Mathematical Models in Landscape Ecology: Stability Analysis and Numerical Tests

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).

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

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

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 [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