# American Institute of Mathematical Sciences

August  2020, 13(8): 2195-2209. doi: 10.3934/dcdss.2020185

## 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  January 2019 Published  August 2020 Early access  November 2019

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).

Citation: Evariste Sanchez-Palencia, Philippe Lherminier. Paradoxes of vulnerability to predation in biological dynamics and mediate versus immediate causality. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2195-2209. doi: 10.3934/dcdss.2020185
##### References:
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##### 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.
Comparison of the present and classical predation functions
Typical pattern of predation (cycle case). System (1), values of the parameters $a = 1$, $p = 4$, $c = 0.7616$, $v = 0.85$
Typical pattern of predation (stable foyer case). System (1), values of the parameters $a = 1$, $p = 4$, $c = 0.7616$, $v = 0.6$
Typical pattern of predation (stable node case). System (1), values of the parameters $a = 1$, $p = 4$, $c = 0.7616$, $v = 0.35$
Evolution of the pattern as a function of the vulnerability $v$
Plot of $x(t)$, $y(t)$, $z(t)$ for vulnerability $v = 1.2$
Plot of $x(t)$, $y(t)$, $z(t)$ for vulnerability $v = 0.7$
Plot of $x(t)$, $y(t)$, $z(t)$ for vulnerability $v = 0.2$
The attractors for vulnerability $v = 1.2$, $v = 0.7$ and$v = 0.2$
Plot of $z(t)$ for vulnerability $v = 0.5(1+Cos(0.03 t))$
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