Skip to main content
SpringerLink
Log in

Small-scale spatial structure affects predator-prey dynamics and coexistence

  • ORIGINAL PAPER
  • Published:
Theoretical Ecology Aims and scope Submit manuscript

Abstract

Small-scale spatial variability can affect community dynamics in many ecological and biological processes, such as predator-prey dynamics and immune responses. Spatial variability includes short-range neighbour-dependent interactions and small-scale spatial structure, such as clustering where individuals aggregate together, and segregation where individuals are spaced apart from one another. Yet, a large class of mathematical models aimed at representing these processes ignores these factors by making a classical mean-field approximation, where interactions between individuals are assumed to occur in proportion to their average density. Such mean-field approximations amount to ignoring spatial structure. In this work, we consider an individual-based model of a two-species community that is composed of consumers and resources. The model describes migration, predation, competition and dispersal of offspring, and explicitly gives rise to varying degrees of spatial structure. We compare simulation results from the individual-based model with the solution of a classical mean-field approximation, and this comparison provides insight into how spatial structure can drive the system away from mean-field dynamics. Our analysis reveals that mechanisms leading to intraspecific clustering and interspecific segregation, such as short-range predation and short-range dispersal, tend to increase the size of the resource species relative to the mean-field prediction. We show that under certain parameter regimes these mechanisms lead to the extinction of consumers whereas the classical mean-field model predicts the coexistence of both species.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  • Abrams PA (2000) The evolution of predator-prey interactions: theory and evidence. Annual Rev Ecol Syst 31:79–105

    Google Scholar 

  • Abrams PA, Ginzburg LR (2000) The nature of predation: prey dependent, ratio dependent or neither?. Trends Ecol Evol 15:337–341

    CAS  PubMed  Google Scholar 

  • Agnew DJG, Green JEF, Brown TM, Simpson MJ, Binder BJ (2014) Distinguishing between mechanisms of cell aggregation using pair-correlation functions. J Theor Biol 352:16–23

    CAS  PubMed  Google Scholar 

  • Akira S, Uematsu S, Takeuchi O (2006) Pathogen recognition and innate immunity. Cell 124:783–801

    CAS  PubMed  Google Scholar 

  • Baker RE, Simpson MJ (2010) Correcting mean-field approximations for birth-death-movement processes. Phys Rev E 82:041905

    Google Scholar 

  • Barraquand F, Murrell DJ (2012) Intense or spatially heterogeneous predation can select against prey dispersal. PLoS One 7:e28924

    CAS  PubMed  PubMed Central  Google Scholar 

  • Binder BJ, Simpson MJ (2015) Spectral analysis of pair-correlation bandwidth: application to cell biology images. Royal Soc Open Sci 2:140494

    Google Scholar 

  • Binny RN, Haridas P, James A, Law R, Simpson MJ, Plank MJ (2016a) Spatial structure arising from neighbour-dependent bias in collective cell movement. PeerJ 4:e1689

  • Binny RN, James A, Plank MJ (2016b) Collective cell behaviour with neighbour-dependent proliferation, death and directional bias. Bull Math Biol 78:2277–2301

  • Binny RN, Law R, Plank MJ (2020) Living in groups: Spatial-moment dynamics with neighbour-biased movements. Ecol Model 415:108825

    Google Scholar 

  • Binny RN, Plank MJ, James A (2015) Spatial moment dynamics for collective cell movement incorporating a neighbour-dependent directional bias. J R Soc Interface 12:20150228

    PubMed  PubMed Central  Google Scholar 

  • Bolker BM, Pacala SW (1999) Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am Natur 153:575–602

    PubMed  Google Scholar 

  • Brigatti E, Oliva M, Nunez-Lopez M, Oliveros-Ramos R, Benavides J (2009) Pattern formation in a predator-prey system characterized by a spatial scale of interaction. Europhys Lett 88:68002

    Google Scholar 

  • Britton N (2003) Essential mathematical biology. Springer, London

    Google Scholar 

  • Browning AP, McCue SW, Binny RN, Plank MJ, Shah ET, Simpson MJ (2018) Inferring parameters for a lattice-free model of cell migration and proliferation using experimental data. J Theor Biol 437:251–260

    PubMed  Google Scholar 

  • Browning AP, Jin W, Plank MJ, Simpson MJ (2020) Identifying density-dependent interactions in collective cell behaviour. J R Soc Interface 17:20200143.

