## Abstract

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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## Appendices

### 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

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.

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

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

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 [12–14, 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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### Cite this article

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). https://doi.org/10.1007/s10440-012-9786-z

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### Keywords

- Landscape ecology
- Mathematical models
- Stability analysis

### Mathematics Subject Classification (2000)

- 34D05
- 92F05