Estabilidad y Complejidad en Ecología (a tale of

the sixties)

José M Pacheco Max-Planck-Institut für

Wissenschaftsgeschichte zu Berlin Universidad

de Las Palmas de Gran Canaria

Heres complexity!!!

Modelling

- 1. Some philosophical remarks on Modelling
- Analysis vs Synthesis
- Simplifying vs Complication
- Explicative Ability vs Predictability
- 2. The haunting of Complexity, or the necessity

of a trade-off.

Whats in a Model?

- A Model is a description (either static or

dynamic, or both) of the real world, or rather

some part thereof, made under incomplete

information - Incompleteness can mean either
- Lack of information, or else
- A deliberate decision in order to deal with an

overwhelming amount of unstructured, redundant

information

Analysis vs Synthesis

- As a rule, scientific endeavour starts by

analysing (from Greek lysis decomposition),

i.e. looking for the smallest parts of the study

object that can convey interesting or useful

informations. - Then, these partial informations are processed

and assembled in order to build a general picture

of the real world. This second procedure is known

as synthesis (from the Greek syn together). - This double route, analysissynthesis, is

possibly the best known paradigm of scientific

discovery.

Simplicity vs Complication

- Actually, Simplifying means translating

informations into more tractable and abstract

languages subjected to the laws of some logical

framework Mathematics provides a number of such

languages. - As a rule, Simplifying is performed by

discarding information on (more or less) sound

hypotheses. - Therefore, in a first instance, Complication

could be considered as the task whose aim is to

recover information and to incorporate it into

the models.

Explicative Ability vs Predictability

- There is a twofold interest in building Models
- To offer understandable and manageable

descriptions of the objects, processes and

patterns under study. - (and most important) To provide means for

prediction or forecasting future behaviour of the

modelled systems. - Both aims are nearly always in conflict Paying

attention to accurate static descriptions

diagnostic- often becomes a burden when trying

to construct a preview of future developments

prognostic-, where general trends are the

primary interest (Greek gnosisknowledge).

The haunting of Complexity, or the necessity of a

trade-off (1)

- Complexity (in the above sense of information

recovery and its incorporation to model building)

haunts the performance of models - In (over-) simplified models aimed to prediction,

supplementary information can act as a noisy

input distorting the output, thus making it

useless. - In extremely detailed models basically aimed to

diagnostic tasks, prediction can be very

difficult and time-consuming if all tiny details

must be taken into account.

The haunting of Complexity, or the necessity of a

trade-off (2)

Ability, Accuracy

Diagnostic accuracy

High cost predictive ability

Predictive ability

Growing Complexity

Trade-off complexity level

On Complexity

- 1. Some philosophical questions
- Why Complexity?
- Is Complexity unavoidable?
- Is Complexity fashionable?
- 2. Several definitions. Which one to choose, and

why?

Why Complexity?

- In a naïve way, Complexity is a natural and

distinctive feature of the development of any

system with a growing number of elements. - Here, elements is used in a very broad sense

The word can encompass actually different

material or intellectual objects, or patterns

observed in an set of objects, or coexistence of

several time scales - Complexity is ubiquitous and puzzling, therefore

the interest on understanding and deciphering it.

Is Complexity unavoidable?

- Perceiving Complexity is deeply rooted on

conscience and psychological grounds, so in some

sense it is unavoidable. Let us offer a couple of

reasons - Counting ability It is difficult to humans to

tell how many elements a small set has when there

are more than four objects, unless they are

presented according to some definite patterns,

soSpatial vision usually simplifies complex

plane figures or graphs, but it must be educated

Is Complexity fashionable?

- Science is indeed subjected to fashion waves,

like anything else. In a sense, studying

complexity amounts to abandoning the usual

two-way-road analysissynthesis in favour of a

new, softer style

Several definitions Which one to choose?

- We have already seen Complication (or Complexity)

as the task whose aim is to consider global

information and to incorporate it into the

scientific discourse. - Complexity deals with those properties modified

by increasing in a system the number of objects,

and relationships between them. - Complexity studies objects and their

relationships across many spatial and temporal

scales.

Our choice for this talk

Why this choice?

