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March  2018, 23(2): 731-747. doi: 10.3934/dcdsb.2018040

Extinction and the Allee effect in an age structured Ricker population model with inter-stage interaction

Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, P.O. Box 842014, Richmond, Virginia 23284-2014, USA

* Corresponding author: Nika Lazaryan

Received  October 2016 Revised  September 2017 Published  March 2018

We study the evolution in discrete time of certain age-structured populations, such as adults and juveniles, with a Ricker fitness function. We determine conditions for the convergence of orbits to the origin (extinction) in the presence of the Allee effect and time-dependent vital rates. We show that when stages interact, they may survive in the absence of interior fixed points, a surprising situation that is impossible without inter-stage interactions. We also examine the shift in the interior Allee equilibrium caused by the occurrence of interactions between stages and find that the extinction or Allee threshold does not extend to the new boundaries set by the shift in equilibrium, i.e. no interior equilibria are on the extinction threshold.

Citation: 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
References:
[1]

A. Ackleh and S. Jang, A discrete two-stage population model: Continuous versus seasonal reproduction, J. Difference Eq.Appl, 13 (2007), 261-274.  doi: 10.1080/10236190601079217.  Google Scholar

[2]

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

[3]

W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, Principles of Animal Ecology, WB Saunders, Philadelphia, 1949. Google Scholar

[4]

L. BerecE. Angulo and F. Courchamp, Multiple Allee effects and population management, TRENDS in Ecol. Evol., 22 (2006), 185-191.  doi: 10.1016/j.tree.2006.12.002.  Google Scholar

[5]

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

[6]

J. M. Cushing, Oscillations in age-structured population models with an Allee effect, J. Comput. Appl. Math., 52 (1994), 71-80.  doi: 10.1016/0377-0427(94)90349-2.  Google Scholar

[7]

J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 1998, SIAM, Philadelphia.  Google Scholar

[8]

J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math Biol, 53 (2006), 520-539.   Google Scholar

[9]

J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations, J. Biol. Dyn., 8 (2014), 57-73.  doi: 10.1007/s00285-006-0382-6.  Google Scholar

[10]

J. M. Cushing and J. T. Hudson, Evolutionary dynamics and strong Allee effects, J. Biol. Dyn., 6 (2012), 941-958.   Google Scholar

[11]

S. N. Elaydi and R. J. Sacker, Basin of attraction of periodic orbits of maps on the real lin, J. Difference Eq. Appl., 10 (2004), 881-888.  doi: 10.1080/10236190410001731443.  Google Scholar

[12]

S. N. Elaydi and R. J. Sacker, Population models with Allee effects: A new model, J. Biol. Dyn., 4 (2010), 397-408.  doi: 10.1080/17513750903377434.  Google Scholar

[13]

S. R. Jang, Allee effects in discrete-time host-parasitoid model, J. Difference Eq. Appl., 12 (2006), 165-181.  doi: 10.1080/10236190500539238.  Google Scholar

[14]

N. Lazaryan and H. Sedaghat Dynamics of planar systems that model stage-structured populations, Discr. Dyn. Nature Society,, (2015), Article ID 137182, 14pp. doi: 10.1155/2015/137182.  Google Scholar

[15]

N. Lazaryan and H. Sedaghat, Extinction, periodicity and multistability in a Ricker model of stage-structured populations, J. Difference Eq. Appl., 22 (2016), 645-670.   Google Scholar

[16]

N. Lazaryan and H. Sedaghat, Periodic and non-periodic solutions in a Ricker-type secondorder equation with periodic parameters, J. Difference Eq. Appl., 22 (2016), 1199-1223.  doi: 10.1080/10236198.2016.1187142.  Google Scholar

[17]

W. Z. Lidicker, The Allee effect: Its history and future importance, Open Ecol. J., 3 (2010), 71-82.   Google Scholar

[18]

G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology, J. Biol. Dyn, 6 (2012), 959-973.  doi: 10.1080/17513758.2012.700075.  Google Scholar

[19]

E. Liz and P. Pilarczyk, Global dynamics in a stage-sturctured discrete-time population model with harvesting, J. Theor. Biol., 297 (2012), 148-165.  doi: 10.1016/j.jtbi.2011.12.012.  Google Scholar

[20]

R. LuisS. N. Elaydi and H. Oliveira, Non-autonomous periodic systems with Allee effects, J. Difference Eq. Appl., 16 (2010), 1179-1196.  doi: 10.1080/10236190902794951.  Google Scholar

