Multi-species aggregation equations: a bridge between movement ecology and spatial population dynamics

Lead Research Organisation: University of Sheffield
Department Name: Mathematics and Statistics


How do ecosystems arrange themselves in space? This is a core question for understanding how to conserve species, maintain biodiversity, and ensure that ecosystems are still functioning to provide services on which humanity depends (such as food, water, and air). Often, ecosystems incorporate a large variety of moving and interacting animals. Think of the various large mammal species moving on the Serengeti Plains, or the myriad animal species on a coral reef. As they move and interact with one another (as well as the more static plant species) they form arrangements in space. These can take the form of aggregations of symbiotic populations, segregations of competitors, or more complex patterns that can fluctuate in time and space.

These spatial arrangements are not, of course, planned in a "top down" fashion. Rather they emerge as a natural consequence of the movements and interactions of individual animals going about their daily lives. By building mathematical models of these movements and interactions, we can understand and predict the spatial distributions that ought to arise from different interaction scenarios. For example, suppose individuals from one species have a tendency to move towards areas where there are members of another, mutualist species, whilst at the same time individuals from the latter species like to move towards areas inhabited by the former. Then mathematical models can answer the question: how strong do these attractive tendencies have to be so that both species aggregate in a smaller part of space than they might otherwise occupy? Or suppose we have a more complicated system, with multiple species, some of which are attracted to one another, some of which repel each other, and others that have asymmetric movement tendencies (e.g. one chases the other and the latter retreats). What sort of spatial distribution of the various species will emerge? Will it stabilise in time, so that certain species occupy one part of space and others occupy different areas? Or will the distributions be in perpetual flux, continually changing over time?

This proposal aims to provide a general theory for answering such questions, using a mathematical formalism called a "multi-species aggregation equation". Present understanding of animal species distributions typically centres around understanding how they are correlated with relatively static environmental features, such as topography and vegetation cover. Here, instead, we will show how between-population movement responses can lead to the spontaneous formation of a wide range of spatio-temporal distributions. We aim to classify these, relating qualitative features of the emergent patterns to underlying movement-and-interaction processes. We will also examine so-called "hysteresis" effects, whereby different patterns can emerge from the same underlying processes, dependent upon the recent history of spatio-temporal patterns.

This work has the potential to change the way the scientific community thinks about how animal species are distributed in space, by shifting focus from static environmental covariates to non-linear feedbacks in animal movement mechanisms. If successful, this could give rise to much better-informed decisions regarding spatial conservation and interventions to maintain biodiversity. The project gives a core example of the vital importance of a mechanistic, mathematical approach in understanding ecological phenomena.

Planned Impact

Understanding the mechanisms behind spatial distributions of species is of core concern for spatial conservation efforts and thus maintenance of biodiversity and ecosystem services. It is also key to understanding spread of biological invasions. Loss of ecosystem services and introduction of biological invaders are both extremely costly, with worldwide bills estimated as running into the 10s or even 100s of billions of pounds per year (see "Pathways to Impact" for detailed figures), as well as associated negative impacts on human health and wellbeing. Therefore insights such as those provided by this proposal feed into a body of knowledge that is used by governmental and non-governmental organisations (e.g. the Centre for Ecology and Hydrology in the UK,; the National Parks Service in the USA,; Fisheries and Oceans Canada,; etc.) to inform policy decisions. Indeed, as a case in point, one of the present project partners, Prof. Mark Lewis, has been directly involved in networks where his mathematical research has been used to make policy decisions regarding invasive species (see

The pathway from fundamental mathematical insights to policy decisions and other non-academic impact is often indirect. Specifically, the path for mathematical ecologists is usually via empirical ecologists within academia who have links to the non-academic beneficiaries. To this end, the PI (Jonathan Potts) has built up a strong network of collaboration with empirical ecologists, publishes regularly in ecological journals, organises interdisciplinary conferences, and has office within the British Ecological Society (see details in "Pathways to Impact"). Furthermore, the present proposal has two objectives (O5 and O6) aimed towards moving the theoretical impacts down the impact pathway, and incorporates organisations of two impact workshops to the same end.

