Coexistence, dispersal and spatial structure in metacommunities: a stochastic model approach

Abstract

We propose a stochastic model for interacting species in a metacommunity in order to study the factors affecting the intensity of the competition/colonization trade-off as a coexistence mechanism in metacommunities. We particularly focus on the role of the number of local communities and the number of refuges for the inferior competitor. The stochastic component is associated with the dispersal process and is represented by Poisson random measures. Thus, this stochastic model includes two dynamic scales: a continuous one, which refers to the interactions among species, and a low frequency one, referring to dispersal following a Poisson scheme. We show the well-posedness of the model and that it is possible to study its long-term behavior using Lyapunov exponents; the extinction of a species is associated with a negative slope in the time trajectory of the Lyapunov exponent, otherwise, it is equal to zero. We show that the competition/colonization trade-off is a function of the dispersal rate of the inferior competitor, and that it becomes less intense as the number of local communities increases, while the opposite is true with an increase in the number of refuges for the inferior competitor. We also show that under a priority effect type of scenario, dispersal can reverse priority effects and generate coexistence. Our results emphasize the importance of coexistence mechanisms related to the topology of the system of local communities, and its relationship with dispersal, in affecting the result of competition in local communities.

This is a preview of subscription content, access via your institution.

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

References

  1. Aiken CM, Navarrete SA (2014) Coexistence of competitors in marine metacommunities: environmental variability, edge effects, and the dispersal niche. Ecol 95(8):2289–2302

    Article  Google Scholar 

  2. Amarasekare P (2003) Competitive coexistence in spatially structured environments: a synthesis. Ecol Lett 6(12):1109–1122

    Article  Google Scholar 

  3. Amarasekare P (2004) Spatial variation and density-dependent dispersal in competitive coexistence. Proc R Soc London Ser B Biological Sci 271(1547):1497–1506

    Article  Google Scholar 

  4. Amarasekare P, Hoopes MF, Mouquet N, Holyoak M (2004) Mechanisms of coexistence in competitive metacommunities. Am Nat 164(3):310–326

    PubMed  Article  Google Scholar 

  5. Amarasekare P, Nisbet RM (2001) Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species. Am Nat 158(6):572–584

    CAS  PubMed  Article  Google Scholar 

  6. Applebaum D (2009) Lévy processes and stochastic calculus. Cambridge University Press, Cambridge

    Google Scholar 

  7. Bansaye V, Méléard S (2015) Stochastic models for structured populations. Springer, Berlin

    Google Scholar 

  8. Bao J, Mao X, Yin G, Yuan C (2011) Competitive lotka–Volterra population dynamics with jumps. Nonlinear Anal Theor Methods Appl 74(17):6601–6616

    Article  Google Scholar 

  9. Barter E, Gross T (2017) Spatial effects in meta-foodwebs. Sci Rep 7(1):1–10

    CAS  Article  Google Scholar 

  10. Benaïm M, Schreiber SJ (2019) Persistence and extinction for stochastic ecological models with internal and external variables. J Math Biol 79(1):393–431

    PubMed  Article  PubMed Central  Google Scholar 

  11. Boettiger C (2018) From noise to knowledge: how randomness generates novel phenomena and reveals information. Ecol Lett 21(8):1255–1267

    PubMed  Article  PubMed Central  Google Scholar 

  12. Bolker BM, Pacala SW (1999) Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am Nat 153(6):575–602

    PubMed  Article  PubMed Central  Google Scholar 

  13. Brechtel A, Gross T, Drossel B (2019) Far-ranging generalist top predators enhance the stability of meta-foodwebs. Sci Rep 9(1):1–15

    CAS  Article  Google Scholar 

  14. Cadotte MW (2006) Dispersal and species diversity: a meta-analysis. Am Nat 167(6):913–924

    PubMed  Article  PubMed Central  Google Scholar 

  15. Calcagno V, Mouquet N, Jarne P, David P (2006) Coexistence in a metacommunity: the competition–colonization trade-off is not dead. Ecol Lett 9(8):897–907

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  16. Cattiaux P, Méléard S (2010) Competitive or weak cooperative stochastic lotka–Volterra systems conditioned on non-extinction. J Math Biol 60(6):797–829

    PubMed  Article  PubMed Central  Google Scholar 

  17. Chesson P (1994) Multispecies competition in variable environments. Theor Popul Biol 45 (3):227–276

    Article  Google Scholar 

  18. Chesson P (2000) General theory of competitive coexistence in spatially-varying environments. Theor Popul Biol 58(3):211–237

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  19. Chesson P (2018) Updates on mechanisms of maintenance of species diversity. J Ecol 106 (5):1773–1794

    Article  Google Scholar 

  20. Chesson PL (1982) The stabilizing effect of a random environment. J Math Biol 15(1):1–36

    Article  Google Scholar 

  21. Cornell HV, Lawton JH (1992) Species interactions, local and regional processes, and limits to the richness of ecological communities: a theoretical perspective. J Anim Ecol 61(1):1–12

    Article  Google Scholar 

  22. Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and conservation. Oxford University Press, Oxford

    Google Scholar 

  23. De Masi A, Galves A, Löcherbach E, Presutti E (2015) Hydrodynamic limit for interacting neurons. J Stat Phys 158(4):866–902

    Article  Google Scholar 

  24. De Meester L, Vanoverbeke J, Kilsdonk LJ, Urban MC (2016) Evolving perspectives on monopolization and priority effects. Trends Ecol Evol 31(2):136–146

    PubMed  Article  PubMed Central  Google Scholar 

  25. Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34(5-6):579–612

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  26. Du NH, Sam VH (2006) Dynamics of a stochastic lotka–Volterra model perturbed by white noise. J Math Anal Appl 324(1):82–97

    Article  Google Scholar 

  27. Durrett R, Levin S (1998) Spatial aspects of interspecific competition. Theor Popul Biol 53 (1):30–43

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  28. Economo EP, Keitt TH (2008) Species diversity in neutral metacommunities: a network approach. Ecol Lett 11(1):52–62

    PubMed  PubMed Central  Google Scholar 

  29. Economo EP, Keitt TH (2010) Network isolation and local diversity in neutral metacommunities. Oikos 119(8):1355–1363

    Article  Google Scholar 

  30. Forbes AE, Chase JM (2002) The role of habitat connectivity and landscape geometry in experimental zooplankton metacommunities. Oikos 96(3):433–440

    Article  Google Scholar 

  31. Fournier N, Löcherbach E (2016) On a toy model of interacting neurons. Annales de l’Institut Henri poincaré Probabilités et Statistiques 52(4):1844–1876

