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Stochastic Lotka–Volterra food chains

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Abstract

We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka–Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka–Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and \(n-1\) predators. The jth predator eats the \(j-1\)th species and is eaten by the \(j+1\)th predator; this way each species only interacts with at most two other species—the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam ’79 to a stochastic setting. As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction. Another novelty of our analysis is the fact that we can describe the behavior of the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.

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References

  • Blath J, Etheridge A, Meredith M (2007) Coexistence in locally regulated competing populations and survival of branching annihilating random walk. Ann Appl Probab 17(5–6):1474–1507

    Article  MathSciNet  MATH  Google Scholar 

  • Benaïm M (2014) Stochastic persistence, preprint

  • Benaïm M, Hofbauer J, Sandholm WH (2008) Robust permanence and impermanence for stochastic replicator dynamics. J Biol Dyn 2(2):180–195

    Article  MathSciNet  MATH  Google Scholar 

  • Benaïm M, Lobry C (2016) Lotka Volterra in fluctuating environment or how switching between beneficial environments can make survival harder. Ann Appl Probab 26(6):3754–3785

    Article  MathSciNet  MATH  Google Scholar 

  • Braumann CA (2002) Variable effort harvesting models in random environments: generalization to density-dependent noise intensities. Math Biosci 177/178:229–245, Deterministic and stochastic modeling of biointeraction (West Lafayette, IN, 2000)

  • Benaïm M, Schreiber SJ (2009) Persistence of structured populations in random environments. Theor Popul Biol 76(1):19–34

    Article  MATH  Google Scholar 

  • Benaïm M, Strickler E (2017) Random switching between vector fields having a common zero, arXiv:1702.03089

  • Cattiaux P, Collet P, Lambert A, Martínez S, Méléard S, Martín J San (2009) Quasi-stationary distributions and diffusion models in population dynamics. Ann Probab 37(5):1926–1969

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Costa M (2016) A piecewise deterministic model for a prey-predator community. Ann Appl Probab 26(6):3491–3530

    Article  MathSciNet  MATH  Google Scholar 

  • Dieu NT, Du NH, Nguyen DH, Yin G (2016) Protection zones for survival of species in random environment. SIAM J Appl Math 76(4):1382–1402

    Article  MathSciNet  MATH  Google Scholar 

  • Du NH, Nguyen DH (2011) Dynamics of kolmogorov systems of competitive type under the telegraph noise. J Differ Equ 250(1):386–409

    Article  MathSciNet  MATH  Google Scholar 

  • Dieu NT, Nguyen DH, Du NH, Yin G (2016) Classification of asymptotic behavior in a stochastic SIR model. SIAM J Appl Dyn Syst 15(2):1062–1084

    Article  MathSciNet  MATH  Google Scholar 

  • Du NH, Nguyen DH, Yin G (2016) Dynamics of a stochastic Lotka–Volterra model perturbed by white noise. J Appl Probab 53(1):187–202

    Article  MathSciNet  Google Scholar 

  • Evans SN, Hening A, Schreiber SJ (2015) Protected polymorphisms and evolutionary stability of patch-selection strategies in stochastic environments. J Math Biol 71(2):325–359

    Article  MathSciNet  MATH  Google Scholar 

  • Evans SN, Ralph PL, Schreiber SJ, Sen A (2013) Stochastic population growth in spatially heterogeneous environments. J Math Biol 66(3):423–476

    Article  MathSciNet  MATH  Google Scholar 

  • Freedman HI, So JWH (1985) Global stability and persistence of simple food chains. Math Biosci 76(1):69–86

    Article  MathSciNet  MATH  Google Scholar 

  • Gard TC (1980) Persistence in food chains with general interactions. Math Biosci 51(1–2):165–174

    Article  MathSciNet  MATH  Google Scholar 

  • Gard TC (1984) Persistence in stochastic food web models. Bull Math Biol 46(3):357–370

    Article  MathSciNet  MATH  Google Scholar 

  • Gard TC (1988) Introduction to stochastic differential equations. Marcel Dekker Inc., New York-Basel, p 234

  • Gard TC, Hallam TG (1979) Persistence in food webs. I. Lotka–Volterra food chains. Bull Math Biol 41(6):877–891

    MathSciNet  MATH  Google Scholar 

  • Geiß C, Manthey R (1994) Comparison theorems for stochastic differential equations in finite and infinite dimensions. Stoch Process Appl 53(1):23–35

    Article  MathSciNet  MATH  Google Scholar 

  • Harrison GW (1979) Global stability of food chains. Am Nat 114(3):455–457

    Article  Google Scholar 

  • Hastings A (1978) Global stability in Lotka–Volterra systems with diffusion. J Math Biol 6(2):163–168

    Article  MathSciNet  MATH  Google Scholar 

  • Hening A, Nguyen DH (2017a) Coexistence and extinction for stochastic Kolmogorov systems. Ann Appl Probab (accepted)

  • Hening A, Nguyen DH (2017b) Persistence in stochastic Lotka-Volterra food chains with intraspecific competition, preprint (2017), arXiv:1704.07501

  • Hening A, Nguyen DH, Yin G (2017) Stochastic population growth in spatially heterogeneous environments: the density-dependent case. J Math Biol. https://doi.org/10.1007/s00285-017-1153-2

