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Memory and adaptive behavior in population dynamics: anti-predator behavior as a case study

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Abstract

Memory allows organisms to forecast the future on the basis of experience, and thus, in some form, is important for the development of flexible adaptive behavior by animal communities. To model memory, we use the concept of hysteresis, which mathematically is described by the Preisach operator. As a case study, we consider anti-predator adaptation in the classic Lotka-Volterra predator-prey model. Despite its simplicity, the model allows us to naturally incorporate essential features of an adaptive system and memory. Our analysis and simulations show that a system with memory can have a continuum of equilibrium states with non-trivial stability properties. The main factor that determines the actual equilibrium state to which a trajectory converges is the maximal number achieved by the population of predator along this trajectory.

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Notes

  1. The neutral patch \(\omega _0\) may play the role of a transport route.

  2. As \(\mu (\alpha _R,\alpha _S)=0\) outside the triangle \(0\le \alpha _R<\alpha _S\le 1\), we can restrict the location of admissible pairs \((\alpha _R,\alpha _S)\) to this triangle only.

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Author information

Authors and Affiliations

  1. Weierstrass Institute, Berlin, Germany

    Alexander Pimenov

  2. University College Cork, Cork, Ireland

    Thomas C. Kelly & Michael J. O’Callaghan

  3. Centre de Recerca Matemática, Barcelona, Spain

    Andrei Korobeinikov

  4. The University of Texas at Dallas, Richardson, USA

    Dmitrii Rachinskii

Authors
  1. Alexander Pimenov
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  2. Thomas C. Kelly
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  3. Andrei Korobeinikov
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  4. Michael J. O’Callaghan
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  5. Dmitrii Rachinskii
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Corresponding author

Correspondence to Dmitrii Rachinskii.

Additional information

AP was supported by SFB-787 of the DFG. AK was supported by the Ministry of Economics of Spain via Ramón y Cajal Fellowship RYC-2011-08061 and Grant MTM2015-71509-C2-1-R, by CONACYT (Mexico) via Grant N219614, and by AGAUR (Generalitat de Catalunya) via Grant 2014SGR-1307. DR was supported by the National Science Foundation Grant DMS-1413223.

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Pimenov, A., Kelly, T.C., Korobeinikov, A. et al. Memory and adaptive behavior in population dynamics: anti-predator behavior as a case study. J. Math. Biol. 74, 1533–1559 (2017). https://doi.org/10.1007/s00285-016-1065-6

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  • DOI: https://doi.org/10.1007/s00285-016-1065-6

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