# 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 & Continuous Dynamical Systems - S, 2020, 13 (8) : 2195-2209. doi: 10.3934/dcdss.2020185
##### References:

show all references

##### References:
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 $v = 0.5(1+Cos(0.03 t))$
 [1] J. M. Cushing, Simon Maccracken Stump. Darwinian dynamics of a juvenile-adult model. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1017-1044. doi: 10.3934/mbe.2013.10.1017 [2] Jinling Zhao, Wei Chen, Su Zhang. Immediate schedule adjustment and semidefinite relaxation. Journal of Industrial & Management Optimization, 2019, 15 (2) : 633-645. doi: 10.3934/jimo.2018062 [3] George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43 [4] Andrey V. Kremnev, Alexander S. Kuleshov. Nonlinear dynamics and stability of the skateboard. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 85-103. doi: 10.3934/dcdss.2010.3.85 [5] Marcelo M. Disconzi. On the existence of solutions and causality for relativistic viscous conformal fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1567-1599. doi: 10.3934/cpaa.2019075 [6] Zhijian Yang, Pengyan Ding, Xiaobin Liu. Attractors and their stability on Boussinesq type equations with gentle dissipation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 911-930. doi: 10.3934/cpaa.2019044 [7] Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Stability of the dynamics of an asymmetric neural network. Communications on Pure & Applied Analysis, 2009, 8 (2) : 655-671. doi: 10.3934/cpaa.2009.8.655 [8] Juan Calvo. On the hyperbolicity and causality of the relativistic Euler system under the kinetic equation of state. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1341-1347. doi: 10.3934/cpaa.2013.12.1341 [9] Takayuki Niimura. Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2561-2591. doi: 10.3934/dcds.2020141 [10] Gerhard Keller. Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 313-334. doi: 10.3934/dcdss.2017015 [11] Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875 [12] Philip Boyland, André de Carvalho, Toby Hall. Statistical stability for Barge-Martin attractors derived from tent maps. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2903-2915. doi: 10.3934/dcds.2020154 [13] Luci H. Fatori, Marcio A. Jorge Silva, Vando Narciso. Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6117-6132. doi: 10.3934/dcds.2016067 [14] Everaldo de Mello Bonotto, Daniela Paula Demuner. Stability and forward attractors for non-autonomous impulsive semidynamical systems. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1979-1996. doi: 10.3934/cpaa.2020087 [15] Baowei Feng. On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4729-4751. doi: 10.3934/dcds.2017203 [16] Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051 [17] Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909 [18] Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115 [19] Mirosław Lachowicz, Andrea Quartarone, Tatiana V. Ryabukha. Stability of solutions of kinetic equations corresponding to the replicator dynamics. Kinetic & Related Models, 2014, 7 (1) : 109-119. doi: 10.3934/krm.2014.7.109 [20] Saul Mendoza-Palacios, Onésimo Hernández-Lerma. Stability of the replicator dynamics for games in metric spaces. Journal of Dynamics & Games, 2017, 4 (4) : 319-333. doi: 10.3934/jdg.2017017

2020 Impact Factor: 2.425