  • Hening A, Strickler E (2017) On a predator-prey system with random switching that never converges to its equilibrium, arXiv:1710.01220

  • Kallenberg O (2002) Foundations of modern probability. Springer, New York

    Book  MATH  Google Scholar 

  • Kendall BE, Bjørnstad ON, Bascompte J, Keitt TH, Fagan WF (2000) Dispersal, environmental correlation, and spatial synchrony in population dynamics. Am Nat 155(5):628–636

    Article  Google Scholar 

  • Liu M, Bai C (2016) Analysis of a stochastic tri-trophic food-chain model with harvesting. J Math Biol 73(3):597–625

    Article  MathSciNet  MATH  Google Scholar 

  • Lande R, Engen S, Saether B-E (2003) Stochastic population dynamics in ecology and conservation. Oxford University Press on Demand, Oxford

    Book  MATH  Google Scholar 

  • Liebhold A, Koenig WD, Bjørnstad ON (2004) Spatial synchrony in population dynamics. Annu Rev Ecol Evol Syst 35:467–490

    Article  Google Scholar 

  • Luo Q, Mao X (2009) Stochastic population dynamics under regime switching. II. J Math Anal Appl 355(2):577–593

    Article  MathSciNet  MATH  Google Scholar 

  • Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, Baltimore

  • Mallik RK (2001) The inverse of a tridiagonal matrix. Linear Algebra Appl 325:1–3

    Article  MathSciNet  MATH  Google Scholar 

  • Massarelli N, Hoffman K, Previte JP (2014) Effect of parity on productivity and sustainability of Lotka–Volterra food chains: bounded orbits in food chains. J Math Biol 69(6–7):1609–1626

    Article  MathSciNet  MATH  Google Scholar 

  • Nguyen DH, Du NH, Yin G (2014) Existence of stationary distributions for kolmogorov systems of competitive type under telegraph noise. J Differ Equ 257(6):2078–2101

    Article  MathSciNet  MATH  Google Scholar 

  • Nguyen DH, Yin G, Zhu C (2017) Certain properties related to well posedness of switching diffusions. Stoch Process Appl 127(10):3135–3158

    Article  MathSciNet  MATH  Google Scholar 

  • Pimm SL (1982) Food webs, food webs. Springer, New York, pp 1–11

    Book  Google Scholar 

  • Polansky P (1979) Invariant distributions for multi-population models in random environments. Theor Popul Biol 16(1):25–34

    Article  MathSciNet  MATH  Google Scholar 

  • Rey-Bellet L (2006) Ergodic properties of Markov processes, Open quantum systems. II, Lecture Notes in Mathematics, vol 1881 Springer, Berlin, pp 1–39

  • Rudnicki R (2003) Long-time behaviour of a stochastic prey-predator model. Stoch Process Appl 108(1):93–107

    Article  MathSciNet  MATH  Google Scholar 

  • Schreiber SJ, Benaïm M, Atchadé KAS (2011) Persistence in fluctuating environments. J Math Biol 62(5):655–683

    Article  MathSciNet  MATH  Google Scholar 

  • Schreiber SJ (2012) Persistence for stochastic difference equations: a mini-review. J Differ Equ Appl 18(8):1381–1403

    Article  MathSciNet  MATH  Google Scholar 

  • Schreiber SJ, Lloyd-Smith JO (2009) Invasion dynamics in spatially heterogeneous environments. Am Nat 174(4):490–505

    Article  Google Scholar 

  • So JWH (1979) A note on the global stability and bifurcation phenomenon of a Lotka–Volterra food chain. J Theor Biol 80(2):185–187

    Article  MathSciNet  Google Scholar 

  • Takeuchi Y, Du NH, Hieu NT, Sato K (2006) Evolution of predator-prey systems described by a Lotka–Volterra equation under random environment. J Math Anal Appl 323(2):938–957

    Article  MathSciNet  MATH  Google Scholar 

  • Tyson R, Lutscher F (2016) Seasonally varying predation behavior and climate shifts are predicted to affect predator-prey cycles. Am Nat 188(5):539–553

    Article  Google Scholar 

  • Turelli M (1977) Random environments and stochastic calculus. Theor Popul Biol 12(2):140–178

    Article  MathSciNet  MATH  Google Scholar 

  • Volterra V (1928) Variations and fluctuations of the number of individuals in animal species living together. J Cons Int Explor Mer 3(1):3–51

    Article  Google Scholar 

Download references

Acknowledgements

We thank two anonymous referees for their comments which helped improve this manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Tufts University, Bromfield-Pearson Hall, 503 Boston Avenue, Medford, MA, 02155, USA

    Alexandru Hening

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Dang H. Nguyen

Authors
  1. Alexandru Hening
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  2. Dang H. Nguyen
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Correspondence to Alexandru Hening.

Additional information

D. Nguyen was in part supported by the National Science Foundation under Grant DMS-1207667.

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Hening, A., Nguyen, D.H. Stochastic Lotka–Volterra food chains. J. Math. Biol. 77, 135–163 (2018). https://doi.org/10.1007/s00285-017-1192-8

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  • DOI: https://doi.org/10.1007/s00285-017-1192-8

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