Paradoxes of vulnerability to predation in biological dynamics and mediate versus immediate causality

Evariste Sanchez-Palencia; Philippe Lherminier

doi:10.3934/dcdss.2020185

Article Contents

2020, Volume 13, Issue 8: 2195-2209. Doi: 10.3934/dcdss.2020185

This issue Previous Article What mathematical models can or cannot do in glioma description and understanding Next Article Global-in-time Gevrey regularity solutions for the functionalized Cahn-Hilliard equation

Paradoxes of vulnerability to predation in biological dynamics and mediate versus immediate causality

Sorbonne Université, CNRS, Institut Jean Le Rond d'Alembert, IJLRD, 75005 Paris, France, Le Fontenil, Saint-Sulpice-sur-Rille, F-61300, France

Received: December 31, 2018

Early access: November 2019

Published: May 2020

Abstract Full Text(HTML) Figure(10) Related Papers Cited by

Abstract

The causality scheme of an (essentially non symmetric) predator-prey system involves automaticaly advantages and disadvantages highly dependent on time. We study systems with one predator and one or two preys furnishing issues which involve mediate and inmediate causality (naturally associated with the attractor and the previous transient). The issues are highly dependent on the parameter accounting for the vulnerability of the preys. When the vulnerability is small, an increase of it implies a (demographic) disadvantage for the preys, but, when it is large (involving periodic cycles) an increase turnes out in an advantage because of the rarefaction of predators (this is associated with average populations on the periodic cycles). When two preys with different vulnerability are present, the most vulnerable may desappear (i. e. the attractor does not contain such prey). This phenomenon only occurs when the less vulnerable prey is nevertheless able to support the predator; otherwise, this one keeps eating anyway the other preys. The mechanism of such patterns are better described in terms of attractors and stability than in terms of advantages versus disadvantages (which are drastically dependent on the viewpoints of the three species).

Keywords:

Mathematics Subject Classification: 92B05, 34A34, 34C15.

Citation:

Full Text(HTML)

Figure 1. Comparison of the present and classical predation functions

Download: Full-size image PowerPoint slide

Figure 2. Typical pattern of predation (cycle case). System (1), values of the parameters $ a = 1 $, $ p = 4 $, $ c = 0.7616 $, $ v = 0.85 $

Download: Full-size image PowerPoint slide

Figure 3. Typical pattern of predation (stable foyer case). System (1), values of the parameters $ a = 1 $, $ p = 4 $, $ c = 0.7616 $, $ v = 0.6 $

Download: Full-size image PowerPoint slide

Figure 4. Typical pattern of predation (stable node case). System (1), values of the parameters $ a = 1 $, $ p = 4 $, $ c = 0.7616 $, $ v = 0.35 $

Download: Full-size image PowerPoint slide

Figure 5. Evolution of the pattern as a function of the vulnerability $ v $

Download: Full-size image PowerPoint slide

Figure 6. Plot of $ x(t) $, $ y(t) $, $ z(t) $ for vulnerability $ v = 1.2 $

Download: Full-size image PowerPoint slide

Figure 7. Plot of $ x(t) $, $ y(t) $, $ z(t) $ for vulnerability $ v = 0.7 $

Download: Full-size image PowerPoint slide

Figure 8. Plot of $ x(t) $, $ y(t) $, $ z(t) $ for vulnerability $ v = 0.2 $

Download: Full-size image PowerPoint slide

Figure 9. The attractors for vulnerability $ v = 1.2 $, $ v = 0.7 $ and$ v = 0.2 $

Download: Full-size image PowerPoint slide

Figure 10. Plot of $ z(t) $ for vulnerability $ v = 0.5(1+Cos(0.03 t)) $

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics, 40, Springer, New York, 2012. doi: 10.1007/978-1-4614-1686-9.
[2]	J-P. Françoise, Oscillations en biologie, Analyse Qualitative et Modèles, Mathématiques & Applications, 46, Springer-Verlag, Berlin, 2005. doi: 10.1007/3-540-37670-4.
[3]	A. Klebanoff and A. Hastings, Chaos in three-species food chains, J. Math. Biol., 32 (1994), 427-451. doi: 10.1007/BF00160167.
[4]	P. Lherminier and E. Sanchez-Palencia, Remarks and examples on transient processes and attractors in biological evolution, Proceedings of the 2014 Madrid Conference on Applied Mathematics, Electron. J. Differ. Equ. Conf., 22, Texas State Univ., San Marcos, TX, 2015, 63 = -77.
[5]	C. Lobry, Modèles déterministes en Dynamique des Populations, Ecole CIMPA Saint Louis du Sénégal, 2001.
[6]	K. S. McCann, The diversity - stability debate, Nature, 405 (2000), 228-233. doi: 10.1038/35012234.
[7]	J. D. Murray, Mathematical Biology. Ⅰ: An Introduction, Interdisciplinary Applied Mathematics, 17, Springer-Verlag, New York, 2002. doi: 10.1007/b98868.
[8]	K. Sigmund, Kolmogorov and population dynamics, in Kolmogorov's Heritage in Mathemetics, Springer, Berlin, 2007,177–186. doi: 10.1007/978-3-540-36351-4_9.
[9]	S. Smale, On the differential equations of species in competition, J. Math. Biol., 3 (1976), 5-7. doi: 10.1007/BF00307854.
[10]	A. Vidal, Periodic orbits of tritrophic slow-fast system and double homoclinic bifurcation, Proceedings of the 6th AIMS International Conference, 2007, 1021–1030.