American Institute of Mathematical Sciences

Advanced
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 & 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. Google Scholar

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Kulenovic and O. Merino, Invariant manifolds for competitive discrete systems in the plane,, Int. J. Bif. Chaos, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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. Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[16]

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

[17]

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

[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.  Google Scholar

[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.  Google Scholar

[20]

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

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

M. Kulenovic and O. Merino, Invariant manifolds for competitive discrete systems in the plane,, Int. J. Bif. Chaos, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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. Google Scholar

[32]

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

[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.  Google Scholar

[34]

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

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  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]

E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010
[3]

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
[4]

M. W. Hirsch, Hal L. Smith. Asymptotically stable equilibria for monotone semiflows. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 385-398. doi: 10.3934/dcds.2006.14.385
[5]

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
[6]

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
[7]

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
[8]

C. Bonanno, G. Menconi. Computational information for the logistic map at the chaos threshold. Discrete & Continuous Dynamical Systems - B, 2002, 2 (3) : 415-431. doi: 10.3934/dcdsb.2002.2.415
[9]

Yi Yang, Robert J. Sacker. Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 589-606. doi: 10.3934/dcdsb.2014.19.589
[10]

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
[11]

Sophia R.-J. Jang. Discrete host-parasitoid models with Allee effects and age structure in the host. Mathematical Biosciences & Engineering, 2010, 7 (1) : 67-81. doi: 10.3934/mbe.2010.7.67
[12]

Yongli Cai, Malay Banerjee, Yun Kang, Weiming Wang. Spatiotemporal complexity in a predator--prey model with weak Allee effects. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1247-1274. doi: 10.3934/mbe.2014.11.1247
[13]

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
[14]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[15]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[16]

Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth. Mathematical Biosciences & Engineering, 2016, 13 (1) : 19-41. doi: 10.3934/mbe.2016.13.19
[17]

Dong-Mei Zhu, Wai-Ki Ching, Robert J. Elliott, Tak-Kuen Siu, Lianmin Zhang. Hidden Markov models with threshold effects and their applications to oil price forecasting. Journal of Industrial & Management Optimization, 2017, 13 (2) : 757-773. doi: 10.3934/jimo.2016045
[18]

Plamen Stefanov, Gunther Uhlmann, Andras Vasy. On the stable recovery of a metric from the hyperbolic DN map with incomplete data. Inverse Problems & Imaging, 2016, 10 (4) : 1141-1147. doi: 10.3934/ipi.2016035
[19]

Yanan Zhao, Daqing Jiang, Xuerong Mao, Alison Gray. The threshold of a stochastic SIRS epidemic model in a population with varying size. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1277-1295. doi: 10.3934/dcdsb.2015.20.1277
[20]

Yun Kang, Carlos Castillo-Chávez. A simple epidemiological model for populations in the wild with Allee effects and disease-modified fitness. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 89-130. doi: 10.3934/dcdsb.2014.19.89

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy