2015, 12(1): 83-97. doi: 10.3934/mbe.2015.12.83

Delayed population models with Allee effects and exploitation

1. 

Departamento de Matemática Aplicada II, E.T.S.E. Telecomunicación, Universidade de Vigo, Campus Marcosende, 36310 Vigo

2. 

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged, Hungary

Received  April 2014 Revised  October 2014 Published  December 2014

Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.
Citation: 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
References:
[1]

D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters,, J. Theoret. Biol., 218 (2002), 375. doi: 10.1006/jtbi.2002.3084. Google Scholar

[2]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model,, Math. Biosci., 248 (2014), 78. doi: 10.1016/j.mbs.2013.12.003. Google Scholar

[3]

C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources,, $2^{nd}$ edition, (1990). Google Scholar

[4]

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

[5]

A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators,, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907. Google Scholar

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar

[7]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, J. Biol. Dyn., 4 (2010), 397. doi: 10.1080/17513750903377434. Google Scholar

[8]

S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?,, J. Math. Biol., 65 (2012), 1411. doi: 10.1007/s00285-011-0496-3. Google Scholar

[9]

K. P. Hadeler, Neutral delay equations from and for population dynamics,, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1. Google Scholar

[10]

C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities,, J. Differential Equations, 256 (2014), 2101. doi: 10.1016/j.jde.2013.12.015. Google Scholar

[11]

A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations,, Differential Equations Dynam. Systems, 11 (2003), 33. Google Scholar

[12]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations,, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164. Google Scholar

[13]

M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect,, Theor. Ecol., 7 (2014), 335. doi: 10.1007/s12080-014-0222-z. Google Scholar

[14]

T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83. doi: 10.1007/s10998-008-5083-x. Google Scholar

[15]

T. Krisztin and E. Liz, Bubbles for a class of delay differential equations,, Qual. Theory Dyn. Syst., 10 (2011), 169. doi: 10.1007/s12346-011-0055-8. Google Scholar

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993). Google Scholar

[17]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting,, Theor. Ecol., 3 (2010), 209. doi: 10.1007/s12080-009-0064-2. Google Scholar

[18]

E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models,, Quart. Appl. Math., 63 (2005), 56. Google Scholar

[19]

E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback,, Discrete Contin. Dyn. Syst., 24 (2009), 1215. doi: 10.3934/dcds.2009.24.1215. Google Scholar

[20]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback,, J. Differential Equations, 255 (2013), 4244. doi: 10.1016/j.jde.2013.08.007. Google Scholar

[21]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596. doi: 10.1137/S0036141001399222. Google Scholar

[22]

J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation,, Ann. Mat. Pura Appl., 145 (1986), 33. doi: 10.1007/BF01790539. Google Scholar

[23]

G. Röst, On the global attractivity controversy for a delay model of hematopoiesis,, Appl. Math. Comput., 190 (2007), 846. doi: 10.1016/j.amc.2007.01.103. Google Scholar

[24]

G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655. doi: 10.1098/rspa.2007.1890. Google Scholar

[25]

S. Ruan, Delay differential equations in single species dynamics,, in Delay differential equations and applications, 205 (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[26]

S. J. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239. doi: 10.1007/s002850000070. Google Scholar

[27]

S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

[28]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-94-015-8897-3. Google Scholar

[29]

A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications,, Kluwer Academic Publishers, (1993). doi: 10.1007/978-94-011-1763-0. Google Scholar

[30]

D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260. doi: 10.1137/0135020. Google Scholar

[31]

A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries,, Math. Biosci., 232 (2011), 66. doi: 10.1016/j.mbs.2011.04.004. Google Scholar

[32]

T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955. doi: 10.1098/rspa.2009.0650. Google Scholar

show all references

References:
[1]

D. S. Boukal and L. Berec, Single-species models of the Allee effect: Extinction boundaries, sex ratios and mate encounters,, J. Theoret. Biol., 218 (2002), 375. doi: 10.1006/jtbi.2002.3084. Google Scholar

[2]

B. Cid, F. M. Hilker and E. Liz, Harvest timing and its population dynamic consequences in a discrete single-species model,, Math. Biosci., 248 (2014), 78. doi: 10.1016/j.mbs.2013.12.003. Google Scholar

[3]

C. W. Clark, Mathematical Bioeconomics. Optimal Management of Renewable Resources,, $2^{nd}$ edition, (1990). Google Scholar

[4]

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

[5]

A. M. De Roos and L. Persson, Size-dependent life-history traits promote catastrophic collapses of top predators,, Proc. Natl. Acad. Sci. USA, 99 (2002), 12907. Google Scholar

[6]

O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis,, Springer-Verlag, (1995). doi: 10.1007/978-1-4612-4206-2. Google Scholar

[7]

S. N. Elaydi and R. J. Sacker, Population models with Allee effect: A new model,, J. Biol. Dyn., 4 (2010), 397. doi: 10.1080/17513750903377434. Google Scholar

[8]

S. A. H. Geritz and E. Kisdi, Mathematical ecology: Why mechanistic models?,, J. Math. Biol., 65 (2012), 1411. doi: 10.1007/s00285-011-0496-3. Google Scholar

[9]

K. P. Hadeler, Neutral delay equations from and for population dynamics,, Electron. J. Qual. Theory Differ. Equ., 11 (2008), 1. Google Scholar

[10]

C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities,, J. Differential Equations, 256 (2014), 2101. doi: 10.1016/j.jde.2013.12.015. Google Scholar

[11]

A. F. Ivanov, E. Liz and S. Trofimchuk, Global stability of a class of scalar nonlinear delay differential equations,, Differential Equations Dynam. Systems, 11 (2003), 33. Google Scholar

