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

• 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).

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

 Citation: • • Figure 1.  Comparison of the present and classical predation functions

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

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

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

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

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

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

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

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

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

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