Manuscript submitted to DCDS Supplement Volume 2007
Website: www.aimSciences.org pp. 1–10
PERIODIC ORBITS OF TRITROPHIC SLOW–FAST SYSTEM AND DOUBLE HOMOCLINIC BIFURCATIONS
Alexandre VIDAL Universit´ e Pierre et Marie Curie-Paris6, Paris, F-75005 France Laboratoire J.L. Lions, UMR 7598 CNRS 175 rue du Chevaleret 75013 Paris, France Abstract. The biological models - particularly the ecological ones - must be understood through the bifurcations they undergo as the parameters vary. However, the transition between two dynamical behaviours of a same system for diverse values of parameters may be sometimes quite involved. For instance, the analysis of the non generic motions near the transition states is the first step to understand fully the bifurcations occurring in complex dynamics. In this article, we address the question to describe and explain a double bursting behaviour occuring for a tritrophic slow–fast system. We focus therefore on the appearance of a double homoclinic bifurcation of the fast subsystem as the predator death rate parameter evolves. The first part of this article introduces the slow–fast system which extends Lotka– Volterra dynamics by adding a superpredator. The second part displays the analysis of singular points and bifurcations undergone by fast dynamics. The third part is devoted to the flow analysis near the homoclinic points. Finally, the fourth part is concerned with the main results about the existence of periodic orbits of different periods as the two homoclinic orbits are close enough to each other.
1. Introduction. Let us consider the following system, which is built from a natural plane dynamics: X P1 Y dX = X R 1− − dT K S1 + X dY P1 X P2 Z = Y E1 − D1 − (1) dT S1 + X S1 + Y dZ P2 Y = εZ E2 − D2 dT S2 + Y where X, Y and Z represent the membership of three populations. The variable X is the prey, Y its predator and Z a superpredator for Y . The threshold constant K and the intrinsic growth rate of the prey R characterize the logistic evolution of X (see [6]).
2000 Mathematics Subject Classification. Prim: 34C05, 34C25, 34C26, 34C37; Second: 92D25. Key words and phrases. Slow–Fast System, Tritrophic System, Homoclinic Bifurcation, Periodic Orbit.
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ALEXANDRE VIDAL
The predator–prey interactions are described by two Holling type II matching defined by the following positive parameters: Pj : the maximum predation rates Sj : the half-saturation constants Dj : the death rates Ej : the efficiencies of predation j = 1 concerning the predator Y , j = 2 concerning the superpredator Z. Finally, it is natural to consider that the evolution of the superpredator is slower than that of the preys. Then we introduce different time scales by means of the constant ε 0 so that the steady point: ! d1 a1 − d1 (b1 + 1) M = (xM , yM , 0) = ,0 (7) , a1 − b1 d1 (a1 − b1 d1 )2 lays in the phase space R3+ . 2.1. Critical Set Analysis. As z, considered as parameter, varies, the number of critical points of the fast dynamics (Fz ) evolves. To explain its dependency in terms of z, it is more convenient to consider the critical set C defined in (6). One obtains easily that C = ∆ ∪ L where: ∆
= {(1, 0, z)|z ∈ R+ } (x, yL (x), zL (x)) ∈ R3+ |x ∈ [0, 1]
(8)
L =
(9)
and: 1 (1 − x) (1 + b1 x) a1 (a1 x − d1 (b1 + 1)) (a1 + b2 (1 − x)(1 + b1 x)) zL (x) = a1 a2 (1 + b1 x)
yL (x) =
(10) (11)
Note that ∆ always intersect L at the point T = (1, 0, zT ) where: zT =
(a1 − d1 (b1 + 1)) a1 d1 = − >0 a2 (1 + b1 ) a2 (1 + b1 ) a2
(12)
Moreover, we assume that d1 is small enough such that: 0 ∃xP > 0, zL (xP ) = 0
(13)
Consequently, L is ∩-shaped in R3+ and we denote: L - for each z ∈ ]0, zP [, Rz ± the unique point (x, yL (x), z) ∈ L such as x < xP , - for each z ∈ ]zT , zP [, RzLS the unique point (x, yL (x), z) ∈ L such as x > xP . ∆ Furthermore, we name Rz∆S the point (1, 0, z) ∈ ∆ for each 0 < z < zT and Rz − for each z > zT . Then, let us pose (see figure 1): S S ∆ ∆− = ∆ ∩ {z > zT } = {Rz − }, ∆S = ∆ ∩ {z < zT } = {Rz∆S } z>zT 0
