# The Allee-type ideal free distribution

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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.

## Keywords

Dispersal Evolutionary game theory Habitat selection game Logistic equation Non-smooth analysis Population dynamics Optimal foraging## Mathematics Subject Classification

26A27 92D25 92D40 92D50## Notes

### Acknowledgments

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

## References

- Courchamp F, Berec L, Gascoigne J (2008) Allee effects. Oxford University Press, OxfordCrossRefGoogle Scholar
- Cressman R, Krivan V (2013) Two-patch population models with adaptive dispersal: the effects of varying dispersal speeds. J Math Biol 67:329–358CrossRefzbMATHMathSciNetGoogle Scholar
- Cressman R, Křivan V (2006) Migration dynamics for the ideal free distribution. Am Nat 168:384–397CrossRefGoogle Scholar
- Cressman R, Křivan V (2010) The ideal free distribution as an evolutionarily stable state in density-dependent population games. Oikos 119:1231–1242CrossRefGoogle Scholar
- Fretwell DS, Lucas HL (1969) On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoret 19:16–32CrossRefGoogle Scholar
- Fretwell SD (1972) Populations in a seasonal environment. Princeton University Press, PrincetonGoogle Scholar
- Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
- Křivan V, Cressman R (2009) On evolutionary stability in prey-predator models with fast behavioral dynamics. Evol Ecol Res 11:227–251Google Scholar
- Křivan V, Cressman R, Schneider S (2008) The ideal free distribution: a review and synthesis of the game-theoretic perspective. Theoret Popul Biol 73:403–425CrossRefzbMATHGoogle Scholar
- Morris DW (1988) Habitat-dependent population regulation and community structure. Evol Ecol 2:253–269CrossRefGoogle Scholar
- Morris DW (2002) Measuring the Allee effect: positive density dependence in small mammals. Ecology 83:14–20CrossRefGoogle Scholar
- Pohley JH, Thomas B (1983) Non-linear ESS models and frequency dependent selection. Biosystems 16:87–100CrossRefGoogle Scholar
- Taylor PD, Jonker LB (1978) Evolutionary stable strategies and game dynamics. Math Biosci 40:145–156CrossRefzbMATHMathSciNetGoogle Scholar
- Tran T, Cressman R (2014) The Ideal Free Distribution and Evolutionary Stability in habitat selection games with linear fitness and Allee effect (Mimeo)Google Scholar