Pseudospectral methods and numerical continuation forthe analysis of structured population models
Doctoral thesis
Author: Julia Sanchez Sanz
Advisor: Philipp Getto
June 7th, 2016
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Institutions and Funding
this thesis was developed at BCAM in the research line M3A
it was funded by the MINECO projects and programs
FPI Ph.D. grant BES-2011-047867
National projects MTM2010-18318 and MTM2013-46553-C3-1-P
Estancias breves internship grants EEBB-I-2013-05933 andEEBB-I-2014-08194
Severo Ochoa excellence accreditation SEV-2013-0323
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part of the work was done during research visits to
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Contents1 Introduction
Structured population dynamicsState of the artMotivation
2 Formulation & analysis of structured population modelsFormulation of the class of modelsEquilibriaLinearized stability analysisBehavior under parameter variation
3 Pseudospectral methods for delay equations & population modelsA method for computing eigenvalues of linear VFE/DDENumerical implementationsThe Daphnia model
4 Continuation of equilibria & bifurcationsOne parameter variation analysisTwo parameter variation analysisValidation with models from ecology 4 / 49
Introduction
Chapter 1
Introduction
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Introduction
Introduction
contribution: development of new numerical methods for analyzingstructured population models
objective: provide new tools for the computation of equilibria andbifurcations. Present the results in an environment that facilitatesinterpretation
applications: biology, ecology, epidemiology, fishery, . . .
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Introduction Structured population dynamics
Introduction to structured population dynamics
population dynamics is the area of mathematical biology that study theevolution in time of populations due to natural processes
continuous time, large populations ⇒ differential equations
individuals equal and independent ⇒ ODE, Lotka-Volterra
maturation time ⇒ DDE, Nicholson’s blowfly
individuals differ due to physiology ⇒ structure, PDE, VFE, DDE
structure (age, size) at the i-level induces complex dynamics at the p-level
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Introduction Structured population dynamics
Daphnia: a size-structured consumer-resource model
resource dynamics:
ddt E = f (E )
E (0) = E0
Et : [−h, 0]→ REt(θ) = E (t + θ)
→ history
consumer dynamics i-level: reproduction, mortality, food intake, growth
β, µ, γ, g(x ,E )size x(α,Et), survival probability F(α,Et)
0, t − αx0, 1
τ , t − α+ τx(τ), F(τ)
age α, time tx(α,Et) := x(α), F(α,Et) := F(α)
ddτ x = g(x ,Et(−α + τ))
x(0) = x0
ddτ F = −µ(x ,Et(−α + τ))FF(0) = 1
β(x(α,Et),E (t)) = 0 if x < xa
γ(x(α,Et),E (t))
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Introduction Structured population dynamics
dynamics at p-level:
B:= p-birth rateF(α,Et)B(t − α) densityxa maturation size ⇒ τa(Et) age s.t. x(τa,Et) = xa
VFE/DDE
B(t) =
∫ h
τa(Et)β(x(α,Et),E (t))F(α,Et)B(t − α)dα
d
dtE (t) = f (E (t))−
∫ h
0γ(x(α,Et),E (t))F(α,Et)B(t − α)dα
(B0,E0) = (φ, ψ) ∈ L1([−h, 0];R)× C ([−h, 0];R)
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Introduction State of the art
linearized stability analysis
numerical linearized stability of positive equilibrium for particular casecharacteristic equation - imaginary solution - compute boundaries - validation
de Roos, Metz, Evers, Leipoldt. A size dependent predator-prey interaction: who pursueswhom? J. Math. Biol. 28:609-643, 1990.
principle of linearized stability proved for VFE/DDE
equilibrium is stable if all roots of the characteristic equation have negative real part andunstable if at least one root has positive real part
Diekmann, Getto, Gyllenberg. Stability and bifurcation analysis of Volterra functionalequations in the light of suns and stars. SIAM J. Math. Anal. 39(4):1023-1069, 2007.
linearized stability analysis for general caselinearization, characteristic equation, stability analysis in examples with simplified rates
Diekmann, Gyllenberg, Metz, Nakaoka, de Roos. Daphnia revisited: local stability andbifurcation theory for physiologically structured population models explained by way of anexample. J. Math. Biol. 61:277-318, 2010.