Once these pathways are traversed, enabled by the PI's collaborative interdisciplinary approach to research, we anticipate some quite specific impacts of our research within a non-academic context. Particularly, our insights will show how a rich range of spatial patterns of species distributions can emerge spontaneously, regardless of underlying (static) environmental covariates. This will enable better modelling of species distributions therefore better predictions of the spatial spread of species under scenarios of anthropogenic change. These will be of direct use for government agencies and NGOs (e.g. those listed above) interested in managing the effects of climate change and other anthropogenic actions on both spatial spread of species (e.g. in the context of biological invasions) and biodiversity loss: for example by informing biological control strategies or designing marine protected areas.
Description University of Alberta 
Organisation University of Alberta
Country Canada 
Sector Academic/University 
PI Contribution Collaboration with Mark Lewis and Thomas Hillen at the University of Alberta; sharing ideas co-authoring manuscripts; co-organising workshop (at University of Sheffield, April 2022)
Collaborator Contribution Collaboration with Mark Lewis and Thomas Hillen at the University of Alberta; sharing ideas co-authoring manuscripts; co-organising workshop (at University of Sheffield, April 2022)
Impact Interdisciplinary maths-ecology collaboration, with multiple publications over the years. Those related to the grant are: - Potts JR, Giunta V, Lewis MA. (in review) Beyond resource selection: emergent spatio-temporal distributions from animal movements and stigmergent interactions. Oikos - Giunta V, Hillen T, Lewis MA, Potts JR. (in press) Local and Global Existence for Non-local Multi-Species Advection-Diffusion Models. SIAM J Appl Dyn Syst Those prior to the grant (dtaing back to 2014): - Klappstein NJ, Potts JR, Michelot T, Börger L, Pilford N, Lewis M, Derocher A. (in press) Energy selection functions: modelling the energetic drivers of animal movement and habitat use. J Anim Ecol - Potts JR, Lewis MA (2019) Spatial memory and taxis-driven pattern formation in model ecosystems. Bull Math Biol 81:2725-2747, doi:10.1007/s11538-019-00626-9 - Bastille-Rousseau G, Murray DL, Schaefer JA, Lewis MA, Mahoney S, Potts JR. (2018) Spatial scales of habitat selection decisions: implications for telemetry-based movement modelling. Ecography 40:001-007, doi:10.1111/ecog.02655 - Potts JR, Lewis MA. (2016) How memory of direct animal interactions can lead to territorial pattern formation. J Roy Soc Interface 13:20160059, doi:10.1098/rsif.2016.0059 - Lewis MA, Petrovskii SV, Potts JR. (2016) The mathematics behind biological invasions. Springer Interdisciplinary Applied Mathematics series, 44. - Potts JR, Lewis MA. (2016) Territorial pattern formation in the absence of an attractive potential. J Math Biol 72:25-46, doi: 10.1007/s00285-015-0881-4 - Avgar T, Potts JR, Boyce MS, Lewis MA. (2016) Integrated step selection analysis: bridging the gap between resource selection and animal movement. Methods Ecol Evol 7:619-630, doi:10.1111/2041-210X.12528 - Potts JR, Hillen T, Lewis MA. (2016) The `edge effect' phenomenon: deriving population abundance patterns from individual movement decisions. Theor Ecol 9:233-247, doi:10.1007/s12080-015-0283-7 - Bastille-Rousseau G, Potts JR , Murray DL, Lewis MA, Ellington H, Rayl ND, Mahoney SP, Fuller TK, Organ JF, Schaefer JA. (2015) Unveiling tradeoffs in resource selection of migratory caribou using a mechanistic movement model of availability. Ecography doi: 10.1111/ecog.01305 - Potts JR, Mokross K, Stouffer PC, Lewis MA. (2014) Step selection techniques uncover the environmental predictors of space use patterns in flocks of Amazonian birds. Evol Ecol 4:4578-4588, preprint at arXiv: 1403.6869 - Potts JR, Auger-Méthé M, Mokross K, Lewis MA. (2014) A generalised residual technique for analysing complex movement models using earth mover's metric. Methods Ecol Evol 5:1012-1022, preprint at arXiv:1402.1805 - Potts JR, Mokross K, Lewis MA. (2014) A unifying framework for quantifying the nature of animal interactions. J Roy Soc Interface 11:20140333 - Potts JR, Lewis MA. (2014) How do animal territories form and change? Lessons from 20 years of mechanistic modelling. Proc Roy Soc B 5:253-262 - Potts JR, Lewis MA. (2014) A mathematical approach to territorial pattern formation. Am Math Mon 121(9):754-770 (Invited publication) - Potts JR, Bastille-Rousseau G, Murray DL, Schaefer JA, Lewis MA. (2014) Predicting local and non-local effects of resources on animal space use using a mechanistic step-selection model. Methods Ecol Evol 5:253-262
Start Year 2021