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

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)