In this chapter, different aspects have been described to investigate the ecological system with respect to several factors that may be responsible for emergence of complex behaviors. Some mathematical models incorporate time delay(s) such as delay due to maturation, gestation and other kinds of negative feedback. These nonlinear delay models have led to complexities in the system. The emphasis is to explore the complex dynamical behaviors including chaos in ecological models with respect to different control parameters. Motivation behind synchronization and stabilization of chaos in the system has also been emphasized. Hence, in this chapter attempts have been made to study order and chaos in variety of models applicable to multi-species ecological systems. Such a study is important since the ecological systems have all the necessary ingredients to be able to support chaos. The attempt is made to brief different mechanism that may bring order into chaotic systems or vice versa.
TopOrder To Chaos
Over the last two decades there has been a race between theoretical and empirical population biologists to look for complex dynamics and chaos in real systems (Aziz-Alaoui, 2002; Gakkhar et al., 2012; Hassell, Lawton, & May, 1976; Hastings & Powell, 1991; Klebano & Hastings, 1994; Ruxton, 1999). It is well accepted that ecosystems possess all the necessary features such as nonlinearity, multidimensionality etc. to display chaotic behavior. Many theoretical studies have demonstrated that ecological chaos, if and when exists, can affect important ecosystem features such as predictability, species persistence and bio-diversity (Allen, Schaffer, & Rosko, 1993; Aziz-Alaoui, 2002; Berryman & Millstein, 1989; Bhattacharya & Begum, 1996; Jorgensen & Bernardi, 1997; Jorgensen, 2002). Unpredictability is everywhere in ecological system. Although Hassell et al. (1976) found no evidence of chaotic behavior in any population, the notion that populations fluctuations in nature may be caused by deterministic chaos has persisted. It was discussed that the ecological systems have seeds of chaos (Hastings & Powell, 1991; Klebano & Hastings, 1994; Rai, 2004). Rai (2004) explained one of the reasons for non-occurrence of chaos lies in the organization of the ecological system. Chaos may be driven by human actions that increase growth rates or induce delays in the regulatory process. The chaotically fluctuating population is prone to extinction, with the consequence that group selection acts to eliminate species and chaos disappears.