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Introduction State of the art
numerical bifurcation: curve continuation
approximate curves implicitly defined
ODE ⇒ MATCONT, COCO, AUTO
DDE ⇒ DDE-BIFTOOL
VFE ⇒ no software
continuation of equilibria
Kirkilionis, Diekmann, Lisser, Nool, Sommejier, de Roos. Numerical continuation ofequilibria of physiologically structured population models. I. Theory Math. Mod. Meth.Appl. Sci. 11(6):1101-1127, 2001.
continuation of imaginary roots of characteristic equation
de Roos, Diekmann, Getto, Kirkilionis. Numerical equilibrium analysis for structuredconsumer resource models Bull. Math. Biol. 72:259-297, 2010.
⇒ λ = ωi necessary but not sufficient condition for changing stability11 / 49
Introduction State of the art
numerical bifurcation: pseudospectral methods
linear DDE, VFE, PDE
⇓linear ODE
⇓eigenvalues
linear discrete & distributed DDE ⇒ TRACE-DDE
Breda, Maset, Vermiglio. Stability of linear delay differential equations. A numericalapproach with MATLAB Springer, 2015.
VFE/DDE structured populations
Breda, Diekmann, Maset, Vermiglio. A numerical approach to investigate the stability ofequilibria for structured population models J. Biol. Dyn. 7(1):4-20, 2013.
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Introduction Motivation
Motivation
competition, cannibalism, trophic chains... ⇒ generalize Daphnia
linearized stability of structured population models ⇒ extendpseudospectral methods
equilibrium and bifurcations analysis under parameter variation ⇒extend numerical continuation
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Formulation & analysis of structured population models
Chapter 2
Formulation & analysis ofstructured population models
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Formulation & analysis of structured population models Formulation of the class of models
Generalization of the Daphnia model
environment:
E := (I ,E ), I := s interactions, E := n unstructured populations
i-dynamics:
m continuous physiological characteristics ⇒ state x ∈ Ω ⊆ Rm
unique state at birth x0
x(α, Et) i-state, F(α, Et) survival probability
d
dτx =g(x , Et(−α + τ)) g : D(g)→ Rm development
x(0) =x0 µ : D(µ)→ R+ mortality
d
dτF =− µ(x , Et(−α + τ))F β : D(β)→ R0+ reproduction
F(0) =1 γ : D(γ)→ Rs impact to the environment
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Formulation & analysis of structured population models Formulation of the class of models
several life stages ⇒ i-rates defined piecewise smooth
di : Ω× Rs+n → R s.t. di (x , E(t)) = 0 partition Ω
trajectories along i-state space satisfy a transversality condition
switches τi solutions of di (x(τ), Et(−α + τ)) = 0
p-dynamics:
VFE/ODE
B(t) =
∫ h
0
β(x(α, Et), E(t))F(α, Et)B(t − α)dαk∑
i=1
∫ τi
τi−1
I (t) =
∫ h
0
γ(x(α, Et), E(t))F(α, Et)B(t − α)dα
d
dtE (t) = F (E(t))
(B0, I0,E0) = (χ, φ, ψ) ∈ L1([−h, 0];R)× L1([−h, 0];Rs)× C ([−h, 0];Rn)
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Formulation & analysis of structured population models Formulation of the class of models
assume a hierarchical structure in γ ⇒ plug in the interactions in F (E) ⇒reformulate as VFE/DDE ⇒ apply analytical formalism
some components in r.h.s. of F (E) admit a factorization F (E) = Gi (E)Ei
F (E) :=
(D(E)EI
F (E)
)I set of factorisable components
Dij(E) := δijGi (E) diagonal matrix, G (E) ∈ Rl
F (E) ∈ Rn−l
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Formulation & analysis of structured population models Equilibria
Equilibrium types and conditions
(B, I ,E ) equilibrium iff for
R0(I ,E ) :=
∫ h
0
β(x(α, I ,E ), I ,E )F(α, I ,E )dα
Θ(I ,E ) :=
∫ h
0
γ(x(α, I ,E ), I ,E )F(α, I ,E )dα
it holds that
B(1− R0(I ,E )) = 0
