hard copy ISSN 1974-3041

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La Matematicae le sue Applicazionin. 11, 2010

A mathematical procedure for the evolution of futurelandscape scenarios

F. Gobattoni, G. Lauro, A. Leone, R. Monaco, R. Pelorosso

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La Matematica e le sue Applicazioni

hard copy ISSN 1974-3041

on-line ISSN 1974-305X

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A mathematical procedure for the evolution of future landscape scenarios. Authors: F. Gobattoni1 , G. Lauro2, A. Leone1 , R. Monaco3, R. Pelorosso1 1 Department of Environment and Forestry, University of Tuscia, Viterbo (Italy) 2 Architecture Faculty, Second University of Naples, 81031 Aversa, (Italy) 3 Department of Mathematics, Politecnico di Torino, Torino (Italy) Corresponding author: f.gobattoni@unitus.it Introduction Since the European Landscape Convention aims to encourage all European countries to define their landscape qualit y objectives, a new frame of mind in management and planning of territory is nowadays required as a point of reference for territorial government, entities and authorities, so as to advance towards conservation and protection of landscapes providing positive impact on the qualit y of li fe of population. Moreover, planning eff iciently at the landscape scale needs a cross-sector cooperation and people involvement with planners and communities working closely in achieving positive landscape change understanding and in characterizing the relations between nature and society, which will integrate landscapes and their dynamics. (C. Petit et al., 2008). The methodologies for successful landscape and environmental planning have been severely challenged when classical concepts and models such as economic and socioeconomic development, ecosystems preservation and sustainabilit y have been questioned (J. E. Vermaat et al., 2005). The actual challenge is to build transparent and flexible decision-making tools to be used in environmental planning and to embrace a broad range of stakeholders needs together with landscape management requirements. Landscape-scale geographical information provides an ideal framework for developing spatial strategies and to implement simulation models to support landscape planning identifying opportunities for change and anticipating and comparing the results of planning decisions. Model outputs can be used to elaborate thematic maps and visualizations of landscape scenarios as an effective and direct way of communicating results and opportunities. In this framework, landscapes can be defined as spatiall y extended heterogenous complex systems organized hierarchicall y into structural arrangements determined by nonlinear interactions among their components through flows of energy and materials (we shall use the overall term "bio-energy" as introduced by Ingegnoli , 2002 ). Human activity has, also, strongly modified recent landscape by means of land use and land cover changes and consuming of natural resources (Pelorosso et al., 2009). Alterations of natural equili briums has pointed out several consequences on landscape capacity to furnish goods and service (Will emen et al, 2008) as biodiversity conservation (Tscharntke et al., 2005) and regulation of water regimes (Lindborg et al., 2008). Environmental available energy has been pointed out as a key factor in the explanation of many ecological processes and theories (metabolic and species energy theory, biodiversity conservation) recurring to different indeces (e.g. Net Primary Productivity, NNP, Actual Evapotraspiration, AET, Potential Evapotraspiration, PET) (Hurlbert and Jetz, 2010; Currie, 1991; Carrara et al. 2010; Brown at al., 2004). A more complete measure of available energy, so-called Biological Territorial Capacity (BTC), was proposed by Ingegnoli (2002) considering a synthetic function of vegetation metabolism. As the importance of energy exchange, although it is rare for a landscape to be in any form of equili brium and the landscape equili brium concept is not yet clear (Perry, 2002; Turner et al., 1993; Bracken and Wainwright, 2006), it’s interesting to focus on which hypothetical energetic equili brium state is going to be realized and the effects of human decisions on that equili brium. Most of mathematical models are strictly related to air dispersion, hydrology and hydrodynamics, water qualit y, ground water qualit y, erosion and sedimentation, and so on, just taking into account each aspect of the environmental system separately and without looking directly at landscape as a

unique system and without understanding its intrinsic evolution mechanisms. All decision making involves an implicit (if not explicit) use of models, since the decision maker invariably has a causal relationship in mind when he makes a decision. So mathematical models able to explain the landscape evolution and to compare the effects of future scenarios on its evolution and equili brium in time, are reall y needed not also to understand environmental system mechanism and behaviour but above all to better plan strategies for natural resources conservation management and landscape preservation. The environment is here considered as composed by several Landscape Units (LU) delimited by natural and/or anthropic barriers. An integrated GIS (Geographic Information System)-based approach was developed (G. Lauro, R. Monaco, 2008) combining an ecological graph model for the analysis of the relationship between spatial pattern and ecological fluxes and a mathematical model, based on a system of two nonlinear differential equations. These equations are mainly based on a balance law between a logistic growth of bio-energy and its reduction due to limiting factors coming from environmental constraints. The energy exchange among them will be more or less strong depending on the degree of permeabilit y of the barriers which can obstruct the energy passage from each LU to the other. A GIS-based mathematical model, based on the ecological graph and on the cited two differential equations, is presented and discussed here. A study case in Central Italy is introduced and discussed here just to show the importance of a such mathematical procedure to figure out planning strategies and environmental conservation and management actions with the support of rigorous methods able to fix and compare the effects of anthropic decisions on landscape. Study area description The study area (Fig. 1) for the model development, is Traponzo River catchment; Traponzo River is located in the northern part of Lazio Region and its watershed covers part of nine municipaliti es, lying completely in the Province of Viterbo. This stream originates in Monti Cimini relief and flows into Marta river so that Traponzo catchment, as a sub-basin of Marta river, has a total area of 475 Km2, with a mean elevation of 526 m a.s.l. and a maximum of 979 m a.s.l.. The climate of this area is quite Mediterranean with a mean annual temperature of about 15°C and a mean annual precipitation of about 970 mm. The total area covered by urban sprawl is about 2.23 km2 that is about 10% of the total urban area.

