On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay

Abstract

Abstract. In this survey, we briefly review some of our recent studies on predator-prey models with discrete delay. We first study the distribution of zeros of a second degree transcendental polynomial. Then we apply the general results on the distribution of zeros of the second degree transcendental polynomial to various predator-prey models with discrete delay, including Kolmogorov-type predator-prey models, generalized Gause-type predator-prey models with harvesting, etc. Bogdanov-Takens bifurcations in delayed predator-prey models with nonmonotone functional response and in delayed predator-prey model with predator harvesting are also introduced.

Long-term transients and complex dynamics of a stage-structured population with time delay and the Allee effect

“…Time delays are widely recognized as an inherent feature of population dynamics (Beretta and Kuang, 1998;Hansen et al, 1998;May, 2001;Murdoch et al, 2003;Ruan, 2006). Triggered by Hutchinson seminal work (Hutchinson, 1948), there have been a large number of theoretical studies concerned with the implications of time delays for populations and communities (Hastings, 1984;Aiello and Freedman, 1990;Murdoch et al, 2003;Ruan, 2009). Time delays are thought to have a destabilizing effect as a sufficiently long delay often turns an otherwise stable positive steady state unstable (Kuang, 1993;May, 2001, Ruan, 2006, and hence converts steady population densities into population cycles, although the generality of this scenario remains controversial, e.g.…”

Photosynthesis rate, salinity induced grazing rate and gestation delay as determinants of the dynamics of a realistic food chain model: a stochastic approach

“…It often represents gestation time (Jana 2014;Jana et al 2014), incubation period, transport delay or can simply lump complicated biological process together, accounting only for the time requires for these process to occur. Delay parameter emphasizes on Hopf-bifurcation (Ruan 2009;Jana 2014;Jana et al 2014Jana et al , 2015 and also it causes some complex dynamical behaviors, such as double Hopf bifurcation, quasi-periodic solutions and chaos (Lü 2002;Ma et al 2008;Matouk 2008;Ding and Jiang 2011;Wang et al 2014;Ding et al 2013;Song and Xu 2013). Along with different studies of environmental stochasticity and biological delays in prey-predator system, influence of noise on a delayed system is very significant for advance study of biological system.…”

“…On a fairly realistic ecological model with generalist top predator, Upadhyay et al (2007) investigate the influence of environmental noise and show the importance of the noise amplitude on the trophic level and the susceptibility of populations to environmental noise. Complex dynamical behavior of a biological system arises as a consequence of time delay (with significant time delay) which may exhibit limit cycle oscillations (Ruan 2009) and chaos (Mackey and Glass 1977;Jana 2014;Jana et al 2014). In reality, time delays occur in almost every ecological situation and assume to be one of the most prominent reason of regular fluctuations on population biomass (Jana 2014;Jana et al 2014Jana et al , 2015.…”

“…In reality, time delays occur in almost every ecological situation and assume to be one of the most prominent reason of regular fluctuations on population biomass (Jana 2014;Jana et al 2014Jana et al , 2015. Ruan (2009) presents a brief review on prey-predator models with descrete delay for both Kolmogorov and non-Kolmogorov type model system. It often represents gestation time (Jana 2014;Jana et al 2014), incubation period, transport delay or can simply lump complicated biological process together, accounting only for the time requires for these process to occur.…”

Temperature dependent growth rate of phytoplankton and salinity induced grazing rate of zooplankton as determinants of realistic multi-delayed food chain model

“…Consumer's (predator's) functional response which is defined as the amount of resource (prey) catch per consumer (predator) per unit of time is assumed to be one of the most important components of resource-consumer/prey-predator interaction because predation intensity may change the shape of the community structure and ecosystem properties (Sih et al 1985). A brief review on prey-predator models with discrete delay for both Kolmogorov and Non-Kolmogorov type model system is presented by Ruan (2009) where he shows different mathematical criterions for different biological assumptions due to delay. Models with delay are much more realistic.…”