematical analysis: dynamical system

∀t ₀ ∈ [0, 1), Y _t

is the set of continuous functions N:

[−A ₁ , 0] 7→ [0, +∞) such that N(0) =

Z A

A

S(a)m _ρ (t ₀ − a)N(−a)m(N(−a))da Phase space:

Y ^] = {(t, N) / t ∈ R / Z , N ∈ Y _t } Semi-group (T ^s ) s≥0 : T ^s (t, N) = (t + s mod. 1, N _t ^s )

N

(−a) =

( N(s − a) if 0 ≤ s ≤ a ≤ A

R

A₀

S (b)N(s − a − b)m(N(s − a − b))m

(t + s − a − b)db otherwise

⇒ well-defined for 0 ≤ s ≤ A ₀ ,

⇒ extended to s ≥ 0 by the semi-group property.

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Mathematical analysis: assumptions

N 7→ Nm(N) is K f Lipschitz m ₀ ≥ m(N) ≥ m ₀ /2 if N ≤ 1

m ₀ N ^−γ ≥ m(N) ≥ min {1/2, N ^−γ } m ₀ 1 ≥ m ρ (t) ≥ 0 for every t ∈ [0, 1]

m ρ (t) = 1 on some interval of length 1 − ρ − ε > 0

large domain of the parameter space:

γ ≥ 1

A

≥ max {2A

, A

+ 1}

m

R

A₀+ρ+ε

S (a)da > 2

⇒ T ^s : Y ^] 7→ Y ^] is max {1, K _f (A ₁ − A ₀ )} Lipschitz w.r.t.

| · | + k·k _∞

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Mathematical analysis: assumptions

N 7→ Nm(N) is K f Lipschitz m ₀ ≥ m(N) ≥ m ₀ /2 if N ≤ 1

m ₀ N ^−γ ≥ m(N) ≥ min {1/2, N ^−γ } m ₀ 1 ≥ m ρ (t) ≥ 0 for every t ∈ [0, 1]

m ρ (t) = 1 on some interval of length 1 − ρ − ε > 0 large domain of the parameter space:

γ ≥ 1

A

≥ max {2A

, A

+ 1}

m

R

A₀+ρ+ε

S (a)da > 2

⇒ T ^s : Y ^] 7→ Y ^] is max {1, K _f (A ₁ − A ₀ )} Lipschitz w.r.t.

| · | + k·k _∞

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Mathematical analysis: assumptions

N 7→ Nm(N) is K f Lipschitz m ₀ ≥ m(N) ≥ m ₀ /2 if N ≤ 1

m ₀ N ^−γ ≥ m(N) ≥ min {1/2, N ^−γ } m ₀ 1 ≥ m ρ (t) ≥ 0 for every t ∈ [0, 1]

m ρ (t) = 1 on some interval of length 1 − ρ − ε > 0 large domain of the parameter space:

γ ≥ 1

A

≥ max {2A

, A

+ 1}

m

R

A₀+ρ+ε

S (a)da > 2

⇒ T ^s : Y ^] 7→ Y ^] is max {1, K _f (A ₁ − A ₀ )} Lipschitz w.r.t.

| · | + k·k _∞

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Mathematical analysis: attractor

K _t

= {t ₀ }×{N ∈ Y t

/ N min ≤ N(t) ≤ N max and N is L-Lipschitz}

(N _min , N _max and L are explicit functions of m ₀ , A ₀ , A ₁ and γ )

∀t ₀ ∈ [0, 1), K _t

is compact w.r.t. the uniform convergence

∀(0, N) ∈ K ₀ , ∀s ≥ 0, T ^s (0, N) ∈ K _s

∀N ∈ Y ₀ , ∃s ₀ (N) ≥ 0, ∀s ≥ s ₀ , T ^s (0, N) ∈ K _s Λ = {(t, N) / t ∈ R / Z , N ∈ Λ _t } where Λ _t = T

n∈ N T ⁿ (K _t ) Λ is compact, stable by T ^s for all s ≥ 0

For any neighborhood U of Λ and any N ∈ Y 0 , T ^s (0, N) ∈ U

for every s ≥ s ₀ (U, N)

