The evolutionary dynamics of a population model with a strong Allee effect

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2015, 12(4): 643-660. doi: 10.3934/mbe.2015.12.643

The evolutionary dynamics of a population model with a strong Allee effect

1.	Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, Tucson, Arizona, 85721, United States

Received May 2014 Revised September 2014 Published April 2015

Abstract
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An evolutionary game theoretic model for a population subject to predation and a strong Allee threshold of extinction is analyzed using, among other methods, Poincaré-Bendixson theory. The model is a nonlinear, plane autonomous system whose state variables are population density and the mean of a phenotypic trait, which is subject to Darwinian evolution, that determines the population's inherent (low density) growth rate (fitness). A trade-off is assumed in that an increase in the inherent growth rate results in a proportional increase in the predator's attack rate. The main results are that orbits equilibrate (there are no cycles or cycle chains of saddles), that the extinction set (or Allee basin) shrinks when evolution occurs, and that the meant trait component of survival equilibria occur at maxima of the inherent growth rate (as a function of the trait).

Keywords: Population dynamics, evolution, evolutionary game theory, Allee effects, evolutionarily stable strategy..

Mathematics Subject Classification: Primary: 92D25, 92D15; Secondary: 37N2.

Citation: Jim M. Cushing. The evolutionary dynamics of a population model with a strong Allee effect. Mathematical Biosciences & Engineering, 2015, 12 (4) : 643-660. doi: 10.3934/mbe.2015.12.643

References:

[1]	P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods,, Ecology Letters, 4 (2001), 166. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar
[2]	W. C. Allee, Animal Aggregations, a Study in General Sociology,, University of Chicago Press, (1931). Google Scholar
[3]	W. C. Allee, The Social Life of Animals,, 3rd edition, (1941). Google Scholar
[4]	W. C. Allee, O. Park, T. Park and K. Schmidt, Principles of Animal Ecology,, W. B. Saunders Company, (1949). Google Scholar
[5]	D. S. Boukal and L. Berec, Single-species Models of the Allee effect: Extinction boundaries, sex Ratios and mate Encounters,, Journal of Theoretical Biology, 218 (2002), 375. doi: 10.1006/jtbi.2002.3084. Google Scholar
[6]	F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect,, TREE, 14 (1999), 405. doi: 10.1016/S0169-5347(99)01683-3. Google Scholar
[7]	F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001. Google Scholar
[8]	J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations,, Journal of Biological Dynamics, 8 (2014), 57. doi: 10.1080/17513758.2014.899638. Google Scholar
[9]	J. M. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects,, Journal of Biological Dynamics, 6 (2012), 941. doi: 10.1080/17513758.2012.697196. Google Scholar
[10]	B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction,, Natural Resource Modeling, 3 (1989), 481. Google Scholar
[11]	F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications,, Princeton University Press, (2008). Google Scholar
[12]	L. Edelstein-Keshet, Mathematical Models in Biology,, Classics in Applied Mathematics 46, (2005). doi: 10.1137/1.9780898719147. Google Scholar
[13]	S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, Journal of Biological Dynamics , 4 (2010), 397. doi: 10.1080/17513750903377434. Google Scholar
[14]	D. S. Falconer and T. F. C. Mackay, Introduction to Quantitative Genetics,, Pearson Education Limited, (1996). Google Scholar
[15]	F. A. Hopf and F. W. Hopf, The role of the Allee effect in species packing,, Theoretical Population Biology, 27 (1985), 27. doi: 10.1016/0040-5809(85)90014-0. Google Scholar
[16]	M. R. S. Kulenovic and A.-A. Yakubu, Compensatory versus overcompensatory dynamics in density-dependent leslie models,, Journal of Difference Equations and Applications, 10 (2004), 1251. doi: 10.1080/10236190410001652711. Google Scholar
[17]	R. Lande, Natural selection and random genetic drift in phenotypic evolution,, Evolution, 30 (1976), 314. Google Scholar
[18]	R. Lande, A quantitative genetic theory of life history evolution,, Ecology, 63 (1982), 607. Google Scholar
[19]	M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theoretical Population Biology, 43 (1993), 141. doi: 10.1006/tpbi.1993.1007. Google Scholar
[20]	J. Lush, Animal Breeding Plans,, Iowa State College Press, (1937). Google Scholar
[21]	S. P. Otto and T. Day, A Biologist's Guide to Mathematical Modeling in Ecology and Evolution,, Princeton University Press, (2007). Google Scholar
[22]	I. Scheuring, Allee effect increases dynamical stability in populations,, Journal of Theoretical Biology, 199 (1999), 407. doi: 10.1006/jtbi.1999.0966. Google Scholar
[23]	S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar
[24]	T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511542633. Google Scholar
[25]	G. Wang, X.-G. Liang and F.-Z. Wang, The competitive dynamics of populations subject to an Allee effect,, Ecological Modelling, 124 (1999), 183. doi: 10.1016/S0304-3800(99)00160-X. Google Scholar

