Diapositiva 1

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WORKSHOP   COMPUTATIONAL LIFE SCIENCES Innsbruck,1
2 15 october 2005
TIME DEPENDENT REGIMES IN A THEORETICAL MODEL ON
TOURISM
Deborah Lacitignola

Department
of Mathematics

University of Lecce-Italy deborah.lacitignola_at_un
ile.it
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THE TWO FACES OF TOURISM
TOURISM IS OFTEN DEEMED AS AN OPPORTUNITY FOR
PROMOTING ECONOMIC AND SOCIAL DEVELOPMENT BUT IT
ALSO REPRESENTS A DRIVING FORCE WHICH COULD
GREATLY AFFECT ENVIRONMENTAL QUALITY AND DEGRADE
NATURAL RESOURCES
FOR THIS REASON, PARTICULAR ATTENTION IS TO BE
GIVEN TO THE AMOUNT BY WHICH NATURAL RESOURCES
ARE EXPLOITED.
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NAMELY

• ECOLOGICAL QUALITY AND INTEGRITY OF RESOURCES
  MUST BE MANTAINED TO WARRENT THEIR
  ATTRACTIVENESS TO TOURISTS AS WELL AS THEIR
  USEFULNESS TO RESIDENTS

• ATTENTION MUST BE GIVEN NOT ONLY TO THE
  QUALITY OF NATURAL ENVIRONMENT BUT ALSO TO THE
  LEVEL AND THE NATURE OF INTERACTIONS BETWEEN
  GROUPS OF USERS

MATHEMATICAL MODELLING CAN BE A USEFUL TOOL AND
A QUALITATIVE HELP IN THIS DIRECTION
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MATHEMATICAL MODELLING OF TOURISM INDUSTRY
CASAGRANDI AND RINALDI() WERE THE FIRST TO
INTRODUCE THE APPROACH OF MINIMAL DESCRIPTIVE
MODELS IN THE CONTEXT OF TOURISM, A FIELD WHICH
HAS TRADITIONALLY BEEN DOMINATED BY DIFFERENT
APPROACHES
THEY PROPOSED A MINIMAL THEORETICAL MODEL, WHICH
REFERS TO A GENERIC TOURISTIC SITE AND DESCRIBES
THE INTERACTIONS AMONG THE TOURISTS PRESENT IN
THE AREA, THE NATURAL ENVIRONMENT AND THE CAPITAL
() R. Casagrandi S.Rinaldi A Theoretical
Approach to Tourism Sustainability
Conservation Ecology 6(1) (2002)
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THE CASAGRANDI-RINALDI MODEL
T
E
negative interaction
C
positive interaction
STATES VARIABLES AND INTERACTIONS BETWEEN THE
TREE COMPONENTS OF THE CASAGRANDI-RINALDI
MINIMAL MODEL
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ENVIRONMENTAL CONSERVATION AND
THE SOCIO-ECOLOGICAL POINT OF VIEW
FROM AN ECOLOGICAL POINT OF VIEW, THE ROLE
PLAYED BY DIFFERENT TYPOLOGIES OF TOURISTS HAS
GREAT RELEVANCE FOR THE STUDY OF TOURISTIC TRENDS
BECAUSE OF THEIR DIFFERENT IMPACTS ON THE
ENVIRONMENT
THE IDEA ()
A MATHEMATICAL MODEL THAT EXPLICITLY TAKES INTO
ACCOUNT THE DIFFERENCES AMONG TOURISTS BASING ON
THEIR UNDERLYING MOTIVATIONAL FEATURES AND ON
THEIR RELATIONSHIP WITH ENVIRONMENT AND
INFRASTRUCTURES.
() D.L. jointly with M.Cataldi, I.Petrosillo and
G. Zurlini, Department of Biological and
Environmental Sciences and Technologies,
University of Lecce-Italy
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WHICH KIND OF TOURISM?
IN LITERATURE, IT IS POSSIBLE TO FIND A NUMBER OF
THEORIES CATEGORIZING TOURISTS BASING ON THEIR
IMPACT ON THE ENVIRONENT. .AMONG THE MANY
DIFFERENT GROUPS, WE HAVE DECIDED TO CONSIDER
THE TWO EXTREME TYPOLOGIES
MASS TOURISM
?
ECOTOURISM
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MASS TOURISM AND ECOTOURISM

• MASS TOURISM
• high density
• strong land and water uses
• causing an increase of tourists pressure
• if not properly managed can have severe
  consequences on environment

• ECOTOURISM
• low density
• less demanding in terms of facilities and
  infrastructures
• low environmental impact
• it is defined as responsible travel to natural
  areas, which improves environmental conservation
  and local people welfare

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THE MODEL
C
T2
T1
E
T1(t) number of ecotourists present in the
specific area at time t T2(t) number of mass
tourists present in the specific area at time t
C(t) capital in the specific area at time t
(intended as structures for tourists activities)
E(t) environmental quality in the specific
area at time t
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THE MODEL
a is an important indicator of socio-ecological
aspects and will be considered as a
bifurcation parameter.
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THE SITE
We choose as touristic site a natural reserve
Torre Guaceto, Puglia-Italy
Puglia
Torre Guaceto
Brindisi
Lecce
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SEARCHING FOR COEXISTENCE

