Ecological Modelling 220 (2009) 3546–3554 Contents lists available at ScienceDirect Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel A phenomenological model without dispersal kernel to model species migration F. Saltré a,b,c,∗ , I. Chuine c , S. Brewer e , C. Gaucherel d a Université Montpellier 2, UMR AMAP, Montpellier F-34000, France b CNRS, UMR AMAP, Montpellier F-34000, France c Centre d’Ecologie Fonctionnelle et Evolutive, Equipe BIOFLUX, CNRS, 1919 route de Mende, 34293 Montpellier Cedex 5, France d INRA, UMR AMAP, Montpellier F-34000, France e Botany Department, 3165, 1000 E. University Ave., University of Wyoming, Laramie, WY 82071, USA article info Article history: Available online 17 July 2009 Keywords: Non-homogeneous Gibbs process Point pattern Spatial analysis Optimization process Paleo-landscape abstract Phenomenological approaches to model species migration are usually based on kernel-based methods. These methods require a good knowledge of the dispersal agent behaviour for a given species. They also calculate the location of individuals independently to each other (except the mother plant) and then suppress some of them according to additional interactions such as competition, facilitation and recruitment. In this paper, we propose to use a new phenomenological method, the Gibbs method, to model tree species migration at large scale. The Gibbs method handles the location of adult individuals in terms of pairwise interactions described by a potential function. This function summarizes the set of known and unknown factors determining the spatial distribution of the individuals (or cohorts). The principle of the Gibbs method is to minimize the sum of all pairwise interactions, also called the cost function, in order to optimize the spatial point pattern according to the chosen potential function. We compared dispersal models based on the non-homogeneous Gibbs method to several models based on kernel methods, and in detail with a leptokurtic kernel-based model. An elasticity test of the Gibbs- based dispersal model showed a strong dependence among the parameters and the key role of the potential of interaction in the dispersion obtained. We found important differences in the resulting pat- terns of migration between Gibbs-based and kernel-based models: Gibbs-based model generated more random point patterns, leading to more diversified migration pathways than kernel-based model. Finally, a semi-realistic application to paleo-landscapes showed that the Gibbs-based model was able to simu- late the migration pathways of Fagus sylvatica during the Holocene more realistically than kernel-based models. © 2009 Elsevier B.V. All rights reserved. 1. Introduction One of the current challenges in ecology is to accurately forecast the changes in species distribution due to future environmental changes. If several models are currently able to forecast species potential distribution, they all lack a major driver of species dis- tribution: species migration (Morin and Lechowicz, 2008; Thuiller et al., 2008). Species migration can be simulated with two main kinds of model: mechanistic models and phenomenological mod- els. Mechanistic models simulate dispersal patterns of seeds as a function of some characteristics of the plants and of the plant dis- persal agents (Nathan and Muller-Landau, 2000), and may be used to simulate animal as well as wind dispersals (Okubo and Levin, 1989; Greene and Johnson, 1996; Kuparinen, 2006). Phenomeno- ∗ Corresponding author at: CEFE, CNRS, 1919 route de Mende, 34293 Montpellier Cedex 5, France. E-mail address: frederik.saltre@cefe.cnrs.fr (F. Saltré). logical models, at present, use dispersal kernel methods (Kot et al., 1996; Clark et al., 1998, 1999). The dispersal kernel is a density prob- ability function that describes the probability for a seed to cover a certain distance from its mother plant. Kernel-based dispersion models efficiently describe the spread of seeds over short distances as well as rarer extreme events, which allow for the important pro- cess of long distance seed dispersal (LDD) (Le Corre et al., 1997; Clark and McLachlan, 2003; Nathan, 2006). There is a wide range of kernel-based models, which may be considered as arbitrary equa- tions that fit data well, or they can be chosen according to their derivation from diffusion theory or other plausible assumptions (Clark et al., 1999; Katul et al., 2005; Snäll et al., 2007). All these methods are based on an accurate description of the behaviour of dispersal agents and on the influence of these behaviours on a given seed. However, even if seed morphologies or chemical traits suggest a particular dispersal agent, recent findings have shown that seeds of any given species are usually dispersed by multiple agents (Nathan et al., 2008). Further, ecological pro- cesses that influence the location of seeds, such as competition, 0304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2009.06.026