- It is of intermediate difficulty.
- The idea of a trophic web fits rather well in its

framework - It has been extensively studied by many authors.
- Starting in the sixties, there is a current

debate on the relationship between Complexity in

this sense and Stability of Ecosystems in some

sense to be defined. - Additional problems, like temporal scales, are

left out.

A simple graph-theoretic notation

Some more explanations

Stability

- Stability the path to predictability
- The need to predict
- How to predict?
- How can we trust predictions?
- 2. The various definitions of Stability

Stability the path to Predictability

- The main aim of Science is Prediction in its

various forms. Usually, predictions are

formulated for the occurrence of future facts,

but e.g. palaeozoological studies can be used to

formulate predictions on past facts, where they

may fill some gaps in fossile records. - All interpretations of Prediction rely heavily

on the idea of some smoothly varying basic

dynamics.

The need to predict

- Prediction is an essential ingredient in

contemporary life. Business, holidays,

travelling, everything depends heavily on good

and accurate predictions of many phenomena, from

weather to fashion trends, to oil prices, to

social and religious movements. - In many affairs, simulations are quite often

considered as predictions. This is usual in

scientific and theoretical analyses, when there

is no immediate need of predictions, but rather

on understanding how things evolve

How to predict?

- A priori any method is a valid one.

Nevertheless, there are a few conditions to be

met - A sound understanding of the physical basics of

the predicted phenomenon. - Knowledge of observed trends.
- Ability to recognise the importance of noise

and/or new information Reanalysis. - Luck.

How trust-worthy can prediction be?

- As a rule, predictions must be formulated in

terms of likelihood. - The usual technique is to run any predictive

method on past events in order to measure

ability and accuracy, the so-called predictive

experiments. These yield a confidence interval

that must be included in actual predictions

What is Stability?

- Any notion of Stability addresses the idea of

small departure from a given, standard state,

which in turn is considered to be stationary in

some sense, e.g. familiar, comfortable,

problem-free, and the like. - If a system has a tendence towards one of these

situations we usually speak of a stronger

condition asymptotic stability,. - A related concept is resilience, we shall

consider it next.

Stability conceptions (1)

Stability conceptions (2)

Stability conceptions (3)

Asymptotic stability in a 2-D system

The Van der Pol 2-D oscillator shows an unstable

state and a stable cycle or attractor

Lyapunovs Stability Criteria

- All definitions offered above are known as

stability à la Lyapunov (First Method) - There exists as well the Second Lyapunov Method,

or energy method, related to the important

concept of attraction basins

An energy surface associated to an unstable point

and a stable cycle. The attraction basin of the

cycle is the plane projection of the mexican

hat wing

One more look at Stability

Potential indeed, because its relationship with

potential energy...

V(x)

Local and Global Dynamics

- Two different potentials and their dynamics

Here we have two singular states One is a stable

one (the well), the other is unstable (the

step). The attraction basin of the well

spans from the unstable point to plus infinity.

The attraction basin for the unstable point spans

from minus infinity to the point itself.

Now, three singular points Two stable ones

(wells) and an unstable one (the peak). Here

the peak is a separatrix between the attraction

basins of the wells.

Resilience From Local Dynamics to Ecosystems

- Resilience measures the effort needed to jump

from one stable state to the other. It can be

expressed in time units, in energy units, or as

the size of the attraction basin. - Therefore, a stable state is a resilient one

when perturbations affect it only transitorily.

whoopss!

hmm!!

Structural Stability (1)

- Structural Stability is a more general concept

where one considers whether the general, global

picture of the dynamics will dramatically change

when the defining principles or equations are

modified (usually when some parameter crosses

through certain values). - The mathematical idea of Bifurcation is the

basic tool in the study of structural Stability.

Structural Stability (2)

Stable cycle

Stable cycle

Unstable origin

Unstable origin

Structural Instability

Negative alpha stable origin

Alpha0 neutral origin

Bifurcation in the linear system

Positive alpha unstable origin

Stability, Nonlinearity, and Bifurcation

UNS

UNS

STA

STA

STA

The qualitative behaviour (modification in the

stability of singular points) changes when the

parameter crosses through 8 We say that a

bifurcation has happened.