    Article  Google Scholar 

  32. Freilich MA, Rebolledo R, Corcoran D, Marquet PA (2020) Reconstructing ecological networks with noisy dynamics. Proc R Soc A 476(2237):20190739

    PubMed  Article  PubMed Central  Google Scholar 

  33. Gravel D, Massol F, Leibold MA (2016) Stability and complexity in model meta-ecosystems. Nat Commun 7(1):1–8

    Article  CAS  Google Scholar 

  34. Ha S-Y, Liu J-G (2009) A simple proof of the cucker-smale flocking dynamics and mean-field limit. Commun Math Sci 7(2):297–325

    Article  Google Scholar 

  35. Hanski I (1982) Dynamics of regional distribution: the core and satellite species hypothesis. Oikos 38(2):210–221

    Article  Google Scholar 

  36. Hanski I, Ovaskainen O (2000) The metapopulation capacity of a fragmented landscape. Nature 404(6779):755–758

    CAS  PubMed  Article  Google Scholar 

  37. Haskovec J (2013) Flocking dynamics and mean-field limit in the cucker–smale-type model with topological interactions. Phys D Nonlinear Phenom 261:42–51

    Article  Google Scholar 

  38. Hastings A (1980) Disturbance, coexistence, history, and competition for space. Theor Popul Biol 18(3):363–373

    Article  Google Scholar 

  39. Hening A, Nguyen DH (2018a) Coexistence and extinction for stochastic Kolmogorov systems. Ann Appl Probab 28(3):1893–1942

    Article  Google Scholar 

  40. Hening A, Nguyen DH (2018b) Stochastic lotka–Volterra food chains. J Math Biol 77(1):135–163

    PubMed  Article  Google Scholar 

  41. Hening A, Nguyen DH, Yin G (2018) Stochastic population growth in spatially heterogeneous environments: the density-dependent case. J Math Biol 76(3):697–754

    PubMed  Article  Google Scholar 

  42. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge

    Google Scholar 

  43. Holt RD (1993) Ecology at the mesoscale: the influence of regional processes on local communities. In: Ricklefs, Schluter (eds) Species Diversity: Historical and Geographical Perspectives. University of Chicago, p 77–88

  44. Holt RD (1996) Food webs in space: an island biogeographic perspective. In: Polis GA, Winemiller KO (eds) Food webs. Springer, pp 313–323

  45. Holt RD (2009) Toward a trophic island biogeography. In: Losos JB, Ricklefs RE (eds) The theory of island biogeography revisited. Princeton University Press Princeton, NJ, pp 143–185

  46. Holt RD, Hoopes MF, Holyoak M, Leibold MA, Holt RD (2005) Food web dynamics in a metacommunity context. In: Metacommunities: spatial dynamics and ecological communities. University of Chicago Press Chicago, Illinois, pp 68–94

  47. Holyoak M (2000) Habitat subdivision causes changes in food web structure. Ecol Lett 3(6):509–515

    Article  Google Scholar 

  48. Hubbell SP (2001) The unified neutral theory of biodiversity and biogeography (MPB-32), volume 32. Princeton University Press

  49. Hubbell SP (2005) Neutral theory in community ecology and the hypothesis of functional equivalence. Funct Ecol 19(1):166–172

    Article  Google Scholar 

  50. Ikeda N, Watanabe S (2014) Stochastic differential equations and diffusion processes. Elsevier, Amsterdam

    Google Scholar 

  51. Jocque M, Field R, Brendonck L, De Meester L (2010) Climatic control of dispersal–ecological specialization trade-offs: a metacommunity process at the heart of the latitudinal diversity gradient? Glob Ecol Biogeogr 19(2):244–252

    Article  Google Scholar 

  52. Keymer JE, Marquet PA, Velasco-Hernández JX, Levin SA (2000) Extinction thresholds and metapopulation persistence in dynamic landscapes. Am Nat 156(5):478–494

    PubMed  Article  PubMed Central  Google Scholar 

  53. Leibold MA, Holyoak M, Mouquet N, Amarasekare P, Chase JM, Hoopes MF, Holt RD, Shurin JB, Law R, Tilman D et al (2004) The metacommunity concept: a framework for multi-scale community ecology. Ecol Lett 7(7):601–613

    Article  Google Scholar 

  54. Levin SA (1974) Dispersion and population interactions. Am Nat 108(960):207–228

    Article  Google Scholar 

  55. Levins R (1969) The effect of random variations of different types on population growth. Proc Natl Acad Sci 62(4):1061–1065

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  56. Levins R, Culver D (1971) Regional coexistence of species and competition between rare species. Natl Acade Sci 68(6):1246–1248

    CAS  Article  Google Scholar 

  57. Li X, Mao X (2009) Population dynamical behavior of non-autonomous lotka-Volterra competitive system with random perturbation. Discrete Contin Dyn Syst-A 24(2):523

    CAS  Article  Google Scholar 

  58. Lions P-L, Sznitman A-S (1984) Stochastic differential equations with reflecting boundary conditions. Commun Pur Appl Math 37(4):511–537

    Article  Google Scholar 

  59. Loreau M, Mouquet N (1999) Immigration and the maintenance of local species diversity. Am Nat 154(4):427–440

    PubMed  Article  PubMed Central  Google Scholar 

  60. Luo Q, Mao X (2007) Stochastic population dynamics under regime switching. J Math Anal Appl 334(1):69–84

    Article  Google Scholar 

  61. MacArthur RH, Wilson EO (1963) An equilibrium theory of insular zoogeography. Evolution 17(4):373–387

    Article  Google Scholar 

  62. Mao X (2007) Stochastic differential equations and applications. Elsevier, Amsterdam

    Google Scholar 

  63. Mao X, Marion G, Renshaw E (2002) Environmental Brownian noise suppresses explosions in population dynamics. Stoch Process Appl 97(1):95–110

    Article  Google Scholar 

  64. Mao X, Sabanis S, Renshaw E (2003) Asymptotic behaviour of the stochastic lotka–Volterra model. J Math Anal Appl 287(1):141–156

    Article  Google Scholar 

  65. Marquet PA (1997) A source-sink patch occupancy metapopulation model. Rev Chil Hist Nat 70:371–380

    Google Scholar 

  66. Marquet PA, Allen AP, Brown JH, Dunne JA, Enquist BJ, Gillooly JF, Gowaty PA, Green JL, Harte J, Hubbell SP et al (2014) On theory in ecology. Bioscience 64(8):701–710

    Article  Google Scholar 

  67. Marquet PA, Espinoza G, Abades SR, Ganz A, Rebolledo R (2017) On the proportional abundance of species: integrating population genetics and community ecology. Sci Rep 7(1):1–10

