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Topological remarks and new examples of persistence of diversity in biological dynamics

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  • There are several definitions of persistence of species, which amount to define interactions between them ensuring the survival of all the species initially present in the system. The aim of this paper is to present a wide family of examples in dimension $n>2$ (very natural in biological dynamics) exhibiting convergence towards a cycle when starting from anywhere with the exception of a zero-measure set of "forbidden" initial positions. The forbidden set is a heteroclinic orbit linking two equilibria on the boundary of the domain. Moreover, such systems have no equilibrium point interior to the domain (which is necessary for classical persistence for topological reasons). Such systems do not enjoy persistence in a strict sense, whereas in practice they do. The forbidden initial set does not matter in practice, but it modifies drastically the topological properties.

    Mathematics Subject Classification: Primary: 34A34, 34C15, 34C37; Secondary: 92B05.


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  • Figure 1.  Plot of orbits on the coordinate planes and of the heteroclinic orbit of system (1)

    Figure 2.  Plot of the attractor of system (1)

    Figure 3.  Plot of a solution of system (1) on the attractor (i.e. longtime after the initial instant

    Figure 10.  Artist view of of an orbit approaching a limit cycle turning around in the case when the period of the "turning around" is smaller than the period of the limit cycle

    Figure 11.  Artist view of of an orbit approaching a limit cycle turning around in the case when the period of the "turning around" is larger than the period of the limit cycle

    Figure 12.  The same orbit of Fig 11 after a diffeomorphism

    Figure 4.  Plot of $z_{2}(t)$ of a solution of system (1) starting with small $z_{2}(0)$ showing a double periodicity (the small period is the attractor, whereas the long period one is the transient, which vanishes slowly)

    Figure 5.  Plot of a solution of system (6) with the parameters (7)

    Figure 6.  Plot of the solution of system (6) with the parameters (6) starting from the point $(1.5,1,0.7,1.5)$: four-dimensional cycle

    Figure 7.  Plot of the solution of system (6) with the parameters (6) starting from the point $(1.5,0.6,0.6,0.8)$: there is a stable equilibrium with extinction of $x_2$ and $z_1$

    Figure 8.  Plot of the limit cycle of system (10) (see text for the values of the parameters)

    Figure 9.  Plot of the periodic solution of system (10) (see text for the values of the parameters) with the parameters)

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