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A new flexible discrete-time model for stable populations

This research has been supported by the Spanish Government and FEDER, under grant MTM2013-43404-P.
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  • We propose a new discrete dynamical system which provides a flexible model to fit population data. For different values of the three involved parameters, it can represent both globally persistent populations (compensatory or overcompensatory), and populations with Allee effects. In the most relevant cases of parameter values, there is a stable positive equilibrium, which is globally asymptotically stable in the persistent case. We study how population abundance depends on the parameters, and identify extinction windows between two saddle-node bifurcations.

    Mathematics Subject Classification: 39A10, 92D25.


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  • Figure 1.  Different graphs of the map $f$ defined in (1.1). (a): $f$ is unimodal for $\gamma<1$; (b): $f$ is increasing for $\gamma = 1$, with a unique positive fixed point if $\beta>1$; (c) and (d): $f$ is increasing for $1<\gamma<2$, and can have 0, 1, or $2$ positive fixed points; (e): $f$ is increasing and convex for $\gamma = 2$, with linear growth at infinity; it has a unique positive fixed point if $\beta>\delta$ and no positive fixed points if $\beta\leq\delta$; (f): $f$ is increasing and convex, with superlinear growth at infinity, if $\gamma>2$. In all cases, the red dashed line represents the graph of $y = x$

    Figure 2.  Graph of the map $\beta = F_{\delta}(\gamma)$ showing the survival/extinction switches for (1.1), which only occur if $\beta<1+\delta$

    Figure 3.  Relative position of the graphs of $f_1(x) = \beta x^{\gamma-1}$ (red color) and $f_2(x) = 1+\delta x$ (blue color) when equation $f_1(x) = f_2(x)$ has two positive solutions

    Figure 4.  Bifurcation diagrams for equation (1.1), using $\gamma$ as the bifurcation parameter. Red dashed lines correspond to unstable equilibria, which, in case of bistability, establish the boundary between the basins of attraction of the extinction equilibrium 0 and the nontrivial attractor $p$. (a): $\beta = 3, \delta = 1$; (b): $\beta = 2, \delta = 1$; (c): $\beta = 2, \delta = 1.5$; (b): $\beta = 2, \delta = 2.5$. Each case is an example of the corresponding case in Theorem 5.1

    Figure 5.  Bifurcation diagrams for equation (1.1), using $\gamma$ as the bifurcation parameter. Red dashed lines correspond to unstable equilibria. (a): $\beta = 0.9, \delta = 1.5$; (b): $\beta = 0.9, \delta = 0.5$

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