    CAS  Article  Google Scholar 

  68. Marquet PA, Tejo M, Rebolledo R (2020) What is the species richness distribution?. In: Dobson A, Holt RD, Tilman D (eds) Unsolved problems in ecology. Princeton University Press, pp 177–188

  69. Mena-Lorca J, Velasco-Hernández JX, Marquet PA (2006) Coexistence in metacommunities: a tree-species model. Math Biosci 202(1):42–56

    PubMed  Article  PubMed Central  Google Scholar 

  70. Mohd MH, Murray R, Plank MJ, Godsoe W (2016) Effects of dispersal and stochasticity on the presence–absence of multiple species. Ecol Model 342:49–59

    Article  Google Scholar 

  71. Murray JD (2007) Mathematical biology: i. an introduction, vol 17. Springer Science & Business Media, Berlin

  72. Murrell DJ, Law R (2003) Heteromyopia and the spatial coexistence of similar competitors. Ecol Lett 6(1):48–59

    Article  Google Scholar 

  73. Nagylaki T (1979) The island model with stochastic migration. Genetics 91(1):163–176

    CAS  PubMed  PubMed Central  Google Scholar 

  74. Nee S, May RM (1992) Dynamics of metapopulations: habitat destruction and competitive coexistence. J Anim Ecol 61(1):37–40

    Article  Google Scholar 

  75. Neuhauser C, Pacala SW (1999) An explicitly spatial version of the lotka-Volterra model with interspecific competition. Ann Appl Probab 9(4):1226–1259

    Article  Google Scholar 

  76. Ovaskainen O, Meerson B (2010) Stochastic models of population extinction. Trends Ecol Evol 25(11):643–652

    PubMed  Article  PubMed Central  Google Scholar 

  77. Pakdaman K, Perthame B, Salort D (2009) Dynamics of a structured neuron population. Nonlinearity 23(1):55

    Article  Google Scholar 

  78. Pakdaman K, Perthame B, Salort D (2013) Relaxation and self-sustained oscillations in the time elapsed neuron network model. SIAM J Appl Math 73(3):1260–1279

    Article  Google Scholar 

  79. Pillai P, Gonzalez A, Loreau M (2011) Metacommunity theory explains the emergence of food web complexity. Proc Natl Acad Sci 108(48):19293–19298

    CAS  PubMed  Article  Google Scholar 

  80. Plitzko SJ, Drossel B (2015) The effect of dispersal between patches on the stability of large trophic food webs. Theor Ecol 8(2):233–244

    Article  Google Scholar 

  81. Quininao C, Touboul J (2015) Limits and dynamics of randomly connected neuronal networks. Acta Appl Math 136(1):167–192

    Article  Google Scholar 

  82. Rebolledo R, Navarrete SA, Kéfi S., Rojas S, Marquet PA (2019) An open-system approach to complex biological networks. SIAM J Appl Math 79(2):619–640

    Article  Google Scholar 

  83. Richter-Dyn N, Goel NS (1972) On the extinction of a colonizing species. Theor Popul Biol 3(4):406–433

    CAS  PubMed  Article  Google Scholar 

  84. Ricklefs RE (1987) Community diversity: relative roles of local and regional processes. Science 235(4785):167–171

    CAS  PubMed  Article  Google Scholar 

  85. Saavedra S, Rohr RP, Bascompte J, Godoy O, Kraft NJ, Levine JM (2017) A structural approach for understanding multispecies coexistence. Ecol Monogr 87(3):470–486

    Article  Google Scholar 

  86. Schreiber SJ (2017) Coexistence in the face of uncertainty. In: Recent progress and modern challenges in applied mathematics, modeling and computational science. Springer, pp 349–384

  87. Schreiber SJ, Killingback TP (2013) Spatial heterogeneity promotes coexistence of rock–paper–scissors metacommunities. Theor Popul Biol 86:1–11

    PubMed  Article  Google Scholar 

  88. Seeley TD, Visscher PK, Schlegel T, Hogan PM, Franks NR, Marshall JA (2012) Stop signals provide cross inhibition in collective decision-making by honeybee swarms. Science 335(6064):108–111

    CAS  PubMed  Article  Google Scholar 

  89. Singh P, Baruah G (2020) Higher order interactions and species coexistence. Theoretical Ecology, 1–13

  90. Skorokhod AV (1961) Stochastic equations for diffusion processes in a bounded region. Theory Probab Its Appl 6(3):264–274

    Article  Google Scholar 

  91. Skorokhod AV (1962) Stochastic equations for diffusion processes in a bounded region. ii. Theory Probab Its Appl 7(1):3–23

    Article  Google Scholar 

  92. Southwood TR (1977) Habitat, the templet for ecological strategies? J Anim Ecol 46(2):337–365

    Article  Google Scholar 

  93. Tanaka H (2002) Stochastic differential equations with reflecting boundary condition in convex regions. In: Stochastic processes: selected papers of Hiroshi Tanaka. World Scientific, pp 157–171

  94. Tejo M, Niklitschek-Soto S, Vásquez C, Marquet PA (2017) Single species dynamics under climate change. Theor Ecol 10(2):181–193

    Article  Google Scholar 

  95. Tilman D (1994) Competition and biodiversity in spatially structured habitats. Ecol 75(1):2–16

    Article  Google Scholar 

  96. Turnbull LA, Rees M, Crawley MJ (1999) Seed mass and the competition/colonization trade-off: a sowing experiment. J Ecol 87:899–912

    Article  Google Scholar 

  97. Venail PA, MacLean RC, Bouvier T, Brockhurst MA, Hochberg ME, Mouquet N (2008) Diversity and productivity peak at intermediate dispersal rate in evolving metacommunities. Nature 452(7184):210–214

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  98. Warren BH, Simberloff D, Ricklefs RE, Aguilée R, Condamine FL, Gravel D, Morlon H, Mouquet N, Rosindell J, Casquet J et al (2015) Islands as model systems in ecology and evolution: prospects fifty years after macArthur-Wilson. Ecol Lett 18(2):200–217

    PubMed  Article  PubMed Central  Google Scholar 

  99. Wilson DS (1992) Complex interactions in metacommunities, with implications for biodiversity and higher levels of selection. Ecol 73(6):1984–2000

    Article  Google Scholar 

  100. Wilson EO, MacArthur RH (1967) The theory of island biogeography. Princeton University Press, Princeton

    Google Scholar 

  101. Xiong J, Li X, Wang H (2019) The survival analysis of a stochastic Lotka-Volterra competition model with a coexistence equilibrium. Math Biosci 1(k1k2):1–k1k2

    Google Scholar 

  102. Zhou Y, Kot M (2011) Discrete-time growth-dispersal models with shifting species ranges. Theor Ecol 4(1):13–25

    Article  Google Scholar 

Download references

Acknowledgments

We acknowledge funding from projects FONDECYT 1161023 and 1200925 and Project EcoDep. We also acknowledge funding from a postdoctoral fellowship to Cristóbal Quiñinao (FONDECYT 3170435), and support from projects CIMFAV-CIDI and Redes CONICYT 180018 to Rolando Rebolledo and AFB170008. We thank Sergio Navarrete for stimulating discussions on the role of dispersal and spatial structure in metacommunities and two anonymous reviewers for their insightful comments, suggestions, and criticisms.

