La complessa dinamica del modello di Gurtin e MacCamy

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1 IASI, Roma, January 26, 29 p. 1/99 La complessa dinamica del modello di Gurtin e MacCamy Mimmo Iannelli Università di Trento

2 IASI, Roma, January 26, 29 p. 2/99 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

3 IASI, Roma, January 26, 29 p. 3/99 Outline of the talk A collaboration with: F. Milner, Arizona University, Tempe, Mathematics Department 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

4 IASI, Roma, January 26, 29 p. 4/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a β(a,s 1 (t),...,s n (t))p(a,t) da, S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n,

5 IASI, Roma, January 26, 29 p. 5/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a β(a,s 1 (t),...,s n (t))p(a,t) da, age-distribution S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n,

6 IASI, Roma, January 26, 29 p. 6/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a 2 β(a,s 1 (t),...,s n (t))p(a,t) da, mortality S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n,

7 IASI, Roma, January 26, 29 p. 7/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a β(a,s 1 (t),...,s n (t))p(a,t) da, S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n,

8 IASI, Roma, January 26, 29 p. 8/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a β(a,s 1 (t),...,s n (t))p(a,t) da, S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n, fertility

9 IASI, Roma, January 26, 29 p. 9/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a β(a,s 1 (t),...,s n (t))p(a,t) da, S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n,

10 IASI, Roma, January 26, 29 p. 1/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a β(a,s 1 (t),...,s n (t))p(a,t) da, S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n,

11 IASI, Roma, January 26, 29 p. 11/99 Gurtin-MacCamy The Gurtin-MacCamy system p p (a,t) + t a (a,t) + µ(a,s 1(t),...,S n (t))p(a,t) =, p(,t) = a β(a,s 1 (t),...,s n (t))p(a,t) da, S i (t) = a p(a, ) = p (a). γ i (a)p(a,t) da, i = 1,...,n,

12 IASI, Roma, January 26, 29 p. 12/99 Gurtin-MacCamy The basic ingredients p(a, t) age-distribution of the population S i (t) = a β(a,s 1 (t),...,s n (t)) µ(a,s 1 (t),...,s n (t)) γ i (a)p(a,t)da weighted selection of the population fertility mortality

13 IASI, Roma, January 26, 29 p. 13/99 Structured logistic growth Logistic growth one single size: S(t) = fertility: mortality: a γ(a)p(a, t)da β(a,x) = R β (a)φ(x) µ(a,x) = µ (a) + m(a)ψ(x)

14 IASI, Roma, January 26, 29 p. 14/99 Structured logistic growth Logistic growth one single size: S(t) = fertility: mortality: with a γ(a)p(a, t)da β(a,x) = R β (a)φ(x) µ(a,x) = µ (a) + m(a)ψ(x) γ(a) Φ(x) Ψ(x) non-decreasing decreasing increasing

15 IASI, Roma, January 26, 29 p. 15/99 Structured logistic growth Logistic growth one single size: S(t) = fertility: mortality: with a γ(a)p(a, t)da β(a,x) = R β (a)φ(x) µ(a,x) = µ (a) + m(a)ψ(x) γ(a) Φ(x) Ψ(x) non-decreasing decreasing increasing β (a) and m(a) describe how crowding impacts on different ages

16 IASI, Roma, January 26, 29 p. 16/99 Structured logistic growth Logistic growth one single size: S(t) = fertility: mortality: with a γ(a)p(a, t)da β(a,x) = R β (a)φ(x) µ(a,x) = µ (a) + m(a)ψ(x) γ(a) Φ(x) Ψ(x) non-decreasing decreasing increasing β (a) and m(a) describe how crowding impacts on different ages R = basic reproduction number

17 IASI, Roma, January 26, 29 p. 17/99 Structured logistic growth Logistic growth one single size: S(t) = fertility: mortality: with a γ(a)p(a, t)da β(a,x) = R β (a)φ(x) µ(a,x) = µ (a) + m(a)ψ(x) γ(a) Φ(x) Ψ(x) non-decreasing decreasing increasing β (a) and m(a) describe how crowding impacts on different ages R = the number of off-springs produced during the whole life

