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January  2011, 15(1): 113-135. doi: 10.3934/dcdsb.2011.15.113

Allee effects in an iteroparous host population and in host-parasitoid interactions

Sophia R.-J. Jang 1,

1. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States

Received  December 2009 Revised  May 2010 Published  October 2010

We investigate a stage-structured model of an iteroparous population with two age classes. The population is assumed to exhibit Allee effects through reproduction. The asymptotic dynamics of the model depend on the maximal reproductive number of the population. The population may persist if the maximal reproductive number is greater than one. There exists a population threshold in terms of the unstable interior equilibrium. The host population will become extinct if its initial distribution lies below the threshold and the host population can persist indefinitely if its initial distribution lies above the threshold. In addition, if the unstable equilibrium is a saddle point and the system has no $2$-cycles, then the stable manifold of the saddle point provides the Allee threshold for the host. Based on this host population system, we construct a host-parasitoid model to study the impact of Allee effects upon the population interaction. The parasitoid population may drive the host to below the Allee threshold so that both populations become extinct. On the other hand, under some conditions on the parameters, the host-parasitoid system may possess an interior equilibrium and the populations may coexist as an interior equilibrium.
Keywords: monotone map, population threshold, stable manifold., Allee effects.
Mathematics Subject Classification: Primary: 92D25; Secondary: 39A1.
Citation: Sophia R.-J. Jang. Allee effects in an iteroparous host population and in host-parasitoid interactions. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 113-135. doi: 10.3934/dcdsb.2011.15.113
References:
[1]

W. C. Allee, "The Social Life of Animals," William Heinemann, London, 1938.

[2]

L. J. S. Allen, "An Introduction to Mathematical Biology," Prentice Hall, New Jersey, 2006.

[3]

M. Begon, J. Harper and C. Townsend, "Ecology: Individuals, Populations and Communities," Blackwell Science Ltd, New York, 1996.

[4]

W. J. Bell and K. G. Adiyodi, "The American Cockroach," Chapman and Hall, London, 1981.

[5]

A. Brännström and D. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B, 272 (2005), 2065-2072. doi: doi:10.1098/rspb.2005.3185.

[6]

J. B. Conway, "Functions of One Complex Variable," 2nd edition, Springer, New York, 1978.

[7]

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

[8]

J. M. Cushing, The Allee effect in age-structured population dynamics, in "Mathematical Ecology" (eds. T. Hallam, L. Gross and S. Levin), (1988), 479-505.

[9]

J. M. Cushing, A strong ergodic theorem for some nonlinear matrix models for the dynamics of structured population, Nat. Res. Mod., 3 (1989), 331-357.

[10]

J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.

[11]

J. M. Cushing, "An Introduction to Structured Population Models," Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, 1998.

[12]

J. M. Cushing, Nonlinear semelparous Leslie models, Math. Bio. Sci. Eng., 3 (2006), 17-36.

[13]

B. Dennis, Allee effects, population growth, critical density, and the chance of extinction, Nat. Res. Mod., 3 (1989), 481-538.

[14]

B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401. doi: doi:10.1034/j.1600-0706.2002.960301.x.

[15]

A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos, 112 (2006), 667-679. doi: doi:10.1111/j.0030-1299.2006.14243.x.

[16]

A. Grove and G. Ladas, "Periodicities in Nonlinear Difference Equations," CRC Press, Boca Raton, 2005.

[17]

J. Hale and H. Kocak, "Dynamics and Bifurcations," Springer-Verlag, New York, 1991.

[18]

F. M. Hilker, M. Langlais, S. V. Petrovskii and H. Malcho, A diffusive SI model with Allee effect and application to FIV, Math. Biosci., 206 (2007), 61-80. doi: doi:10.1016/j.mbs.2005.10.003.

[19]

F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Bio. Dyn., 4 (2010), 86-101. doi: doi:10.1080/17513750903026429.

[20]

M. Hirsch and H. Smith, Monotone maps: A review, J. Diff. Equ. & Appl., 11 (2005), 379-398.

[21]

J. Hofbauer and J. So, Uniform persistence and repellors for maps, Proc. Am. Math. Soc., 107 (1989), 1137-1142.

[22]

S. R.-J. Jang, Allee effects in a discrete-time host parasitoid model with stage structure in the host, Dis. Con. Dyn. Sys. Ser. B, 8 (2007), 145-159. doi: doi:10.3934/dcdsb.2007.8.145.

[23]

S. R.-J. Jang, Discrete host-parasitoid models with Allee effects and age structure in the host, Math. Biosci. & Eng., 7 (2010), 67-81. doi: doi:10.3934/mbe.2010.7.67.

