“TIME DEPENDENT REGIMES IN A THEORETICAL MODEL ON TOURISM ”

1. WORKSHOP   COMPUTATIONAL LIFE SCIENCES Innsbruck,12 – 15 october 2005 “TIME DEPENDENT REGIMES IN A THEORETICAL MODEL ON TOURISM” Deborah Lacitignola Department of Mathematics University of Lecce-Italy deborah.lacitignola@unile.it

2. 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.

3. 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

4. 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)

5. 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

6. 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

7. 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

8. 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

9. 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

10. THE MODEL ais an important indicator of socio-ecological aspects and will be considered as abifurcation parameter.

11. THE SITE We choose as touristic site a natural reserve: Torre Guaceto, Puglia-Italy Puglia Torre Guaceto Brindisi Lecce

12. 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 T1>>T2 • Coexistence equilibrium P1*( T1,T2,E,C) with T2>>T1

13. WHICH FORM FOR COEXISTENCE ? If for 2.8139 < a < 2.86 (case 7) coexistence is assured by the presence of the stable equilibrium P*…. ….for 2.4 < a < 2.8139 (case 6) coexistence can be obtained through both periodic and chaotic patterns.

14. 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.

15. 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.

16. 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

17. COEXISTENCE THROUGH PERIODIC PATTERNS a=2.81 COEXISTENCE THROUGH CHAOTIC PATTERNS a=2.797335

18. WHAT’S 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

19. THE PERIOD DOUBLING SCENARIO A first period doubling bifurcationis found to occur ata = 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

20. The 2-cycle The4-cycle The 8-cycle

21. TESTING “UNIVERSALITY” The five calculated period doubling points, allow to determine three values of the sequence equation Namely: d1=7.6047 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

22. 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

23. 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