Food chain chaos due to junction-fold point

Abstract

Consideration is given to a basic food chain model satisfying the trophic time diversification hypothesis which translates the model into a singularly perturbed system of three time scales. It is demonstrated that in some realistic system parameter region, the model has a unimodal or logistic-like Poincare return map when the singular parameter for the fastest variable is at the limiting value 0. It is also demonstrated that the unimodal map goes through a sequence of period-doubling bifurcations to chaos. The mechanism for the creation of the unimodal criticality is due to the existence of a junction-fold point [B. Deng, J. Math. Biol. 38, 21-78 (1999)]. The fact that junction-fold points are structurally stable and the limiting structures persist gives us a rigorous but dynamical explanation as to why basic food chain dynamics can be chaotic. (c) 2001 American Institute of Physics.

Recurrent and synchronous insect pest outbreaks in forests

“…Of course the analysis simplifies if each population evolves at radically different speeds. Although there are a number of contributions along this line (Muratori, 1991;Rinaldi and Muratori, 1992;Muratori and Rinaldi, 1992;De Feo and Rinaldi, 1998;Mehidi, 2001;Deng, 2001Deng, , 2004Brons and Kaasen, 2010) none of them fits with the present case. Moreover, the slow fast analysis can become quite sophisticated if the phenomenon known as "canard explosion" is involved (Deng, 2004;Brons and Kaasen, 2010).…”

Geometric singular perturbation theory in biological practice

“…To allow straightforward application of singular perturbation techniques, it is very important that ε and δ are two independent small parameters (0 < ε, δ 1). Examples of this type are for instance found in food chain models with a third class of so-called super or top-predators Deng 2001;Deng and Hines 2002) or in hormone secretion models (Kunpasuruang et al 2002). Models with three or more time-scales are also used to study neuronal behaviour, in particular to explain firing of neurons or so-called mixed mode oscillations.…”

“…4a. Independent of the choice of f and g, (3.2) has for ε = 0 a 1-parameter family of homoclinic orbits (Rosenzweig and MacArthur 1963) as proposed by Deng (2001). The food chain model is of the type (1.6) and comprises a logistic prey x, a Holling type II predator y and a Holling type II top-predator z (Holling 1959).…”

“…We do not intend to study this system thoroughly, but only to demonstrate the use of Fenichel's first two theorems in this example. For a much completer analysis, including existence proofs for various types of chaotic behaviour in both the 0 < ε, δ 1 regime and in a regime with larger δ, we refer to Deng (2001) and Deng andHines (2002, 2003). Muratori and Rinaldi (1992) essentially studied the same model, but they imposed the three time-scales by inserting small parameters in appropriate places, whereas Deng really rescaled the system so that it is clear what the assumption 0 < ε, δ 1 means for the original parameters in Rosenzweig and MacArthur (1963).…”

“…The orbit γ 2 keeps following this intermediate manifold until the fast flow becomes dominant again at y = y P . The Pontryagin value y P is calculated in Deng (2001).…”