[24]

S. R.-J. Jang, Dynamics of an age-structured population with Allee effects and harvesting, J. Bio. Dyn., 4 (2010), 409-427. doi: doi:10.1080/17513750903389082.

[25]

M. Kulenovic and M. Nurkanovic, Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation, J. Diff. Equ. & Appl., 9 (2003), 149-159.

[26]

M. Kulenovic and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Int. J. Bif. Chaos, to appear.

[27]

R. M. May, H. P. Hassell, R. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, J. Ani. Ecol., 50 (1981), 855-865. doi: doi:10.2307/4142.

[28]

C. C. McCluskey and J. S. Muldowney, Bendixson-Dulac criteria for difference equations, J. Dyn. Dif. Eq., 10 (1998), 567-575. doi: doi:10.1023/A:1022677008393.

[29]

A. Morozov, S. Petrovskii and B.-L. Li, Bifurcations and chaos in a predator-prey system with the Allee effect, Proc. Roy. Soc., Ser. B, 271 (2004), 1407-1414. doi: doi:10.1098/rspb.2004.2733.

[30]

A. J. Nicholson and V. A. Bailey, The balance of animal population: Part I, Proc. Zool. Soc. Lond., 3 (1935), 551-598.

[31]

W. F. Patterson, J. H. Cowan, G. R. Fitzhugh and D. L. Nieland, "Population Ecology and Fisheries of U. S. Gulf of Mexico Red Snapper," American Fisheries Society, Bethesda, Maryland, 2007.

[32]

C. Robinson, "Stability, Symbolic Dynamics, and Chaos," CRC Press, Boca Raton, 1995.

[33]

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

[34]

H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. & Appl., 3 (1998), 335-357.

[35]

D. R. Suiter, Biological suppression of synanthropic cockroaches, J. Agri. Entomo., 14 (1997), 259-270.

[36]

H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious diseases and Allee effect, J. Biol. Dyn., 3 (2009), 305-323. doi: doi:10.1080/17513750802376313.

[37]

G. Voorn, L. Hemerik, M. Boer and B. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469. doi: doi:10.1016/j.mbs.2007.02.006.

[38]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer, New York, 2003.

[39]

S. Zhou and G. Wang, Allee-like effects in metapopulation dynamics, Math. Biosci., 189 (2004), 103-113. doi: doi:10.1016/j.mbs.2003.06.001.

[40]

S. Zhou, Y. Liu and G. Yang, The stability of predator-prey systems subject to the Allee effects, Theo. Pop. Biol., 67 (2005), 23-31. doi: doi:10.1016/j.tpb.2004.06.007.

show all references

References:
[1]

W. C. Allee, "The Social Life of Animals," William Heinemann, London, 1938.

[2]

L. J. S. Allen, "An Introduction to Mathematical Biology," Prentice Hall, New Jersey, 2006.

[3]

M. Begon, J. Harper and C. Townsend, "Ecology: Individuals, Populations and Communities," Blackwell Science Ltd, New York, 1996.

[4]

W. J. Bell and K. G. Adiyodi, "The American Cockroach," Chapman and Hall, London, 1981.

[5]

A. Brännström and D. Sumpter, The role of competition and clustering in population dynamics, Proc. R. Soc. B, 272 (2005), 2065-2072. doi: doi:10.1098/rspb.2005.3185.

[6]

J. B. Conway, "Functions of One Complex Variable," 2nd edition, Springer, New York, 1978.

[7]

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

[8]

J. M. Cushing, The Allee effect in age-structured population dynamics, in "Mathematical Ecology" (eds. T. Hallam, L. Gross and S. Levin), (1988), 479-505.

[9]

J. M. Cushing, A strong ergodic theorem for some nonlinear matrix models for the dynamics of structured population, Nat. Res. Mod., 3 (1989), 331-357.

[10]

J. M. Cushing and Z. Yicang, The net reproductive value and stability in matrix population models, Nat. Res. Mod., 8 (1994), 297-333.

[11]

J. M. Cushing, "An Introduction to Structured Population Models," Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, 1998.

[12]

J. M. Cushing, Nonlinear semelparous Leslie models, Math. Bio. Sci. Eng., 3 (2006), 17-36.

[13]

B. Dennis, Allee effects, population growth, critical density, and the chance of extinction, Nat. Res. Mod., 3 (1989), 481-538.

[14]

B. Dennis, Allee effects in stochastic populations, Oikos, 96 (2002), 389-401. doi: doi:10.1034/j.1600-0706.2002.960301.x.

