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Permanence and universal classification of discrete-time competitive systems via the carrying simplex

  • * Corresponding author: Lei Niu

    * Corresponding author: Lei Niu 

This work is supported by the National Natural Science Foundation of China (NSFC) under Grant No. 11371252 and Grant No. 11771295, Shanghai Gaofeng Project for University Academic Program Development, and the Academy of Finland

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  • We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of $ 33 $ stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes $ 1-25 $ and $ 33 $; there is always a heteroclinic cycle in class $ 27 $; Neimark-Sacker bifurcations may occur in classes $ 26-31 $ but cannot occur in class $ 32 $. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes $ 29, 31, 33 $ and those in class $ 27 $ with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.

    Mathematics Subject Classification: Primary: 37B25, 37Cxx, 37N25; Secondary: 92D25.


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  • Figure 1.  A carrying simplex $ \Sigma $ with a repelling heteroclinic cycle $ \partial\Sigma $

    Figure 2.  The phase portrait on $ \Sigma $ replaced by $ \Delta^1 $. A closed dot $ \bullet $ denotes a fixed point which attracts on $ \Sigma $, and an open dot $ \circ $ denotes the one which repels on $ \Sigma $. Each $ \Sigma $ stands for an equivalence class. Class $ 1 $ corresponds to Proposition 4.7 (a) and (b); class $ 2 $ corresponds to Proposition 4.7 (c); class $ 3 $ corresponds to Proposition 4.7 (d)

    Figure 3.  The phase portrait on $ \Sigma $ for class $ 33 $. Every orbit in the interior of $ \Sigma $ converges to $ p $. The fixed point notation is as in Table 1

    Figure 4.  The phase portrait on $ \Sigma $ for class $ 29 $. The fixed point notation is as in Table 1

    Figure 5.  The orbit emanating from $ x_0 = (1, 0.0667, 0.0667) $ for the map $ T\in\mathrm{CLG}(3) $ with the parameter matrix $ U $ given in Example 5.1 and $ r_1 = 1, r_2 = 0.2, r_3 = 1 $ leads away from $ \partial \Sigma $ and tends to an attracting invariant closed curve, and the orbit emanating from $ x_0 = (0.2151, 0.746, 0.0173) $ also tends to an attracting invariant closed curve

    Figure 6.  The orbit emanating from $ x_0 = (1, 0.0667, 0.0667) $ for the map $ T\in\mathrm{CGAA}(3) $ with the parameter matrix $ U $ given in Example 5.1 and $ r_1 = r_2 = r_3 = 1 $, $ c_1 = \frac{1}{10}, c_2 = \frac{1}{5}, c_3 = \frac{1}{5} $ leads away from $ \partial \Sigma $ and tends to an attracting invariant closed curve, and the orbit emanating from $ x_0 = (0.7, 0.1642, 0.1685) $ also tends to an attracting invariant closed curve

    Figure 7.  The orbit emanating from $ x_0 = (0.04, 0.12, 0.36) $ for the map $ T\in\mathrm{CGAA}(3) $ with the parameter matrix $ U $ given in Example 5.3 and $ r_1 = r_2 = r_3 = 1 $, $ c_1 = 0.1, c_2 = 0.79, c_3 = 0.1 $ tends to an attracting invariant closed curve, while the orbit emanating from $ x_0 = (0.0002, 0.023, 0.486) $ approaches the heteroclinic cycle $ \partial \Sigma $

    Figure 8.  The orbit emanating from $ x_0 = (0.427, 0.8574, 0.014) $ for the map $ T\in\mathrm{MFC}(3) $ with the parameter matrix $ U $ given in Example 5.4, $ c = \frac{4}{5} $ and $ r_1 = r_3 = 1, r_2 = 0.03 $ tends to an attracting invariant closed curve

    Figure 9.  The orbit emanating from $ x_0 = (0.5962, 0.4857, 0.193) $ for the map $ T\in\mathrm{MFC}(3) $ with the parameter matrix $ U $ given in Example 5.5, $ c = \frac{4}{5} $ and $ r_1 = r_3 = 1, r_2 = 0.02 $ tends to an attracting invariant closed curve

    Figure 10.  The orbit emanating from $ x_0 = (0.3128, 0.8347, 0.0199) $ for the map $ T\in\mathrm{CRC}(3) $ with the parameter matrix $ U $ given in Example 5.4 and $ r_1 = \frac{1}{11}, r_2 = 0.01, r_3 = \frac{2}{7} $ tends to an attracting invariant closed curve

    Table 1.  The $33$ equivalence classes in $\mathrm{DCS}(3, f)$, where $\gamma_{ij} = \mu_{ii}-\mu_{ji}$, $\beta_{ij} = \frac{\mu_{jj}-\mu_{ij}}{\mu_{ii}\mu_{jj}-\mu_{ij}\mu_{ji}}$ ($\beta_{ij}$ is well defined; see Remark 4.6), $i, j = 1, 2, 3$ and $i\neq j$, and each $\Sigma$ is given by a representative map of that class. A fixed point is represented by a closed dot $\bullet$ if it attracts on $\Sigma$, by an open dot $\circ$ if it repels on $\Sigma$, and by the intersection of its stable and unstable manifolds if it is a saddle on $\Sigma$. For classes $1-25$ and $33$, every orbit converges to some fixed point; for classes $26-31$, Neimark-Sacker bifurcations might occur; for class $27$, $\partial \Sigma$ is a heteroclinic cycle; for class $32$, the unique positive fixed point is a repeller and Neimark-Sacker bifurcation cannot occur in this class

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