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     [13]Journal of Mathematical Biology

     Article

                          The Allee-type ideal free distribution

     * Published: 04 December 2013

     * Volume 69, pages 1497-1513, (2014)
     * [14]Cite this article

   [15]Journal of Mathematical Biology [16]Aims and scope [17]Submit manuscript

     * [18]Vlastimil Krivan^[19]1

     * 565 Accesses
     * 10 Citations
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Abstract

   The ideal free distribution (IFD) in a two-patch environment where individual
   fitness is positively density dependent at low population densities is studied.
   The IFD is defined as an evolutionarily stable strategy of the habitat selection
   game. It is shown that for low and high population densities only one IFD exists,
   but for intermediate population densities there are up to three IFDs. Population
   and distributional dynamics described by the replicator dynamics are studied. It
   is shown that distributional stability (i.e., IFD) does not imply local stability
   of a population equilibrium. Thus distributional stability is not sufficient for
   population stability. Results of this article demonstrate that the Allee effect
   can strongly influence not only population dynamics, but also population
   distribution in space.

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Acknowledgments

   I thank Ross Cressman and two anonymous reviewers for their thoughtful
   suggestions. Institutional support RVO:60077344 is acknowledged.

Author information

Authors and Affiliations

    1. Biology Center, Academy of Sciences of the Czech Republic, and Faculty of
       Science, University of South Bohemia, Branisovská 31, 370 05 , Ceske
       Budejovice, Czech Republic
       Vlastimil Krivan

   Authors
    1. Vlastimil Krivan
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Corresponding author

   Correspondence to [61]Vlastimil Krivan.

Appendices

Appendix A: The IFD

   First, I study under which conditions strategy \(u=(1,0)\) or \((0,1)\) is a
   strict NE at overall population abundance \(x\). These two strategies correspond
   to population distribution \((x_1^1,x_2^1)=(x,0)\) and \((x_1^2,x_2^2)=(0,x)\),
   respectively. A strict NE \(u^*=(u_1^*,u_2^*)\) satisfies
   \(W(u^*,u^*;x)>W(u,u^*;x)\) for any other strategy \(u=(u_1,u_2)\ne u^*.\) Thus,
   \(u^*=(1,0)\) is a strict NE provided \(V_1(x)=\min \{r_1 (1-x/K_1),a_1
   (x-A_1)\}>V_2(0)=\min \{r_2,-a_2 A_2\}=-a_2 A_2\) due to assumption ([62]1). For
   \(x\le x_1^{\max }=\frac{ K_1 (a_1 A_1+r_1)}{a_1 K_1+ r_1},\) \(V_1(x)=a_1
   (x-A_1)\) which implies that \(u^*=(1,0)\) is a strict NE when \(x>\frac{a_1
   A_1-a_2 A_2}{a_1}.\) For \(x>x_1^{\max },\) \(V_1(x)=r_1 (1-x/K_1)\) which
   implies that \(u_1^*=1\) is a strict NE when \(x0, \end{aligned}$$

   where \(V_1'(x_{1+}^*)\) denotes the right derivative. Thus, distribution
   \((x_1^{\max },x_2^*)\) where \(x_2^*>x_2^{\max }\) is an ESS provided
   \(a_1