From individual-based models to partial differential equations ☆: An application to the upstream movement of elvers
TL;DR: Two individual-based models for the movement of elvers in the French river ‘Adour’ are built and a set of stochastic differential equations and a partial differential equation for the elvers’ density are rigorously obtained.
About: This article is published in Ecological Modelling.The article was published on 2005-10-25. It has received 25 citations till now. The article focuses on the topics: Partial differential equation & Stochastic differential equation.
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TL;DR: In this article, the authors study a class of swarming problems where particles evolve dynamically via pairwise interaction potentials and a velocity selection mechanism and find that the swarming system undergoes various changes of state as a function of the self-propulsion and interaction potential parameters.
308 citations
303 citations
01 Jun 2012
TL;DR: In this paper, a review of the well-known methods of reducing spatially structured population models to mean-field models is provided, and the importance of spatial scales in the reduction of spatially explicit models and briefly consider the inverse problem of scaling up local ecological interactions from microscales to macroscales.
Abstract: In this paper, we provide a brief review of the well-known methods of reducing spatially structured population models to mean-field models. First, we discuss the terminology of mean-field approximation which is used in the ecological modelling literature and show that the various existing interpretations of the mean-field concept can imply different meanings. Then we classify and compare various methods of reducing spatially explicit models to mean-field models: spatial moment approximation, aggregation techniques and the mean-field limit of IBMs. We emphasize the importance of spatial scales in the reduction of spatially explicit models and briefly consider the inverse problem of scaling up local ecological interactions from microscales to macroscales. Then we discuss the current challenges and limitations for construction of mean-field population models. We emphasize the need for developing mixed methods based on a combination of various reduction techniques to cope with the spatio-temporal complexity of real ecosystems including processes taking place on multiple time and space scales. Finally, we argue that the construction of analytically tractable mean-field models is becoming a key issue to provide an insight into the major mechanisms of ecosystem functioning. We complete this review by introducing the contributions to the current special issue of Ecological Complexity.
41 citations
TL;DR: This paper presents a uniform approach to micro, macro and state-based population modelling so that these different types of models produce consistent results and conclusions and demonstrates the procedures necessary to keep these three type of models consistent.
Abstract: A population system can be modelled using a micro model focusing on the individual entities, a macro model where the entities are aggregated into compartments, or a state-based model where each possible discrete state in which the system can exist is represented. However, the concepts, building blocks, procedural mechanisms and the time handling for these approaches are very different. For the results and conclusions from studies based on micro, macro and state-based models to be consistent (contradiction-free), a number of modelling issues must be understood and appropriate modelling procedures be applied. This paper presents a uniform approach to micro, macro and state-based population modelling so that these different types of models produce consistent results and conclusions. In particular, we demonstrate the procedures (distribution, attribute and combinatorial expansions) necessary to keep these three types of models consistent. We also show that the different time handling methods usually used in micro, macro and state-based models can be regarded as different integration methods that can be applied to any of these modelling categories. The result is free choice in selecting the modelling approach and the time handling method most appropriate for the study without distorting the results and conclusions.
41 citations
TL;DR: This paper forms an individual-based model for navigation within a flowing field and applies scaling to derive its corresponding macroscopic and continuous model, which is applied to various movement classes, from drifters that simply go with the flow to navigators that respond to environmental orienteering cues.
Abstract: Navigation for aquatic and airborne species often takes place in the face of complicated flows, from persistent currents to highly unpredictable storms. Hydrodynamic models are capable of simulating flow dynamics and provide the impetus for much individual-based modelling, in which particle-sized individuals are immersed into a flowing medium. These models yield insights on the impact of currents on population distributions from fish eggs to large organisms, yet their computational demands and intractability reduce their capacity to generate the broader, less parameter-specific, insights allowed by traditional continuous approaches. In this paper, we formulate an individual-based model for navigation within a flowing field and apply scaling to derive its corresponding macroscopic and continuous model. We apply it to various movement classes, from drifters that simply go with the flow to navigators that respond to environmental orienteering cues. The utility of the model is demonstrated via its application to ‘homing’ problems and, in particular, the navigation of the marine green turtle Chelonia mydas to Ascension Island.
38 citations
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TL;DR: An antithesis to the view that gregarious behaviour is evolved through benefits to the population or species is presented, and simply defined models are used to show that even in non-gregarious species selection is likely to favour individuals who stay close to others.
3,343 citations
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23 Apr 19742,537 citations
TL;DR: In this paper, the authors used the random walk problem as a starting point for the analytical study of dispersal in living organisms and applied the law of diffusion to the understanding of the spatial distribution of population density in both linear and two-dimensional habitats.
Abstract: The random-walk problem is adopted as a starting point for the analytical study of dispersal in living organisms. The solution is used as a basis for the study of the expanson of a growing population, and illustrative examples are given. The law of diffusion is deduced and applied to the understanding of the spatial distribution of population density in both linear and two-dimensional habitats on various assumptions as to the mode of population growth or decline. For the numerical solution of certain cases an iterative process is described and a short table of a new function is given. The equilibrium states of the various analytical models are considered in relation to the size of the habitat, and questions of stability are investigated. A mode of population growth resulting from the random scattering of the reproductive units in a population discrete in time, is deduced and used as a basis for study on interspecific competition. The extent to which the present analytical formulation is applicable to biological situations, and some of the more important biological implications are briefly considered.
2,212 citations
TL;DR: In this paper, a mini-review of 50 individual-based animal population models shows that the majority of them are driven by pragmatic motivation, which does not explicitly refer to theoretical ecology.
978 citations