%0 Book Section %1 statphys23_1071 %A Yoshino, Y. %A Galla, T. %A Tokita, K. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K competition dynamics eco-system functional generating interaction random replicator resource statphys23 topic-10 %T A model eco-system with resource competition %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1071 %X We study a simple model of eco-systems described by the replicator dynamics equationeq:def n_i(t)n_i(t)=f(n_i(t))+1N\sum_\mu=1^ N Q^\mu(t)\xi_i^\mu+\sum_j J_ij n_j(t)-\nu(t), equation where $n_i(t)$ dentes the abandance of species $i=1,,N$ at time $t$, $Q^\mu(t)$ the abundance of resource $\mu=1,,P=N$ and $\xi_i^\mu$ consumption of resource $\mu$ by species $i$. We assume the diversity $P$ of the resources is comparable to the number of species $N$ , i.e. $\alpha=O(1)$. Although it is similar to the model proposed in (De Martino and Marsili, 2006), but we newly consider effects by random inter-species interactions $\J_ ij\$ obeying a gaussian distribution with mean zero, variance $w/N $ and correlation $J_ijJ_ji\rangle_J=\Gamma w/N$. By the parameter $\Gamma$, the model represents a variety of ecological interactions, for example, antisymmetric interaction corresponds to prey-predator interactions in a food web. The first term $f(n_i(t))$ represents co-operation pressure or carrying capacities in a Lotka-Volterra system. The last term $\nu(t) $ is a Lagrange multiplier to keep normalisation. Time evolution of the resource $\mu$ is given by equation Q^\mu(t)=Q_0^\mu-\sum_j=1^N \xi_j^n_j(t) equation where $Q_0^\mu$ is steady resource abundance in absence of species, which obeys a Gaussian distribution with mean $P$ and variance $\sigma^2 P$. To study this system we apply the generating functional (GF) method based on path-integrals,which gives us a powerful tool to analyze systems with asymmetric disordered interactions. For such a system, there is no Lyapnov function and static approach like replica method are inapplicable. The GF allows one to formulate a closed theory for macroscopic order parameters of the system, such as the correlation and response functions, which are determined as self-consistent averages over an effective single- species stochastic process, subject to non-Markovian interaction and coloured noise. From the effective process, we can discuss the stability of this eco-system and the composition of surviving species as a function of the model parameters. We focused on the effects of variability of resources, direct interaction between species, co-operation pressure and dilution on the stability and the diversity of the ecosystem.

@incollection{statphys23_1071, abstract = {We study a simple model of eco-systems described by the replicator dynamics \begin{equation}\label{eq:def} \frac{\dot n_i(t)}{n_i(t)}=f(n_i(t))+\frac{1}{N}\sum_{\mu=1}^{\alpha N} Q^{\mu}(t)\xi_i^\mu+\sum_{j} J_{ij} n_j(t)-\nu(t), \end{equation} where $n_i(t)$ dentes the abandance of species $i=1,\dots ,N$ at time $t$, $Q^\mu(t)$ the abundance of resource $\mu=1,\dots ,P=\alpha N$ and $\xi_i^\mu$ consumption of resource $\mu$ by species $i$. We assume the diversity $P$ of the resources is comparable to the number of species $N$ , i.e. $\alpha={\cal O}(1)$. Although it is similar to the model proposed in (De Martino and Marsili, 2006), but we newly consider effects by random inter-species interactions $\{J_ {ij}\}$ obeying a gaussian distribution with mean zero, variance $w/N $ and correlation $\langle J_{ij}J_{ji}\rangle_{J}=\Gamma w/N$. By the parameter $\Gamma$, the model represents a variety of ecological interactions, for example, antisymmetric interaction corresponds to prey-predator interactions in a food web. The first term $f(n_i(t))$ represents co-operation pressure or carrying capacities in a Lotka-Volterra system. The last term $\nu(t) $ is a Lagrange multiplier to keep normalisation. Time evolution of the resource $\mu$ is given by \begin{equation} Q^\mu(t)=Q_0^\mu-\sum_{j=1}^N \xi_j^\mu n_j(t) \end{equation} where $Q_0^{\mu}$ is steady resource abundance in absence of species, which obeys a Gaussian distribution with mean $P$ and variance $\sigma^2 P$. To study this system we apply the generating functional (GF) method based on path-integrals,which gives us a powerful tool to analyze systems with asymmetric disordered interactions. For such a system, there is no Lyapnov function and static approach like replica method are inapplicable. The GF allows one to formulate a closed theory for macroscopic order parameters of the system, such as the correlation and response functions, which are determined as self-consistent averages over an effective single- species stochastic process, subject to non-Markovian interaction and coloured noise. From the effective process, we can discuss the stability of this eco-system and the composition of surviving species as a function of the model parameters. We focused on the effects of variability of resources, direct interaction between species, co-operation pressure and dilution on the stability and the diversity of the ecosystem.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Yoshino, Y. and Galla, T. and Tokita, K.}, biburl = {https://www.bibsonomy.org/bibtex/22afba85e34cdfe351b4e924192b43485/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {6eacb8151373be3a45390129978661aa}, intrahash = {2afba85e34cdfe351b4e924192b43485}, keywords = {competition dynamics eco-system functional generating interaction random replicator resource statphys23 topic-10}, month = {9-13 July}, timestamp = {2007-06-20T10:16:38.000+0200}, title = {A model eco-system with resource competition}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1071}, year = 2007 }