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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. [2] W. C. Allee, The Social Life of Animals, William Heinman, London, 1938. [3] W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, Principles of Animal Ecology, WB Saunders, Philadelphia, 1949. [4] L. Berec, E. 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. [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. [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. [7] J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 1998, SIAM, Philadelphia. [8] J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math Biol, 53 (2006), 520-539. [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. [10] J. M. Cushing and J. T. Hudson, Evolutionary dynamics and strong Allee effects, J. Biol. Dyn., 6 (2012), 941-958. [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. [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. [13] S. R. Jang, Allee effects in discrete-time host-parasitoid model, J. Difference Eq. Appl., 12 (2006), 165-181. doi: 10.1080/10236190500539238. [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. [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. [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. [17] W. Z. Lidicker, The Allee effect: Its history and future importance, Open Ecol. J., 3 (2010), 71-82. [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. [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. [20] R. Luis, S. N. Elaydi and H. Oliveira, Non-autonomous periodic systems with Allee effects, J. Difference Eq. Appl., 16 (2010), 1179-1196. doi: 10.1080/10236190902794951. [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. [22] H. Sedaghat, Folding, cycles and chaos in planar systems, J. Difference Eq. Appl., 21 (2015), 1-15. doi: 10.1080/10236198.2014.974585. [23] A. Yakubu, Multiple attractors in juvenile-adult single species models, J. Difference Eq. Appl., 9 (2007), 1083-1098. doi: 10.1080/1023619031000146887. [24] E. F. Zipkin, C. E. Kraft, E. 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.

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. [2] W. C. Allee, The Social Life of Animals, William Heinman, London, 1938. [3] W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, Principles of Animal Ecology, WB Saunders, Philadelphia, 1949. [4] L. Berec, E. 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. [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. [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. [7] J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 1998, SIAM, Philadelphia. [8] J. M. Cushing, A juvenile-adult model with periodic vital rates, J. Math Biol, 53 (2006), 520-539. [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. [10] J. M. Cushing and J. T. Hudson, Evolutionary dynamics and strong Allee effects, J. Biol. Dyn., 6 (2012), 941-958. [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. [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. [13] S. R. Jang, Allee effects in discrete-time host-parasitoid model, J. Difference Eq. Appl., 12 (2006), 165-181. doi: 10.1080/10236190500539238. [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. [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. [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. [17] W. Z. Lidicker, The Allee effect: Its history and future importance, Open Ecol. J., 3 (2010), 71-82. [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. [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. [20] R. Luis, S. N. Elaydi and H. Oliveira, Non-autonomous periodic systems with Allee effects, J. Difference Eq. Appl., 16 (2010), 1179-1196. doi: 10.1080/10236190902794951. [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. [22] H. Sedaghat, Folding, cycles and chaos in planar systems, J. Difference Eq. Appl., 21 (2015), 1-15. doi: 10.1080/10236198.2014.974585. [23] A. Yakubu, Multiple attractors in juvenile-adult single species models, J. Difference Eq. Appl., 9 (2007), 1083-1098. doi: 10.1080/1023619031000146887. [24] E. F. Zipkin, C. E. Kraft, E. 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.
$E_{0}$ with $\lambda =3$, $a = 0.7936$, $b = 0.0891$, $s^{\prime} = 1$
$E$ (shaded) and its complement for $\lambda =3$, $a = 0.7936$, $b = 0.0891$, $s^{\prime} = 1$
$E$ for $\lambda =2$, $a = 1.1$, $s^{\prime} = 1$ and two different values of $b$
A summary of results
 Conditions Outcomes and Comments References General $x_{0}, x_{1}<\rho$ Extinction for all possible parameter values if Thrm 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. survival Cor 15(b) is possible with some positive initial values! (31), (34) Extinction occurs from some initial values, survival Open problems from others; nontrivial basins (see Figures 1-3)
 Conditions Outcomes and Comments References General $x_{0}, x_{1}<\rho$ Extinction for all possible parameter values if Thrm 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. survival Cor 15(b) is possible with some positive initial values! (31), (34) Extinction occurs from some initial values, survival Open problems from others; nontrivial basins (see Figures 1-3)
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