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October  2019, 12(6): 1775-1789. doi: 10.3934/dcdss.2019117

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

Evariste Sanchez-Palencia 1, and  Jean-Pierre Françoise 2,

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.

Keywords: Darwinian dynamics, persistence of species, systems without equilibrium.
Mathematics Subject Classification: Primary: 34A34, 34C15, 34C37; Secondary: 92B05.
Citation: Evariste Sanchez-Palencia, Jean-Pierre Françoise. Topological remarks and new examples of persistence of diversity in biological dynamics. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1775-1789. doi: 10.3934/dcdss.2019117
References:
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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.   Google Scholar

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E. Sanchez-Palencia and J.-P. Françoise, Structural stability and emergence of biodiversity, Acta Biotheoretica, 61 (2013), 397-412.   Google Scholar

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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.  Google Scholar

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S. J. Schreiber, Criteria for Cr robust permanence, Jour Diff Equations, 162 (2000), 400-426.  doi: 10.1006/jdeq.1999.3719.  Google Scholar

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Hal. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, vol 111, Amer. Math. Soc., 2011.  Google Scholar

show all references

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. Google Scholar

[2]

J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, London Math. Soc. Student Texts, 7, Cambridge University Press, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[6]

C. Lobry, Modèles Déterministes en Dynamique des Populations, Ecole CIMPA Saint Louis du Sénégal, 2001. Google Scholar

[7]

K. S. McCann, The diversity - stability debate, Nature, 405 (2000), 228-230.   Google Scholar

[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.  Google Scholar

[9]

J. Milnor, Topology from the Differential Viewpoint, The University Press of Virginia, Charlottesville, 1965. Google Scholar

[10] V. A. Pliss, Nonlocal Problems in the Theory of Oscillations, Academic Press, 1966.   Google Scholar
[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.   Google Scholar

[12]

E. Sanchez-Palencia and J.-P. Françoise, Structural stability and emergence of biodiversity, Acta Biotheoretica, 61 (2013), 397-412.   Google Scholar

[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.  Google Scholar

[14]

S. J. Schreiber, Criteria for Cr robust permanence, Jour Diff Equations, 162 (2000), 400-426.  doi: 10.1006/jdeq.1999.3719.  Google Scholar

[15]

Hal. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, vol 111, Amer. Math. Soc., 2011.  Google Scholar

Figure 1.  Plot of orbits on the coordinate planes and of the heteroclinic orbit of system (1)
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Figure 2.  Plot of the attractor of system (1)
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Figure 3.  Plot of a solution of system (1) on the attractor (i.e. longtime after the initial instant
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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
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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
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Fig 11 after a diffeomorphism">Figure 12.  The same orbit of Fig 11 after a diffeomorphism
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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)
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Figure 5.  Plot of a solution of system (6) with the parameters (7)
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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
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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 $z_1$
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Figure 8.  Plot of the limit cycle of system (10) (see text for the values of the parameters)
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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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