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Mathematical Biosciences and Engineering

2008, Volume 5, Issue 4: 859-875. doi: 10.3934/mbe.2008.5.859
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Variation in risk in single-species discrete-time models

  • 1.
    Department of Ecology, Evolution and Marine Biology, University of California at Santa Barbara, CA 93106-9610
  • Received: 01 January 2008 Accepted: 29 June 2018 Published: 01 October 2008

  • MSC : Primary: 92D25, 92D40; Secondary: 92B05

  • Simple, discrete-time, population models typically exhibit complex dynamics, like cyclic oscillations and chaos, when the net reproductive rate, R, is large. These traditional models generally do not incorporate variability in juvenile "risk,'' defined to be a measure of a juvenile's vulnerability to density-dependent mortality. For a broad class of discrete-time models we show that variability in risk across juveniles tends to stabilize the equilibrium. We consider both density-independent and density-dependent risk, and for each, we identify appropriate shapes of the distribution of risk that will stabilize the equilibrium for all values of R. In both cases, it is the shape of the distribution of risk and not the amount of variation in risk that is crucial for stability.

    Citation: Abhyudai Singh, Roger M. Nisbet. Variation in risk in single-species discrete-time models[J]. Mathematical Biosciences and Engineering, 2008, 5(4): 859-875. doi: 10.3934/mbe.2008.5.859

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  • Simple, discrete-time, population models typically exhibit complex dynamics, like cyclic oscillations and chaos, when the net reproductive rate, R, is large. These traditional models generally do not incorporate variability in juvenile "risk,'' defined to be a measure of a juvenile's vulnerability to density-dependent mortality. For a broad class of discrete-time models we show that variability in risk across juveniles tends to stabilize the equilibrium. We consider both density-independent and density-dependent risk, and for each, we identify appropriate shapes of the distribution of risk that will stabilize the equilibrium for all values of R. In both cases, it is the shape of the distribution of risk and not the amount of variation in risk that is crucial for stability.


  • This article has been cited by:

    1. Abhyudai Singh, Attack by a common parasitoid stabilizes population dynamics of multi-host communities, 2021, 531, 00225193, 110897, 10.1016/j.jtbi.2021.110897
    2. Abhyudai Singh, Brooks Emerick, 2020, Hybrid systems framework for modeling host-parasitoid population dynamics, 978-1-7281-7447-1, 4628, 10.1109/CDC42340.2020.9304450
    3. Abhyudai Singh, Brooks Emerick, Generalized stability conditions for host–parasitoid population dynamics: Implications for biological control, 2021, 456, 03043800, 109656, 10.1016/j.ecolmodel.2021.109656
    4. Abhyudai Singh, Stochasticity in host-parasitoid models informs mechanisms regulating population dynamics, 2021, 11, 2045-2322, 10.1038/s41598-021-96212-y
    5. Abhyudai Singh, Lucas D. B. Faria, A comparative approach to stabilizing mechanisms between discrete- and continuous-time consumer-resource models, 2022, 17, 1932-6203, e0265825, 10.1371/journal.pone.0265825
    6. Abhyudai Singh, Brooks Emerick, Coexistence conditions in generalized discrete-time models of insect population dynamics, 2022, 474, 03043800, 110148, 10.1016/j.ecolmodel.2022.110148
    7. Abhyudai Singh, Lucas D. B. Faria, Fundamental limits of parasitoid-driven host population suppression: Implications for biological control, 2023, 18, 1932-6203, e0295980, 10.1371/journal.pone.0295980
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  • © 2008 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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