Land cover class Area (km 2 ) %

Urban 21.02 4.4Forest 126.36 26.6Non irrigated crops 203.04 42.8Pasture 22.01 4.6Orchards 88.17 18.6Irrigated crops 2.49 0.5Hedges 11.77 2.5Water bodies 0.02 0Total area 474.89 100

Table 1.

Fig.1. Study area

Model description and methodologies Gathering all the available information on the climatic, topographic, land cover, geological and pedological characteristics describing our study area, a GIS dataset has been created not only to represent the watershed but also to identify and derive the LU structure. Taking into account urban areas distribution, primary and secondary roads network, soil features and elevation and weighting them through the application of Saaty matrix, a standard heuristic methodology has been fixed to split the study area into well defined LU. A number of 46 LU has been derived for this study area and an ecological graph for representing the energy flows between them has been elaborated taking into account the Biological Territorial Capacity (BTC) proposed by Ingegnoli (2002). The ecological graph, at first introduced by Fabbri (2003), has been applied here to quantify and correlate the energy production from a landscape unit as a flow of energy to its neighbours according to the permeabilit y of the boundaries. The study area GIS dataset also allow to set up all the parameters needed to run the mathematical model structured as follows. It consists in two ordinary differential equations, of evolution in time, of the following two state variables, M(t) and V(t) : 1) M(t), the mean value of the biological energy of the whole system, t being the time, given by

∑=

=m

jjM

mtM

1

1)( , Mj = (1 + Kj)Bj (1), (2)

where Bj is an index measuring the BTC of the j LU, that is the magnitude representing the energy (Mcal/m2/year) that system needs to dissipate in order to maintain its equili brium state and its organizational level, and where the parameter Kj depends on several properties related to unit’s composition. In particular, in this paper we shall consider that it depends on 5 features and it is given by

5/)( Ej

Cj

Dj

Pj

Fjj KKKKKK ++++= (3)

where FjK takes into account the shape of the patch borders, P

jK their permeabilit y, DjK the bio-

diversity of the LU. The last two parameters CjK and E

jK are related, respectively, to the relative

humidity of the soil and to the sun exposition of the LU, and are introduced for the first time in this paper. We recall that all these parameters are defined in such a way that jK ∈ [0; 1].

For the computation of the parameters FjK , P

jK and DjK , see Finotto et al. (2010).

Conversely CjK and E

jK are defined by the following formulae:

,21

j

sj

hjC

j A

AwAwK

+=

j

NEj

Wj

SWEjE

j A

AwAwAwK 543 ++

= (4), (5)

where the wk are suitable weights, jA is the total area of the j LU, and hjA , s

jA , SWEjA , W

jA , NEjA

are the fractions of soil surfaces characterized by humidity, sub-humidity, south-west-east, west and north-west exposition. Moreover:

maxmax 2BM = , { }jmj

BB,....,1

max max=

= (6), (7)

in other words Mmax is the maximum value of biological energy that the environment under consideration can exhibit according to the maximum value of bio-potentiality expressed by the m different LU. These last definitions lead to fix the bio-potentialit y index as

maxB

BbT = (8)

which, thus, turns out to be confined in the range [0; 1]. 2) V (t), the fraction of the total territory’s surface occupied by areas characterized by high values of Bj ( high values correspond, for example, to wooden areas).

The proposed equations read :

[ ] ),()(1)(

1)()(max

' tMtVkM

tMtcMtM −−

−= (9)

[ ] ),()(1)()(' 00 tVUhtVtVbtV T −−= (10)

They read as a balance law between a logistic growth of bio-energy M (t) and its reduction due to limiting factors coming from environmental constraints expressed by the coeff icients c, k, bT , h, Uo. Having already given the definition of bT , we now proceed to deal with the other parameters. c∈ [0; 1] is the connectivity index which is a function of the energy fluxes Fij between each pair of confining LU, namely

pPP

LijMMF

ji

Sj

Si

ij ⋅+

⋅+

=2

(11)

