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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:

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##### References:
$E_{0}$ 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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