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Mathematical analysis: attractor

K _t

= {t ₀ }×{N ∈ Y t

/ N min ≤ N(t) ≤ N max and N is L-Lipschitz}

(N _min , N _max and L are explicit functions of m ₀ , A ₀ , A ₁ and γ )

∀t ₀ ∈ [0, 1), K _t

is compact w.r.t. the uniform convergence

∀(0, N) ∈ K ₀ , ∀s ≥ 0, T ^s (0, N) ∈ K _s

∀N ∈ Y ₀ , ∃s ₀ (N) ≥ 0, ∀s ≥ s ₀ , T ^s (0, N) ∈ K _s Λ = {(t, N) / t ∈ R / Z , N ∈ Λ _t } where Λ _t = T

n∈ N T ⁿ (K _t ) Λ is compact, stable by T ^s for all s ≥ 0

For any neighborhood U of Λ and any N ∈ Y 0 , T ^s (0, N) ∈ U

for every s ≥ s ₀ (U, N)

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Mathematical analysis: attractor

K _t

= {t ₀ }×{N ∈ Y t

/ N min ≤ N(t) ≤ N max and N is L-Lipschitz}

(N _min , N _max and L are explicit functions of m ₀ , A ₀ , A ₁ and γ )

∀t ₀ ∈ [0, 1), K _t

is compact w.r.t. the uniform convergence

∀(0, N) ∈ K ₀ , ∀s ≥ 0, T ^s (0, N) ∈ K _s

∀N ∈ Y ₀ , ∃s ₀ (N) ≥ 0, ∀s ≥ s ₀ , T ^s (0, N) ∈ K _s Λ = {(t, N) / t ∈ R / Z , N ∈ Λ _t } where Λ _t = T

n∈ N T ⁿ (K _t ) Λ is compact, stable by T ^s for all s ≥ 0

For any neighborhood U of Λ and any N ∈ Y 0 , T ^s (0, N) ∈ U

for every s ≥ s ₀ (U, N)

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Mathematical analysis: attractor

K _t

= {t ₀ }×{N ∈ Y t

/ N min ≤ N(t) ≤ N max and N is L-Lipschitz}

(N _min , N _max and L are explicit functions of m ₀ , A ₀ , A ₁ and γ )

∀t ₀ ∈ [0, 1), K _t

is compact w.r.t. the uniform convergence

∀(0, N) ∈ K ₀ , ∀s ≥ 0, T ^s (0, N) ∈ K _s

∀N ∈ Y ₀ , ∃s ₀ (N) ≥ 0, ∀s ≥ s ₀ , T ^s (0, N) ∈ K _s Λ = {(t, N) / t ∈ R / Z , N ∈ Λ _t } where Λ _t = T

n∈ N T ⁿ (K _t ) Λ is compact, stable by T ^s for all s ≥ 0

For any neighborhood U of Λ and any N ∈ Y 0 , T ^s (0, N) ∈ U

for every s ≥ s ₀ (U, N)

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Mathematical analysis: attractor

K _t

= {t ₀ }×{N ∈ Y t

/ N min ≤ N(t) ≤ N max and N is L-Lipschitz}

(N _min , N _max and L are explicit functions of m ₀ , A ₀ , A ₁ and γ )

∀t ₀ ∈ [0, 1), K _t

is compact w.r.t. the uniform convergence

∀(0, N) ∈ K ₀ , ∀s ≥ 0, T ^s (0, N) ∈ K _s

∀N ∈ Y ₀ , ∃s ₀ (N) ≥ 0, ∀s ≥ s ₀ , T ^s (0, N) ∈ K _s Λ = {(t, N) / t ∈ R / Z , N ∈ Λ _t } where Λ _t = T

n∈ N T ⁿ (K _t ) Λ is compact, stable by T ^s for all s ≥ 0

For any neighborhood U of Λ and any N ∈ Y 0 , T ^s (0, N) ∈ U

for every s ≥ s ₀ (U, N)

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Mathematical analysis: attractor

K _t

= {t ₀ }×{N ∈ Y t

/ N min ≤ N(t) ≤ N max and N is L-Lipschitz}

(N _min , N _max and L are explicit functions of m ₀ , A ₀ , A ₁ and γ )

