Chapter 6: Differential Equations - mced-ecology.org...Material for MCED by Broder Breckling, Hauke Reuter, and Uta Berger supplementing Chapter 6: Differential Equations The change

Modelling Complex Ecological DynamicsModelling Complex Ecological Dynamics

Material for MCED by Material for MCED by BroderBroder Breckling, Breckling, HaukeHauke Reuter, and Reuter, and UtaUta Berger supplementing Chapter 6: Differential EquationsBerger supplementing Chapter 6: Differential Equations

Presentation material accompanying

Chapter 6:

Differential Equations



MModelling CComplex

EEcological DDynamics

Adding to

Chapter 6:

Part 3: Differential Equations(Continuous Systems)



Preface

• The presentation is free for use in non-commercial teaching context.

• The content was based on lecture material developed (among others) for the course “Systems Analysis” in the Master of Science Programme “International Studies in Aquatic Tropical Ecology” at the University of Bremen during the years 1999 –2011

• Because of the page restriction this presentation partially extends the content covered in MCED Chapter 6 Differential Equations.

Broder Breckling

Hauke Reuter

Uta Berger




(Continuous Systems)

plus Group Exercises



Basic Properties of Continuous Dynamic Systems

Contents

Basic terms of classical system dynamics: differential equations, variables, trajectories,

direction field, phase space

Predator-Prey modelling

What can happen in a continuous system?Explosion, collapse, flow-equilibrium,

limit cycle, chaos, bifurcations, phase transitions

Group Exercises



Differential Equations (continuous systems)

State Variables ( ~ elements, compartments):- basic entities in system dynamics- represent a homogeneous quantity which may change

during time

Differential equations: - determine the state of the system for each point in time

( )Nfdt

dN

tt

NN

t

Nt

tt = →−

−=

∆

∆→∆ 0lim

12

12



A form of display to visualize system dynamicsIt has one dimension for each variable:

- a 1D system can be displayed along a line

- a 2D system on a plane

- a 3D system as …

- higher dimensional systems as n-dimensional (nD) vectors

prey

predator

grassrabbits

foxes

Phase Space (~state space)



The direction field puts a grid onto a (usually 2D) phase space and indicates for each point the changes of the variables as small vectors.

The direction field allows an overall orientation about the system dynamics.

It allows to see, to where the stateof the system will move depending on the particular initial conditions. prey

predator

Direction field



Trajectories are lines (paths) of system development drawn within the state space.

A consequence of the deterministic paradigm: Trajectories are not allowed to cross(They are allowed to approximate an attractor-point, which is reached in infinity (i.e. never).)

prey

predator

Trajectory

Trajectories



What can happen in a continuous system?

The condition, that trajectories cannot cross in deterministic systems implies, that the types of possible dynamics are limited according to the number of variables.

(in discrete systems successive states are defined only point-wise; lines drawn to connect the points can cross …)

Types of Dynamics in Continuous Systems



• trivial "dynamics": nothing happens (i.e. change is zero)

• explosion or collapse: the trajectories of the system may approximate + or - infinity (explosion) or zero (collapse)

t

dN

dt

t

dN

dt

0

0

What can happen in1-dimensional Systems



• a flow-equilibrium can occur The system approximates a non-zero equilibrium point. There can be stable and unstable equilibria. The number of possible equilibria is not limited

• multiple stable points (and domains of attraction) can occur. In this case, the initial conditions determine which equilibrium point is approximated

• as a result of parameter-shifts: emergence and disappearance of stable and instable equilibria can occur

Note:

here is a a display of change over stock – not stock over time!




… Predator Prey

Modelling



Example: Lotka-Volterra Model of predator-prey interaction

(It is not biologically realistic, but it is a simple starting point

to develop more "realistic" and more complex derivates.)

