Versión Máster ULPGC / Noviembre 2008 1 On Stability and Complexity (a story of the sixties)

Versión Máster ULPGC / Noviembre 2008

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On Stability and Complexity(a story of the sixties)


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José M PachecoMPIWG / ULPGC


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The Importance of the History of Science…


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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.


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What’s in a Model?A Model a description of the real world, or rather some part thereof, made under incomplete information

Incompleteness can mean either:1) Lack of information, or else2) A deliberate decision in order to deal with an

overwhelming amount of unstructured, redundant information.


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Analysis vs Synthesis

As a rule, scientific endeavour starts by analysing (from the 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, analysis+synthesis, is only one of many possible paradigms of scientific discovery.


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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.


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Explicative Ability vs PredictabilityThere is a twofold interest in building Models:

1. To offer understandable and manageable descriptions of the objects under study.

2. (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 “gnosis”=“knowledge”).


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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:

1. In (over-) simplified models aimed to prediction, supplementary information can act as a noisy input distorting the output, thus making it useless.

2. 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.


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The haunting of Complexity, or the necessity of a trade-off (2)

Growing Complexity

Ability,

AccuracyDiagnostic accuracy

Predictive ability

Trade-off complexity level

High cost predictive ability


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An apparent contradiction: Chaos

If an observed signal “looks complicated”, then we can think about its “being chaotic”. This means that there may exist some deterministic underlying mechanism to the phenomenon which gave rise to the signalTherefore, in a first instance, identification of chaos in a signal means predictability, at least to a certain extent…


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Some technicalities on Chaos (1)Predictability of a time series x(t) amounts to the existence of some rule F providing x(t) as a function of previous values:

( ) ( ( 1), ( 2),..., ( ))x t F x t x t x t k

( ) ( ( ), ( 1),..., ( )),x t T F x t x t x t k k T


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Some technicalities on Chaos (2)


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Some technicalities on Chaos (3)


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ComplexityComplexity

1. Some philosophical questions:1. Some philosophical questions: “ “Why Complexity?”Why Complexity?” “ “Is Complexity unavoidable?”Is Complexity unavoidable?” “ “Is Complexity fashionable?”Is Complexity fashionable?”

2. Several definitions. Which one to choose, and why?2. Several definitions. Which one to choose, and why?


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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.


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Is Complexity unavoidable?

Perceiving Complexity is deeply rooted on conscience and psychological grounds, so in some sense it is unavoidable. Here are 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, so…Spatial vision usually simplifies complex plane figures or graphs, but it must be educated…


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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 analysis+synthesis in favour of a new, softer style.In my opinion, this is something of a backward step…


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Several definitions: Which one to choose?

1. 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.

2. Complexity deals with those properties modified by increasing the number of objects and relationships between them in a system.

3. Complexity studies objects and their relationships across many spatial and temporal scales.

Our choice for this talk


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Why this choice?

1. It is of intermediate difficulty.2. The idea of a trophic web fits rather well in its framework3. It has been extensively studied by many authors.4. There is a current debate on the relationship between

Complexity in this sense and Stability of Ecosystems (in a sense to be defined).

5. Additional problems, like temporal scales, are left out.


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Some explanations


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Some more explanations


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Stability

1. Stability: the path to predictability“The need to predict”“How to predict?”“How can we trust predictions?”

2. The various definitions of Stability


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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. Let us see an example of foremost interest: Climate.


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The need to predict

Prediction is an essential ingredient in contemporary life. Business, holidays, travelling, stock markets… 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 clear in scientific and theoretical analyses, where there is no immediate need of predictions, but rather on understanding how things evolve…


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How to predict?

A priori any method is a valid one. Nevertheless, there are a few conditions to be met:

1. A sound understanding of the physical basics of the predicted phenomenon.

2. Knowledge of observed trends.3. Ability to recognise the importance of noise and/or new

information gathered during the forecasting process.4. Luck.


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How can we trust predictions?

