Eur. Phys. J. B 66, 97–106 (2008) DOI: 10.1140/epjb/e2008-00373-8 T HE EUROPEAN P HYSICAL JOURNAL B Dissipative oscillations in spatially restricted ecosystems due to long range migration N. Kouvaris 1, 2 and A. Provata 1, a 1 Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, 15310 Athens, Greece 2 Department of Mathematical, Physical and Computational Science, Faculty of Engineering, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece Received 21 April 2008 / Received in final form 1st August 2008 Published online 27 September 2008 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2008 Abstract. An ecosystem containing three interacting species is studied using both Mean Field approach and Kinetic Monte Carlo simulations on a lattice substrate. The so called 3rd order LLV model involves birth, death and reaction processes with 3rd order nonlinearities and feedbacks. At the mean field level this system exhibits conservative oscillations; the analytic form of the constant of motion is presented. The stochastic simulations show that the density oscillations disappear for sufficiently large lattices, while they are present locally, on small lattice windows. Introduction of mixing via long range migration in the two reacting species changes this picture. For small migration rates p, the behavior remains as with p =0 and the system is divided into local asynchronous oscillators. As p increases the system passes through a phase transition and exhibits a weak disorder limit cycle through a supercritical Hopf-like bifurcation. The amplitude of the limit cycle depends on the rate p, on the range of migration r and on the system kinetic rates k1, k2 and k3. PACS. 82.40.Bj Oscillations, chaos, and bifurcations – 05.45.Xt Synchronization; coupled oscillators – 92.20.jp Ecosysystems, structure, dynamics and modeling – 02.70.Uu Applications of Monte Carlo methods – 05.65.+b Self-organized systems – 05.45.-a Nonlinear dynamics and chaos 1 Introduction Ecosystems research started with the pioneer works of Lotka in 1920’s [1] and Volterra in 1930’s [2]. It was ob- served by Volterra that the population of two species of fish in the Adriatic Sea varies with some specific period. He proposed a model to explain these oscillations: ˙ N =(a - bP ) N (1a) ˙ P =(dN - c) P (1b) where a, b, c, d are constants characterizing the popula- tions of prey N and predators P . This set of equations is known as the Lotka-Volterra model (LV) since Lotka arrived at similar equations from a theoretical chemical reaction which could exhibit periodic oscillations in the chemical concentrations [1,3]. The LV model is the simplest model for describing the competition between interacting species and shows that a nonlinear predator-prey system can result in oscillatory behavior of the populations. This result is not unexpected. Equations (1) produce conservative oscillations which at- tempt to describe the following natural process: the preda- tors survive by consuming the prey. Due to this process a e-mail: aprovata@limnos.chem.demokritos.gr the predators grow and their numbers increase. As time grows, the population of prey decreases to such a level that some of the predators can not survive as they are starving. So their numbers begin to decline. With the de- cline in the number of predators, the population of prey begins to increase again. However with more prey becom- ing available, the population of predators begins to grow and the cycle repeats. The LV model has some obvious drawbacks. First, using Ordinary Differential Equations (ODEs) the sys- tem’s variables (such as populations density or birth and death rates) are treated as continuous [3]. Second, the populations are assumed to be homogeneously mixed. Third, the closed, conservative, periodic trajectories in the phase space (predator-prey density space) are some- what unrealistic; in nature such a strict requirement will be difficult to occur. The LV equations are struc- turally unstable, which means that a small parameter change can cause drastic change in the global stabil- ity. To overcome this problem more realistic predator- prey models were developed which are structurally sta- ble and exhibit limit cycles. Nevertheless, even in this case the models are space independent and determinis- tic and one can not study local interactions and fluctua- tions. It is known that complexity in ecosystems is gov- erned by local interactions and intrinsic characteristics like