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Title: Estabilidad y Complejidad en Ecolog


1
Estabilidad y Complejidad en Ecología (a tale of
the sixties)
2
José M Pacheco Max-Planck-Institut für
Wissenschaftsgeschichte zu Berlin Universidad
de Las Palmas de Gran Canaria
3
Heres complexity!!!
4
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.

5
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

6
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.

7
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.

8
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).

9
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.

10
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
11
On Complexity
  • 1. Some philosophical questions
  • Why Complexity?
  • Is Complexity unavoidable?
  • Is Complexity fashionable?
  • 2. Several definitions. Which one to choose, and
    why?

12
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.

13
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

14
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

15
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 in a system the number of objects,
    and relationships between them.
  3. Complexity studies objects and their
    relationships across many spatial and temporal
    scales.

Our choice for this talk
16
Why this choice?
  1. It is of intermediate difficulty.
  2. The idea of a trophic web fits rather well in its
    framework
  3. It has been extensively studied by many authors.
  4. 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.
  5. Additional problems, like temporal scales, are
    left out.

17
A simple graph-theoretic notation
18
Some more explanations
19
Stability
  • Stability the path to predictability
  • The need to predict
  • How to predict?
  • How can we trust predictions?
  • 2. The various definitions of Stability

20
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.

21
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

22
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.

23
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

24
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.

25
Stability conceptions (1)
26
Stability conceptions (2)
27
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
28
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
29
One more look at Stability


Potential indeed, because its relationship with
potential energy...
V(x)
30
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.
31
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!!
32
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.

33
Structural Stability (2)

Stable cycle
Stable cycle
Unstable origin
Unstable origin
34
Structural Instability
Negative alpha stable origin
Alpha0 neutral origin
Bifurcation in the linear system
Positive alpha unstable origin
35
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.
36
More Nonlinearity
A double well quartic potential
37
and a bit more Nonlinearity

38
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

39
The IKEA bookstand, or more Complexity implies
enhanced Stability
40
Gadgets in your car or, more Complexity implies
loss of Stability
41
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

42
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

43
More on complex foodwebs
44
Back to Lyapunovs 1st Method
45
Some more Math

46
Even more Math

47
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?

48
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

49
Lets play a bit

50
a bit more
51
and let us make a conjecture

52
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

53
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

54
Graphs and Matrices

55
Graphs and Matrices (contd.1)

56
Graphs and Matrices (contd.2)

57
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?

58
Increasing Connectance (1)
59
Increasing Connectance (2)
60
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

61
Three final remarks
(This is the first one)
62
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,

63
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.

64
Gracias!
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