Funding

FONDECYT 1161023, 1200925, FONDECYT 3170435), CIMFAV-CIDI, AFB170008, and Redes CONICYT 180018.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Pablo A. Marquet.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Authors’ contributions

All authors participated in the design and analysis, and jointly wrote the manuscript.

Data availability

Not applicable.

Code availability

All codes used in the simulations are available upon request.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix 1: Proofs

Proof of Proposition 1

Let ∥⋅∥ be the usual Euclidean norm in \(\mathbb {R}^{d}\), and for real matrices Q = (qlk)l,k= 1,..,d define the norm \(\parallel \cdot \parallel _{\max \limits }\) as \(\parallel Q\parallel _{\max \limits }=\underset {1\leq l,k\leq d}{\max \limits }\mid q_{lk}\mid \). Notice that for an \(\mathbb {R}^{d}\)-vector Y, we have \(\parallel QY\parallel \leq \sqrt {d}\parallel Q\parallel _{\max \limits }\parallel Y\parallel \). Let \(\boldsymbol {x}\in \mathbb {R}^{IJ}\). Then, we have the following inequalities: \(\parallel {\Lambda }(\boldsymbol {x})\boldsymbol {x}\parallel \leq \sqrt {IJ}\parallel {\Lambda }(\boldsymbol {x})\parallel _{\max \limits }\parallel \boldsymbol {x}\parallel \) and \(\parallel B(\boldsymbol {x},z)\parallel =\parallel \text {vec}(B(\boldsymbol {x},z))\parallel \leq \sqrt {IJ}\parallel B(\boldsymbol {x},z)\parallel _{\max \limits }\leq \sqrt {IJ}\parallel \beta (z)\parallel _{\max \limits }\parallel \boldsymbol {x}\parallel \), where

$$ \beta(z) = \left( \!\begin{array}{ccccc} \varepsilon_{11}1_{\left\{ z\leq b_{1}\right\} }\cdots\varepsilon_{1I}1_{\left\{ z\leq b_{1}\right\} } & 0 & \cdot & \cdot & 0\\ 0 & \cdot & & & \vdots\\ {\vdots} & & \cdot & & 0\\ 0 & \cdot & \cdot & 0 & \varepsilon_{I1}1_{\left\{ z\leq b_{J}\right\} }\cdots\varepsilon_{II}1_{\left\{ z\leq b_{J}\right\} } \end{array}\!\right). $$
(14)

Using those inequalities, we have that:

$$ D(\boldsymbol{x})=\parallel F(\boldsymbol{x})\parallel^{2}+\parallel{\Lambda}(\boldsymbol{x})\boldsymbol{x}\parallel^{2}+\int\limits_{0}^{\infty}\parallel B(\boldsymbol{x},z)\parallel^{2}dz $$
$$ \begin{array}{@{}rcl@{}} &\leq&\parallel F(\boldsymbol{x})\parallel^{2}+IJ\parallel{\Lambda}(\boldsymbol{x})\parallel_{\max}^{2}\parallel\boldsymbol{x}\parallel^{2}\\&&+IJ\int\limits_{0}^{\infty}\parallel\beta(z)\parallel_{\max}^{2}dz\parallel\boldsymbol{x}\parallel^{2} \end{array} $$
$$ \leq K(\boldsymbol{x})(1+\parallel\boldsymbol{x}\parallel^{2}), $$

where \(K(\boldsymbol {x})=\max \limits \{\parallel F(\boldsymbol {x})\parallel ^{2},IJ\parallel {\Lambda }(\boldsymbol {x})\parallel _{\max \limits }^{2},IJ{\int \limits }_{0}^{\infty }\parallel \beta (z)\parallel _{\max \limits }^{2}dz\}\). By (A.1), ∥X(0) ∥ is bounded, so we can choose a large K0K(X(0)) such that D(x) ≤ K0(1+ ∥x2), for all x belonging to the set \(\{\boldsymbol {x}\in \mathbb {R}^{IJ}:K(\boldsymbol {x})\leq K_{0}\}\). This means that (4) has a local linear growth.

On the other hand, by (A.2) F(⋅) is a locally Lipschitz function. Let L(x) = Λ(x)x, for all \(\boldsymbol {x}\in \mathbb {R}^{IJ}\). Now, note that for any \(\boldsymbol {x},\boldsymbol {y}\in \mathbb {R}^{IJ}\) we have that L(x) − L(y) = [Λ(x) + Λ(y)][xy]/2 + [Λ(x) −Λ(y)][x + y]/2, and then, \(\parallel L(\boldsymbol {x})-L(\boldsymbol {y})\parallel \leq (\sqrt {IJ}/2)\parallel {\Lambda }(\boldsymbol {x})+{\Lambda }(\boldsymbol {y})\parallel _{\max \limits }\parallel \boldsymbol {x}-\boldsymbol {y}\parallel +(\sqrt {IJ}/2)\parallel {\Lambda }(\boldsymbol {x})-{\Lambda }(\boldsymbol {y})\parallel _{\max \limits }\parallel \boldsymbol {x}+\boldsymbol {y}\parallel \), where

$$ \begin{array}{@{}rcl@{}} \parallel{\Lambda}(\boldsymbol{x})-{\Lambda}(\boldsymbol{y})\parallel_{\max}&=&\underset{i,j}{\max}\mid{\sum}_{j^{\prime}=1}^{J}(x_{ij^{\prime}}-y_{ij^{\prime}})\lambda_{jj^{\prime}}\mid\\&\leq&{\sum}_{i^{\prime}=1}^{I}{\sum}_{j^{\prime}=1}^{J}\mid x_{ij^{\prime}}-y_{ij^{\prime}}\mid\mid\lambda_{jj^{\prime}}\mid \end{array} $$
$$ \leq\parallel\boldsymbol{x}-\boldsymbol{y}\parallel\sqrt{I\sum\limits_{j^{\prime}=1}^{J}\mid\lambda_{jj^{\prime}}\mid^{2}}, $$

