# American Institute of Mathematical Sciences

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

 1 Sorbonne Université, UPMC Univ Paris 06, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005, Paris, France 2 Sorbonne Université, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-louis Lions, F-75005, Paris, France

Received  November 2017 Revised  January 2018 Published  November 2018

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.

Citation: Evariste Sanchez-Palencia, Jean-Pierre Françoise. Topological remarks and new examples of persistence of diversity in biological dynamics. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2019117
##### References:
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##### References:
 [1] R. Arditi and J. Michalski, Nonlinear food web models and their response to increased basal productivity, in food webs; integration of patterns and dynamics, G.A. Polis and K.O. Winemiller eds. Chapman and Hall, New York, (1996), 122-133. [2] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, London Math. Soc. Student Texts, 7, Cambridge University Press, 1988. [3] V. Hutson and K. Schmitt, Permanence and the dynamics of biological systems, Mathematical Biosciences, 111 (1992), 1-71. doi: 10.1016/0025-5564(92)90078-B. [4] G. Kirlinger, Permanence of some ecological systems with several predators and one prey species, Jour Mathematical Biol, 26 (1988), 217-232. doi: 10.1007/BF00277734. [5] Ph. Lherminier and E. Sanchez-Palencia, Remarks and examples on transient processes and attractors in biological evolution, Elec. Jour. Diff. Equat. Conference, 22 (2015), 63-77. [6] C. Lobry, Modèles Déterministes en Dynamique des Populations, Ecole CIMPA Saint Louis du Sénégal, 2001. [7] K. S. McCann, The diversity - stability debate, Nature, 405 (2000), 228-230. [8] R. McGehee and R. A. Armstrong, Some mathematical problems concerning the ecological principle of competetive exclusion, Jour Diff Equations, 23 (1977), 30-52. doi: 10.1016/0022-0396(77)90135-8. [9] J. Milnor, Topology from the Differential Viewpoint, The University Press of Virginia, Charlottesville, 1965. [10] V. A. Pliss, Nonlocal Problems in the Theory of Oscillations, Academic Press, 1966. [11] A. Rapaport, D. Dochain and J. Harmand, Practical coexistence in the chemostat with arbitrarily close growth functions, Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées, 9 (2008), 231-243. [12] E. Sanchez-Palencia and J.-P. Françoise, Structural stability and emergence of biodiversity, Acta Biotheoretica, 61 (2013), 397-412. [13] E. Sanchez-Palencia and J.-P. Françoise, Constrained evolution processes and emergence of organized diversity, Math Meth Applied Sci., 39 (2016), 104-133. doi: 10.1002/mma.3463. [14] S. J. Schreiber, Criteria for Cr robust permanence, Jour Diff Equations, 162 (2000), 400-426. doi: 10.1006/jdeq.1999.3719. [15] Hal. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, vol 111, Amer. Math. Soc., 2011.
Plot of orbits on the coordinate planes and of the heteroclinic orbit of system (1)
Plot of the attractor of system (1)
Plot of a solution of system (1) on the attractor (i.e. longtime after the initial instant
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
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
The same orbit of Fig 11 after a diffeomorphism
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)
Plot of a solution of system (6) with the parameters (7)
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
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 $z_1$
Plot of the limit cycle of system (10) (see text for the values of the parameters)
Plot of the periodic solution of system (10) (see text for the values of the parameters) with the parameters)
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