In real world or real complex networks this may be necessary as Internet network has become ubiquitous and system segment, at peak hours could forward heterogeneous data packets with variable QoS (quality of service) requirements such as voice data, multimedia data etc 22. This model not only emphasises predator survival dependency on prey population size via the functional response, but also offers more possibilities in adjusting parameters during low network traffic. If we observe the underlying relationship in this particular situation at a leaky bucket for instance, where all packets are mixed before being moved to the output link, it becomes clear that a ratio-dependant type model like Rosenzweig-MacArthur predator-prey system is a logical choice. While a classical competitive model focuses mainly on the outcome of competition, this can be limiting for tuning or adjusting parameters. In our approach, instead of modelling the underlying relationship between network users using a classical predator-prey competitive system as in the previous published article 21, we have chosen a Rosenzweig-MacArthur type model for its accuracy in capturing density dependency phenomenon and sensitivity to small perturbations. Particularly, one can find several modified Rosenzweig-MacArthur models studied in the related literature 3, 6, 14, 15, 16, 17, 18, 19, 20. Many authors have studied the dynamic of classical Rosenzweig-MacArthur predator-prey model and there are numerous published articles on this subject. Analyzing network packets forwarding to depict the performance of a particular node or segment is important in understanding users’ behaviour impact on the overall performance of the network during peak hours for informing decisions made locally at certain given segments 10, 11, 12, 13. If we consider network users’ behaviour to be stochastic and the accommodating segment to have limited buffering space then, in rush hours, when users interact intensively, forwarding generated data packets can be assimilated to a predator-prey type interaction with limited resources characteristics. This can be applied to network users that share bandwidth and resources at a bottleneck node or a leaky bucket set up to monitor traffic flows for example. In nature, it is admitted that most of interactions occur in delayed or discrete fashion, as both predator and prey act stochastically in consuming available resources. In our approach, we separate predator searching time and handling time by introducing a delay parameter in the differential equations. In most of ecosystems, predator searching and handling efficiency is strongly dependent on prey density or resources availability. Holling Type II supposes maximum mortality of predator at low prey density 5, 6, 7, 8, 9. Predator growth is proportional to its prey population size or density. Adding Holling Type II terms in the modified version allows better control of populations’ density and handling time of the predator, which is important to control its growth whereas the classical model assumes searches are random and that predator search rate and handling time are constant. It is mostly used to study bifurcation and chaotic behaviour in predator-prey interactions. Rosenzweig-MacArthur predator-prey model is one of such model that presents the advantage of being simple and yet exhibits very rich dynamics. Examining bifurcation, especially the supercritical ones, is very common in population dynamics, as one can determine a set of periodic solutions that may lead to system stabilization or to chaos 1, 2, 3, 4. For such systems, it is necessary to analyze stability of some given nonhyperbolic trajectories around equilibrium points and determine whether these systems exhibit rich dynamic or not. In applied engineering and complex system sciences, mathematical models that display deterministic chaotic dynamical behaviour are of interest.
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