by Hölder inequality. That is,

$$ \begin{array}{@{}rcl@{}} &&\parallel L(\boldsymbol{x}) - L(\boldsymbol{y})\parallel\!\leq\!\frac{\sqrt{IJ}}{2}\left( \parallel{\Lambda}(\boldsymbol{x})+{\Lambda}(\boldsymbol{y})\parallel_{\max}\vphantom{\sqrt{I{\sum}_{j^{\prime}=1}^{J}\mid\lambda_{jj^{\prime}}\mid^{2}}}\right.\\&&\left.\!+\parallel\boldsymbol{x} + \boldsymbol{y}\parallel\sqrt{I{\sum}_{j^{\prime}=1}^{J}\mid\lambda_{jj^{\prime}}\mid^{2}}\right)\parallel\boldsymbol{x} - \boldsymbol{y}\parallel, \end{array} $$

and thus for any \(\boldsymbol {x},\boldsymbol {y}\in \mathbb {R}^{IJ}\) such that ∥x ∥∨∥y ∥≤ M, where M is a constant such that M ≫∥X(0) ∥, there exists a constant HM such that ∥ L(x) − L(y) ∥≤ HMxy ∥. So, it means that L(⋅) is locally Lipschitz as well. Therefore, conditions of Theorem 9.1 in Chapter IV from Ikeda and Watanabe (2014) (linear growth and Lipschitz conditions) are locally satisfied, and hence, we have existence and uniqueness of a local solution. \(\square \)

Proof of Proposition 2

Recalling the formulation Xij(t) of (3), we define the following quantity:

$$ Y_{j}(t) := \sum\limits_{i=1}^{I}X_{ij}(t), $$
(15)

which models the total biomass of species j over the whole system (i.e., archipelago) at time t. We recall that no biomass is transferred from outside the archipelago and all the emigrant biomass of any species in any community at most lands over some of the other communities. Therefore, the stochastic part of (3) must satisfy:

$$ \begin{array}{@{}rcl@{}} &&{\kern-9.5pt}\sum\limits_{i=1}^{I}\sum\limits_{i^{\prime}=1}^{I}\varepsilon_{ii^{\prime}}{{\int}_{0}^{t}}X_{i^{\prime}j}(s-){\int}_{0}^{\infty}1_{\{z\leq b_{j}\}}N_{i^{\prime}j}(dz,ds)\\ &&{} =\sum\limits_{i^{\prime}=1}^{I}\left( \sum\limits_{i=1}^{I}\varepsilon_{ii^{\prime}}\right){{\int}_{0}^{t}}X_{i^{\prime}j}(s-){\int}_{0}^{\infty}1_{\{z\leq b_{j}\}}N_{i^{\prime}j}(dz,ds)\! \leq\! 0,\!\!\\ \end{array} $$
(16)

almost surely, since \({\sum }_{i=1}^{I}\varepsilon _{ii^{\prime }}\leq 0\) for any \(i^{\prime }\). Let Y = (Y1,...,YJ)T. Now, in (6), let Ξ be the J × IJ matrix such that ΞX = Y. We have that:

$$ \boldsymbol{Y}(t)\leq\boldsymbol{Y}(0)+{\int\limits_{0}^{t}}{\Xi}{\Gamma}(\boldsymbol{X}(s))\boldsymbol{X}(s)ds, $$

by (16). Then we get:

$$ \begin{array}{@{}rcl@{}} \boldsymbol{Y}(t)&\leq&\boldsymbol{Y}(0)+{\int\limits_{0}^{t}}{\Xi}{\Gamma}(\boldsymbol{X}(s))\boldsymbol{X}(s)ds\leq\boldsymbol{Y}(0)\\&&+{\int\limits_{0}^{t}}\max_{i=1,..,I}\eta_{i}(\boldsymbol{X(s)}){\Xi}\boldsymbol{X}(s)ds \end{array} $$
$$ =\boldsymbol{Y}(0)+{\int\limits_{0}^{t}}\max_{i=1,..,I}\eta_{i}(\boldsymbol{X(s)})\boldsymbol{Y}(s)ds. $$

By (A.3), it is clear that \(\sup _{t\in \mathbb {R}_{+}}\max \limits _{j=1,..,J}Y_{j}(t)<\infty \), almost surely, and (7) follows from the fact that \(\max \limits _{j=1,..,J}Y_{j}(\cdot )\geq \max \limits _{i=1,..,I;j=1,..,J}X_{ij}(\cdot )\), almost surely.

Now, we shall prove that extinction occurs in the long term. First, notice that under structural assumptions made in (6), we can write in (3) fij(x) in the form \(\tilde {f}_{ij}(x)x\). Therefore, for any \(t\in \mathbb {R}_{+}\), i = 1,...,I and j = 1,...,J we have:

$$ \begin{array}{@{}rcl@{}} &&X_{ij}(t)\geq X_{ij}(0)+{{\int}_{0}^{t}}\tilde{f}_{ij}(X_{ij}(s))X_{ij}(s)ds\\ &&+{{\int}_{0}^{t}}{\sum}_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}X_{ij^{\prime}}(s)X_{ij}(s)ds\\&&-{{\int}_{0}^{t}}X_{ij}(s){\int}_{0}^{\infty}1_{\left\{z\leq b_{j}\right\}}N_{ij}(dz,ds) \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&=X_{ij}(0)+{{\int}_{0}^{t}}\tilde{f}_{ij}(X_{ij}(s))X_{ij}(s)ds\\&&+{{\int}_{0}^{t}}{\sum}_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}X_{ij^{\prime}}(s)X_{ij}(s)ds\\ &&-{{\int}_{0}^{t}}X_{ij}(s)b_{j}ds+{{\int}_{0}^{t}}X_{ij}(s)\mathcal{M}_{ij}([0,b_{j}],ds), \end{array} $$

where \({\mathscr{M}}_{ij}([0,b_{j}],t)=-({\int \limits }_{0}^{\infty }1_{\left \{z\leq b_{j}\right \}}N_{ij}(dz,t)-tb_{j})=tb_{j}-N_{ij}([0,b_{j}],t)\) is a square-integrable martingale. This implies that:

$$ \begin{array}{@{}rcl@{}} X_{ij}(t)&\geq& X_{ij}(0)\exp\left\{{{\int}_{0}^{t}}\left[\tilde{f}_{ij}(X_{ij}(s))\vphantom{\sum\limits_{j^{\prime}\neq j}^{J}}\right.\right.\\&&\left.\left.+\sum\limits_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}X_{ij^{\prime}}(s)-b_{j}\right]ds+\mathcal{M}_{ij}([0,b_{j}],t)\right\} \end{array} $$
$$ \geq X_{ij}(0)\exp\left\{t\underset{\mathsf{x}\in R^{J}}{\inf}G_{ij}(\mathsf{x})+\mathcal{M}_{ij}([0,b_{j}],t)\right\}, $$

