Nonlinear Studies Nonlinear Studies Vol. 15, No. 2, pp. 163-176, 2008 c I&S Publishers Florida, USA 2008 Oscillatory phenomena of nutrient-plant-herbivore system with time lag: a mathematical approach Nurul Huda Gazi 1 Kalyan Das 2∗ 1 Department of Mathematics, B. P. Poddar Institute of Management and Tech- nology, Poddar Vihar: 137, VIP Road, Kolkata 700 052, India. E-mail: nursha@rediffmail.com 2 Department of Mathematics, B. P. Poddar Institute of Management and Tech- nology, Poddar Vihar: 137, VIP Road, Kolkata 700 052, India. E-mail : kalyandas70@rediffmail.com Abstract In the present study we consider a nutrient-based model of nutrient- plant-herbivore interaction with a delayed nutrient recycling and incorporate the number of herbivores attacking the plants follows Holling type III functional re- sponse. We have derived the conditions for asymptotic stability of the steady state and also estimated the length of the delay preserving the stability. The criterion for existence of Hopf-type small amplitude periodic oscillations of plant biomass and herbivore numbers are derived. Finally, all the analytical results are interpreted ecologically and compared with the computer simulation. 1 Introduction Food webs describe the transfer of energy between species in an ecosystem. While a food chain examines one, linear, energy pathway through an ecosystem, a food web is more complex and illustrates all of the potential pathways. Much of this transferred energy comes from the sun. Plants use the sun’s energy to convert inorganic compounds into energy-rich, organic compound, turning carbon dioxide and minerals into plant material by photosynthesis. Plants are called autotroph because they make their own energy; they are also called producers because they produce energy available for other organisms to eat. Heterotrophs are consumers that cannot make their own food. In order to obtain energy they eat plants or other heterotrophs. In above ground food webs, energy moves from producers (plants) to primary consumers (herbivores) and then to secondary consumers (predators). * Corresponding author 2000 Mathematics Subject Classification: 92D40; 34A34; Keywords: nutrient, plant, herbivore, recycling, stability, discrete time delay, Hopf-bifurcation. 163