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Predation with indirect effects in fluctuating environments

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Abstract

We investigate the long-term dynamics for a predation model of Plankton community with indirect effects, under fluctuating environments. A random version and a stochastic version with multiplicative noise of the model are discussed and compared. We prove that the solutions to both versions are nonnegative and bounded given any nonnegative positive initial conditions. We also prove that both the random system and the stochastic system possess a unique random attractor under the same set of assumptions, by using the classical theory of random dynamical systems. In addition, we provide conditions under which coexistence of species exists for the random system.

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References

  1. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  2. Asai, Y., Kloeden, P.E.: Numerical schemes for random ODEs via stochastic differential equations. Commun. Appl. Anal. 17(3&4), 521–528 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bolker, B., Holyoak, M., Krivan, V., Rowe, L., Schmitz, O.: Connecting theoretical and empirical studies of trait-mediated interactions. Ecology 84, 1101–1114 (2003)

    Article  Google Scholar 

  4. Brooks, J.L., Stanley, I.D.: Predation, body size, and composition of plankton. Science 150(3692), 23–35 (1965)

    Article  Google Scholar 

  5. Caraballo, T., Garrido-Atienza, M.J., Schmalfuß, B., Valero, J.: Non-autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete Contin. Dyn. Syst. 21(2), 415–443 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cariveau, D., Irwin, R.E., Brody, A.K., Garcia-Mayeya, S.L., Von der Ohe, A.: Direct and indirect effects of pollinators and seed predators to selection on plant and floral traits. OIKOS 104, 15–26 (2004)

    Article  Google Scholar 

  7. Caraballo, T., Colucci, R., Han, X.: Non-autonomous dynamics of a Semi-Kolmogorov population model with periodic forcing. (under review)

  8. Caraballo, T., Lu, K.: Attractors for stochastic lattice dynamical systems with a multiplicative noise. Front. Math. China 3(3), 317–335 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Caraballo, T., Lukaszewicz, G., Real, J.: Pullback attractors for asymptotically compact nonautonomous dynamical systems. Nonlinear Anal. TMA 6, 484–498 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Colucci, R.: Coexistence in a one-predator, two-prey system with indirect effects. J. Appl. Math. (2013). doi:10.1155/2013/625391

  11. Colucci, R., Nunez, D.: Periodic Orbits for a Three-Dimensional Biological Differential Systems. Abstr. Appl. Anal. (2013). doi:10.1155/2013/465183

  12. Indirect effects affects ecosystem dynamics. http://www.ictp-saifr.org (2011)

  13. Flandoli, F., Schmalfuß, B.: Random attractors for the 3D stochastic Navier-Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59(1–2), 21–45 (1996)

    Article  MATH  Google Scholar 

  14. Kloeden, P.E., Rasmussen, M.: Nonautonomous Dynamical Systems. American Mathematical Society, Providence (2011)

    Book  MATH  Google Scholar 

  15. Menge, B.A.: Indirect effects in marine rocky intertidal interaction webs: patterns and importance. Ecol. Monogr. 65, 21 (1995)

    Article  Google Scholar 

  16. Rohde, K.: Nonequilibrium Ecology. Cambridge University Press, Cambridge (2005)

    Google Scholar 

  17. Walsh, M.R., Reznick, D.N.: Interactions between the direct and indirect effects of predators determine life history evolution in a killifish. Pnas, www.pnas.org/cgi. doi: 10.1073/pnas.0710051105

  18. Wootton, J.T.: Indirect effects, prey susceptibility, and habitat selection: impacts of birds on limpets and algae. Ecology 73(3), 981–991 (1992)

    Article  Google Scholar 

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Acknowledgments

We sincerely thank the referees for the helpful and interesting comments which allowed us to improve the presentation of this paper.

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Authors and Affiliations

  1. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo. de Correos 1160, 41080, Sevilla, Spain

    Tomás Caraballo

  2. Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, 111 Ren’ai Road, SIP, Suzhou, 215123, People’s Republic of China

    Renato Colucci

  3. Department of Mathematics and Statistics, Auburn University Auburn, 221 Parker Hall, Auburn, AL, 36849, USA

    Xiaoying Han

Authors
  1. Tomás Caraballo
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  2. Renato Colucci
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  3. Xiaoying Han
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Corresponding author

Correspondence to Tomás Caraballo.

Additional information

This work has been partially supported by FEDER and the Spanish Ministerio de Economía y Competitividad project MTM2011-22411 and the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492.

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Caraballo, T., Colucci, R. & Han, X. Predation with indirect effects in fluctuating environments. Nonlinear Dyn 84, 115–126 (2016). https://doi.org/10.1007/s11071-015-2238-3

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  • DOI: https://doi.org/10.1007/s11071-015-2238-3

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