∀t ₀ ∈ [0, 1), K _t

is compact w.r.t. the uniform convergence

∀(0, N) ∈ K ₀ , ∀s ≥ 0, T ^s (0, N) ∈ K _s

∀N ∈ Y ₀ , ∃s ₀ (N) ≥ 0, ∀s ≥ s ₀ , T ^s (0, N) ∈ K _s Λ = {(t, N) / t ∈ R / Z , N ∈ Λ _t } where Λ _t = T

n∈ N T ⁿ (K _t ) Λ is compact, stable by T ^s for all s ≥ 0

For any neighborhood U of Λ and any N ∈ Y 0 , T ^s (0, N) ∈ U

for every s ≥ s ₀ (U, N)

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Mathematical results (J.-C. Yoccoz & Birkeland 1998)

Theorem: Λ is an attractor for (T ^s ) s≥0 , which attracts any initial condition (0, N) with N ∈ Y 0 .

Unseasonal model: an equilibrium exists, that is stable when

1 ≤ γ ≤ γ ₀ ≈ 6.2 . Hopf bifurcation when γ crosses γ ₀ ?

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Mathematical results (J.-C. Yoccoz & Birkeland 1998)

Theorem: Λ is an attractor for (T ^s ) s≥0 , which attracts any initial condition (0, N) with N ∈ Y 0 .

Unseasonal model: an equilibrium exists, that is stable when

1 ≤ γ ≤ γ ₀ ≈ 6.2 . Hopf bifurcation when γ crosses γ ₀ ?

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Plan

1 The model

Voles in the Arctic Description of the model Mathematical analysis

2 Simulation study Principle

Exploring the parameter space

Study of an attractor: (0.15 ; 0.30 ; 8.25)

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Discretization of the dynamical system

Each year is divided into p = 100 intervals

Define I k = [(k − 1)/p; k /p) for k ≥ −pA ₁ , k ∈ N n k : number of births in I k

N k : average number of mature females during I k

e _k : average of m _ρ,ε over I _k

s _k : fraction of surviving mature females among females born k time intervals before

n _i = m(N _i ) × N _i × e _i p N _i =

A

p

X

k =1

(s _k × n i−k )

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Discretization of the dynamical system

Each year is divided into p = 100 intervals

Define I k = [(k − 1)/p; k /p) for k ≥ −pA ₁ , k ∈ N n k : number of births in I k

N k : average number of mature females during I k

e _k : average of m _ρ,ε over I _k

s _k : fraction of surviving mature females among females born k time intervals before

n _i = m(N _i ) × N _i × e _i p N _i =

A

p

X

k =1

(s _k × n i−k )

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Discretization of the dynamical system

Each year is divided into p = 100 intervals

Define I k = [(k − 1)/p; k /p) for k ≥ −pA ₁ , k ∈ N n k : number of births in I k

N k : average number of mature females during I k

e _k : average of m _ρ,ε over I _k

s _k : fraction of surviving mature females among females born k time intervals before

n _i = m(N _i ) × N _i × e _i p N _i =

A

p

X

k =1

(s _k × n i−k )

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Principle of the simulation experiments

A ₁ = 2 m ₀ = 50 ε = 0.1 (A ₀ ; ρ ; γ) varying

Stationary behaviour: t ≥ t 0 = 10000 (for one arbitrary initial condition).

t = 0 at the end of summer Bifurcation diagram: (x, N(t)) .

Projection on R ³ : (N(t ), N(t + 1), N(t + 2)) (Whitney

theorem, assuming a fractal dimension D < 3/2 for ω(N)).

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Principle of the simulation experiments

A ₁ = 2 m ₀ = 50 ε = 0.1 (A ₀ ; ρ ; γ) varying

Stationary behaviour: t ≥ t 0 = 10000 (for one arbitrary initial condition).

t = 0 at the end of summer Bifurcation diagram: (x, N(t)) .

Projection on R ³ : (N(t ), N(t + 1), N(t + 2)) (Whitney

theorem, assuming a fractal dimension D < 3/2 for ω(N)).