Frame:

decreasepredincreasepreddt

dpred

decreasepreyincreasepreydt

dprey

__

__

−=

−=

2-dimensional SystemsPredator-Prey example



Assumptions made in the Lotka Volterra model:

• The model calculates averages, no spatially explicit

representation• All predators have the same properties, all prey have the

same properties (no internal population structure)

• Prey have unlimited resources (exponential growth -> explosion)

• Predators decrease exponentially (-> collapse)

• The equations are coupled through a "capture"-term. Capture depends on the densities of both populations

(-> meeting probabilities).




The simplest form is to specify the terms just by (positive) constants:

k1: exponential growth rate of preyk2: probability of predator and prey to meet with lethal

consequence for preyb: conversion factor from prey to predator biomass k3: mortality factor of predator (exponential decay)

predkpredpreykbdt

dpred

predpreykpreykdt

dprey

*3**2*

**2*1

−=

−=




Investigation of the dynamics of the system:

-a look at the phase space ...

-calculating the 0-isoclines

ISOCLINES*: all the points in the phase space, where the change of

one of the variables is zero.

(* in more precise terms: 0-growth isocline)




Calculating the Isoclines for the Lotka-Volterra model

prey-isocline

predator-isocline

2

1

*210

**2*10

k

kpred

predkk

predpreykpreyk

=

−=

−=

2*

3

3*2*0

*3**2*0

kb

kprey

kpreykb

predkpredpreykb

=

−=

−=







�The separation of different areas through the isoclines suggests, that there will be an oscillatory component in the dynamics of the system.

�Integration of the equations - or simulation brings about that the system oscillates in a marginally stable way (i.e. the amplitude depends on the initial condition (see computer exercises).

time

Prey,

predator




The same types of dynamics as in 1D systems can occur:

�- Trivial dynamics (no change)�- Explosion�- Collapse�- Stable and unstable equilibrium points

�… and in addition




and in addition… limit cycles:

increased oscillations around an unstable equilibrium, which are damped at a certain amplitude

balanced by a dominance of a damping process at higher oscillation amplitudes

(limit cycles are not possible in continuous 1-d systems)

p

N

continuous oscillation




�example: limit cycle

See MCED textbook page 81




The change of one form of qualitative behavior to another can happen in a nonlinear system, if a parameter p is changed. Frequent examples are:

A stable point

turns unstable

and a limit cycle

emerges

(Hopf bifurcation)

A stable point

turns unstable

and 2 (or more)

new stable points

emerge

A stable point

turns unstable

(or vice versa)

Phase transitions



What can happen in3 (and higher) -dimensional Systems

The same types of dynamics as in 1D and 2D systems can occur:

�- Trivial dynamics (no change)�- Explosion�- Collapse�- Stable and unstable equilibrium points�- Limit cycles

�… and in addition



ChaosIn chaotic systems the trajectories do not cross, but initially closely related trajectories tend to loose

correlation (singularity of dynamics).

Chaotic dynamics in physical systems were discovered

during 1960ies (Ed Lorenz).

In early 70ies, also in predator-prey systems

What can happen in3 (and higher) -dimensional Systems



Meteorologist Edward N. Lorenz (1917 – 2008)

Dept. of Earth, Atmospheric, and Planetary SciencesMassachusetts Institute of Technology77 Massachusetts Ave.Cambridge, MA 02139-4307 USA

Discovered chaotic dynamics in a comparatively simple differential equation model

Chaos

Source: http://photos.aip.org/history/Thumbnails/lorenz_edward_a1.jpg

Edward N. Lorenz



The Lorenz (1963) equations (fluid flow on thermal gradients, believed to play a role in the development of weather pattern):

zxydt

dz

yzxdt

dy

xydt

dx

3

8

)28(

)(10

−=

−−=

−=

x(0) = -15.80

y(0) = -17.48

z(0) = 35.64

Chaos

The simulation used the Interaction Engine of the free software POPULUS, see http://www.cbs.umn.edu/populus/



x(0) = -15.80

y(0) = -17.48

z(0) = 35.64

x(0) = -12.80

y(0) = -17.48

z(0) = 35.64

deterministic / sensitive to initial conditions („butterfly effect“)

ChaosSensitivity to initial conditions



Lorenz. 1963. New York Academy of Sciences : “One

meteorologist remarked that if the theory were correct,

one flap of a seagull’s wings would be enough to alter

the course of the weather forever.”