    PubMed  Google Scholar 

  • Cantrell RS, Cosner C (2004) Deriving reaction–diffusion models in ecology from interacting particle systems. J Math Biol 48:187–217

    CAS  PubMed  Google Scholar 

  • Cuddington KM, Yodzis P (2000) Diffusion-limited predator-prey dynamics in euclidean environments: an allometric individual-based model. Theor Popul Biol 58:259–278

    CAS  PubMed  Google Scholar 

  • Dini S, Binder BJ, Green JEF (2018) Understanding interactions between populations: individual based modelling and quantification using pair correlation functions. J Theor Biol 439:50–64

    CAS  PubMed  Google Scholar 

  • Dobramysl U, Tauber UC (2013) Environmental versus demographic variability in two-species predator-prey models. Phys Rev Lett 110:048105

    PubMed  Google Scholar 

  • Edelstein-Keshet L (2005) Mathematical models in biology (classics in applied mathematics). Society for Industrial and Applied Mathematics, New York

    Google Scholar 

  • Fadai NT, Johnston ST, Simpson MJ (2019) Unpacking the Allee effect: determining individual-level mechanisms that drive population dynamics. https://doi.org/10.1101/774000v1https://doi.org/10.1101/774000v1bioRxiv.Accessed. December 2019

  • Galetti M, Moleon M, Jordano P, Pires MM, Guimaraes PR Jr, Pape T, Nichols E, Hansen D, Olesen JM, Munk M, de Mattos JS, Schweiger AH, Owen-Smith N, Johnson CN, Marquis RJ, Svenning JC (2018) Ecological and evolutionary legacy of megafauna extinctions. Biol Rev 93:845–862

    PubMed  Google Scholar 

  • Gerum R, Richter S, Fabry B, Bohec CL, Bonadonna F, Nesterova A, Zitterbart DP (2018) Structural organisation and dynamics in king penguin colonies. J Phys D:, Appl Phys 51:164004

    Google Scholar 

  • Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81:2340–2361

    CAS  Google Scholar 

  • Grunbaum D (2012) The logic of ecological patchiness. Inter Focus 2:150–155

    Google Scholar 

  • Hillen T, Painter KJ (2009) A user’s guide to PDE models for chemotaxis. J Math Biol 58:183–217

    CAS  PubMed  Google Scholar 

  • Hosseini PR (2003) How localized consumption stabilizes predator-prey systems with finite frequency of mixing. Am Natur 161:567–585

    PubMed  Google Scholar 

  • Hosseini PR (2006) Pattern formation and individual-based models: the importance of understanding individual-based movement. Ecol Model 194:357–371

    Google Scholar 

  • Hunt VM, Brown JS (2018) Coexistence and displacement in consumer-resource systems with local and shared resources. Theor Ecol 11:83–93

    Google Scholar 

  • Jin W, McCue SW, Simpson MJ (2018) Extended logistic growth model for heterogeneous populations. J Theor Biol 445:51–61

    PubMed  Google Scholar 

  • Johnston ST, Baker RE, McElwain DLS, Simpson MJ (2017) Co-operation, competition and crowding: a discrete framework linking Allee kinetics, nonlinear diffusion, shocks and sharp-fronted travelling waves. Sci Report 7:42134

    CAS  Google Scholar 

  • Kuperman MN, Laguna MF, Abramson G, Monjeau JA (2019) Meta-population oscillations from satiation of predators. Phys A 527:121288

    Google Scholar 

  • Law R, Dieckmann U (2000) A dynamical system for neighbourhoods in plant communities. Ecology 81:2137–2148

    Google Scholar 

  • Law R, Illian J, Burslem DFRP, Gratzer G, Gunatilleke CVS, Gunatilleke IAUN (2009) Ecological information from spatial patterns of plants: insights from point process theory. J Ecol 97:616–628

    Google Scholar 

  • Law R, Murrell DJ, Dieckmann U (2003) Population growth in space and time: spatial logistic equations. Ecology 84:252–262

    Google Scholar 

  • Markham DC, Simpson MJ, Maini PK, Gaffney EA, Baker RE (2013) Incorporating spatial correlations into multispecies mean-field models. Phys Rev E 88:052713

    Google Scholar 

  • Mathworks (2019) https://www.mathworks.com/help/matlab/ref/ode45.htmlhttps://www.mathworks.com/help/matlab/ref/ode45.html Solve nonstiff differential equations — medium order method. Accessed December 2019

  • Mobilia M, Georgiev IT, Tauber UC (2006) Fluctuations and correlations in lattice models for predator-prey interaction. PhysRev E 73(R):040903