where

$$ \underset{\mathsf{x}\in R^{J}}{\inf}G_{ij}(\mathsf{x})\!:=\!\underset{(x_{1},...,x_{J})^{\mathrm{T}}\in R^{J}}{\inf}\left[\tilde{f}_{ij}(x_{j}) + \sum\limits_{j^{\prime}\neq j}^{J}\lambda_{jj^{\prime}}x_{j^{\prime}}-b_{j}\right], $$

and RJ is the bounded subset of \(\mathbb {R}^{J}_{+}\) which delimits the state space of (Xi1(⋅),...,XiJ(⋅))T for any i = 1,...,I. On the other hand, we have that:

$$ \mathbb{E}\left( \exp\left\{\mathcal{M}_{ij}([0,b_{j}],t)\right\}\right)=\exp\left\{b_{j}t\exp\{-1\}\right\}, $$

and thus,

$$ \mathbb{E}(X_{ij}(t))\!\geq\!\mathbb{E}(X_{ij}(0))\exp\left\{t(\underset{\mathsf{x}\in R^{J}}{\inf}G_{ij}(\mathsf{x})+b_{j}\exp\{-1\})\right\}. $$
(17)

Notice that it must satisfy \(b_{j}\exp \{-1\}+\inf _{\mathsf {x}\in R^{J}}G_{ij}(\mathsf {x})\leq 0\) if Xij(0) > 0, since otherwise \(\mathbb {E}(X_{ij}(t))\rightarrow \infty \) when \(t\rightarrow \infty \), which contradicts the fact that Xij(⋅) is bounded. So, if Xij(0) > 0, \(\exp \left \{t\inf _{\mathsf {x}\in R^{J}}G_{ij}(\mathsf {x})+{\mathscr{M}}_{ij}([0,b_{j}],t)\right \}\) never reaches zero in finite time, and then neither does Xij(⋅). \(\square \)

Proof of proposition 3

First notice that the vector b(x) can be decomposed into b(x) = βx, where \(\boldsymbol {\beta }={\int \limits }_{0}^{\infty }\beta (z)dz\) and β(z) is the IJ × IJ matrix defined in (14). Also, we have that \(\parallel B(\boldsymbol {x},z)\parallel \leq \sqrt {IJ}\parallel \beta (z)\parallel _{\max \limits }\parallel \boldsymbol {x}\parallel \).

We start by showing the existence of a Lyapunov exponent for (6). Consider its representation given by (11). According to Theorem 6.8.2 in Applebaum (2009), we have to show that the alternative condition to the Assumption 6.8.1, given in Equation (6.42), is satisfied for our system (11) for all xR. That is, the existence of a positive constant \(L^{\prime }\) such that \(\boldsymbol {x}^{\mathrm {T}}{\Gamma }(\boldsymbol {x})\boldsymbol {x}+\boldsymbol {x}^{\mathrm {T}}\boldsymbol {b}(\boldsymbol {x})+{\int \limits }_{0}^{\infty }\parallel B(\boldsymbol {x},z)\parallel ^{2}dz\leq L^{\prime }(1+\parallel \boldsymbol {x}\parallel ^{2})\). In fact, by the above, we have that:

$$ \begin{array}{@{}rcl@{}} \boldsymbol{x}^{\mathrm{T}}{\Gamma}(\boldsymbol{x})\boldsymbol{x}+\boldsymbol{x}^{\mathrm{T}}\boldsymbol{b}(\boldsymbol{x})+{\int}_{0}^{\infty}\parallel B(\boldsymbol{x},z)\parallel^{2}dz\leq\boldsymbol{x}^{\mathrm{T}}{\Gamma}(\boldsymbol{x})\boldsymbol{x}\\+\boldsymbol{x}^{\mathrm{T}}\boldsymbol{\beta}\boldsymbol{x}+IJ\parallel\boldsymbol{x}\parallel^{2}{\int}_{0}^{\infty}\parallel\beta(z)\parallel_{\max}^{2}dz\leq L^{\prime\prime}\parallel\boldsymbol{x}\parallel^{2}, \end{array} $$

where \(L^{\prime \prime }=\max \limits \{\sup _{\boldsymbol {x}\in \boldsymbol {R}}{\Gamma }({\boldsymbol {x}}),\parallel \boldsymbol {\beta }\parallel ,IJ{\int \limits }_{0}^{\infty }\parallel \beta (z)\parallel _{\max \limits }^{2}dz\}\). Clearly it also implies that the process (12) has a Lyapunov exponent.

Now, consider that there exists a constant c > 0 such that \({\mathscr{L}}\langle u_{IJ},\boldsymbol {x}\rangle \leq -c\langle u_{IJ},\boldsymbol {x}\rangle \) for all xR. Define \(\mathbb {E}_{\boldsymbol {x}}(\langle u_{IJ},\boldsymbol {X}(t)\rangle ):=\mathbb {E}(\langle u_{IJ},\boldsymbol {X}(t)\rangle \mid \boldsymbol {X}(0)=\boldsymbol {x})\), for all \(t\in \mathbb {R}_{+}\). Applying it to (12) we obtain:

$$ \begin{array}{@{}rcl@{}} \mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},\boldsymbol{X}(t)\rangle)&=&\langle u_{IJ},\boldsymbol{x}\rangle+{\int\limits_{0}^{t}}\mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},{\Gamma}(\boldsymbol{X}(s))\boldsymbol{X}(s)\\&&+\boldsymbol{b}(\boldsymbol{X}(s))\rangle) ds \end{array} $$
$$ \leq \langle u_{IJ},\boldsymbol{x}\rangle-c{\int\limits_{0}^{t}}\mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},\boldsymbol{X}(s)\rangle)ds. $$

Then, by Gronwall’s inequality we get:

$$ \mathbb{E}_{\boldsymbol{x}}(\langle u_{IJ},\boldsymbol{X}(t)\rangle)\leq \langle u_{IJ},\boldsymbol{x}\rangle\exp\{-ct\}, $$

which implies that 〈uIJ,X(⋅)〉 goes exponentially fast to 0. \(\square \)

Proof of Lemma 4

For a chosen uIJ, there exists \(i^{\prime }=1,...,I\) and \(j^{\prime }=1,...,J\) such that \(\mathbb {E}_{\boldsymbol {x}}(X_{i^{\prime }j^{\prime }}(\cdot ))\leq \mathbb {E}_{\boldsymbol {x}}(\langle u_{IJ},\boldsymbol {X}(\cdot )\rangle )\), where \(\mathbb {E}_{\boldsymbol {x}}(X_{i^{\prime }j^{\prime }}(\cdot ))\) satisfies (17) for \(i=i^{\prime }\) and \(j=j^{\prime }\) given \(X_{i^{\prime }j^{\prime }}(0)=\boldsymbol {x}\). Therefore, extinction can only occur in an exponentially fast way. \(\square \)