I − BΘ(I ,E ) = 0
D(I ,E )EI = 0
F (I ,E ) = 0
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Formulation & analysis of structured population models Equilibria
K-triviality
given K ⊆ I, a vector E is K-trivial iff
Ei
= 0, ∀i ∈ K6= 0, ∀i ∈ I \ K
Equilibrium type Definition
trivial B,E = 0(B,K)-trivial B = 0, E K-trivialB-trivial B = 0, Ei 6= 0 ∀i ∈ IE -trivial B 6= 0, E = 0K-trivial B 6= 0, E K-trivialnontrivial B 6= 0, Ei 6= 0 ∀i ∈ I
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Formulation & analysis of structured population models Linearized stability analysis
Linearized stability
linearize ⇒ characteristic equation ⇒ principle of linearized stability
sketch of linearization:
consider small perturbations of equilibrium
linearize VFE/ODE
state dependent limits of integration ⇒ additional terms
define Dϕτi (E)εt ⇐ differentiate di (x(τ), Et(−α + τ)) = 0
define Dϕx(α, E)εt ⇐ differentiate ODE i-state, variation of constantsformula for linear systems
define DϕF(α, E)εt ⇐ differentiate solution of ODE survival probability
exchange limits of integration20 / 49
Formulation & analysis of structured population models Linearized stability analysis
characteristic equation:
exponential trial solutions eλt(B, I ,E )T of linearized system
define function
f (λ,B, I ,E ) :=
∣∣∣∣∣∣ 1
IsλIn
−M(λ,B, I ,E )
∣∣∣∣∣∣characteristic equation
f (λ,B, I ,E ) = 0
determine linearized stability:
given equilibrium (B, I ,E ), solve characteristic equation in λ andapply principle of linearized stability
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Formulation & analysis of structured population models Behavior under parameter variation
Stability diagrams: 1 parameter p-variation
equilibrium branch
a tuple (B, I ,E , p) s.t. for p fixed (B, I ,E ) is an equilibrium
implicit function theorem ⇒ curve in equilibrium-parameter space
bifurcation
the change in the qualitative behavior of a dynamical system
equilibrium intersects bifurcation ⇒ point in equilibrium-parameter space
transcritical
trivial and nontrivial branches intersect transversally and exchange stability
B-transcritical: trivial & E -trivial, (B,K)-trivial & K-trivial, B-trivial & nontrivialEi -transcritical: K′ = K \ i , (B,K)-trivial & (B,K′)-trivial, K-trivial & K′-trivial
saddle-node & Hopf classical definitions22 / 49
Formulation & analysis of structured population models Behavior under parameter variation
Stability charts: 2 parameter q-variation
bifurcations projected in q-parameter plane ⇒ curves that partition planein regions where behavior does not change
bifurcation curve
a tuple (B, I ,E , q) s.t. for q fixed (B, I ,E ) is an equilibrium and it satisfiesa bifurcation condition φ(B, I ,E , q) = 0 where φ : D(φ) ⊂ Rs+n+3 → R
existence & stability boundaries
invasion & persistence thresholds
0.02 0.04 0.06 0.08
q1
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
q2
(1)
(2)
(3)
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Pseudospectral methods for delay equations & population models
Chapter 3
Pseudospectral methods for delayequations & structured population
models
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Pseudospectral methods for delay equations & population models
analyze linearized stability of equilibria in structured population models
nonlinear VFE/DDE → linearize → characteristic equation
characteristic equation has infinitely many solutions
not easy to obtain λ for complex models
⇒ extend pseudospectral method by Breda et al. to VFE/DDE
⇒ adapt to structured population models
Breda, Getto, Sanchez Sanz, Vermiglio. Computing the eigenvalues of realistic Daphniamodels by pseudospectral methods SIAM J. Sci. Comput. 37(6):2607-2629, 2015.