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Principle of the simulation experiments

A ₁ = 2 m ₀ = 50 ε = 0.1 (A ₀ ; ρ ; γ) varying

Stationary behaviour: t ≥ t 0 = 10000 (for one arbitrary initial condition).

t = 0 at the end of summer Bifurcation diagram: (x, N(t)) .

Projection on R ³ : (N(t ), N(t + 1), N(t + 2)) (Whitney

theorem, assuming a fractal dimension D < 3/2 for ω(N)).

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(A ₀ = 0.18 ; ρ = 0.41 ; γ ∈ [2; 16])

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(A ₀ = 0.18 ; ρ = 0.41 ; γ ∈ [2; 16]) : Fractal dimension

2 4 6 8 10 12 14 16

−0.2 0 0.2 0.4 0.6 0.8 1 1.2

γ df(Λγ)

Evolution de la dimension fractale : A₀ = 0.18 ; ρ = 0.41 ; ε = 0

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(A ₀ = 0.18 ; ρ = 0.41 ; γ = 8.61) : d _f ≈ 1.06

1 2

3 4

5 6

1 2 3 4 5 6 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

N(t) A₀ = 0.18 A₁ = 2 ; ρ = 0.410 ε = 0 ; m₀ = 50 γ = 8.61 m(N) C¹ 10001 ≤ t ≤ 19998 ; 1.2935 ≤ N ≤ 5.1125 ; 100 pas de temps par an

N(t+1)

N(t+2)

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(A ₀ = 0.18 ; ρ = 0.41 ; γ = 9.89) : d _f ≈ 0.92

1 2

3 4

5

1 2 3 4 5 1 1.5 2 2.5 3 3.5 4 4.5 5

6

2 10 8 4

N(t) A₀ = 0.18 A₁ = 2 ; ρ = 0.410 ε = 0 ; m₀ = 50 γ = 9.89 m(N) C¹ 10001 ≤ t ≤ 19998 ; 1.2334 ≤ N ≤ 4.6427 ; 100 pas de temps par an

9

N(t+1) 3 7 1 5

N(t+2)

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(A ₀ = 0.18 ; ρ ∈ [0 ; 0.5] ; γ = 8.25) [see video]

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(A ₀ = 0.18 ; ρ = 0.16 ; γ = 8.25) : d _f ≈ 0.99

1.6 1.8 2 2.2 2.4 2.6 2.8 3

1.5 2 2.5 3 1.6 1.8 2 2.2 2.4 2.6 2.8 3

N(t) A₀ = 0.18 A₁ = 2 ; ρ = 0.160 ε = 0.1 ; m₀ = 50 γ = 8.25 m(N) C¹ 10002 ≤ t ≤ 19998 ; 1.6909 ≤ N ≤ 2.6275 ; 100 pas de temps par an

N(t+1)

N(t+2)

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(A ₀ = 0.15 ; ρ = 0.78; γ = 8.25) : d _f ≈ 1.25

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(A ₀ ∈ [0 ; 0.4] ; ρ = 0.30 ; γ = 8.25)

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(A ₀ = 0.18 ; ρ = 0.30 ; γ = 8.25) : d _f ≈ 1.19

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(A ₀ = 0.15 ; ρ = 0.30 ; γ = 8.25) : d _f ≈ 1.33

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One connex component (physical measure)

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Injectivity of the projection R ²⁰¹ 7→ R ³

0 10 20 30 40 50 60 70 80

0 10 20 30

j = n^o sur la carte sup des rapports des normes LInf(R201) / LInf(R3) sur la boule B(xj,0.1) de LInf(R201)

0 10 20 30 40 50 60 70 800

500 1000 1500

nombre de points dans la boule B(xj,0.1) de LInf(R201) Test de l’injectivité de la projection L^Inf(R²⁰¹) → L^Inf(R³) ; δ_t = 0

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Dynamics (1): equilibrium, period-2, period-3

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

A B

N(2t+1)

C2

N(2t+2)

C3 C1

N(2t+3)

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Dynamics (2)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

2 eq 1

0

N(2t+1)

6

3

4

N(2t+2)

5

N(2t+3)

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Dynamics (3) [+ see video]