18 IASI, Roma, January 26, 29 p. 18/99 Structured logistic growth The search for a stationary state p (a) p a (a) + µ(a,s )p (a) = = p (a) = Π(a,S )p (), Π(a,S) = e a µ(σ,s)dσ 1 = a β(a,s )Π(a,S )da, p () = a S γ i (a)π(a,s )da

19 IASI, Roma, January 26, 29 p. 19/99 Structured logistic growth a 1 = β(a, S )Π(a, S )da

20 IASI, Roma, January 26, 29 p. 2/99 Structured logistic growth 1 = R Φ(S ) a β (a)e a µ (σ)dσ e Ψ(S ) a m(σ)dσ da

21 IASI, Roma, January 26, 29 p. 21/99 Structured logistic growth 1 = R Φ(S ) a β (a)e a µ (σ)dσ e Ψ(S ) a m(σ)dσ da decreasing as a function of S

22 IASI, Roma, January 26, 29 p. 22/99 Structured logistic growth 1 = R Φ(S ) a β (a)e a µ (σ)dσ e Ψ(S ) a m(σ)dσ da bifurcation graph

23 IASI, Roma, January 26, 29 p. 23/99 Structured logistic growth 1 = R Φ(S ) a β (a)e a µ (σ)dσ e Ψ(S ) a m(σ)dσ da bifurcation graph trivial state

24 IASI, Roma, January 26, 29 p. 24/99 Structured logistic growth 1 = R Φ(S ) a β (a)e a µ (σ)dσ e Ψ(S ) a m(σ)dσ da bifurcation graph non trivial state

25 IASI, Roma, January 26, 29 p. 25/99 Structured logistic growth Stability by linearization at deviation from the steady state p (a) v(a, t) = p(a, t) p (a)

26 IASI, Roma, January 26, 29 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 v (a, t) + t a (a, t) + µ(a, S )v(a, t)+ +p (a) µ a S (a, S ) γ(a)v(a, t)da = v(, t) = a β(a, S )v(a, t)da+ + a p (σ) β S (σ, S )dσ a γ(a)v(a, t)da

27 IASI, Roma, January 26, 29 p. 27/99 Structured logistic growth Stability by linearization at p (a) deviation from the steady state v(a, t) = p(a, t) p (a) v v (a, t) + t a (a, t) + µ(a, S )v(a, t)+ +p (a) µ a S (a, S ) γ(a)v(a, t)da = v(, t) = a β(a, S )v(a, t)da+ + a p (σ) β S (σ, S )dσ a γ(a)v(a, t)da

28 IASI, Roma, January 26, 29 p. 28/99 Structured logistic growth Characteristic equation det 1 K (λ) b K 1 (λ) K 1 (λ) 1 K 11 (λ) = K (t) = β(t, S )Π(t, S ) K 1 (t) = γ(t)π(t, S ) a K 1 (t) = p µ () S (σ, S )K (t + σ)dσ a K 11 (t) = p µ () S (σ, S )K 1 (t + σ)dσ a b = p β () S (σ, S )Π(σ, S )dσ

29 IASI, Roma, January 26, 29 p. 29/99 Structured logistic growth Characteristic equation det 1 K (λ) b K 1 (λ) K 1 (λ) 1 K 11 (λ) = K (t) = β(t, S )Π(t, S ) K 1 (t) = γ(t)π(t, S ) a K 1 (t) = p µ () S (σ, S )K (t + σ)dσ a K 11 (t) = p µ () S (σ, S )K 1 (t + σ)dσ a b = p β () S (σ, S )Π(σ, S )dσ

30 IASI, Roma, January 26, 29 p. 3/99 Structured logistic growth Characteristic equation det 1 K (λ) b K 1 (λ) K 1 (λ) 1 K 11 (λ) = K (t) = β(t, S )Π(t, S ) K 1 (t) = γ(t)π(t, S ) a K 1 (t) = p µ () S (σ, S )K (t + σ)dσ a K 11 (t) = p µ () S (σ, S )K 1 (t + σ)dσ a b = p β () S (σ, S )Π(σ, S )dσ

