Model ecosystem with variable interspecies interactions

Model ecosystems with quenched, symmetric interspecies interactions have been extensively studied using the replica method of the statistical mechanics of disordered systems. Here, we consider a more general scenario in which both the species abundances and the interspecies interactions change with time, albeit in widely separated timescales. The equilibrium of the coupled dynamics is studied analytically within the partial annealing framework, in which the number of replicas n takes on positive as well as negative values. In the case n > 0, which describes ecosystems characterized by the cooperative interspecies interactions, we find a discontinuous transition to a regime of zero diversity, whereas in the case where competition prevails, n < 0, we find that the species diversity is maximum.

[1]  E. H. Simpson.  Measurement of Diversity , 1949, Nature.

[2]  D. Thouless,et al.  Stability of the Sherrington-Kirkpatrick solution of a spin glass model , 1978 .

[3]  J. Fontanari,et al.  Complementarity and diversity in a soluble model ecosystem. , 2002, Physical review letters.

[4]  Sherrington,et al.  Coupled dynamics of fast spins and slow interactions: An alternative perspective on replicas. , 1993, Physical review. B, Condensed matter.

[5]  R. W. Penney,et al.  Coupled dynamics of fast spins and slow interactions in neural networks and spin systems , 1993 .

[6]  Tobias Galla,et al.  Dynamics of random replicators with Hebbian interactions , 2005, cond-mat/0507473.

[7]  J. Fontanari,et al.  Controlling species richness in spin-glass model ecosystems. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  R. Palmer,et al.  The replica method and solvable spin glass model , 1979 .

[9]  J. Fontanari,et al.  The random replicator model at nonzero temperature , 2005 .

[10]  Opper,et al.  Replicators with random interactions: A solvable model. , 1989, Physical review. A, General physics.

[11]  N. Hairston.  Species Abundance and Community Organization , 1959 .

[12]  R. Fisher,et al.  The Relation Between the Number of Species and the Number of Individuals in a Random Sample of an Animal Population , 1943 .

[13]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[14]  G Szabó,et al.  Phase transition in a spatial Lotka-Volterra model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Tobias Galla,et al.  Random replicators with asymmetric couplings , 2005, cond-mat/0508174.

[16]  G. Szabó,et al.  Competing associations in six-species predator–prey models , 2004, q-bio/0408005.

[17]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[18]  Fontanari,et al.  Random replicators with high-order interactions , 2000, Physical review letters.

[19]  P. Parisi.  Replica symmetry breaking in the random replicant model , 1995, cond-mat/9501137.

[20]  J. Fontanari,et al.  Extinctions in the random replicator model. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Robert M. May,et al.  Lifting the veil on abundance patterns , 1991, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[22]  M. Mézard,et al.  Partial annealing and overfrustration in disordered systems , 1994 .

[23]  Slow interaction dynamics in the spherical spin-glass model , 2003 .

[24]  Daniel J. Amit,et al.  Modeling brain function: the world of attractor neural networks, 1st Edition , 1989 .