Pseudospectral methods and numerical continuation for the ... · Breda, Diekmann, Maset,...

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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

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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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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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) =

τa(Et)β(x(α,Et),E (t))F(α,Et)B(t − α)dα

dtE (t) = f (E (t))−

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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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

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

dτx =g(x , Et(−α + τ)) g : D(g)→ Rm development

x(0) =x0 µ : D(µ)→ R+ mortality

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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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) =

β(x(α, Et), E(t))F(α, Et)B(t − α)dαk∑

∫ τi

τi−1

I (t) =

γ(x(α, Et), E(t))F(α, Et)B(t − α)dα

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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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

)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 ) :=

β(x(α, I ,E ), I ,E )F(α, I ,E )dα

Θ(I ,E ) :=

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

= 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

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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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(yt , zt) = (u(t), v(t))⇔ VFE/DDE equivalent to the abstract Cauchyproblem

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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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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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

∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗ ∗ ∗

∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗∗ ∗ ∗ ∗ ∗

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Interpolation, differentiation and integration

Chebyshev extremal nodes on [−τm,−τm−1]

θmi =τm − τm−1

)− τm − τm−1

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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Test and validation

y(t) = y(t − 1) + (3 − 2e)

y(t + θ)dθ +

z(t + θ)dθ

z ′(t) = 2

y(t + θ)dθ +

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

β(x(α,E ),E )F(α,E )dα = 0

f (E )− B

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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Daphnia: linearization and extra implementations

y(t) =

β(α)F(α)yt(−α)dα+

F(α)β2(α)dα

g−F(τa)

∫ τa

0M(τa, τa − α)zt(−α)dα

[∫ mh,τa+α

Mτa,αF(σ)

(µ− − µ+

g−β(σ)M(τa, σ − α)

g−− 1

)(β1(σ)M(σ, σ − α)− β(σ)

∫ σ

µ1(θ)M(θ, σ − α)dθ

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

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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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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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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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

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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0.01 0.02 0.03 0.04 0.05 0.06

×10-5

(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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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

incorporate cannibalism by increasing the cannibalistic voracity β0

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lifeboat mechanism: cannibalistic populations survive whereas noncannibalistic go extinct

bistability: two stable nontrivial equilibria coexist

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under two parameter variation

lifeboat mechanism (5)bistability (2) & (4)

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Thank you for your attention

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