2 2.5 3 3.5 4 4.5 5

1.2 1.4 1.6 1.8

2 2.5 3 3.5 4 4.5

eq D3 B3

B0 A0 C0 A1 D0 C3

B1

N(2t+1)

C1 D1 A2 B2 A3 D2 C2

N(2t+2)

N(2t+3)

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Remarkable points: local unstable manifolds

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

A B

N(2t+1)

C2

N(2t+2)

C3 C1

N(2t+3)

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Equilibrium: expansion of the unstable manifold (1)

2 2.5 3 3.5 4 4.5 5 5.5

1.2 1.4 1.6 1.8 2 2 3 4 5 6

N(2t+1) Variéte instable : f

(W

(x

eq

))

N(2t+2)

N(2t+3)

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Equilibrium: expansion of the unstable manifold (2)

2 2.5 3 3.5 4 4.5 5 5.5

1.2 1.4 1.6 1.8 2 2 3 4 5 6

N(2t+1) Variéte instable : f

(W

(x

eq

))

N(2t+2)

N(2t+3)

36/73

Equilibrium: expansion of the unstable manifold (3)

2 2.5 3 3.5 4 4.5 5 5.5

1.2 1.4 1.6 1.8 2 2 3 4 5 6

N(2t+1) Variéte instable : f

(W

(x

eq

))

N(2t+2)

N(2t+3)

37/73

Equilibrium: expansion of the unstable manifold (4)

2 2.5 3 3.5 4 4.5 5 5.5

1.2 1.4 1.6 1.8 2 2 3 4 5 6

N(2t+1) Variéte instable : f

(W

(x

eq

))

N(2t+2)

N(2t+3)

38/73

Equilibrium: global unstable manifold

2 3

4 5

6

1.2 1.4 1.6 1.8 2 2 2.5 3 3.5 4 4.5 5 5.5

N(2t+1) Variéte instable : f

(W

(x

))

N(2t+2)

N(2t+3)

39/73

Period 2: expansion of the unstable manifold (1)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(eq. num 2) − step 07

N(2t+2)

N(2t+3)

40/73

Period 2: expansion of the unstable manifold (2)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(eq. num 2) − step 08

N(2t+2)

N(2t+3)

41/73

Period 2: expansion of the unstable manifold (3)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(eq. num 2) − step 09

N(2t+2)

N(2t+3)

42/73

Period 2: expansion of the unstable manifold (4)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(eq. num 2) − step 10

N(2t+2)

N(2t+3)

43/73

Period 3: expansion of the unstable manifold (1)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 15

N(2t+2)

N(2t+3)

44/73

Period 3: expansion of the unstable manifold (2)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 16

N(2t+2)

N(2t+3)

45/73

Period 3: expansion of the unstable manifold (3)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 17

N(2t+2)

N(2t+3)

46/73

Period 3: expansion of the unstable manifold (4)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 18

N(2t+2)

N(2t+3)

47/73

Period 3: expansion of the unstable manifold (5)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 19

N(2t+2)

N(2t+3)

48/73

Period 3: expansion of the unstable manifold (6)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 20

N(2t+2)

N(2t+3)

49/73

Period 3: expansion of the unstable manifold (7)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 21

N(2t+2)

N(2t+3)

50/73

Period 3: expansion of the unstable manifold (8)

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 22

N(2t+2)

N(2t+3)

51/73

Period 3: global unstable manifold

2 2.5

3 3.5

4 4.5

5

1.2 1.3 1.4 1.5 1.6 1.7 1.8

2 2.5 3 3.5 4 4.5

N(2t+1) Var_u(per. 3) − step 23

N(2t+2)

N(2t+3)

52/73

Dynamics in continuous time: equilibrium

0 0.5 1 1.5 2 2.5 3 3.5 4

0 1 2 3 4 5 6 7

L’équilibre : (N(t),N(t+1),N(t+2))=(4.5381,1.2352,4.5381)

t

N(t)

53/73

Dynamics in continuous time: birth at equilibrium

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Naissances à l’équilibre : (N(t),N(t+1),N(t+2))=(4.5381,1.2352,4.5381)

t

naiss(t)

54/73

Dynamics in continuous time: period-3 orbit (triangle)