Lorenz. 1972. American Association for the Advancement of

Science in Washington, D.C.: “Predictability: Does the Flap of a

Butterfly’s Wings in Brazil set off a Tornado in Texas?”

“Butterfly effect”



��Deterministicchaos

��Limit cycles

��Multiple equilibria

��Attractive points(i.e. dynamicequilibria)

��Explosions, Extinctions (appr. Infinity or zero)

Three variables

(3D system)

Two variables

(2D system)

One variable

(1D system)

Summary: Types of Dynamicsoccurring in continuous systems



Example Direction Fieldof the 2-dimensional differential equation system

dX/dt = (2*(- 8*X3 + 12*X2 - 5*X * 0.5) + 0.5) * X – X * Y

dY/dt = X * Y - 0.5 * Y

in the unit sqare (0 < x <1, 0 < y <1)

Trajectories can be coarsely anticipated from a direction field.

Differential equations (continuous systems)



dX/dt = (2*(- 8*X3 + 12*X2 - 5*X * 0.5) + 0.5) * X – X * Y dY/dt = X * Y - 0.5 * Y

Direction Field

Try

to e

stim

ate

the c

ontinuatio

n o

f a t

raje

cto

ry

sta

rtin

g a

t th

e a

rro

ws



dY/dt = X * Y - 0.5 * Y

Isoclines Trajectories

dX/dt = (2*(- 8*X3 + 12*X2 - 5*X * 0.5) + 0.5) * X – X * Y



Exercises:

Predator Prey

Population Modelling with




predkpredpreykbdt

dpred

predpreykpreykdt

dprey

*3**2*

**2*1

−=

−=

Computer Exercises: Predator-Prey population modelling

parameters:k1= 0.1k2= 0.001b = 0.1k3=0.1

initial conditions to be used bydifferent working groups:

prey = 100 300 500 700 900 1100 1300 pred = 100 100 100 100 100 100 100

Predator-prey modelling exerciseTask



Simulationtool:

See: http://www.cbs.umn.edu/populus/index.html




Use the„interaction engine“instead of any of thepredefined models




predkpredpreykbdt

dpred

predpreykpreykdt

dprey

*3**2*

**2*1

−=

−=

parameter:

k1= 0.1

k2= 0.001

b = 0.1

k3=0.1

Tasks for all groups:

• Type the equations given above into the Populus

Interaction Engine (overwrite the Populus default example)

• Select an initial value and run the model

for 250 simulation steps

• Draw the overall behaviour qualitatively in a

ransparency using the provided task sheet

- Population sizes over time: note the times for maxima

and minima, and the population sizes at that time and

draw it.

- Predator over prey: note the combination of values when

one of them has its maximum or minimum (i.e. 4 per cycle)

How to interpret the results?

Predator-prey modelling exercise

Task



predkpredpreykbdt

dpred

predpreykpreykdt

dprey

*3**2*

**2*1

−=

−=

parameter:

k1= 0.1

k2= 0.001

b = 0.1

k3=0.1

Initial conditions

• Group 1: Prey population 100

Predator population 100







Predator-prey modelling exercise

Task



Predator-prey modelling exerciseSolut

ion

Display predator over prey (x/y-plot)



preyini = 100

predini = 100


ion

Display predator and prey over time



preyini = 300

predini = 100


ion




preyini = 900

predini = 100


ion




This is it

for now,

but the fun

continues

…

Documents

Chapter 6: Differential Equations - mced-ecology.org...Material for MCED by Broder Breckling, Hauke Reuter, and Uta Berger supplementing Chapter 6: Differential Equations The change