Spectral control for ecological stability

A system made up of N interacting species is considered. Self-reaction terms are assumed of the logistic type. Pairwise interactions take place among species according to different modalities, thus yielding a complex asymmetric disordered graph. A mathematical procedure is introduced and tested to stabilise the ecosystem via an {\it ad hoc} rewiring of the underlying couplings. The method implements minimal modifications to the spectrum of the Jacobian matrix which sets the stability of the fixed point and traces these changes back to species-species interactions. Resilience of the equilibrium state appear to be favoured by predator-prey interactions

Spatiotemporal dynamics in a spatial plankton system

In this paper, we investigate the complex dynamics of a spatial plankton-fish system with Holling type III functional responses. We have carried out the analytical study for both one and two dimensional system in details and found out a condition for diffusive instability of a locally stable equilibrium. Furthermore, we present a theoretical analysis of processes of pattern formation that involves organism distribution and their interaction of spatially distributed population with local diffusion. The results of numerical simulations reveal that, on increasing the value of the fish predation rates, the sequences spots $\rightarrow$ spot-stripe mixtures$\rightarrow$ stripes$\rightarrow$ hole-stripe mixtures holes$\rightarrow$ wave pattern is observed. Our study shows that the spatially extended model system has not only more complex dynamic patterns in the space, but also has spiral waves.Comment: Published Pape

Modelling Food Webs

We review theoretical approaches to the understanding of food webs. After an overview of the available food web data, we discuss three different classes of models. The first class comprise static models, which assign links between species according to some simple rule. The second class are dynamical models, which include the population dynamics of several interacting species. We focus on the question of the stability of such webs. The third class are species assembly models and evolutionary models, which build webs starting from a few species by adding new species through a process of "invasion" (assembly models) or "speciation" (evolutionary models). Evolutionary models are found to be capable of building large stable webs.Comment: 34 pages, 2 figures. To be published in "Handbook of graphs and networks" S. Bornholdt and H. G. Schuster (eds) (Wiley-VCH, Berlin

Complex dynamics in coevolution models with ratio-dependent functional response

We explore the complex dynamical behavior of two simple predator-prey models of biological coevolution that on the ecological level account for interspecific and intraspecific competition, as well as adaptive foraging behavior. The underlying individual-based population dynamics are based on a ratio-dependent functional response [W.M. Getz, J. Theor. Biol. 108, 623 (1984)]. Analytical results for fixed-point population sizes in some simple communities are derived and discussed. In long kinetic Monte Carlo simulations we find quite robust, approximate 1/f noise in species diversity and population sizes, as well as power-law distributions for the lifetimes of individual species and the durations of periods of relative evolutionary stasis. Adaptive foraging enhances coexistence of species and produces a metastable low-diversity phase and a stable high-diversity phase.Comment: 19 page

Chaos to Permanence - Through Control Theory

Work by Cushing et al. [18] and Kot et al. [60] demonstrate that chaotic behavior does occur in biological systems. We demonstrate that chaotic behavior can enable the survival/thriving of the species involved in a system. We adopt the concepts of persistence/permanence as measures of survival/thriving of the species [35]. We utilize present chaotic behavior and a control algorithm based on [66, 72] to push a non-permanent system into permanence. The algorithm uses the chaotic orbits present in the system to obtain the desired state. We apply the algorithm to a Lotka-Volterra type two-prey, one-predator model from [30], a ratio-dependent one-prey, two-predator model from [35] and a simple prey-specialist predator-generalist predator (for ex: plant-insect pest-spider) interaction model [67] and demonstrate its effectiveness in taking advantage of chaotic behavior to achieve a desirable state for all species involved