Transient behavior of a population dynamical model

The transient behavior of an ecosystem with N random interacting species in the presence of a multiplicative noise is analyzed. The multiplicative noise mimics the interaction with the environment. We investigate different asymptotic dynamical regimes and the role of the external noise on the probability distribution of the local field. Population dynamics attracted a lot of attention in recent years and became the object of many studies as well by biologists as by physicists. 1), 2), 3), 4), 5) Tools developed in the context of nonequilibrium statistical physics to analyze nonequi-librium nonlinear physical systems provide new insights and at the same time new approaches to study biological systems. Biological population dynamics has many interesting, and still not solved, problems such as pattern formation, 6), 7), 8), 9) the role of the noise on complex ecosystem behaviour, and the noise-induced effects, such as stochastic resonance, noise delayed extinction, quasi periodic oscillations etc... The dynamical behavior of ecological systems of interacting species evolves towards the equilibrium states through the long, slow, and complex process of nonlinear relaxation, which is strongly dependent on the random interaction between the species, the initial conditions and the random interaction with environment. A good mathematical model to analyze the dynamics of N biological species with spatially homogeneous densities is the generalized Lotka-Volterra system with a Malthus-Verhulst modelization of the self regulation mechanism, in the presence of a multiplicative noise. 19), 20), 21) By neglecting the fluctuations of the local field we derive a quasi-stationary probability of the populations. We obtain the asymptotic analytical expressions for different nonlinear relaxation regimes, and we analyze the role of the multiplicative noise on the probability distribution of the local field. §2. The model and results The dynamical evolution of our ecosystem composed by N interacting species in a noisy environment and in the presence of an absorbing barrier is described by