−5 −4 −3 −2 −1 0 1 2 3 4 5

0 1 2 3 4 5 6 7

A₀ = 0.15 ; point 78 [ 73 → 78 → 60 ]

t−197440

N(t)

55/73

Attractor: there is a “fold”

2 3

4 5

6

1.2 1.4 1.6 1.8 2 2 2.5 3 3.5 4 4.5 5 5.5

N(2t+1) Variéte instable : f

(W

(x

))

N(2t+2)

N(2t+3)

56/73

Folding (1): before the fold

57/73

Folding (2)

58/73

Folding (3): curvature

0 0.5 1 1.5 2 2.5 3 3.5 4

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

t log10(Maximum de courbure a l’instant t)

Formation du pli sur 4 ans

59/73

Fold: dynamics in continuous time

−20 −1 0 1 2 3 4

1 2 3 4 5 6 7

t N

(t)

Population mature au niveau du pli

j = 1

j = 690

j = 999

60/73

Global dynamics: summary

shrinking stretching

folding (stable in the parameter space, see video)

“branching” (from equilibrium/period-2 to period-3)

60/73

Global dynamics: summary

shrinking stretching

folding (stable in the parameter space, see video)

“branching” (from equilibrium/period-2 to period-3)

60/73

Global dynamics: summary

shrinking stretching

folding (stable in the parameter space, see video)

“branching” (from equilibrium/period-2 to period-3)

61/73

Local geometry: zoom on a “filament”

23

62/73

Local geometry: zoom on a peak

63/73

Local geometry: zoom on a peak

54

64/73

Dynamics in continuous time: in a “peak”

−5 −4 −3 −2 −1 0 1 2 3 4 5

0 1 2 3 4 5 6 7

A₀ = 0.15 ; point 54 [ 69 → 54 → 01−05 ]

t−182848

N(t)

65/73

Sensitivity to initial conditions around the equilibrium

66/73

Conclusions: conjectures

For some of the parameter values:

periodic orbits, doubling period bifurcations, sub-harmonic cascade.

Hopf bifurcations.

Chaotic dynamics, Henon-type attractors.

For the attractor (0.15 ; 0.30 ; 8.25) :

Strange attractor. Persistent chaotic dynamics.

Fractal dimension in (1 , 3/2) . Quasi-hyperbolicity.

Spectral decomposition.

66/73

Conclusions: conjectures

For some of the parameter values:

periodic orbits, doubling period bifurcations, sub-harmonic cascade.

Hopf bifurcations.

Chaotic dynamics, Henon-type attractors.

For the attractor (0.15 ; 0.30 ; 8.25) :

Strange attractor. Persistent chaotic dynamics.

Fractal dimension in (1 , 3/2) . Quasi-hyperbolicity.

Spectral decomposition.

67/73

Biological conclusions

Simple model, unrealistic.

Sufficient to induce chaotic dynamics, not well understood mathematically.

Qualitatively looks like real data.

68/73

69/73

Smoothing of the fertility rate

0 0.5 1 1.5 2 2.5

0 10 20 30 40 50 60

N = effectif mature

m(N) = fécondite individuelle

Comparaison des fonctions de fécondité ; m

0

= 50 ; γ = 8.25

70/73

Smoothing of the seasonal factor

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.2 0.4 0.6 0.8 1

t en années m

(N) = facteur saisonnier

Régularisation du facteur saisonnier (ρ = 0.41)

Saisons non−continues : ε_ete = 0 Saisons C¹ : ε_ete = 0.1

71/73

Estimation of the fractal dimension

−51 −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0

1.5 2 2.5 3 3.5 4 4.5 5

log₁₀(r) log10(N(r))

A₀=0.15 ; ρ=0.3 ; γ=8.25 ; dimension fractale de l’attracteur

log₁₀(N(r)) = −1.3289× log₁₀(r)+1.1915

72/73

Dynamics on the attractor: expansion areas

73/73

Fold: unstable direction

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

−3

−2

−1 0 1 2 3 4 5 6

t

dT

(t)

Premier vecteur propre de la différentielle de T

en x=x

j

j = 1 ; λ = −3.493

j = 686 ; λ = −5.2058

j = 999 ; λ = −4.8505