j - Dipartimento di Matematica - Politecnico di Torino

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(1) (1) (a) M = 0 ; V = 0 (2) (b) M = 0 ; ( 2) bT − hU 0 V = b T (c) (d) M M M max ( c k) (3) = , V = 0 c ( 3) − (4) = M max [ c − k( 1 −V )] c e , b = hU b ( 4) T − 0 V T As it can be easily understood the first equilibrium is quite negative since it prevents an environment where production and diffusion of biological energy is negligible and no areas of high ecological quality 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 quality vegetation is still present. The third equilibrium corresponds to a territory characterized by some production and diffusion of bio-energy but low vegetation quality. Finally 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-potentiality. The stability analysis (Finotto et al., 2010) has shown that the above equilibrium solutions can be obtained, respectively, when the model parameters satisfy the following inequalities: c < k and b T < hU 0 (I) khU 0 > cb T and b T > hU 0 (II) c > k and b T < hU 0 (III) khU 0 > cb T and b T > hU 0 (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 equilibrium 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 permeability of the barriers which can obstruct the energy passage from each LU to the other.
Fig. 2 Ecological graph representation for the study area.
Page 1 and 2: hard copy ISSN 1974-3041 on-line IS
Page 3 and 4: A mathematical procedure for the ev
Page 5 and 6: Fig.1. Study area
Page 7: in other words M max is the maximum
Page 11: References Bracken L. J and Wainwri

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