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Pseudospectral methods for delay equations & population models A method for computing eigenvalues of linear VFE/DDE
Construction of an abstract Cauchy problem
linear VFE/DDE
y(t) = L11yt + L12zt , t ≥ 0
z ′(t) = L21yt + L22zt , t ≥ 0
(y0, z0) = (φ, ψ) ∈ L1([−τ, 0],Rd1 )× C ([−τ, 0],Rd2 ) := Y × Z
solution operator T (t)
for t ≥ 0, T (t) : Y × Z → Y × Z defined by T (t)(φ, ψ) = (yt , zt)
infinitesimal generator AT (t)t≥0 is a C0-semigroup with IG A : D(A) ⊆ Y × Z → Y × Z with action
A(φ, ψ) = (φ′, ψ′)
and domain
D(A) =
(φ, ψ) ∈ Y × Z : (φ′, ψ′) ∈ Y × Z ,
φ(0) = L11φ+ L12ψψ′(0) = L21φ+ L22ψ
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Pseudospectral methods for delay equations & population models A method for computing eigenvalues of linear VFE/DDE
(yt , zt) = (u(t), v(t))⇔ VFE/DDE equivalent to the abstract Cauchyproblem
d
dt(u(t), v(t)) = A(u(t), v(t)), t > 0
(u(0), v(0)) = (φ, ψ) ∈ D(A)
we are interested in σ(A)
point spectrum
every λ ∈ σ(A) has finite algebraic multiplicity
every right half plane contains finitely many eigenvalues
the spectrum consists of zeros of the characteristic equation
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Pseudospectral methods for delay equations & population models A method for computing eigenvalues of linear VFE/DDE
Going from infinite to finite dimension
A → approximate with AM → compute eigenvalues
[−τ, 0] → mesh ΩM = θ0, θ1, ..., θM s.t. 0 := θ0 > θ1 > ... > θM := −τ
Y ×Z → YM ×ZM∼= Rd1M ×Rd2(M+1) space of discrete functions on ΩM
(Φ,Ψ) ∈ YM × ZM , Φ := (Φ1, ...,ΦM) and Ψ := (Ψ0,Ψ1, ...,ΨM)
(PM ,QM) ∈ Y × Z polynomials of degree at most M determined by
PM(θ0) = L11PM + L12QM QM(θi ) = Ψi , i = 0, 1, ...,M
PM(θi ) = Φi , i = 1, ...,M
construct AM : YM × ZM → YM × ZM with action AM(Φ,Ψ) = (ξ, η) by
ξi = P ′M(θi ), i = 1, ...,M η0 = L21PM + L22QM
ηi = Q ′M(θi ), i = 1, ...,M
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Pseudospectral methods for delay equations & population models A method for computing eigenvalues of linear VFE/DDE
Convergence
Theorem
the approximated eigenvalues of AM converge to the exact ones of A
idea: compare solutions of continuous and discrete characteristic equations
give expressions for continuous and discrete characteristic equations
bound the error of the collocation polynomial
bound the error of the characteristic equation
apply Rouche’s theorem to prove convergence
no ghost roots
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Pseudospectral methods for delay equations & population models Numerical implementations
The piecewise case
functionals Lijϕ =∑k
m=0 Amij ϕ(−τm) +
∑km=1
∫ −τm−1
−τm Bmij (θ)ϕ(θ)dθ
shape of operator AM for k = 2, d1 = d2 = 1
M1 M2 M1 + 1 M2
M1
M2
1
M1
M2
∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗
∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗
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Pseudospectral methods for delay equations & population models Numerical implementations
Interpolation, differentiation and integration
Chebyshev extremal nodes on [−τm,−τm−1]
θmi =τm − τm−1
2cos
(iπ
Mm
)− τm − τm−1
2
Lagrange interpolant polynomials
evaluation of derivatives of Lagrange coefficients at nodes are entries ofthe Chebyshev differentiation matrix
integral terms of Lijϕ approximated with Clenshaw-Curtis quadrature rule
Trefethen Spectral Methods in Matlab Software, Environment and Tools. SIAM (2000).
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Pseudospectral methods for delay equations & population models Numerical implementations
Test and validation
y(t) = y(t − 1) + (3 − 2e)
∫ 0
−1
y(t + θ)dθ +
∫ 0
−1
z(t + θ)dθ
z ′(t) = 2
∫ 0
−1
y(t + θ)dθ +
∫ 0
−1
z(t + θ)dθ
eigenvalues computed with MATLAB (left) and convergence of the error (right). • and × forthe method in this work and and ∗ for the method in
Breda, Diekmann, Maset, Vermiglio A numerical approach to investigate the stability ofequilibria for structured population models J. Biol. Dyn. 7-1:4-20 (2013).
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Pseudospectral methods for delay equations & population models The Daphnia model
Daphnia: numerical equilibrium analysis
equilibria
B-trivial B = 0, f (E ) = 0
positive s.t.
