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Qualitative properties and bifurcations of a leaf-eating herbivores model

The paper was partially supported by the Characteristic innovation projects of colleges and universities in Guangdong Province (2019KTSCX088), the National Natural Science Foundation of China (11771197) and the Key Subject Program of Lingnan Normal University (1171518004)
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  • In this paper, we discuss the dynamics of a discrete-time leaf-eating herbivores model. First of all, to investigate the bifurcations of the model, we study the qualitative properties of a fixed point, including hyperbolic and non-hyperbolic. Secondly, applying the center manifold theorem, we give the conditions that the model produces a supercritical flip bifurcation and a subcritical flip bifurcation respectively, from which we find a generalized flip bifurcation point. And then, we prove rigorously that the model undergoes a generalized flip bifurcation and give three parameter regions that the model possesses two period-two cycles, one period-two cycles and none respectively. Next, computing the normal form, we prove that the model undergoes a subcritical Neimark-Sacker bifurcation and produces a unique unstable invariant circle near the fixed point. Finally, by numerical simulations, we not only verify our results but also show a saddle period-five cycle and a saddle period-six cycle on the invariant circle.

    Mathematics Subject Classification: Primary: 37G10, 39A28; Secondary: 58K50.


    \begin{equation} \\ \end{equation}
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  • Figure 1.  Diagram of bifurcation for system (2)

    Figure 2.  Bifurcation diagram of system (2) near the point $ GF $

    Figure 3.  Bifurcation diagram of system (31) for small $ |\beta| $

    Figure 4.  Flip bifurcation route to chaos for $ k = 0.3 $

    Figure 5.  An invariant circle $ \Gamma $ produced from the Neimark-Sacker bifurcation

    Figure 6.  A saddle period-five cycle on the invariant circle $ \Gamma $

    Figure 7.  A saddle period-six cycle on the invariant circle $ \Gamma $

    Table 1.  Topological types of fixed point $ E $ in the hyperbolic case

    Conditions $ E_1 $ Cases
    $ 0<k<1 $ $ 0<b< 4k $ stable focus $ \mathfrak{D}_5 $
    $ 4k\leq b<4/(2-k) $ stable node $ \mathfrak{D}_1 $
    $ b>4/(2-k) $ saddle point $ \mathfrak{D}_2 $-I
    $ k=1 $ $ b>4 $ saddle point $ \mathfrak{D}_2 $-II
    $ 1<k<2 $ $ 0<b<4k $ unstable focus $ \mathfrak{D}_4 $-I
    $ 4k\leq b<4/(2-k) $ unstable node $ \mathfrak{D}_3 $-I
    $ b>4/(2-k) $ saddle point $ \mathfrak{D}_2 $-III
    $ k\geq2 $ $ 0< b<4k $ unstable focus $ \mathfrak{D}_4 $-II
    $ b\geq 2k $ unstable node $ \mathfrak{D}_3 $-II
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