More Nonlinearity

A double well quartic potential

and a bit more Nonlinearity

The Conflict Stability vs Complexity

- Two real life examples
- The IKEA bookstand, or more Complexity implies

enhanced Stability - Gadgets in your car, or more Complexity implies

loss of Stability - Back to Biology / Ecology

The IKEA bookstand, or more Complexity implies

enhanced Stability

Gadgets in your car or, more Complexity implies

loss of Stability

Back to Biology / Ecology

- The IKEA concept is nearly a tradition in both

fields More complex systems are believed to be

more stablebut, what do we really mean here by

stable? - On the other hand, the gadget concept is also

very common just think of tropical forests and

their weakness under perturbations

Measures of Complexity for Foodwebs

- The mathematical representation of a foodweb is

a directed digraph with S nodes (the number of

species ). - The maximum number of links between nodes is SxS

including canibalism as well. If L is the actual

number of links, then C L/(SxS) is called the

connectance of the web. - The quotient L/S is the mean number of links of

any species, (remeber that 2L/S is the

graph-theoretical mean degree of a node). - An interesting parameter is Om, the number of

omnivorous species, i.e. the number of nodes a

linked with any other node b in the directional

way a eats b

More on complex foodwebs

Back to Lyapunovs 1st Method

Some more Math

Even more Math

Translating all this Math

- In fact, what we have done is just expliciting

our choice of stability à la Lyapunov as the one

we shall use in the study of the conflict between

Stability and Complexity. - This choice translates Stability into matrix

properties, but let us now remember that

Complexity in Foodwebs was also represented by

some matrix properties - The immediate question is Where is the missing

link between both theories?

The missing Link

- The missing link is that
- matrices representing stationary states of

foodwebs (under the mass action interaction

hypothesis) can be interpreted as jacobian

matrices of Lotka-Volterra interaction models! - Proof
- Just write down the Jacobian matrix of the

general Lotka-Volterra model

Lets play a bit

a bit more

and let us make a conjecture

Enter Topology

- The 2x2 example is very easy to work out and

shows that under the adopted definition of

stability for foodwebs, addition of just one link

(thus connectance grows from ½ to ¾) does not

change stability. The Topology of this web is so

symmetric that it plays no role whatsoever. - Nevertheless, if connectance grows to its

maximum value 4/41, then some non-topological

supplementary conditions, presumably related to

real-world foodweb observations, must be

fulfilled in order to drive the system to

stability or instability

Enter Asymmetry

- Topology is much richer in the S3 case more

asymmetric relationships are allowed, the

analysis becomes much more involved, though in

many interesting cases it boils down to the S2

case. - Maybe some relationship could be found between

the even- or oddness of the species number S and

the complexity / stability behaviour of the

foodweb. - Let us dwell on the S3 case for a while

Graphs and Matrices

Graphs and Matrices (contd.1)

Graphs and Matrices (contd.2)

First Insights

- The previous analyses show the importance of

several facts in the conservation of stability - 1. Very small or small connectance are good for

stability, butwhats the interest of

disconnected webs? - 2. Increasing connectance can open the way to

instability, but under topological and/or

numerical constraints stability can be preserved. - 3. An apparently open question Do optimal

topologies exist?

Increasing Connectance (1)

Increasing Connectance (2)

More Insights

- As the S3 study suggest, increasing connectance

in an arbitrary way can lead webs to instability.

It may be even worse, for adding parameters can

send stable points to unfeasible regions in phase

space a pure nosense. - Real-world webs rarely show high connectance.

Rather, they appear to be hyerarchically ordered

from top predators (nearly isolated species) to a

dearth of lower level species where more complex

structures can be observed. Nevertheless,

omnivorous species (Om) seem to play a role in

simplifying webs into separate components

Three final remarks

(This is the first one)

Heres the second one Some History

- The decade-lasting complexity vs stability

debate seems to be a storm in a fishpond. - It may look funny, but back in the sixties

computers were not usual gadgets in labs,

theoretical studies relied heavily on purely

mathematical techniques, statistical simulations

were performed with old random number tables,

and the third one Some Future

- Future studies on the complexity-stability field

may address the following topics - Inclusion of spatial distribution effects

(already very popular in some schools). - Consideration of time-lags as instability

generators. - Determination of optimal length and complexity

for feasible foodwebs.

Gracias!