1−∫ h
τa
β(x(α,E ),E )F(α,E )dα = 0
f (E )− B
∫ h
0
γ(X (α,E ),E )F(α,E )dα = 0
compute equilibrium
size x , survival probability F and maturation age τa with ODE solverRunge-Kutta
numerical integration with same quadrature rule than Runge-Kutta
apply Newton method
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Pseudospectral methods for delay equations & population models The Daphnia model
Daphnia: linearization and extra implementations
y(t) =
∫ h
τa
β(α)F(α)yt(−α)dα+
(B
∫ h
τa
F(α)β2(α)dα
)z(t)
+Bβ+
g−F(τa)
∫ τa
0M(τa, τa − α)zt(−α)dα
+ B
∫ h
0
[∫ mh,τa+α
Mτa,αF(σ)
(µ− − µ+
g−β(σ)M(τa, σ − α)
+
(g+
g−− 1
)(β1(σ)M(σ, σ − α)− β(σ)
∫ σ
τa
µ1(θ)M(θ, σ − α)dθ
))dσ
+
∫ h
Mτa,α[F(σ)β1(σ)M(σ, σ − α) + β(σ)H(σ, σ − α)]dσ
]zt(−α)dα
x and F evaluated at Runge-Kutta nodes and not at Chebyshev nodes
Bmij given by integrals which limits of integration depend on α
Bmij contain inner integrals which limits of integration depend on σ
M(a, b) and H(a, b) piecewise defined w.r.t. second argument
need to know dense solution for x and F ⇒ DOPRI534 / 49
Pseudospectral methods for delay equations & population models The Daphnia model
Daphnia: computational results
stability boundary for trivial equilibrium (existence boundary for positive)
de Roos, Diekmann, Getto, Kirkilionis Numerical Equilibrium Analysis for StructuredConsumer Resource Models Bull. Math. Biol. 72:259-297 (2010).
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Pseudospectral methods for delay equations & population models The Daphnia model
stability boundary for positive equilibrium
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Continuation of equilibria & bifurcations
Chapter 4
Continuation of equilibria &bifurcations in structured
population models
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Continuation of equilibria & bifurcations
extend continuation methods proposed in
Kirkilionis, Diekmann, Lisser, Nool, Sommejier, de Roos. Numerical continuation ofequilibria of physiologically structured population models. I. Theory Math. Mod. Meth.Appl. Sci. 11(6):1101-1127, 2001.
de Roos, Diekmann, Getto, Kirkilionis. Numerical equilibrium analysis for structuredconsumer resource models Bull. Math. Biol. 72:259-297, 2010.
to compute equilibria and bifurcations under parameter variation for theclass in Chapter 2
determine stability properties with pseudospectral method in Chapter 3
apply the technique to consumer-resource, trophic and cannibalistic models
Sanchez Sanz, Getto. Numerical equilibrium and bifurcation analysis for physiologicallystructured populations: consumer-resource, cannibalistic and trophic models Resubmittedto Bull. Math. Biol. in May 2016.
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Continuation of equilibria & bifurcations
Curve continuation
curve implicitly defined by f (u) = 0, f : D(f ) ⊂ RN+1 → RN smooth
initial u0 s.t. f (u0) = 0, rank(f ′(u0)) = N
⇒ find u1, u2, . . . that approximate f −1(0) with predictor-corrector
tangent prediction: vi+1 = ui + εt(f ′(ui ))
quasi-Newton corrector:
f (u) = 0
〈u − vi+1, t(f ′(ui ))〉 = 0
f (u) = 0 equilibrium or bifurcation conditions ⇒ Integrals, ODE, switches
⇒ at each step of the predictor-corrector use DOPRI5
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Continuation of equilibria & bifurcations One parameter variation analysis
Equilibrium branches
u := (B, I ,E , p) equilibrium branch ⇒ f : D(f ) ⊂ Rs+n+2 → Rs+n+1
f (B, I ,E , p) :=
B(1− R0(I ,E , p))
I − BΘ(I ,E , p)D(I ,E , p)EI
F (I ,E , p)
⇒ transcritical bifurcations are not regular points
use linearities & define for each equilibrium new H : D(H) ⊂ Rs+n+2 → Rs+n+1
vanishing components ⇒ H has not maximum rank ⇒ delete
define new u0 and H : D(H) ⊂ Rr+1 → Rr for continuation
determineequilibrium
typefrom u0
simplifyf &
obtain H
reducedimension
obtainu0 & H
applycontinuation
& obtainu1, . . .
extendsolutionto u1, . . .