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 forecasting experiments. These yield a confidence interval that must be included in actual predictions…


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Stability conceptions (1)


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Asymptotic stability in a 2-D systemThe Van der Pol 2-D oscillator shows an

unstable state and a stable cycle or attractor


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Lyapunov’s 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 with 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


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The Conflict: Stability vs Complexity

Two real life examples: a) The IKEA bookstand, or more Complexity implies

enhanced Stabilityb) Gadgets in your car, or more Complexity implies loss of

StabilityBack to Biology / Ecology


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The IKEA bookstand, or more Complexity implies enhanced Stability


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Gadgets in your car or, more Complexity implies loss of Stability


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Back to Biology / Ecology

The IKEA concept is nearly a tradition in these fields: More complex systems are believed to be more stable…but, 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…


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Measures of Complexity for Foodwebs

The mathematical representation of a foodweb is a directed digraph with S nodes (the number of species ) and the maximum number of links between nodes is SxS (this includes 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, (2L/S is the graph-theoretical mean degree of a node). An interesting parameter is Om, the number of omnivore species, i.e. the number of nodes “a” linked with any other node “b” in the unidirectional way “a” eats “b”


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More on complex foodwebs


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Some Math…


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Some more Math…


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Even more Math…

0

The usual interpretation of the linearisation ' in diagonal

form is that all equations can be written as ' , or

exp( ).i i i

i i i

h Jh

h h

h h t

0 0

This means that the time evolution of ( ) in the neighbourhood of

the singular point is: ( ) exp( ).

If at least one of the is positive, then the dynamics will fly away

from the singula

i i

i

X t

X X t X h t

r point and it will be unstable: This establishes the

relationship between Lyapunov exponents and stability.


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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: Which is the link between both theories?


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The missing Link

The missing link is that… matrices representing stationary states of foodwebs can be interpreted as jacobian matrices of Lotka-Volterra interaction models!

Proof:

Just write down the Jacobian matrix of the general Lotka-Volterra model

'i i ii ij ji j

x x A A x


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The Proof

, if

, if

ii ij ji i j

kij i

A A x k iF

x A x k j i

1 2' ( , ,..., )i i ii ij j i si j

x x A A x F x x x

1 2, ,...,E E E EsX x x x, if

( ), if

ii ij Ej iii i j

Ek

ij Ei ij

A A x a k iFX

x A x a k j i

The model

The general Jacobian

The singular point

The Jacobian at the singular point


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Eigenvalues

1 0 ... 0

0 1 ... 0

... ... ... ...

0 0 ... 1

ijI

11 12 1

21 22 2

1 2

...

...

... ... ... ...

...

s

sij

s s ss

a a a

a a aA a

a a a

1

2

0 ... 0

0 ... 0

... ... ... ...

0 0 ...

i

s

( ) 0 yields values

for , the so-calle eigenvad of lues

p I A s

A


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Let’s play Math…


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…a bit more…


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…and let us make a conjecture


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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 4/4=1, then some non-topological supplementary conditions, presummably related to real-world foodweb observations, must be fulfilled in order to drive the system to stability or instability…


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Enter Asymmetry

Topology is much richer in the S=3 case, because more asymmetric relationships are allowed, so the analysis becomes much more involved, though in many interesting cases it boils down to the S=2 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 S=3 case for a while…


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Graphs and Matrices


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Graphs and Matrices (contd.1)


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Graphs and Matrices (contd.2)


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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, but…what’s 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?


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Increasing Connectance (1)


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Increasing Connectance (2)


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More Insights

As the S=3 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…


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Two final remarks


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Some History…

The decade-lasting complexity vs stability debate seems to be “a storm in a glass of water”. It may look funny, but back in the sixties computers were not usual gadgets in labs, theoretical studies relied heavily on pure mathematical techniques, and statistical simulations were performed with old random number tables…


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Some Future…

Future studies on the complexity-stability field may address the following topics:

1. Inclusion of spatial distribution effects (already very popular in some schools).

2. Consideration of time-lags as instability generators.3. Determination of optimal length and complexity for

feasible foodwebs.4. Introduction of other complexity concepts


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Thank you!

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Versión Máster ULPGC / Noviembre 2008 1 On Stability and Complexity (a story of the sixties)