Skip to main content

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).

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Council of Europe: European Landscape Convention. Treaty Series no. 176, Florence, 2000 October 20th (2000)

  2. 2.

    Naveh, Z., Lieberman, A.: Landscape Ecology: Theory and Applications. Springer, New York (1984)

    Google Scholar 

  3. 3.

    Turner, M.G., Gardnerl, R.H.: Quantitative Methods in Landscape Ecology. Springer, New York (1990)

    Google Scholar 

  4. 4.

    Ingegnoli, V.: Landscape Ecology: A Widening Foundation. Springer, New York (2002)

    Book  Google Scholar 

  5. 5.

    Fabbri, P.: Principi Ecologici per la Progettazione del Paesaggio. Franco Angeli Editore, Milano (2007)

    Google Scholar 

  6. 6.

    Turner, M.G., Romme, V.H., Gardnerl, R.H., O’Neil, R.V., Kratz, T.K.: A revised concept of landscape equilibrium: disturbance and stability on scaled landscapes. Landsc. Ecol. 8(3), 213–227 (1993)

    Article  Google Scholar 

  7. 7.

    Pelorosso, R., Della Chiesa, S., Tappeiner, U., Leone, A., Rocchini, D.: Stability analysis for defining management strategies in abandoned mountain landscapes of the Mediterranean basin. Landsc. Urban Plan. 103, 335–346 (2011)

    Article  Google Scholar 

  8. 8.

    Petit, C., et al. (eds.): Landscape Analysis and Visualisation-Spatial Models for Natural and Resources an Planning. Springer, New York (2008)

    Google Scholar 

  9. 9.

    Vermaat, J.E., Eppink, F., Van den Bergh, J.C., Barendregt, A., Van Belle, J.: Matching of scales in spatial economic and ecological analysis. Ecol. Econ. 52, 229–237 (2005)

    Article  Google Scholar 

  10. 10.

    Fabbri, P.: Paesaggio, Pianificazione, Sostenibilità. Alinea, Firenze (2003)

    Google Scholar 

  11. 11.

    Urban, D.L., Minor, E.S., Treml, E.A., Schick, R.S.: Graph models of habitat mosaics. Ecol. Lett. 12, 260–273 (2009)

    Article  Google Scholar 

  12. 12.

    Lauro, G., Monaco, R., Servente, G.: A model for the evolution of bioenergy in an environmental system. In: Sammartino, M., Ruggeri, T. (eds.) Asymptotic Methods in Non-linear Wave Phenomena, pp. 96–106. World Scientific, Singapore (2007)

    Chapter  Google Scholar 

  13. 13.

    Lauro, G., Lisi, M., Monaco, R.: A modeling framework for analysis of landscape stability and bifurcation phenomena. Rend. Sem. Mat. Univ. Politec. Torino 68(4), 399–413 (2010)

    MathSciNet  Google Scholar 

  14. 14.

    Finotto, F., Monaco, R., Servente, G.: Un modello per la valutazione della produzione e della diffusività di energia biologica in un sistema ambientale. Sci. Reg. 9(3), 61–84 (2010)

    Google Scholar 

  15. 15.

    Gobattoni, F., Lauro, G., Leone, A., Monaco, R., Pelorosso, R.: A procedure for mathematical analysis of landscape evolution and equilibrium scenarios assessment. Landsc. Urban Plan. 103, 289–302 (2011)

    Article  Google Scholar 

  16. 16.

    Monaco, R., Servente, G.: Introduzione ai Modelli Matematici nelle Scienze Territoriali, 2nd edn. Celid, Torino (2011)

    Google Scholar 

  17. 17.

    Ingegnoli, V., Giglio, E.: Ecologia del Paesaggio. Manuale per conservare, gestire e pianificare l’ambiente. Esselibri, Napoli (2005)

    Google Scholar 

  18. 18.

    Jordan, D.W., Smith, P.: Nonlinear Ordinary Differential Equations. Clarendon Press, Oxford (1977)

    MATH  Google Scholar 

  19. 19.

    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions. Dover, New York (1980)

    Google Scholar 

  20. 20.

    O’Neill, R.V., Johnson, A.R., King, A.W.: A hierarchical framework for the analysis of scale. Landsc. Ecol. 3, 193–205 (1989)

    Article  Google Scholar 

  21. 21.

    Navino, D.: Valutazione della BTC mediante un modello dinamico relativo al grafo ecologico. Graduate Thesis in Urban Planning, Politecnico di Torino (2010)

  22. 22.

    Li, B.: Why is the holistic approach becoming so important in landscape ecology? Landsc. Urban Plan. 50, 27–41 (2000)

    Article  Google Scholar 

  23. 23.

    Naveh, Z.: Ten major premises for a holistic conception of multifunctional landscapes. Landsc. Urban Plan. 57, 269–284 (2001)

    Article  Google Scholar 

  24. 24.

    Finotto, F.: Landscape assessment: the ecological profile. In: Cassatella, C., Peano, A. (eds.) Landscape Indicators. Assessing and Monitoring Landscape Quality, pp. 47–75. Springer, Dordrecht (2011)

    Chapter  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Roberto Monaco.

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 [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

Rights and permissions

Reprints and Permissions

About this article

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

Download citation

Keywords

  • Landscape ecology
  • Mathematical models
  • Stability analysis

Mathematics Subject Classification (2000)

  • 34D05
  • 92F05