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Ecological implications of global bifurcations. For the occasion of the promotion of. George van Voorn 17 July 2009, Oldenburg. Overview. Laymen-friendly (hopefully) introduction 2D Allee-model 3D Rosenzweig-MacArthur model 3D Letellier-Aziz-Alaoui model Discussion. Ecology.

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slide1
Ecological implications

of global bifurcations

For the occasion of the promotion of

George van Voorn

17 July 2009, Oldenburg

overview
Overview
  • Laymen-friendly (hopefully) introduction
  • 2D Allee-model
  • 3D Rosenzweig-MacArthur model
  • 3D Letellier-Aziz-Alaoui model
  • Discussion
ecology
Ecology
  • Study of dynamics of populations of species
  • Interactions with other species and physical world
  • Obvious issues temporal and spatial scale
modeling
Modeling
  • Modeling can help in understanding
  • Common tool selection:
  • Ordinary differential equations (ODEs)

+ Ease in use and analysis, explicit in time

− Homogeneous space

example allee
Example: Allee
  • Density-dependency affects population

Variables (time-dependent):

X(t) = # (= number of)

Parameters:

β = interspecific growth rate (no explicit nutrient modeling)

K = carrying capacity (= maximum sustainable # of carrots)

ζ = Allee threshold

dynamics allee
Dynamics: Allee
  • Time dynamics of the model:

#

Too little carrots  extinction

Enough carrots  growth to carrying capacity

Too many carrots  decline to carrying capacity

dynamics allee7
Dynamics: Allee
  • Asymptotic behaviour:

X = K

#

X = ζ

X = 0

Stable equilibria: X = 0, X = K

Unstable equilibria: X = ζ

allee with predator
Allee with predator

x2

  • We add a “predator”

x1

x1 = prey population

x2 = predator population

l = extinction threshold, no fixed value (bifurcation parameter)

k = carrying capacity, by default 1

c = conversion ratio, by default 1

m = predator mortality rate, no fixed value (bifurcation parameter)

Note: dimensionless

functional response
Functional response
  • Predator-prey interaction

Functional response  linear

x1 = prey population

x2 = predator population

c = conversion ratio, by default 1

analysis
Analysis
  • Asymptotic behaviour (equilibria)
  • Stability (local info)

 Jacobian matrix  eigenvalues

  • Variation of parameter (e.g, l and m)
  • Switch in asymptotic behaviour = bifurcation point
  • Numerical package AUTO
equilibria
Equilibria
  • 2D Allee model has the following equilibria:

E0 = (0,0), stable

E1 = (l,0), unstable

E2 = (k,0), with k≥l, depends

E3 = (m,(m-l)(k-m)), depends

analysis 2d allee
Analysis 2D Allee

Two-parameter plot of equilibria depending on m vs l

Mortality

rate of

rabbits

Allee threshold for carrots

Plot has several regions: different asymptotic behaviour

analysis13
Analysis

Equilibrium:

Only prey

m > 1

Equilibrium:

Predator-prey

Transcritical bifurcationTC2: transition to a positive equilibrium

analysis14
Analysis

Predator-prey

Equilibrium

Predator-prey

Cycles

Hopf bifurcationH3: transition from equilibrium to stable cycle

periodic behaviour
Periodic behaviour
  • Also: limit cycle, oscillations
  • Hopf bifurcation, also local info

#

#

phase plot
Phase plot

Orbits starting here go to (0,0)

 Allee effect

Attracting region

#

Bistability:

Depending on initial conditions

to E0 or E3/Cycle

#

l = 0.5, m = 0.74837

problem
Problem…

Time-integrated simulations  extinction of both species

What do we miss?

Local info not sufficient

Predator-prey

Cycles

Extinction

Prey AND predator !!

extinction
Extinction

All orbits go to extinction!

“Tunnel”

#

Bistability lost;

Allee-threshold gone

#

l = 0.5, m = 0.735

what happens is
What happens is …

Manifolds of two equilibria connect:

Limit cycle “touches” E1/E2

#

Heteroclinic orbit connecting saddle point to saddle point

#

l = 0.5, m = 0.73544235…

new phenomenon
New phenomenon
  • Explains transition to extinction
  • NOT local info  global bifurcation
    • Heteroclinic connection between two saddle equilibria
homotopy technique
Homotopy technique
  • Need new technique(s): global info
  • Take an educated guess
  • Formulate criteria
  • Convert fault to continuation parameter
  • Change parameter to match criteria

 find connection

method
Method

Δx1 = 0

ξ*w

ε*v

E1

E2

l = 0.5, m = 0.7 (shot in direction unstable eigenvector)

l = 0.5, m = 0.7354423495 (connecting orbit)

global bifurcation in allee
Global bifurcation in Allee

Using developed homotopy method

  • Regions:
  • Only prey
  • Predator –prey
  • 0. Extinct
counter intuitive
Counter-intuitive