31 IASI, Roma, January 26, 29 p. 31/99 Structured logistic growth Characteristic equation det 1 K (λ) b K 1 (λ) K 1 (λ) 1 K 11 (λ) = K (t) = β(t, S )Π(t, S ) K 1 (t) = γ(t)π(t, S ) a K 1 (t) = p µ () S (σ, S )K (t + σ)dσ a K 11 (t) = p µ () S (σ, S )K 1 (t + σ)dσ a b = p β () S (σ, S )Π(σ, S )dσ

32 IASI, Roma, January 26, 29 p. 32/99 Structured logistic growth Characteristic equation det 1 K (λ) b K 1 (λ) K 1 (λ) 1 K 11 (λ) = K (t) = β(t, S )Π(t, S ) K 1 (t) = γ(t)π(t, S ) a K 1 (t) = p µ () S (σ, S )K (t + σ)dσ a K 11 (t) = p µ () S (σ, S )K 1 (t + σ)dσ a b = p β () S (σ, S )Π(σ, S )dσ

33 IASI, Roma, January 26, 29 p. 33/99 Structured logistic growth Characteristic equation det 1 K (λ) b K 1 (λ) K 1 (λ) 1 K 11 (λ) = If all characteristic roots have negative real part then the steady state p (a) is stable. If at least one of the characteristic roots has a positive real part then the state is unstable.

34 IASI, Roma, January 26, 29 p. 34/99 Structured logistic growth stable 1 R

35 IASI, Roma, January 26, 29 p. 35/99 Structured logistic growth stable unstable 1 R

36 IASI, Roma, January 26, 29 p. 36/99 Structured logistic growth stable unstable 1 R

37 IASI, Roma, January 26, 29 p. 37/99 Structured logistic growth stable unstable 1 R

38 IASI, Roma, January 26, 29 p. 38/99 Structured logistic growth stable unstable 1 R

39 IASI, Roma, January 26, 29 p. 39/99 Structured logistic growth stable unstable 1 R

40 IASI, Roma, January 26, 29 p. 4/99 Structured logistic growth stable unstable bifurcation point: two complex conjugate roots cross the imaginary axis and a periodic solution arises by Hopf bifurcation 1 R

41 IASI, Roma, January 26, 29 p. 41/99 Juveniles-adult dynamics The example of juveniles-adults dynamics a two selected groups J(t) = a p(a, t) da, juveniles A(t) = a p(a,t) da, adults

42 IASI, Roma, January 26, 29 p. 42/99 Juveniles-adult dynamics The example of juveniles-adults dynamics a two selected groups J(t) = a p(a, t) da, juveniles A(t) = a p(a,t) da, adults a is the maturation age

43 IASI, Roma, January 26, 29 p. 43/99 Juveniles-adult dynamics The example of juveniles-adults dynamics a two selected groups J(t) = a p(a, t) da, juveniles A(t) = a p(a,t) da, adults separated niches Allee effect cannibalism

44 IASI, Roma, January 26, 29 p. 44/99 Juveniles-adult dynamics The case of two different ecological niches β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a

45 IASI, Roma, January 26, 29 p. 45/99 Juveniles-adult dynamics The case of two different ecological niches β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a

46 IASI, Roma, January 26, 29 p. 46/99 Juveniles-adult dynamics The case of two different ecological niches β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a

47 IASI, Roma, January 26, 29 p. 47/99 Juveniles-adult dynamics The case of two different ecological niches β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a

48 IASI, Roma, January 26, 29 p. 48/99 Juveniles-adult dynamics The case of two different ecological niches β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a

49 IASI, Roma, January 26, 29 p. 49/99 Juveniles-adult dynamics The case of two different ecological niches β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a

