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

  • * Corresponding author: Nika Lazaryan

    * Corresponding author: Nika Lazaryan 
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  • 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.

    Mathematics Subject Classification: Primary: 39A10, 39A23, 39A33; Secondary: 92D25.


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  • 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)
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