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Continuation of equilibria & bifurcations One parameter variation analysis
Bifurcation points
during the continuation... test functions φ : D(φ) ⊂ Rr+1 → Rdetection: check φ(ui )φ(ui+1) < 0
computation: quasi-Newton applied to
H(u) = 0
φ(u) = 0
how we define φ?
transcritical: consider...H for branch that we continue & with which it intersectsdelete components 2 - s + 1 and identical in both mapsthe unique component in the intersecting branch defines φ
saddle-node:detection: last component of t
computation: u := (y , p) ⇒ φ(y , p) = det(∂H(y ,p)∂y
)Hopf: real part rightmost conjugate λ of characteristic equation
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Continuation of equilibria & bifurcations Two parameter variation analysis
Bifurcation curves
u := (B, I ,E , q) bifurcation curve ⇒ f : D(f ) ⊂ Rs+n+3 → Rs+n+2
define for each bifurcation new L : D(L) ⊂ Rs+n+3 → Rs+n+2
define new u0 and L : D(L) ⊂ Rr+1 → Rr for continuation
determinebifurcation
type
simplify f& obtain L
reducedimension
obtainu0 & L
applycontinuation
& obtainu1, . . .
extendsolutionto u1, . . .
transcritical: L and L in terms of intersecting branches H and H
saddle-node: L(u) := (H(u), ψ(u)), ψ : D(ψ) ⊂ Rr+2 → Rbordering technique
Hopf: in the original dimension, continue a pure imaginary solution ofcharacteristic equation
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Continuation of equilibria & bifurcations Validation with models from ecology
Three trophic chain for invasive dynamics
de Roos & Persson. Size-dependent life-history traits promote catastrophic collapses of toppredators Proc. Natl. Acad. Sciences 99(20):12907-12912 (2002).
unstructured predator - size structured prey - unstructured resourcePerca fluviatilis Rutilus rutilus Daphnia magna3 stages: juveniles susceptible, juveniles not susceptible, adultsdynamics for predator & resource ⇒ 2 ODEgrowth, mortality, reproduction, ingestion, predation ⇒ 3 VFE
B :=p-birth rate, E1 resource, E2 predator, ρ :=productivity, K = 2, (B,K)-trivial, K-trivial,
two nontrivial equilibria, B-transcritical supercritical, E2-transcritical subcritical, saddle-node
bifurcations
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Continuation of equilibria & bifurcations Validation with models from ecology
µ
0.01 0.02 0.03 0.04 0.05 0.06
ρ
×10-5
0
0.5
1
1.5
2
(B,K)-trivialK-trivial
(B,K)-trivialK-trivialpositive
(B,K)-trivial,K-trivialtwopositive
(B,K)-trivial
ρ := productivity, µ :=consumer mortality, K = 2, existence boundary consumer(dotted),invasion threshold predator (dashed), persistence threshold predator (continuous)
switch subcritical to supercritical ⇒ invasion & persistence coincide
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Continuation of equilibria & bifurcations Validation with models from ecology
Cannibalism in fish populations
Getto, Diekmann, de Roos. On the (dis) advantages of cannibalism J. Math. Biol.51:695-712 (2005).
Claessen & de Roos. Bistability in a size-structured population model of cannibalistic fish -a continuation study Theor. Pop. Biol. 64:49-65 (2003).
size structured Perca fluviatilis - 2 unstructured resources Daphnia2 stages: juveniles feed on E1, growadults feed on E2 & juveniles, reproducedynamics for 2 resources ⇒ 2 ODEgrowth, mortality, reproduction, 2 ingestion ⇒ 6 VFE
WITHOUT CANNIBALISM: K2 :=carrying capacity for resource E2, B-trivial & nontrivial
equilibria, B-transcritical supercritical & two saddle-node bifurcations 45 / 49
Continuation of equilibria & bifurcations Validation with models from ecology
incorporate cannibalism by increasing the cannibalistic voracity β0
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Continuation of equilibria & bifurcations Validation with models from ecology
lifeboat mechanism: cannibalistic populations survive whereas noncannibalistic go extinct
bistability: two stable nontrivial equilibria coexist
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Continuation of equilibria & bifurcations Validation with models from ecology
under two parameter variation
lifeboat mechanism (5)bistability (2) & (4)
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Continuation of equilibria & bifurcations Validation with models from ecology
Thank you for your attention
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