50 IASI, Roma, January 26, 29 p. 5/99 Juveniles-adult dynamics The case of two different ecological niches J 1 R,1 R,2 R

51 IASI, Roma, January 26, 29 p. 51/99 Juveniles-adult dynamics The Allee effect β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a+ [θ 1 µ (a) + θ 2 m 1 J] χ [,a ](a)α(a) A positive effect (a decrease of mortality) on juveniles, due to adults presence

52 IASI, Roma, January 26, 29 p. 52/99 Juveniles-adult dynamics The Allee effect β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a+ [θ 1 µ (a) + θ 2 m 1 J] χ [,a ](a)α(a) A positive effect (a decrease of mortality) on juveniles, due to adults presence

53 IASI, Roma, January 26, 29 p. 53/99 Juveniles-adult dynamics The Allee effect β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a+ [θ 1 µ (a) + θ 2 m 1 J] χ [,a ](a)α(a) A positive effect (a decrease of mortality) on juveniles, due to adults presence

54 IASI, Roma, January 26, 29 p. 54/99 Juveniles-adult dynamics The Allee effect β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) µ(a,j,a) = µ (a) + m 1 χ [,a ](a)j + m 2 χ [a,a ](a)a+ [θ 1 µ (a) + θ 2 m 1 J] χ [,a ](a)α(a) A positive effect (a decrease of mortality) on juveniles, due to adults presence

55 IASI, Roma, January 26, 29 p. 55/99 Juveniles-adult dynamics The Allee effect

56 IASI, Roma, January 26, 29 p. 56/99 Juveniles-adult dynamics The Allee effect

57 IASI, Roma, January 26, 29 p. 57/99 Juveniles-adult dynamics Cannibalism (of adults on juveniles) β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) A µ(a,j,a) = µ (a) + m 1 χ [,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

58 IASI, Roma, January 26, 29 p. 58/99 Juveniles-adult dynamics Cannibalism (of adults on juveniles) β(a,j,a) = R bχ [a,a ](a)e (b 1J+b 2 A) A µ(a,j,a) = µ (a) + m 1 χ [,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

59 IASI, Roma, January 26, 29 p. 59/99 Juveniles-adult dynamics Cannibalism (of adults on juveniles) J 1 R,2 R,1 R

60 IASI, Roma, January 26, 29 p. 6/99 A numerical method for stability analysis The starting point: linearization at a steady state p (a) v v (a,t) + t a (a,t) + µ(a,s )v(a,t)+ v(,t) = a +p (a) µ S (a,s ) β(a,s )v(a,t)da+ a γ(a)v(a,t)da = + a p (σ) β S (σ,s )dσ a γ(a)v(a, t)da

61 IASI, Roma, January 26, 29 p. 61/99 A numerical method for stability analysis The starting point: the resulting characteristic equation 1 K (λ) b K 1 (λ) det = K 1 (λ) 1 K 11 (λ) The goal: to approximate the roots

62 IASI, Roma, January 26, 29 p. 62/99 A numerical method for stability analysis The starting point: the resulting characteristic equation 1 K (λ) b K 1 (λ) det = K 1 (λ) 1 K 11 (λ) The goal: to approximate the roots reformulation of the linearization as an abstract Cauchy problem discrete approximation of the generator computation of the spectrum of the approximated generator

63 IASI, Roma, January 26, 29 p. 63/99 A numerical method for stability analysis Reformulation as an abstract Cauchy problem v v (a,t) + t a (a,t) + µ(a,s )v(a,t)+ v(,t) = a +p (a) µ S (a,s ) β(a,s )v(a,t)da+ a γ(a)v(a,t)da = v(a,) = v (a) + a p (σ) β S (σ,s )dσ a γ(a)v(a, t)da

64 IASI, Roma, January 26, 29 p. 64/99 A numerical method for stability analysis Reformulation as an abstract Cauchy problem v v (a,t) + t a (a,t) + µ(a,s )v(a,t)+ v(,t) = a +p (a) µ S (a,s ) β(a,s )v(a,t)da+ a γ(a)v(a,t)da = v(a,) = v (a) + a p (σ) β S (σ,s )dσ a γ(a)v(a, t)da