• Extinction equilibrium O(0,0,0,0)
• Extinction equilibrium with environment at its
  carrying capacity E0(0,0,k,0)
• Extinction equilibrium for the mass tourists
  E3(T1,0,E,C)
• Coexistence equilibrium P( T1,T2,E,C) with
  T1gtgtT2
• Coexistence equilibrium P1( T1,T2,E,C) with
  T2gtgtT1

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WHICH FORM FOR COEXISTENCE ?
If for 2.8139 lt a lt 2.86 (case 7) coexistence
is assured by the presence of the stable
equilibrium P.
.for 2.4 lt a lt 2.8139 (case 6) coexistence can
be obtained through both periodic and chaotic
patterns.
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TOWARDS PERIODIC PATTERNS
For a 2.8139 the equilibrium P becomes a sink
(stable focus) because of a Hopf bifurcation
Investigations in the time dependent regimes,
allow to be more precise on the features of such
a Hopf bifurcation
The equilibrium P becomes in fact a stable focus
because of a supercritical Hopf bifurcation a
stable closed orbit OP in the neighbouring of
P collapses on this unstable fixed point causing
its change of stability according to the Hopf
Theorem.
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COMPUTATIONAL ASPECTS
COMPUTATIONS ARE MADE USING MATLAB W ITH DOUBLE
PRECISION
SIMULATIONS ARE PERFORMED USING MATLAB CODES

• THE USED SOLVERS ARE
• in some case ode45
• in other cases ode15s since, for some values of
  the bifurcation parameter, the problem turned out
  to be moderately stiff

• WITH THE FOLLOWING OPTIONS
• the scalar relative error tolerance 'RelTol' was
  set to 1e-4
• the vector of absolute error tolerances
  'AbsTol' was set to 1e-9 for all the
  components.

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THE HOPF BIFURCATION OF P
Hopf bifurcation for the equilibrium P in the
phase space T1T2. (a) a
2.81 (b) a 2.812 (c) a 2.8135 (d) a
2.817
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COEXISTENCE THROUGH PERIODIC PATTERNS
a2.81
COEXISTENCE THROUGH CHAOTIC PATTERNS
a2.797335
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WHATS THE BRIDGE FOR CHAOS?
INVESTIGATIONS ARE PERFORMED ON THE SO CALLED
ROUTES TO CHAOS, BIFURCATION SEQUENCES WHICH
CULMINATE IN CHAOTIC
IT IS SHOWN THAT TRANSITION TO CHAOS OCCURS HERE
THROUGH THE WELL KNOWN PERIOD DOUBLING SCENARIO
WHICH ACHIEVED PROMINENCE AS A RESULT OF
PIONEERING STUDIES BY MAY AND FEIGENBAUM
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THE PERIOD DOUBLING SCENARIO
A first period doubling bifurcation is found to
occur at a 2.801 when the stable single period
oscillation splits into stable double period
oscillations..
The 2-cycle
..and a sequence of period doubling bifurcations
has been found and, through numerical
simulations, up to the 32-cycle was shown, since
trajectories of the next doubling cycles are very
closed each other and therefore difficult to be
distinguished
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The 2-cycle
The 4-cycle
The 8-cycle
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TESTING UNIVERSALITY
The five calculated period doubling points, allow
to determine three values of the sequence equation
Namely
d17.6047 d2 5.0167 d3 4.9669
The smaller and smaller range of bifurcation
intervals and the increasingly rich structure of
the oscillations make it difficult to calculate
further period doubling points.
The involved bifurcations fit the Feigenbaum
scenario of the approach to chaos and the ratio
of successive a intervals is coming to approach
the universal Feigenbaum number of 4.66920
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WHAT AFTER CHAOS?
AN ALTERNANCE OF PERIODIC AND CHAOTIC BEHAVIORS
THROUGH PERIOD DOUBLING CASCADE
1st periodicity window
chaos
THROUGH PERIOD DOUBLING CASCADE
2nd periodicity window
chaos
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CONCLUSIONS
THIS MODEL IS FOUND TO EXHIBIT A VARIETY OF
DYNAMICAL PATTERNS
COEXISTENCE BETWEEN ECOTOURISTS AND MASS TOURISTS
IN THE FORM OF THE EQUILIBRIUM P IS CONSIDERED
IMPORTANT FROM A SOCIO-ECOLOGICAL POINT.
COEXISTENCE THROUGH THE STABLE EQUILIBRIUM P
OCCURS FOR A SMALL RANGE OF THE BIFURCATION
PARAMETER
ALSO COEXISTENCE THROUGH PERIODIC BEHAVIORS (WITH
NOT PRONOUNCED OSCILLATIONS) COULD BE A GOOD
COMPROMISE WHEREAS CHAOTIC PATTERNS SHOULD BE
AVOIDED OR CAREFULLY MANAGED