European Society of Computational Methods in Sciences and Engineering (ESCMSE) Journal of Numerical Analysis, Industrial and Applied Mathematics (JNAIAM) vol. 3, no. 3-4, 2008, pp. 193-209 ISSN 1790–8140 Modelling Environmental Influences on Wanderer Spiders in the Langhe Region (Piemonte – NW Italy) 1 Samrat Chatterjee ∗ , Marco Isaia † , Francesca Bona † , Guido Badino † , Ezio Venturino ∗2 † Dipartimento di Biologia Animale e dell’Uomo, via Accademia Albertina 13, ∗ Dipartimento di Matematica, via Carlo Alberto 10, Universita’ di Torino, 10123 Torino, Italia Received 7 March, 2007; accepted in revised form 10 June, 2008 Abstract: We develop a mathematical model for understanding the effect of wanderer spiders as biological controllers of the insects infesting vineyards, thus accounting also for the role played by residual wood and green patches as spiders habitat in the otherwise homogeneuous landscape of the Langhe region. We then extend the deterministic model allowing random fluctuations around the coexistence equilibrium. The stochastic stability properties of the model are investigated both analytically and numerically. c  2008 European Society of Computational Methods in Sciences and Engineering Keywords: predator-prey model, equilibria, stochastic differential equation, stochastic sta- bility. Mathematics Subject Classification: 92D25, 92D40 1 Background information The earliest historical evidence of wine production dates from between 6000 and 5000 B.C. Wine making production improved considerably at the time of the ancient Greeks. Only toward the end of the Roman Empire however, fifth century A.D., modern cultivation techniques became common throughout Europe. During the instabilities of the Middle Ages, the Christian monasteries maintained and developed viticultural practices. They indeed owned and tended the best vineyards in Europe. This in view of the fact that these secure and stable environments had the resources and the interest to make the quality of their vines better. At the time, vinum theologium was considered superior to all others. A wide variety of the Vitis vinifera grapes has constituted for centuries the bulk of the European vineyards. In the late 19th century however, the plant louse phylloxera was accidentally introduced from North America into European vineyards. Phylloxera is a North American insect that kills the vine by feeding on its roots. This pest caused the failure of the first plantations of European grapes in the Eastern United States, but was identified only in the late 19th century. Since around 1860, it spread around the world, perhaps traveling on 1 Published electronically October 15, 2008 2 Corresponding author. E-mail: ezio.venturino@unito.it