    Google Scholar 

  • Mobilia M, Georgiev IT, Tauber UC (2007) Phase transitions and spatio-temporal fluctuations in stochastic lattice Lotka–Volterra models. J Statis Phys 128:447–483

    Google Scholar 

  • Murray JD (1989) Mathematical biology. Springer, New York

    Google Scholar 

  • Murrell DJ (2005) Local spatial structure and predator-prey dynamics: counterintuitive effects of prey enrichment. Am Natur 166:354–367

    PubMed  Google Scholar 

  • Ovaskainen O, Finkelshtein D, Kutoviy O, Cornell S, Bolker B, Kondratiev Y (2014) A general mathematical framework for the analysis of spatiotemporal point processes. Theor Ecol 7:101–113

    Google Scholar 

  • Penczykowski RM, Laine A, Koskella B (2016) Understanding the ecology and evolution of host–parasite interactions across scales. Evol Appl 9:37–52

    PubMed  Google Scholar 

  • Plank MJ, Law R (2015) Spatial point processes and moment dynamics in the life sciences: a parsimonious derivation and some extensions. Bull Math Biol 77:586–613

    PubMed  Google Scholar 

  • Plank MJ, Simpson MJ, Binny RN (2019) Small-scale spatial structure influences large-scale invasion rates. Theoretical Ecology. https://doi.org/10.1007/s12080-020-00450-1

  • Rincon DF, Canas LA, Hoy CW (2017) Modeling changes in predator functional response to prey across spatial scales. Theor Ecol 10:403–415

    Google Scholar 

  • Santora JA, Reiss CS, Loeb VJ, Veit RR (2010) Spatial association between hotspots of baleen whales and demographic patterns of Antarctic krill Euphausia superba suggests size-dependent predation. Mar Ecol Prog Ser 405:255–269

    Google Scholar 

  • Soehnlein O, Steffens S, Hidalgo A, Weber C (2017) Neutrophils as protagonists and targets in chronic inflammation. Nature Rev Immunol 17:248–261

    CAS  Google Scholar 

  • Stephens PA, Sutherland WJ, Freckleton RP (1999) What is the Allee effect?. Oikos 87:185–190

    Google Scholar 

  • Surendran A, Plank MJ, Simpson MJ (2018) Spatial moment description of birth-death-movement processes incorporating the effects of crowding and obstacles. Bull Math Biol 80:2828–2855

    PubMed  Google Scholar 

  • Surendran A, Plank MJ, Simpson MJ (2019) Spatial structure arising from chase-escape interactions with crowding. Sci Report 9:14988

    Google Scholar 

  • Tobin P, Bjornstad ON (2003) Spatial dynamics and cross-correlation in a transient predator-prey system. J Animal Ecol 72:460–467

    Google Scholar 

  • Treloar KK, Simpson MJ, Binder BJ, McElwain DLS, Baker RE (2015) Assessing the role of spatial correlations during collective cell spreading. Sci Report 4:5713

    Google Scholar 

  • Vijay K (2018) Toll-like receptors in immunity and inflammatory diseases: past, present, and future. Int Immunopharmacol 59:391–412

    CAS  PubMed  PubMed Central  Google Scholar 

  • Wang H, Nagy JD, Gilg O, Kuang Y (2009) The roles of predator maturation delay and functional response in determining the periodicity of predator–prey cycles. Math Biosci 221:1–10

    PubMed  Google Scholar 

  • Wilson WG (1998) Resolving discrepancies between deterministic population models and individual-based simulations. Am Natur 151:116–134

    CAS  PubMed  Google Scholar 

Download references

Acknowledgements

We thank the Associate Editor and anonymous referees for their helpful suggestions.

Funding

This work is supported by the Australian Research Council (DP170100474). MJP is partly supported by Te Pūnaha Matatini, a New Zealand Centre of Research Excellence.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Queensland University of Technology, Brisbane, QLD, Australia

    Anudeep Surendran & Matthew J. Simpson

  2. School of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand

    Michael J. Plank

  3. Te Pūnaha Matatini, A New Zealand Centre of Research Excellence, Auckland, New Zealand

    Michael J. Plank

Authors
  1. Anudeep Surendran
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael J. Plank
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Matthew J. Simpson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew J. Simpson.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Data availability

MATLAB code used to generate results are available on Github at https://github.com/Anudeep-Surendran/Surendran2020

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Surendran, A., Plank, M.J. & Simpson, M.J. Small-scale spatial structure affects predator-prey dynamics and coexistence. Theor Ecol 13, 537–550 (2020). https://doi.org/10.1007/s12080-020-00467-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12080-020-00467-6

Keywords

Search

Navigation