[15]

A. Deredec and F. Courchamp, Combined impacts of Allee effects and parasitism, Oikos, 112 (2006), 667-679. doi: doi:10.1111/j.0030-1299.2006.14243.x.

[16]

A. Grove and G. Ladas, "Periodicities in Nonlinear Difference Equations," CRC Press, Boca Raton, 2005.

[17]

J. Hale and H. Kocak, "Dynamics and Bifurcations," Springer-Verlag, New York, 1991.

[18]

F. M. Hilker, M. Langlais, S. V. Petrovskii and H. Malcho, A diffusive SI model with Allee effect and application to FIV, Math. Biosci., 206 (2007), 61-80. doi: doi:10.1016/j.mbs.2005.10.003.

[19]

F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, J. Bio. Dyn., 4 (2010), 86-101. doi: doi:10.1080/17513750903026429.

[20]

M. Hirsch and H. Smith, Monotone maps: A review, J. Diff. Equ. & Appl., 11 (2005), 379-398.

[21]

J. Hofbauer and J. So, Uniform persistence and repellors for maps, Proc. Am. Math. Soc., 107 (1989), 1137-1142.

[22]

S. R.-J. Jang, Allee effects in a discrete-time host parasitoid model with stage structure in the host, Dis. Con. Dyn. Sys. Ser. B, 8 (2007), 145-159. doi: doi:10.3934/dcdsb.2007.8.145.

[23]

S. R.-J. Jang, Discrete host-parasitoid models with Allee effects and age structure in the host, Math. Biosci. & Eng., 7 (2010), 67-81. doi: doi:10.3934/mbe.2010.7.67.

[24]

S. R.-J. Jang, Dynamics of an age-structured population with Allee effects and harvesting, J. Bio. Dyn., 4 (2010), 409-427. doi: doi:10.1080/17513750903389082.

[25]

M. Kulenovic and M. Nurkanovic, Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation, J. Diff. Equ. & Appl., 9 (2003), 149-159.

[26]

M. Kulenovic and O. Merino, Invariant manifolds for competitive discrete systems in the plane, Int. J. Bif. Chaos, to appear.

[27]

R. M. May, H. P. Hassell, R. M. Anderson and D. W. Tonkyn, Density dependence in host-parasitoid models, J. Ani. Ecol., 50 (1981), 855-865. doi: doi:10.2307/4142.

[28]

C. C. McCluskey and J. S. Muldowney, Bendixson-Dulac criteria for difference equations, J. Dyn. Dif. Eq., 10 (1998), 567-575. doi: doi:10.1023/A:1022677008393.

[29]

A. Morozov, S. Petrovskii and B.-L. Li, Bifurcations and chaos in a predator-prey system with the Allee effect, Proc. Roy. Soc., Ser. B, 271 (2004), 1407-1414. doi: doi:10.1098/rspb.2004.2733.

[30]

A. J. Nicholson and V. A. Bailey, The balance of animal population: Part I, Proc. Zool. Soc. Lond., 3 (1935), 551-598.

[31]

W. F. Patterson, J. H. Cowan, G. R. Fitzhugh and D. L. Nieland, "Population Ecology and Fisheries of U. S. Gulf of Mexico Red Snapper," American Fisheries Society, Bethesda, Maryland, 2007.

[32]

C. Robinson, "Stability, Symbolic Dynamics, and Chaos," CRC Press, Boca Raton, 1995.

[33]

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

[34]

H. L. Smith, Planar competitive and cooperative difference equations, J. Diff. Equ. & Appl., 3 (1998), 335-357.

[35]

D. R. Suiter, Biological suppression of synanthropic cockroaches, J. Agri. Entomo., 14 (1997), 259-270.

[36]

H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: Synergy of infectious diseases and Allee effect, J. Biol. Dyn., 3 (2009), 305-323. doi: doi:10.1080/17513750802376313.

[37]

G. Voorn, L. Hemerik, M. Boer and B. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469. doi: doi:10.1016/j.mbs.2007.02.006.

[38]

S. Wiggins, "Introduction to Applied Nonlinear Dynamical Systems and Chaos," 2nd edition, Springer, New York, 2003.

[39]

S. Zhou and G. Wang, Allee-like effects in metapopulation dynamics, Math. Biosci., 189 (2004), 103-113. doi: doi:10.1016/j.mbs.2003.06.001.

[40]

S. Zhou, Y. Liu and G. Yang, The stability of predator-prey systems subject to the Allee effects, Theo. Pop. Biol., 67 (2005), 23-31. doi: doi:10.1016/j.tpb.2004.06.007.

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