American Institute of Mathematical Sciences

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

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

Jim M. Cushing 1,

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).
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), 166-175. 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, Chicago, 1931. Google Scholar

[3]

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

[4]

W. C. Allee, O. Park, T. Park and K. Schmidt, Principles of Animal Ecology, W. B. Saunders Company, Philadelphia, 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), 375-394. 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), 405-410. 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, Oxford, Great Britain, 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), 57-73. 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), 941-958. 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), 481-538.  Google Scholar

[11]

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

[12]

L. Edelstein-Keshet, Mathematical Models in Biology, Classics in Applied Mathematics 46, SIAM, Philadelphia, USA, 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), 397-408. doi: 10.1080/17513750903377434.  Google Scholar

[14]

D. S. Falconer and T. F. C. Mackay, Introduction to Quantitative Genetics, Pearson Education Limited, Prentice Hall, Essex, England, 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), 27-50. 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), 1251-1265. doi: 10.1080/10236190410001652711.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

I. Scheuring, Allee effect increases dynamical stability in populations, Journal of Theoretical Biology, 199 (1999), 407-414. 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), 201-209. 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, New York, 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), 183-192. doi: 10.1016/S0304-3800(99)00160-X.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[4]

W. C. Allee, O. Park, T. Park and K. Schmidt, Principles of Animal Ecology, W. B. Saunders Company, Philadelphia, 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), 375-394. 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), 405-410. 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, Oxford, Great Britain, 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), 57-73. 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), 941-958. 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), 481-538.  Google Scholar

[11]

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

[12]

L. Edelstein-Keshet, Mathematical Models in Biology, Classics in Applied Mathematics 46, SIAM, Philadelphia, USA, 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), 397-408. doi: 10.1080/17513750903377434.  Google Scholar

[14]

D. S. Falconer and T. F. C. Mackay, Introduction to Quantitative Genetics, Pearson Education Limited, Prentice Hall, Essex, England, 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), 27-50. 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), 1251-1265. doi: 10.1080/10236190410001652711.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[22]

I. Scheuring, Allee effect increases dynamical stability in populations, Journal of Theoretical Biology, 199 (1999), 407-414. 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), 201-209. 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, New York, 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), 183-192. doi: 10.1016/S0304-3800(99)00160-X.  Google Scholar

[1]

Eduardo Liz, Alfonso Ruiz-Herrera. Delayed population models with Allee effects and exploitation. Mathematical Biosciences & Engineering, 2015, 12 (1) : 83-97. doi: 10.3934/mbe.2015.12.83
[2]

Astridh Boccabella, Roberto Natalini, Lorenzo Pareschi. On a continuous mixed strategies model for evolutionary game theory. Kinetic & Related Models, 2011, 4 (1) : 187-213. doi: 10.3934/krm.2011.4.187
[3]

Anna Lisa Amadori, Astridh Boccabella, Roberto Natalini. A hyperbolic model of spatial evolutionary game theory. Communications on Pure & Applied Analysis, 2012, 11 (3) : 981-1002. doi: 10.3934/cpaa.2012.11.981
[4]

Alex Potapov, Ulrike E. Schlägel, Mark A. Lewis. Evolutionarily stable diffusive dispersal. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3319-3340. doi: 10.3934/dcdsb.2014.19.3319
[5]

Dianmo Li, Zhen Zhang, Zufei Ma, Baoyu Xie, Rui Wang. Allee effect and a catastrophe model of population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 629-634. doi: 10.3934/dcdsb.2004.4.629
[6]

Sophia R.-J. Jang. Allee effects in an iteroparous host population and in host-parasitoid interactions. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 113-135. doi: 10.3934/dcdsb.2011.15.113
[7]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, 2021, 14 (1) : 115-148. doi: 10.3934/krm.2020051
[8]

J. Leonel Rocha, Danièle Fournier-Prunaret, Abdel-Kaddous Taha. Strong and weak Allee effects and chaotic dynamics in Richards' growths. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2397-2425. doi: 10.3934/dcdsb.2013.18.2397
[9]

Yun Kang, Sourav Kumar Sasmal, Amiya Ranjan Bhowmick, Joydev Chattopadhyay. Dynamics of a predator-prey system with prey subject to Allee effects and disease. Mathematical Biosciences & Engineering, 2014, 11 (4) : 877-918. doi: 10.3934/mbe.2014.11.877
[10]

William H. Sandholm. Local stability of strict equilibria under evolutionary game dynamics. Journal of Dynamics & Games, 2014, 1 (3) : 485-495. doi: 10.3934/jdg.2014.1.485
[11]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3419-3440. doi: 10.3934/dcdss.2020426
[12]

King-Yeung Lam. Dirac-concentrations in an integro-pde model from evolutionary game theory. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 737-754. doi: 10.3934/dcdsb.2018205
[13]

Jing-Jing Xiang, Yihao Fang. Evolutionarily stable dispersal strategies in a two-patch advective environment. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1875-1887. doi: 10.3934/dcdsb.2018245
[14]

Emile Franc Doungmo Goufo. Bounded perturbation for evolution equations with a parameter & application to population dynamics. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2137-2150. doi: 10.3934/dcdss.2020177
[15]

Jia Li. Modeling of mosquitoes with dominant or recessive Transgenes and Allee effects. Mathematical Biosciences & Engineering, 2010, 7 (1) : 99-121. doi: 10.3934/mbe.2010.7.99
[16]

Erika T. Camacho, Christopher M. Kribs-Zaleta, Stephen Wirkus. Metering effects in population systems. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1365-1379. doi: 10.3934/mbe.2013.10.1365
[17]

Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169
[18]

Caichun Chai, Tiaojun Xiao, Eilin Francis. Is social responsibility for firms competing on quantity evolutionary stable?. Journal of Industrial & Management Optimization, 2018, 14 (1) : 325-347. doi: 10.3934/jimo.2017049
[19]

Donald L. DeAngelis, Bo Zhang. Effects of dispersal in a non-uniform environment on population dynamics and competition: A patch model approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3087-3104. doi: 10.3934/dcdsb.2014.19.3087
[20]

Yannick Viossat. Game dynamics and Nash equilibria. Journal of Dynamics & Games, 2014, 1 (3) : 537-553. doi: 10.3934/jdg.2014.1.537

2018 Impact Factor: 1.313

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences