Mass extinction in a dynamical system of evolution with variable dimension

Temporal inactivation enhances robustness in an evolving system

We study the robustness of an evolving system that is driven by successive inclusions of new elements or constituents with m random interactions to older ones. Each constitutive element in the model stays either active or is temporarily inactivated depending upon the influence of the other active elements. If the time spent by an element in the inactivated state reaches T _W , it gets extinct. The phase diagram of this dynamic model as a function of m and T _W is investigated by numerical and analytical methods and as a result both growing (robust) as well as non-growing (volatile) phases are identified. It is also found that larger time limit T _W enhances the system's robustness against the inclusion of new elements, mainly due to the system's increased ability to reject 'falling-together' type attacks. Our results suggest that the ability of an element to survive in an unfavourable situation for a while, either as a minority or in a dormant state, could improve the robustness of the entire system.

Enhanced robustness of evolving open systems by the bidirectionality of interactions between elements

Living organisms, ecosystems, and social systems are examples of complex systems in which robustness against inclusion of new elements is an essential feature. A recently proposed simple model has revealed a general mechanism by which such systems can become robust against inclusion of elements with totally random interactions when the elements have a moderate number of links. The interaction is, however, in many systems often intrinsically bidirectional like for mutual symbiosis and competition in ecology. This study reports the strong reinforcement effect of the bidirectionality of the interactions on the robustness of evolving systems. We show that the system with purely bidirectional interactions can grow with twofold average degree, in comparison with the purely unidirectional system. This drastic shift of the transition point comes from the reinforcement of each node, not from a change in structure of the emergent system. For systems with partially bidirectional interactions we find that the regime of the growing phase gets expanded. In the dense interaction regime, there exists an optimum proportion of bidirectional interactions for the growth rate at around 1/3. In the sparsely connected systems, small but finite fraction of bidirectional links can change the system's behaviour from non-growing to growing.

A universal transition in the robustness of evolving open systems

Can the structure of a system that consists of many elements interacting with each other grow in complexity when new elements are added to it? This is an essential question for understanding various real, open, complex systems, such as living organisms, ecosystems, and social systems. Using a very simple model, this study demonstrates that such systems can grow only when the elements have a moderate number of interactions on average. This behaviour comes from a balance between two opposing effects: although an increase in the number of interactions makes each individual element more robust against disturbances, it also increases the net impact of the loss of any element on the system.

SPECIES-AREA RELATIONSHIP FOR POWER-LAW SPECIES ABUNDANCE DISTRIBUTION

We studied the mathematical relations between species abundance distributions (SADs) and species-area relationships (SARs) and found that a power-law SAR can be generally derived from a power-law SAD without a special assumption such as the "canonical hypothesis". In the present analysis, an SAR-exponent is obtained as a function of an SAD-exponent for a finite number of species. We also studied the inverse problem, from SARs to SADs, and found that a power-SAD can be derived from a power-SAR under the condition that the functional form of the corresponding SAD is invariant for changes in the number of species. We also discuss general relationships among lognormal SADs, the broken-stick model (exponential SADs), linear SARs and logarithmic SARs. These results suggest the existence of a common mechanism for SADs and SARs, which could prove a useful tool for theoretical and experimental studies on biodiversity and species coexistence.

Emergence of a complex and stable network in a model ecosystem with extinction and mutation

We propose a minimal model of the dynamics of diversity-replicator equations with extinction, invasion and mutation. We numerically study the behavior of this simple model and show that it displays completely different behavior from the conventional replicator equation and the generalized Lotka-Volterra equation. We reach several significant conclusions as follows: (1) a complex ecosystem can emerge when mutants with respect to species-specific interaction are introduced; (2) such an ecosystem possesses strong resistance to invasion; (3) a typical fixation process of mutants is realized through the rapid growth of a group of mutualistic mutants with higher fitness than majority species; (4) a hierarchical taxonomic structure (like family-genus-species) emerges; and (5) the relative abundance of species exhibits a typical pattern widely observed in nature. Several implications of these results are discussed in connection with the relationship of the present model to the generalized Lotka-Volterra equation.