Appendix 2. Some results of a classical competitive Lotka-Volterra system

In the present appendix we recall some results of a classical competitive Lotka-Volterra system and its corresponding parameter space, in order to enlighten the long-term solutions and the reader realize how the presence of dispersal can transform such parameter space and bifurcation diagrams. Notice that as we will start treating the classic case, in a local community without dispersal, first we do not consider the subscript i. Consider then the equations:

$$ \begin{array}{@{}rcl@{}} {x}_{1}^{\prime}(t) & = x_{1}(t)\left( r_{1}+\alpha_{11}x_{1}(t)+\alpha_{12}x_{2}(t)\right), \\ {x}_{2}^{\prime}(t) & = x_{2}(t)\left( r_{2}+\alpha_{21}x_{1}(t)+\alpha_{22}x_{2}(t)\right). \end{array} $$

For any steady state (x1,x2), the eigenvalues of the Jacobian matrix are given by:

$$ \frac12\left( \text{tr}(J)\pm\sqrt{\text{tr}(J)^{2}-4\det(J)}\right), $$

where tr(J) and \(\det (J)\) stand for the trace and determinant of the matrix

$$ J = \left( \begin{array}{cc} r_{1}+2\alpha_{11}x_{1}+\alpha_{12}x_{2} & \alpha_{12}x_{1} \\ \alpha_{21}x_{2} & r_{2}+\alpha_{21}x_{1}+2\alpha_{22}x_{2} \end{array}\right). $$

In the following we simply write T = tr(J) and \(D=\det (J)\). The standard trace-determinant analysis tells us that

  • Case \(4\det (J)>\text {tr}(J)^{2}\): the system has two complex eigenvalues, If tr(J) < 0 then the steady state is stable. On the other hand, if tr(J) > 0 then the steady state is unstable.

  • Case \(\det (J)<0\): in this scenario, both eigenvalues are positive and have different sign, thus the steady state is unstable.

  • Case \(0<4\det (J)<\text {tr}(J)^{2}\): if tr(J) > 0 then the two eigenvalues are positive and the steady state is unstable. Otherwise, tr(J) < 0 then the configuration is asymptotically stable.

Recall also that steady states are either (0,0), (0,−r2/ α22), (−r1/α11, 0) or

$$ \bar x_{1} = \frac{-r_{1}\alpha_{22}+r_{2}\alpha_{12}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}},\quad \bar x_{2} = \frac{-r_{2}\alpha_{11}+r_{1}\alpha_{21}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}. $$

Then we have:

  • (0,0): it follows that T = r1 + r2 and D = r1r2, in which case the extinction scenario is unconditionally unstable.

  • (0,−r2/α22): we have

    $$ T=r_{1}-r_{2}\frac{\alpha_{12}}{\alpha_{22}}-r_{2}\text{ and }D=-r_{2}\left( r_{1}-r_{2}\frac{\alpha_{12}}{\alpha_{22}}\right). $$

    Stability of this fixed point is reduced to the condition

    $$ r_{1}<r_{2}\frac{\alpha_{12}}{\alpha_{22}}. $$
  • (−r1/α11, 0): similarly,

    $$ T=-r_{1}+r_{2}-r_{1}\frac{\alpha_{22}}{\alpha_{11}}\text{ and }D=-r_{1}\left( r_{2}-r_{1}\frac{\alpha_{21}}{\alpha_{11}}\right), $$

    by consequence, the stability of this fixed point holds true if and only if

    $$ r_{2}<r_{1}\frac{\alpha_{21}}{\alpha_{11}}. $$
  • \((\bar x_{1},\bar x_{2})\): it follows that

    $$ T=\alpha_{11}\bar x_{1}+\alpha_{22}\bar x_{2}\text{ and }D=\big(\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}\big)\bar x_{1}\bar x_{2}, $$

    the existence and stability of this final fixed point is only under the conditions

    $$ \alpha_{11}\alpha_{22}>\alpha_{12}\alpha_{21},\quad r_{1}\alpha_{22}<r_{2}\alpha_{12}\quad\text{and}\quad r_{2}\alpha_{11}<r_{1}\alpha_{21}. $$

    Notice that, since αii < 0 then second and third conditions are equivalent to

    $$ r_{1}>r_{2}\frac{\alpha_{12}}{\alpha_{22}}\quad\text{and}\quad r_{2}>\frac{r_{1}\alpha_{21}}{\alpha_{11}}, $$

    thus we have the instability of all previous scenarios.

Analysis of the phase plane

If we come back to the numerical example with 2 species in a local community without dispersal, the system of equations are:

$$ \begin{array}{@{}rcl@{}} {x}_{1}^{\prime}(t) & = x_{1}(t)\left( r_{1}-\frac{r_{1}}{K_{1}}x_{1}(t)+\lambda_{12}x_{2}(t)\right), \\ {x}_{2}^{\prime}(t) & = x_{2}(t)\left( r_{2}+\lambda_{21}x_{1}(t)-\frac{r_{1}}{K_{2}}x_{2}(t)\right), \end{array} $$

thus defining the interacting matrix

$$ \left( \begin{array}{cc} -{r_{1}}/{K_{1}} &\lambda_{12} \\ \lambda_{21} & -{r_{1}}/{K_{2}} \end{array}\right). $$

Parameters rj’s are intrinsic for each species, thus we can consider them as fixed. Therefore, we can make an analysis of the stability of the fixed points under the two nonnegative quantities:

$$ p_{1} = \frac{\alpha_{21}}{\alpha_{11}},\qquad p_{2}=\frac{\alpha_{12}}{\alpha_{22}}, $$

where α21 = λ21, α12 = λ12, α11 = −r1/K1 and α22 = −r2/K2. It follows that the coexistence of the two population can happen only under the restriction

$$ p_{1}p_{2}<1,\quad \frac{r_{1}}{r_{2}}>p_{2},\quad \frac{r_{2}}{r_{1}}>p_{1}, $$

and similar conditions appear for any of the other fixed points. We summarize this parameter dependence of fixed points, in Fig. 8, where without loss of generality we have assumed that r1r2. Long-term behavior correspond to the classical Lotka-Volterra competition: a) coexistence, b) species 1 wins, c) species 2 wins, and d) strong competition (where the species with the largest initial biomass wins), that correspond to each of the zones in Fig. 8. The conditions for each scenario are:

  1. (a)

    p1 < r2/r1 and p2 < r1/r2. Since in this case p1p2 < 1, coexistence of species is expected.

