Ecological Modelling 259 (2013) 10–15 Contents lists available at SciVerse ScienceDirect Ecological Modelling jo u r n al hom ep age : www.elsevier.com/locate/ecolmodel Study of chaotic behavior in predator–prey interactions in a chemostat Emad Ali, Mohammed Asif, AbdelHamid Ajbar ∗ Department of Chemical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia a r t i c l e i n f o Article history: Received 3 November 2012 Received in revised form 26 February 2013 Accepted 27 February 2013 Keywords: Predator Prey Chaos Monod Variable yield coefficient Bifurcation a b s t r a c t This paper investigates the complex dynamics resulting from interactions between one predator and one prey in a chemostat. A standard model is extended by allowing the yield coefficient associated with the prey to vary linearly with the substrate concentration. When this dependence is negligible, the proposed model is reduced to the classical constant yield model which was shown in the literature to produce periodic behavior for a wide range of parameters. In this paper we analyze the proposed model and we show that while the static behavior is relatively simple, the dynamics are complex and involve limit cycles and period doubling sequences leading to chaos. Numerical simulations are also presented to analyze the model equations and to determine the effect of its parameters on the resulting dynamics. The proposed model could serve as a basis to re-examine the importance of variable yield coefficients in predicting complex behavior in predator–prey interactions in the chemostat. © 2013 Elsevier B.V. All rights reserved. 1. Introduction Predator–prey interactions are quite common in many natural ecosystems as well as in waste treatment bioreactors. Protozoa, for instance, is known to play a valuable role in activated sludge reac- tors by preying on unflocculated bacteria and thereby clarifying the reactor effluent (Bailey and Ollis, 1986). The study of predation is therefore important for understanding the dynamics of ecosys- tems and is also important for the optimization of the biological treatment of wastes. The study of interactions between predators and preys has long been one of the main themes in ecological sys- tems. The development and analysis of predator–prey models dates back to the foundational work of Volterra (1926). His interest in this subject was prompted by the work of the zoologist Umberto D’Ancona on the proportion of cartilaginous fish caught over the years 1905–1923 (Kot, 2001). Volterra’s model is now known as the Lotka–Volterra model in recognition that an identical model had been published slightly earlier by Lotka (1920); Lotk’s motiva- tion was the possibility of oscillations in chemical systems. Since these early papers there has been a large number of investigations into extensions of these models. A key feature of these models is the assumption that the environment in which the predator and prey live is static, i.e. the food source for the prey is assumed to ∗ Corresponding author. Tel.: +966 1 467 6850; fax: +966 1 467 8770. E-mail addresses: amkamal@ksu.edu.sa (E. Ali), masif@ksu.edu.sa (M. Asif), aajbar@ksu.edu.sa (A. Ajbar). be unlimited. These classical ecological models focused on two- species systems with one predator and one prey. These models consist generally of two differential equations along with a func- tional response (i.e. specific growth rate) representing the prey consumption per unit time. Extensive studies were carried out in the literature on such continuous systems for various types of functional responses (Ruan and Xia, 2001; Zhu et al., 2002; Ajbar and Alhumaizi, 2011). These studies recognized that since two- variables models, under time invariant conditions, can exhibit only two main types of behavior (static points or limit cycles), these models can therefore describe only a small number of phenomena that are commonly observed in real life. An alternative approach to modeling predator–prey interactions was developed in the early 1970s, based on the use of the chemo- stat. The environment in which the predator and prey live is no longer static as the amount of food available for the prey changes in response to changes in the density of the prey. An important feature of predator–prey experiments carried out in a chemostat is the relative ease with which it is possible to obtain large amounts of reproducible experimental data. In turn the existence of good data sets mean that mathematical models can be developed and validated. This situation is in complete contrast to that of the major- ity of models based upon the Lotka–Volterra approach for which little experimental data exist. Moreover the number and type of microbial species in the chemostat can be well controlled, and the system can be isolated from other interactions that may occur between the populations. The chemostat is also important in eco- logical studies because the chemostat model is the threshold for 0304-3800/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ecolmodel.2013.02.029