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Mathematical Models in Landscape Ecology: Stability Analysis and Numerical Tests


In the present paper a review of some mathematical models for the ecological evaluation of environmental systems is considered. Moreover a new model, capable to furnish more detailed information at the level of landscape units, is proposed. Numerical tests are then performed for a case study in the province of Viterbo (central Italy).

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Correspondence to Roberto Monaco.


Appendix A

In the following table (see [4]) the BTC classes and indexes, considered in this paper, are reported for each land cover.

Land cover BTC class BTC index
continuous and dense urban fabric A 0.0
sprawl urban fabric A 0.0
industrial, commercial, transport units A 0.0
mineral extraction sites A 0.0
dump sites and mine deposits A 0.0
highways and freeways A 0.0
rivers and streams A 0.1
cemeteries A 0.3
leisure and sport facilities B 0.4
non-irrigated arable land B 1.0
nurseries in non-irrigated areas B 0.8
areas of glass or plastic greenhouses B 0.8
irrigated arable land B 1.2
nurseries in irrigated areas B 1.0
pastures B 1.0
annual crops and permanent crops B 1.0
natural grassland B 0.8
vineyards C 1.8
fruit trees and berries plantations C 1.8
olive groves C 1.8
complex cultivation patterns C 1.8
agricultural and natural areas C 1.8
moors and heath-land C 1.8
recolonization areas D 3.2
broad-leaved forests E 6.5
coniferous forests E 6.5

Appendix B

In this Appendix the computation of the parameters \(\mathcal {K}_{i}\) defined in (3) and necessary to determine the initial data \(\mathcal{M}_{i0}\) for (24) is given.

As already mentioned the parameter \(\mathcal{K}_{i}\) takes into account several features of the LU border and of the biotopes belonging to the LU itself. Here we define six parameters [10, 15] that are included in \(\mathcal{K}_{i}\) and have been used throughout several papers. For a complete and specific list of indicators characterizing a landscape the reader may be addressed to paper [24].

The first one \(\mathcal{K}_{i}^{sh}\) takes into account the shape of the LU through the formula

$$\mathcal{K}^{sh}_i=1-P^c_i/P_i=1-2 \sqrt{\pi A_i}/P_i, $$

where \(P^{c}_{i}\) is the perimeter of a circle having the same area A i of the LU. In such a way if the ratio \(P^{c}_{i}/P_{i}\) is very small the parameter \(\mathcal{K}^{sh}_{i}\) tends to one. Thus, the larger is the LU perimeter the larger is the bio-energy transmitted to the neighbor LUs.

The second parameter \(\mathcal{K}^{pe}_{i}\) is referred to the permeability of the LUs border, i.e.

$$\mathcal{K}^{pe}_i=\frac{1}{P_i} \sum _{k\in I_i}\sum_{r=1}^s L_{ik}^r p^r, \quad\sum _{k\in I_i}\sum_{r=1}^s L_{ik}^r=P_i, $$

so that if the border is completely permeable (p r=1, ∀r) then \(\mathcal{K}^{pe}_{i}=1\).

The third parameter \(\mathcal{K}^{ld}_{i}\) is relevant to landscape diversity which takes into account that the biotopes are defined to belong to the afore mentioned five classes of BTC, A,…,E. Then \(\mathcal{K}^{ld}_{i}\) is computed by a Shannon-type entropy formula given by

$$\mathcal{K}^{ld}_i= \Biggl( \sum _{\kappa={\text {A}}}^{\text {E}} \frac{m_i^\kappa}{m_i}\log\frac{m_i^\kappa}{m_i} \Biggr) / \log(1/5), $$

where \(m_{i}^{\kappa}\) are the number of biotopes of class κ in the i-th LU. The last expression must be computed by setting the log equal to zero if \(m_{i}^{\kappa}=0\), so that \(\mathcal{K}^{ld}_{i}=0\) when all the biotopes in the LU are of the same class and \(\mathcal{K}^{ld}_{i}=1\) if the biotopes are therein equally distributed.

The fourth parameter \(\mathcal{K}^{ec}_{i}\) takes into account the length of the ecotone, that is the land cover along the biotope borders. The length of the ecotones has a relevant influence on bio-diversity and we will take it into account by means of the following formula

$$\mathcal{K}^{ec}_i=1 -P_i / \sum _{j=1}^{m_i}P_{ji}, $$

where P ji is the perimeter of the j-th biotope belonging to the i-th LU. From the above computation, however, the biotope perimeter tracts composed by anthrop barriers must be excluded. Obviously \(\mathcal{K}^{ec}_{i}\) must be put equal to zero if the whole LU includes only land cover types of BTC class A.

The last two parameters \(\mathcal{K}^{hu}_{i}\) and \(\mathcal{K}^{se}_{i}\) refer, respectively, to climate condition (De Martonne aridity index) and sun exposition. They are defined by

where \(A^{h}_{i}\), \(A^{s}_{i}\), \(A^{SES}_{i}\), \(A^{W}_{i}\) and \(A^{NE}_{i}\) are, respectively, the fractions of land characterized by humid and sub-humid climate classification, south-east/south, west and north/north-east exposition; the coefficients w are suitable weights such that w 1+w 2=1 and w 3+w 4+w 5=1.

Once the above six parameters have been determined, then the global one \(\mathcal{K}_{i}\) can be computed as their average.

In papers [1214, 21] the average has been computed taking into account the parameters \(\mathcal{K}^{sh}_{i}\), \(\mathcal{K}^{pe}_{i}\), \(\mathcal{K}^{ld}_{i}\), whereas in article [15] also the parameters \(\mathcal{K}^{hu}_{i}\) and \(\mathcal {K}^{se}_{i}\) have been included in the average.

In this paper for the case study of Sect. 5 only the parameters \(\mathcal{K}^{ld}_{i}\), \(\mathcal{K}^{ec}_{i}\), \(\mathcal {K}^{hu}_{i}\), \(\mathcal{K}^{se}_{i}\) have been considered, since, in authors’ opinion, it is more correct to include in the parameter \(\mathcal{K}_{i}\) only quantities related to biotopes. In fact shape and permeability of the LUs border are already taken into account in the formula of the total connectivity indexes c i .

Appendix C

In the following table (see [10]) the permeability indexes of the different types of anthrop and natural barriers considered in this paper are reported.

Layers Barrier type Permeability
edified areas & infrastructures compact urban texture 0.05
linear urban texture 0.4
diffuse urban texture 0.5
freeway 0.05
state road 0.05
provincial road 0.4
secondary road 0.5
high-speed railway 0.05
railway 0.5
viaduct 0.5
small roads and channels 0.7
dirt roads 0.9
pedology volcanic/alluvial soil change 0.9
altimetry hill/mountain zones change 0.95
structurally defined ridges 0.7
rivers main rivers 0.85
rivers with cemented banks 0.4
rivers with riparian vegetation 0.5

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Gobattoni, F., Lauro, G., Monaco, R. et al. Mathematical Models in Landscape Ecology: Stability Analysis and Numerical Tests. Acta Appl Math 125, 173–192 (2013).

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  • Landscape ecology
  • Mathematical models
  • Stability analysis

Mathematics Subject Classification (2000)

  • 34D05
  • 92F05