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

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).
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), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x.

[2]

W. C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, 1931.

[3]

W. C. Allee, The Social Life of Animals, 3rd edition, William Heineman Ltd, London and Toronto, 1941.

[4]

W. C. Allee, O. Park, T. Park and K. Schmidt, Principles of Animal Ecology, W. B. Saunders Company, Philadelphia, 1949.

[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), 375-394. doi: 10.1006/jtbi.2002.3084.

[6]

F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, TREE, 14 (1999), 405-410. doi: 10.1016/S0169-5347(99)01683-3.

[7]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, Great Britain, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[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), 57-73. doi: 10.1080/17513758.2014.899638.

[9]

J. M. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects, Journal of Biological Dynamics, 6 (2012), 941-958. doi: 10.1080/17513758.2012.697196.

[10]

B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.

[11]

F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications, Princeton University Press, Princeton, New Jersey, 2008.

[12]

L. Edelstein-Keshet, Mathematical Models in Biology, Classics in Applied Mathematics 46, SIAM, Philadelphia, USA, 2005. doi: 10.1137/1.9780898719147.

[13]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, Journal of Biological Dynamics , 4 (2010), 397-408. doi: 10.1080/17513750903377434.

[14]

D. S. Falconer and T. F. C. Mackay, Introduction to Quantitative Genetics, Pearson Education Limited, Prentice Hall, Essex, England, 1996.

[15]

F. A. Hopf and F. W. Hopf, The role of the Allee effect in species packing, Theoretical Population Biology, 27 (1985), 27-50. doi: 10.1016/0040-5809(85)90014-0.

[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), 1251-1265. doi: 10.1080/10236190410001652711.

[17]

R. Lande, Natural selection and random genetic drift in phenotypic evolution, Evolution, 30 (1976), 314-334.

[18]

R. Lande, A quantitative genetic theory of life history evolution, Ecology, 63 (1982), 607-615.

[19]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158. doi: 10.1006/tpbi.1993.1007.

[20]

J. Lush, Animal Breeding Plans, Iowa State College Press, Ames, Iowa, USA, 1937.

[21]

S. P. Otto and T. Day, A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, Princeton University Press, Princeton, New Jersey, USA, 2007.

[22]

I. Scheuring, Allee effect increases dynamical stability in populations, Journal of Theoretical Biology, 199 (1999), 407-414. doi: 10.1006/jtbi.1999.0966.

[23]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.

[24]

T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511542633.

[25]

G. Wang, X.-G. Liang and F.-Z. Wang, The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124 (1999), 183-192. doi: 10.1016/S0304-3800(99)00160-X.

show all references

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), 166-175. doi: 10.1046/j.1461-0248.2001.00199.x.

[2]

W. C. Allee, Animal Aggregations, a Study in General Sociology, University of Chicago Press, Chicago, 1931.

[3]

W. C. Allee, The Social Life of Animals, 3rd edition, William Heineman Ltd, London and Toronto, 1941.

[4]

W. C. Allee, O. Park, T. Park and K. Schmidt, Principles of Animal Ecology, W. B. Saunders Company, Philadelphia, 1949.

[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), 375-394. doi: 10.1006/jtbi.2002.3084.

[6]

F. Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, TREE, 14 (1999), 405-410. doi: 10.1016/S0169-5347(99)01683-3.

[7]

F. Courchamp, L. Berec and J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, Oxford, Great Britain, 2008. doi: 10.1093/acprof:oso/9780198570301.001.0001.

[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), 57-73. doi: 10.1080/17513758.2014.899638.

[9]

J. M. Cushing and J. Hudson, Evolutionary dynamics and strong Allee effects, Journal of Biological Dynamics, 6 (2012), 941-958. doi: 10.1080/17513758.2012.697196.

[10]

B. Dennis, Allee effects: Population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.

[11]

F. Dercole and S. Rinaldi, Analysis of Evolutionary Processes: The Adaptive Dynamics Approach and Its Applications, Princeton University Press, Princeton, New Jersey, 2008.

[12]

L. Edelstein-Keshet, Mathematical Models in Biology, Classics in Applied Mathematics 46, SIAM, Philadelphia, USA, 2005. doi: 10.1137/1.9780898719147.

[13]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model, Journal of Biological Dynamics , 4 (2010), 397-408. doi: 10.1080/17513750903377434.

[14]

D. S. Falconer and T. F. C. Mackay, Introduction to Quantitative Genetics, Pearson Education Limited, Prentice Hall, Essex, England, 1996.

[15]

F. A. Hopf and F. W. Hopf, The role of the Allee effect in species packing, Theoretical Population Biology, 27 (1985), 27-50. doi: 10.1016/0040-5809(85)90014-0.

[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), 1251-1265. doi: 10.1080/10236190410001652711.

[17]

R. Lande, Natural selection and random genetic drift in phenotypic evolution, Evolution, 30 (1976), 314-334.

[18]

R. Lande, A quantitative genetic theory of life history evolution, Ecology, 63 (1982), 607-615.

[19]

M. A. Lewis and P. Kareiva, Allee dynamics and the spread of invading organisms, Theoretical Population Biology, 43 (1993), 141-158. doi: 10.1006/tpbi.1993.1007.

[20]

J. Lush, Animal Breeding Plans, Iowa State College Press, Ames, Iowa, USA, 1937.

[21]

S. P. Otto and T. Day, A Biologist's Guide to Mathematical Modeling in Ecology and Evolution, Princeton University Press, Princeton, New Jersey, USA, 2007.

[22]

I. Scheuring, Allee effect increases dynamical stability in populations, Journal of Theoretical Biology, 199 (1999), 407-414. doi: 10.1006/jtbi.1999.0966.

[23]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theoretical Population Biology, 64 (2003), 201-209. doi: 10.1016/S0040-5809(03)00072-8.

[24]

T. L. Vincent and J. S. Brown, Evolutionary Game Theory, Natural Selection, and Darwinian Dynamics, Cambridge University Press, New York, 2005. doi: 10.1017/CBO9780511542633.

[25]

G. Wang, X.-G. Liang and F.-Z. Wang, The competitive dynamics of populations subject to an Allee effect, Ecological Modelling, 124 (1999), 183-192. doi: 10.1016/S0304-3800(99)00160-X.

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