  2. (b)

    p1 > r2/r1 and p2 < r1/r2. It follows that species 1 kicks in and at large times it is expected to converge asymptotically towards (−r1/α11, 0) = (K1, 0).

  3. (c)

    p1 < r2/r1 and p2 > r1/r2. Species 2 dominates and the solution is expected to converge asymptotically towards (0,−r2/α22, 0) = (0,K2). Moreover, since p1 < r2/r1 and p1p2 > 1 we get that

    $$ \bar x_{1} = \frac{-r_{1}\alpha_{22}+r_{2}\alpha_{12}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}} = \frac{1}{\alpha_{11}}\frac{r_{2}p_{2}-r_{1}}{1-p_{1}p_{2}}>0, $$

    and

    $$ \bar x_{2} = \frac{-r_{2}\alpha_{11}+r_{1}\alpha_{21}}{\alpha_{11}\alpha_{22}-\alpha_{12}\alpha_{21}}=\frac{1}{\alpha_{22}}\frac{r_{1}p_{1}-r_{2}}{1-p_{1}p_{2}}<0, $$

    i.e., the second coordinate of the coexistence scenario vanishes and

  4. d)

    p1 > r2/r1 and p2 > r1/r2. In this last zone conditions imply that both winner-takes-all scenario are possible, and that the coexistence fixed point \((\bar x_{1},\bar x_{2})\) exists but it is unstable. Therefore, according to initial condition, we expect to converge at large times towards either (0,K2) or (K1, 0).

Fig. 8
figure8

Outcomes of the model for a single community and two populations when r1 > r2 and on the parameter space p1 = −λ21K1/r1 v/s p2 = −λ12K2/r2. Long-term behavior on each zone: (a) coexistence, (b) species 1 wins, (c) species 2 wins and (d) strong competition. Points 1 and 3 are the coordinates for the analyzes carried out in “Coexistence and the competition/colonization trade-off” and “Effect of the metacommunity spatial structure” and Appendix C, respectively

Appendix 3. Additional results for two species competition with priority effects

Results for the dynamics of two species in a central community within a metacommunity architecture with three local communities

For each pair (b1,b2) with bi taking values from 0.1 to 0.9 with incremental step 0.1, we performed 150 simulations. We report the empirical average of biomass of both species on the central community.

The results of the model are shown in the upper subplots of Fig. 9. One observes that the species with larger dispersal rates dominates the long-term solutions in the central community. In the bottom panel of Fig. 9 we plot the probability that species 1 wins as a function of the difference b1b2. We find a shape very similar to the ones described in literature (see, e.g., Calcagno et al. 2006).

Fig. 9
figure9

Simulations made for parameters r1 = r2 = 2, Ki1 = Ki2 = 10 (for all i = 1,2,3). All εij are 0 except for ε21 = ε23 = −ε11 = −ε33 = 0.1 and λ21 = λ12 = − 0.21. (top) We show the empirical mean calculated for species 1 and 2 in a central community under a strong competition scenario where species with higher initial biomass will win according to the classical Lotka-Volterra model of competition without stochastic dispersal. (down) We show the probability that species 1 wins as a function of the difference between colonization rates b1b2. As expected, under a symmetric dynamics the species with larger bi dominates at long times. Dots represents the empirical mean of 150 simulations of the model, for the three-community architecture where the dispersion parameter of species 1 varies. The solid line corresponds to a nonlinear statistical fit

Results for two species competition under priority effects in a metacommunity with two local communities

The first case is a symmetrical one, in which the outcome is the expected one for the classical Lotka-Volterra model (i.e., for each community, the species with the larger initial biomass will win). In our example, species 1 wins in community 2 due to its larger initial biomass condition and species 2 wins in community 2 for analogous reasons (Fig. 10 upper panel). Here, species 1 cannot disperse and species 2 dispersal rate b2 is low. In the second scenario, a small increase in b2 and no dispersal of species 1, will reverse the competence in community 1, and species 2 will win in both communities (Fig. 10, middle panels). And in the third scenario, species 1 only lives in community 1, and a stronger dispersal intensity of species 2 and no dispersal of species 1, will cause coexistence in community 1 (Fig. 10, bottom panel). This last outcome is due to the rapid and massive exchange of biomass of species 2 from one community to another, which can maintain its biomass levels in both communities, specifically causing in community 1 that the biomass condition for winning frequently changes: sometimes species 1 locally dominates due to a temporary larger biomass and sometimes species 2 locally dominates due to the same reason. In Fig. 11 we can visualize the final outcome of the dynamics of both species in community 1 and under each scenario, through the slope (in the long term) of the graph \(\log (X_{ij}(t))\) vs t, which corresponds to the (estimated) Lyapunov exponent.

Fig. 10
figure10

Here we show the biomass dynamics of two species in two communities under a priority effect scenario whereby the species that wins in competition is the one with higher initial biomass. The three panels represents three cases of the dynamics of the two species in the two communities, where the dispersal parameter of the inferior competitor species (the one with the lowest initial biomass) varies in each of these columns. The simulations where made over a mean of 50 trajectories and we considered the common parameters r1 = 1.6, r2 = 1, Ki1 = 12, Ki2 = 10 (for all i = 1.2), ε12 = ε21 = −ε11 = −ε22 = 0.5, λ12 = − 0.168 and λ21 = − 0.0875 in (8) for the three cases, with initial conditions given by X11(0) = 10, X12(0) = 1, X21(0) = 1 and X22(0) = 10. The first case is one where the dispersal rate of species 2 is low, b2 = 0.003, and no dispersal of species 1. In the second case we consider only a small increase in the dispersal rate of species 2 (b2 = 0.05). Notice that this subtle change makes this species to increase in biomass in community 1 reversing the priority effect. Finally, in the third case, in local community 2 only species 2 exists but as we increase its dispersal rate (b2 = 1) the priority effect coexistence in local community 1 is now possible

Fig. 11
figure11

Convergence of Lyapunov exponents of species 1 and species 2 in community 1 for the corresponding cases of Fig. 10, from top to bottom. In continuous line is depicted one typical trajectory and in dashed line the mean of 50 trajectories, where we can see a softened version of \(\log (X_{ij}(t))\) vs t. The slopes of the long-term tendencies tell us the destination of the species: slope ≈ 0 will mean persistence and negative slope indicates extinction

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Tejo, M., Quiñinao, C., Rebolledo, R. et al. Coexistence, dispersal and spatial structure in metacommunities: a stochastic model approach. Theor Ecol (2021). https://doi.org/10.1007/s12080-020-00496-1

Download citation

Keywords

  • Metacommunity
  • Island biogeography
  • Lotka-Volterra model
  • Poisson measure
  • Lyapunov exponent
  • Trade-off
  • Priority effect