[12]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations,, Dynam. Report. Expositions Dynam. Systems (N.S.), 1 (1992), 164. Google Scholar

[13]

M. Jankovic and S. Petrovskii, Are time delays always destabilizing? Revisiting the role of time delays and the Allee effect,, Theor. Ecol., 7 (2014), 335. doi: 10.1007/s12080-014-0222-z. Google Scholar

[14]

T. Krisztin, Global dynamics of delay differential equations,, Period. Math. Hungar., 56 (2008), 83. doi: 10.1007/s10998-008-5083-x. Google Scholar

[15]

T. Krisztin and E. Liz, Bubbles for a class of delay differential equations,, Qual. Theory Dyn. Syst., 10 (2011), 169. doi: 10.1007/s12346-011-0055-8. Google Scholar

[16]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993). Google Scholar

[17]

E. Liz, Complex dynamics of survival and extinction in simple population models with harvesting,, Theor. Ecol., 3 (2010), 209. doi: 10.1007/s12080-009-0064-2. Google Scholar

[18]

E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk, A global stability criterion for a family of delayed population models,, Quart. Appl. Math., 63 (2005), 56. Google Scholar

[19]

E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback,, Discrete Contin. Dyn. Syst., 24 (2009), 1215. doi: 10.3934/dcds.2009.24.1215. Google Scholar

[20]

E. Liz and A. Ruiz-Herrera, Attractivity, multistability, and bifurcation in delayed Hopfield's model with non-monotonic feedback,, J. Differential Equations, 255 (2013), 4244. doi: 10.1016/j.jde.2013.08.007. Google Scholar

[21]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596. doi: 10.1137/S0036141001399222. Google Scholar

[22]

J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation,, Ann. Mat. Pura Appl., 145 (1986), 33. doi: 10.1007/BF01790539. Google Scholar

[23]

G. Röst, On the global attractivity controversy for a delay model of hematopoiesis,, Appl. Math. Comput., 190 (2007), 846. doi: 10.1016/j.amc.2007.01.103. Google Scholar

[24]

G. Röst and J. Wu, Domain decomposition method for the global dynamics of delay differential equations with unimodal feedback,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655. doi: 10.1098/rspa.2007.1890. Google Scholar

[25]

S. Ruan, Delay differential equations in single species dynamics,, in Delay differential equations and applications, 205 (2006), 477. doi: 10.1007/1-4020-3647-7_11. Google Scholar

[26]

S. J. Schreiber, Chaos and population disappearances in simple ecological models,, J. Math. Biol., 42 (2001), 239. doi: 10.1007/s002850000070. Google Scholar

[27]

S. J. Schreiber, Allee effect, extinctions, and chaotic transients in simple population models,, Theor. Popul. Biol., 64 (2003), 201. doi: 10.1016/S0040-5809(03)00072-8. Google Scholar

[28]

A. N. Sharkovsky, S. F. Kolyada, A. G. Sivak and V. V. Fedorenko, Dynamics of One-Dimensional Maps,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-94-015-8897-3. Google Scholar

[29]

A. N. Sharkovsky, Y. L. Maistrenko and E. Y. Romanenko, Difference Equations and Their Applications,, Kluwer Academic Publishers, (1993). doi: 10.1007/978-94-011-1763-0. Google Scholar

[30]

D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260. doi: 10.1137/0135020. Google Scholar

[31]

A.-A. Yakubu, N. Li, J. M. Conrad and M.-L. Zeeman, Constant proportion harvest policies: Dynamic implications in the Pacific halibut and Atlantic cod fisheries,, Math. Biosci., 232 (2011), 66. doi: 10.1016/j.mbs.2011.04.004. Google Scholar

[32]

T. Yi and X. Zou, Maps dynamics versus dynamics of associated delay reaction-diffusion equation with a Neumann condition,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955. doi: 10.1098/rspa.2009.0650. Google Scholar

[1]

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

[2]

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

[3]

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

[4]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689

[5]

Anatoli F. Ivanov, Sergei Trofimchuk. Periodic solutions and their stability of a differential-difference equation. Conference Publications, 2009, 2009 (Special) : 385-393. doi: 10.3934/proc.2009.2009.385

[6]

Elena Braverman, Alexandra Rodkina. Stochastic difference equations with the Allee effect. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 5929-5949. doi: 10.3934/dcds.2016060

[7]

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

[8]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268

[9]

Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139

[10]

Jan Čermák, Jana Hrabalová. Delay-dependent stability criteria for neutral delay differential and difference equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4577-4588. doi: 10.3934/dcds.2014.34.4577

[11]

Nika Lazaryan, Hassan Sedaghat. Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 731-747. doi: 10.3934/dcdsb.2018040

[12]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[13]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045

[14]

Azmy S. Ackleh, Keng Deng. Stability of a delay equation arising from a juvenile-adult model. Mathematical Biosciences & Engineering, 2010, 7 (4) : 729-737. doi: 10.3934/mbe.2010.7.729

[15]

Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451

[16]

Pierre Magal. Global stability for differential equations with homogeneous nonlinearity and application to population dynamics. Discrete & Continuous Dynamical Systems - B, 2002, 2 (4) : 541-560. doi: 10.3934/dcdsb.2002.2.541

[17]

Cuilian You, Yangyang Hao. Stability in mean for fuzzy differential equation. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1375-1385. doi: 10.3934/jimo.2018099

[18]

Rui Hu, Yuan Yuan. Stability, bifurcation analysis in a neural network model with delay and diffusion. Conference Publications, 2009, 2009 (Special) : 367-376. doi: 10.3934/proc.2009.2009.367

[19]

P. Dormayer, A. F. Ivanov. Symmetric periodic solutions of a delay differential equation. Conference Publications, 1998, 1998 (Special) : 220-230. doi: 10.3934/proc.1998.1998.220

[20]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]