65 IASI, Roma, January 26, 29 p. 65/99 A numerical method for stability analysis Reformulation as an abstract Cauchy problem v v (a,t) + (a,t) + (Hv(,t))(a) = t a v(,t) = K v(,t) v(a,) = v (a)

66 IASI, Roma, January 26, 29 p. 66/99 A numerical method for stability analysis Reformulation as an abstract Cauchy problem v v (a,t) + (a,t) + (Hv(,t))(a) = t a v(,t) = K v(,t) v(a,) = v (a) d u(t) = Au(t), t, dt u() = u X.

67 IASI, Roma, January 26, 29 p. 67/99 A numerical method for stability analysis Reformulation as an abstract Cauchy problem v v (a,t) + (a,t) + (Hv(,t))(a) = t a v(,t) = K v(,t) v(a,) = v (a) d u(t) = Au(t), t, dt u() = u X. L 1 ([,a ], R)

68 IASI, Roma, January 26, 29 p. 68/99 A numerical method for stability analysis Reformulation as an abstract Cauchy problem v v (a,t) + (a,t) + (Hv(,t))(a) = t a v(,t) = K v(,t) v(a,) = v (a) d u(t) = Au(t), t, dt u() = u X. u(t) v(,t) u v ( )

69 IASI, Roma, January 26, 29 p. 69/99 A numerical method for stability analysis Reformulation as an abstract Cauchy problem v v (a,t) + (a,t) + (Hv(,t))(a) = t a v(,t) = K v(,t) v(a,) = v (a) Aϕ = ϕ Hϕ d u(t) = Au(t), D (A) = {ϕ X ϕ X, ϕ() = K ϕ} t, dt u() = u X.

70 IASI, Roma, January 26, 29 p. 7/99 A numerical method for stability analysis Discrete approximation of the generator

71 IASI, Roma, January 26, 29 p. 71/99 Recent results: a numerical method for stability an Discrete approximation of the generator [,a ] Ω N = { θ i = a 2 cos ( N i N π) + a 2 : i =,...,N}

72 IASI, Roma, January 26, 29 p. 72/99 A numerical method for stability analysis Discrete approximation of the generator [,a ] Ω N = { θ i = a 2 cos ( N i N π) + a 2 ϕ X y X N = C N set y i = ϕ(θ i ), i = 1,...,N : i =,...,N}

73 IASI, Roma, January 26, 29 p. 73/99 A numerical method for stability analysis Discrete approximation of the generator [,a ] Ω N = { θ i = a 2 cos ( N i N π) + a 2 ϕ X y X N = C N set y i = ϕ(θ i ), i = 1,...,N : i =,...,N} A A N : X N X N, build ϕ N an interpolating polynomial through y i such that ϕ N () = K ϕ N compute z i = ϕ N (θ i) (Hϕ N ) (θ i ), i = 1,...,N set (A N y) i = z i

74 IASI, Roma, January 26, 29 p. 74/99 A numerical method for stability analysis Discrete approximation of the generator the eigenvalues of A N approximate the eigenvalues of A If λ is an eigenvalue of A with multiplicity ν, then for N sufficiently large, A N has exactly ν eigenvalues λ i, i = 1,...,ν, such that max λ λ i 1 i ν ( C2 C 3 ) 1/ν ( ε N + 1 N ( C1 N ) N ) 1/ν

75 IASI, Roma, January 26, 29 p. 75/99 Exploration of juveniles-adults dinamics Back to adults-juveniles competition: the case of separate niches J 1 R,2 R,1 R

76 IASI, Roma, January 26, 29 p. 76/99 Exploration of juveniles-adults dinamics Separate niches: exploring the bifurcation graph J 1 R,2 R,1 R

77 IASI, Roma, January 26, 29 p. 77/99 Exploration of juveniles-adults dinamics Separate niches: exploring the bifurcation graph J A K A 1 I(λ) R,2 R,1 R R(λ)

78 IASI, Roma, January 26, 29 p. 78/99 Exploration of juveniles-adults dinamics Separate niches: exploring the bifurcation graph J A stable K A 1 I(λ) R,2 R,1 R R(λ)

79 IASI, Roma, January 26, 29 p. 79/99 Exploration of juveniles-adults dinamics Separate niches: exploring the bifurcation graph J stable A B bifurcation J B I(λ) R,2 R,1 R R(λ)

80 IASI, Roma, January 26, 29 p. 8/99 Exploration of juveniles-adults dinamics Separate niches: exploring the bifurcation graph J stable B bifurcation unstable A C I(λ) I C R,2 R,1 R R(λ)

81 IASI, Roma, January 26, 29 p. 81/99 Exploration of juveniles-adults dinamics Separate niches: exploring the bifurcation graph J stable B bifurcation unstable A C D bifurcation I(λ) H D 1 R,2 R,1 R R(λ)

82 IASI, Roma, January 26, 29 p. 82/99 Exploration of juveniles-adults dinamics Separate niches: exploring the bifurcation graph J unstable two complex roots E stable B bifurcation unstable A C D bifurcation I(λ) E E 3 1 R,2 R,1 R R(λ)

83 IASI, Roma, January 26, 29 p. 83/99 Exploration of juveniles-adults dinamics A complete pattern 5 4 J 3 I(λ) R(λ) 1 R,1 R,2 R

84 IASI, Roma, January 26, 29 p. 84/99 Exploration of juveniles-adults dinamics A complete pattern 5 J 4 3 I(λ) R(λ) 1 R,1 R,2 R

85 IASI, Roma, January 26, 29 p. 85/99 Exploration of juveniles-adults dinamics A complete pattern 5 J 4 3 I(λ) R(λ) 1 R,1 R,2 R

86 IASI, Roma, January 26, 29 p. 86/99 Exploration of juveniles-adults dinamics A complete pattern 5 4 J 3 I(λ) R(λ) 1 R,1 R,2 R

87 IASI, Roma, January 26, 29 p. 87/99 Exploration of juveniles-adults dinamics A complete pattern 5 4 J 3 I(λ) R(λ) 1 R,1 R,2 R

88 IASI, Roma, January 26, 29 p. 88/99 Exploration of juveniles-adults dinamics A complete pattern 5 4 J 3 I(λ) R(λ) 1 R,1 R,2 R

89 IASI, Roma, January 26, 29 p. 89/99 Exploration of juveniles-adults dinamics Orbits by numerical computation of the solution

90 IASI, Roma, January 26, 29 p. 9/99 Exploration of juveniles-adults dinamics Orbits by numerical computation of the solution.6 R = R =34 R =3 A.3.2 R =24.1 R =3 R =35 R =5 R = J R =1 R =15 R =2

91 Future work IASI, Roma, January 26, 29 p. 91/99

92 IASI, Roma, January 26, 29 p. 92/99 Future work systematic use of the numerical method for a complete analysis of some specific population models

93 IASI, Roma, January 26, 29 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

94 IASI, Roma, January 26, 29 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

95 IASI, Roma, January 26, 29 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 including

96 IASI, Roma, January 26, 29 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 including numerical methods for the computation of the solution

97 IASI, Roma, January 26, 29 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 including numerical methods for the computation of the solution computation of steady states

98 IASI, Roma, January 26, 29 p. 98/99 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 including numerical methods for the computation of the solution computation of steady states stability analysis via numerical computation of characteristic roots

99 IASI, Roma, January 26, 29 p. 99/99 THANK YOU FOR YOUR ATTENTION

Farkas JZ (2004) Stability conditions for the non-linear McKendrick equations, Applied Mathematics and Computation, 156 (3), pp

Farkas JZ (2004) Stability conditions for the non-linear McKendrick equations, Applied Mathematics and Computation, 156 (3), pp Farkas JZ (24) Stability conditions for the non-linear McKendrick equations, Applied Mathematics and Computation, 156 (3), pp. 771-777. This is the peer reviewed version of this article NOTICE: this is