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References:

[1]	P. A. Abrams, Modelling the adaptive dynamics of traits involved in inter- and intraspecific interactions: An assessment of three methods,, Ecology Letters, 4 (2001), 166. doi: 10.1046/j.1461-0248.2001.00199.x. Google Scholar
[2]	W. C. Allee, Animal Aggregations, a Study in General Sociology,, University of Chicago Press, (1931). Google Scholar
[3]	W. C. Allee, The Social Life of Animals,, 3rd edition, (1941). Google Scholar
[4]	W. C. Allee, O. Park, T. Park and K. Schmidt, Principles of Animal Ecology,, W. B. Saunders Company, (1949). Google Scholar
[5]	D. S. Boukal and L. Berec, Single-species Models of the Allee effect: Extinction boundaries, sex Ratios and mate Encounters,, Journal of Theoretical Biology, 218 (2002), 375. doi: 10.1006/jtbi.2002.3084. Google Scholar
[6]	F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect,, TREE, 14 (1999), 405. doi: 10.1016/S0169-5347(99)01683-3. Google Scholar
[7]	F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation,, Oxford University Press, (2008). doi: 10.1093/acprof:oso/9780198570301.001.0001. Google Scholar
[8]	J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations,, Journal of Biological Dynamics, 8 (2014), 57. doi: 10.1080/17513758.2014.899638. Google Scholar
[9]	J. M. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects,, Journal of Biological Dynamics, 6 (2012), 941. doi: 10.1080/17513758.2012.697196. Google Scholar
[10]	B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction,, Natural Resource Modeling, 3 (1989), 481. Google Scholar
[11]	F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications,, Princeton University Press, (2008). Google Scholar
[12]	L. Edelstein-Keshet, Mathematical Models in Biology,, Classics in Applied Mathematics 46, (2005). doi: 10.1137/1.9780898719147. Google Scholar
[13]	S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, Journal of Biological Dynamics , 4 (2010), 397. doi: 10.1080/17513750903377434. Google Scholar
[14]	D. S. Falconer and T. F. C. Mackay, Introduction to Quantitative Genetics,, Pearson Education Limited, (1996). Google Scholar
[15]	F. A. Hopf and F. W. Hopf, The role of the Allee effect in species packing,, Theoretical Population Biology, 27 (1985), 27. doi: 10.1016/0040-5809(85)90014-0. Google Scholar
[16]	M. R. S. Kulenovic and A.-A. Yakubu, Compensatory versus overcompensatory dynamics in density-dependent leslie models,, Journal of Difference Equations and Applications, 10 (2004), 1251. doi: 10.1080/10236190410001652711. Google Scholar
[17]	R. Lande, Natural selection and random genetic drift in phenotypic evolution,, Evolution, 30 (1976), 314. Google Scholar
[18]	R. Lande, A quantitative genetic theory of life history evolution,, Ecology, 63 (1982), 607. Google Scholar
[19]	M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms,, Theoretical Population Biology, 43 (1993), 141. doi: 10.1006/tpbi.1993.1007. Google Scholar
[20]	J. Lush, Animal Breeding Plans,, Iowa State College Press, (1937). Google Scholar
[21]	S. P. Otto and T. Day, A Biologist's Guide to Mathematical Modeling in Ecology and Evolution,, Princeton University Press, (2007). Google Scholar
[22]	I. Scheuring, Allee effect increases dynamical stability in populations,, Journal of Theoretical Biology, 199 (1999), 407. doi: 10.1006/jtbi.1999.0966. Google Scholar
[23]	S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models,, Theoretical Population Biology, 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar
[24]	T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics,, Cambridge University Press, (2005). doi: 10.1017/CBO9780511542633. Google Scholar
[25]	G. Wang, X.-G. Liang and F.-Z. Wang, The competitive dynamics of populations subject to an Allee effect,, Ecological Modelling, 124 (1999), 183. doi: 10.1016/S0304-3800(99)00160-X. Google Scholar

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