Bizar: lower mortality rate kills the whole population…

  • Regions:
  • Only prey
  • Predator –prey
  • 0. Extinct

Mortality

rate of

rabbits

2

Overharvesting or ecological suicide

add another
Add another…
  • Rosenzweig-MacArthur 3D food chain model, no Allee-effect

where (Holling type II)

x = variable

d = death rate

note: dimensionless

equilibria27
Equilibria
  • There are 4 equilibria:
chaos
Chaos
  • New type of behaviour possible

#

A-periodic, but still “stable”

bifurcation diagram
Bifurcation diagram

Extreme values for top predator are plotted as function of one parameter

#

d1=0.25

d2

bifurcation diagram30
Bifurcation diagram

Chaotic Extinct PeriodicStable coexistence

#

d1=0.25

d2

global bifurcations
Global bifurcations

Region of extinction marked by global bifurcation

#

d1=0.25

d2

Saddle limit cycle

new technique
New technique
  • This is a homoclinic cycle-to-cycle connection
  • No technique thusfar for detection and continuation
  • Formulation of new criteria
  • Adaptation of homotopy method
global bifurcation
Global bifurcation
  • Using new technique:

#

d1=0.25

d2 = 0.0125

#

#

Connecting orbit from saddle limit cycle to itself

bifurcation diagram34
Bifurcation diagram

Two parameters

0: no top predator

SE: stable existence

P: periodic solutions

C: Chaos

“Eye”: extinction

0

SE

d2

P

P

C

d1

Family of tangencies of connecting orbit

 boundary of chaotic behaviour (boundary crisis)

different model
Different model
  • Letellier & Aziz-Alaoui (2002)
different model36
Different model
  • Letellier & Aziz-Alaoui (2002)

Identical to

Rosenzweig-MacArthur

Biological interpretation:

- No dependence prey density

- Different dependence predator density

one parameter diagram
One-parameter diagram

c0 = 0.038

#

a1

As compared to RM: two chaotic attractors

Two different global bifurcations

one parameter diagram38
One-parameter diagram

c0 = 0.038

#

a1

First globif bifurcation  boundary crisis

No stable equilibrium, shift, but… survival

one parameter diagram39
One-parameter diagram

c0 = 0.038

#

a1

Second global bifurcation  interior crisis

Change of chaotic attractor

one parameter diagram40
One-parameter diagram

Chaos

c0 = 0.038

#

Low period limit cycle

a1

Disappearance of one chaotic attractor

Hysteresis (Scheffer) & simplification of system

discussion
Discussion
  • Connection types and ramifications
    • Allee: heteroclinic point-to-point  overharvesting
    • RM: homoclinic saddle cycle  chaos disappears, extinction top predator
    • L&AA: two homoclinic saddle cycle  hysteresis, persistence of top predator
discussion42
Discussion
  • Global bifurcations mark different transitions than local
  • Required development new method
    • Implemented in AUTO
  • Essential for analysis
  • No obvious coupling connection type with biological consequences
acknowledgements
Acknowledgements
  • Bas Kooijman, Bob Kooi
  • Dirk Stiefs, Ulrike Feudel, Thilo Gross
  • Yuri Kuznetsov, Eusebius Doedel
  • Martin Boer, Lia Hemerik
  • Funding: NWO
slide44
George.vanVoorn@wur.nl
  • www.biometris.wur.nl/
  • www.bio.vu.nl/thb/research/project/globif/

Thank you for your attention!

equilibria46
Equilibria
  • The relevant equilibria now are

E0 = (0,0,0)

E1 = (1,0,0)

E3 = (X1*,X2*,0)

 No stable equilibrium all 3 species

Default parameter values:

proof maps
Proof: maps

#

#

T+1

T+2

#

#

T

T

At the point where chaos disappears we plot the number of bears at time T+n as function of number at time T

proof maps48
Proof: maps

#

#

T+1

T+2

#

#

T

T

First globif (upper chaotic attractor) is homoclinic period 1

Second (lower chaotic attractor) homoclinic period 2