[21]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.  doi: 10.1016/S0040-5809(03)00072-8.  Google Scholar

[22]

H. Sedaghat, Folding, cycles and chaos in planar systems, J. Difference Eq. Appl., 21 (2015), 1-15.  doi: 10.1080/10236198.2014.974585.  Google Scholar

[23]

A. Yakubu, Multiple attractors in juvenile-adult single species models, J. Difference Eq. Appl., 9 (2007), 1083-1098.  doi: 10.1080/1023619031000146887.  Google Scholar

[24]

E. F. ZipkinC. E. KraftE. G. Cooch and P. J. Sullivan, When can efforts to control nuisance and invasive species backfire?, Ecol. Appl., 19 (2009), 1585-1595.  doi: 10.1890/08-1467.1.  Google Scholar

show all references

References:
[1]

A. Ackleh and S. Jang, A discrete two-stage population model: Continuous versus seasonal reproduction, J. Difference Eq.Appl, 13 (2007), 261-274.  doi: 10.1080/10236190601079217.  Google Scholar

[2]

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

[3]

W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, Principles of Animal Ecology, WB Saunders, Philadelphia, 1949. Google Scholar

[4]

L. BerecE. Angulo and F. Courchamp, Multiple Allee effects and population management, TRENDS in Ecol. Evol., 22 (2006), 185-191.  doi: 10.1016/j.tree.2006.12.002.  Google Scholar

[5]

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

[6]

J. M. Cushing, Oscillations in age-structured population models with an Allee effect, J. Comput. Appl. Math., 52 (1994), 71-80.  doi: 10.1016/0377-0427(94)90349-2.  Google Scholar

[7]

J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 1998, SIAM, Philadelphia.  Google Scholar

[8]

J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math Biol, 53 (2006), 520-539.   Google Scholar

[9]

J. M. Cushing, Backward bifurcations and strong Allee effects in matrix models for the dynamics of structured populations, J. Biol. Dyn., 8 (2014), 57-73.  doi: 10.1007/s00285-006-0382-6.  Google Scholar

[10]

J. M. Cushing and J. T. Hudson, Evolutionary dynamics and strong Allee effects, J. Biol. Dyn., 6 (2012), 941-958.   Google Scholar

[11]

S. N. Elaydi and R. J. Sacker, Basin of attraction of periodic orbits of maps on the real lin, J. Difference Eq. Appl., 10 (2004), 881-888.  doi: 10.1080/10236190410001731443.  Google Scholar

[12]

S. N. Elaydi and R. J. Sacker, Population models with Allee effects: A new model, J. Biol. Dyn., 4 (2010), 397-408.  doi: 10.1080/17513750903377434.  Google Scholar

[13]

S. R. Jang, Allee effects in discrete-time host-parasitoid model, J. Difference Eq. Appl., 12 (2006), 165-181.  doi: 10.1080/10236190500539238.  Google Scholar

[14]

N. Lazaryan and H. Sedaghat Dynamics of planar systems that model stage-structured populations, Discr. Dyn. Nature Society,, (2015), Article ID 137182, 14pp. doi: 10.1155/2015/137182.  Google Scholar

[15]

N. Lazaryan and H. Sedaghat, Extinction, periodicity and multistability in a Ricker model of stage-structured populations, J. Difference Eq. Appl., 22 (2016), 645-670.   Google Scholar

[16]

N. Lazaryan and H. Sedaghat, Periodic and non-periodic solutions in a Ricker-type secondorder equation with periodic parameters, J. Difference Eq. Appl., 22 (2016), 1199-1223.  doi: 10.1080/10236198.2016.1187142.  Google Scholar

[17]

W. Z. Lidicker, The Allee effect: Its history and future importance, Open Ecol. J., 3 (2010), 71-82.   Google Scholar

[18]

G. Livadiotis and S. Elaydi, General Allee effect in two-species population biology, J. Biol. Dyn, 6 (2012), 959-973.  doi: 10.1080/17513758.2012.700075.  Google Scholar

[19]

E. Liz and P. Pilarczyk, Global dynamics in a stage-sturctured discrete-time population model with harvesting, J. Theor. Biol., 297 (2012), 148-165.  doi: 10.1016/j.jtbi.2011.12.012.  Google Scholar

[20]