Lij being the length of the boundary between the i and j LU; Pi and Pj being their perimeters, while p is the permeabilit y index of such a boundary. Moreover k is the ratio between the sum of the perimeters of the impermeable barriers and the perimeter of the whole environment. h is another environment impact parameter defined as the ratio between the sum of the edified areas perimeter (both compact and spread) and the total perimeter P of the whole environment. Finall y U0 ∈ [0; 1] is the ratio between the surface of the edified areas and the total area S of the system. U0 will be computed as the weighted average:

S

UwUwU sc 76

0

+= (12)

where Uc and Us are the surface fractions of compact and spread edified areas, and w6, w7 suitable weights. Note that conversely to the other parameters of the model, k and h may assume values greater than one (Finotto et al., 2010). In order to get a detailed evolution of the environment, one can integrate the equations (9-10) from suitable initial data M(0) and V(0) which can be recovered by the ecological graph. The model provide these scenarios corresponding to the following four equili brium solutions of the differential equation system:

(a) 0)1( =M ; 0)1( =V

(b) 0)2( =M ; T

T

b

hUbV 0)2( −=

(c) c

kcMM

)(max)3( −= , 0)3( =V

(d) ( )[ ]

c

VkcMM e−−= 1max)4( ,

T

T

b

hUbV 0)4( −=

As it can be easil y understood the first equili brium is quite negative since it prevents an environment where production and diffusion of biological energy is negligible and no areas of high ecological qualit y are present. The second scenario is that of a territory strongly fragmented where diffusion of biological energy between the LU is again negligible but some area of high qualit y vegetation is still present. The third equili brium corresponds to a territory characterized by some production and diffusion of bio-energy but low vegetation qualit y. Finall y the last scenario is that more favourable since strong production and diffusion of biological energy between the LU allows to guarantee an ecological settlement of high level of bio-potentialit y. The stabilit y analysis (Finotto et al., 2010) has shown that the above equili brium solutions can be obtained, respectively, when the model parameters satisfy the following inequaliti es:

kc < and 0hUbT < (I)

TcbkhU >0 and 0hUbT > (II )

kc > and 0hUbT < (III )

TcbkhU >0 and 0hUbT > (IV)

The ecological graph representation together with the solutions of the mathematical procedure can be derived through a NetLogo model application that give us an easy tool to model the equili brium states of landscape evolution starting from the initial, actual conditions as described by the ecological graph itself (Gobattoni F. et al., 2010). Results and discussion As an assessment of the ecological behaviour for an environment, the ecological graph assigns a node dimension proportional to the available energy and a link between LU with a dimension proportional to the flux of energy. The energy exchange among them will be more or less strong depending on the degree of permeabilit y of the barriers which can obstruct the energy passage from each LU to the other.

Fig. 2 Ecological graph representation for the study area.

Applying the proposed LU identification methodology and deriving all the required parameters needed to set up the mathematical model, this summarizing table can be obtained:

Total area (km2) 475 High bio-potentialit y area (km2) 124 V0 0.26 bT 0.1606 Mmax (Mcal/m2/year) 2.3 E+08 Bmax (Mcal/m2/year) 1.5 E+08 U0 0.02 h 1.96 k 0.75 c 0.0374 Ve 0.72 Me 0

Table 2. The values in Table 2, have been entered in NetLogo model elaborating the proposed differential system. In particular, Tb , h, 0U , Mmax, k and c seems to satisfy the inequalit y (II ) so that the

equili brium corresponding to he second scenario turns out to be stable and the system evolves asymptoticall y to this equili brium state. The system tends to keep areas with high value of bio-potentialit y but they are isolated in landscape pattern and the fluxes of bio-energy between them seem to be limited. As a consequence, the dynamic evolution of this environmental system shows a trend toward a good production of bio-energy but with a limited diffusion of it, and, then, with limited fluxes between LU. A low connectivity index (Table 2), discloses to this equili brium state as a solution of the differential equations system, since the lack of connectivity prejudices the energy exchange between LU even containing high BTC values areas. The landscape fragmentation provoked by a large urban sprawl phenomenon and by the rich and structured roads network is reflected on the confined and obstructed energy fluxes between LU. The need of opportune planning strategies and actions to reduce fragmentation favouring the energy fluxes between ecosystems and preserving biodiversity, has to be underlined since the actual landscape pattern for the study area shows a low response in terms of connectivity and energy fluxes. Conclusions If the ecological graph is a powerful tool to represent connections eff iciency between LU and to identify high ecological values areas to be protected and compromised ones limiting energy fluxes, on the other side a mathematical model evaluating the potential effectiveness of natural resources on the long term, is essential to assess the diachronic evolution of landscape as a unique system. An integrated approach combining the ecological graph and the mathematical model, allows to face the great challenge of planning and management under sustainable environmental and economical conditions since it can represent a powerful decision system support to compare effects and impacts of alternative scenarios and actions (evaluation of new roads and urban development plans). Not only for a description of the available energy at LU scale but also as a tool for evaluating the equili brium trend in landscape evolution, this mathematical and GIS interfaced method can help in understanding environment response and dynamic change in time to correctly manage and preserve natural resources and ecosystems.

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