R. LuisS. N. Elaydi and H. Oliveira, Non-autonomous periodic systems with Allee effects, J. Difference Eq. Appl., 16 (2010), 1179-1196.  doi: 10.1080/10236190902794951.  Google Scholar

[21]

S. J. Schreiber, Allee effects, extinctions, and chaotic transients in simple population models, Theor. Popul. Biol., 64 (2003), 201-209.  doi: 10.1016/S0040-5809(03)00072-8.  Google Scholar

[22]

H. Sedaghat, Folding, cycles and chaos in planar systems, J. Difference Eq. Appl., 21 (2015), 1-15.  doi: 10.1080/10236198.2014.974585.  Google Scholar

[23]

A. Yakubu, Multiple attractors in juvenile-adult single species models, J. Difference Eq. Appl., 9 (2007), 1083-1098.  doi: 10.1080/1023619031000146887.  Google Scholar

[24]

E. F. ZipkinC. E. KraftE. G. Cooch and P. J. Sullivan, When can efforts to control nuisance and invasive species backfire?, Ecol. Appl., 19 (2009), 1585-1595.  doi: 10.1890/08-1467.1.  Google Scholar

Figure 1.  $E_{0}$ with $\lambda =3$, $a = 0.7936$, $b = 0.0891$, $s^{\prime} = 1$
Figure 2.  $E$ (shaded) and its complement for $\lambda =3$, $a = 0.7936$, $b = 0.0891$, $s^{\prime} = 1$
Figure 3.  $E$ for $\lambda =2$, $a = 1.1$, $s^{\prime} = 1$ and two different values of $b$
Table 1.  A summary of results
ConditionsOutcomes and CommentsReferences
General$x_{0}, x_{1}<\rho $Extinction for all possible parameter values ifThrm 1(b)
initial values are bounded by $\rho $; $ \lbrack 0, \rho )^{2}\subset E_{0}$
(9)Extinction for all positive initial values; $ E_{0}=[0, \infty )^{2}$Thrm 1(c)
No inter-stage interactions(23)Extinction for all positive initial values; $E_{0}=[0, \infty )^{2}$Cor 9(a)
(24)Extinction with $ E_{0}\subset \lbrack 0, u^{\ast })\times \lbrack 0, u^{\ast }/s^{\prime })$Cor 9(b)
(24), (25)Survival for $x_{0}, x_{1}\in \lbrack u^{\ast }, \bar{u}]^{2}$Cor 9(c)
Survival if $x_{0}=u^{\ast }, x_{1}=0$ or $ x_{1}=u^{\ast }, x_{0}=0$Cor 9(d)
With inter-stage interactions(23)Extinction for all positive initial values; $E_{0}=[0, \infty )^{2}$Cor 15(a)
(33)No positive equilibria but $E_{0}\not=[0, \infty )^{2}$; i.e. survivalCor 15(b)
is possible with some positive initial values!
(31), (34)Extinction occurs from some initial values, survivalOpen problems
from others; nontrivial basins (see Figures 1-3)
ConditionsOutcomes and CommentsReferences
General$x_{0}, x_{1}<\rho $Extinction for all possible parameter values ifThrm 1(b)
initial values are bounded by $\rho $; $ \lbrack 0, \rho )^{2}\subset E_{0}$
(9)Extinction for all positive initial values; $ E_{0}=[0, \infty )^{2}$Thrm 1(c)
No inter-stage interactions(23)Extinction for all positive initial values; $E_{0}=[0, \infty )^{2}$Cor 9(a)
(24)Extinction with $ E_{0}\subset \lbrack 0, u^{\ast })\times \lbrack 0, u^{\ast }/s^{\prime })$Cor 9(b)
(24), (25)Survival for $x_{0}, x_{1}\in \lbrack u^{\ast }, \bar{u}]^{2}$Cor 9(c)
Survival if $x_{0}=u^{\ast }, x_{1}=0$ or $ x_{1}=u^{\ast }, x_{0}=0$Cor 9(d)
With inter-stage interactions(23)Extinction for all positive initial values; $E_{0}=[0, \infty )^{2}$Cor 15(a)
(33)No positive equilibria but $E_{0}\not=[0, \infty )^{2}$; i.e. survivalCor 15(b)
is possible with some positive initial values!
(31), (34)Extinction occurs from some initial values, survivalOpen problems
from others; nontrivial basins (see Figures 1-3)
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