Bulletin of Mathematical Biology

Liu, Meng; Wang, Ke; Wu, Qiong

doi: 10.1007/s11538-010-9569-5pmid: 20725797

Stochastic competitive models with pollution and without pollution are proposed and studied. For the first system with pollution, sufficient criteria for extinction, nonpersistence in the mean, weak persistence in the mean, strong persistence in the mean, and stochastic permanence are established. The threshold between weak persistence in the mean and extinction for each population is obtained. It is found that stochastic disturbance is favorable for the survival of one species and is unfavorable for the survival of the other species. For the second system with pollution, sufficient conditions for extinction and weak persistence are obtained. For the model without pollution, a partial stochastic competitive exclusion principle is derived.

Gorban, Alexander; Pokidysheva, Lyudmila; Smirnova, Elena; Tyukina, Tatiana

doi: 10.1007/s11538-010-9597-1pmid: 21088995

The “Law of the Minimum” states that growth is controlled by the scarcest resource (limiting factor). This concept was originally applied to plant or crop growth (Justus von Liebig, 1840, Salisbury, Plant physiology, 4th edn., Wadsworth, Belmont, 1992) and quantitatively supported by many experiments. Some generalizations based on more complicated “dose-response” curves were proposed. Violations of this law in natural and experimental ecosystems were also reported. We study models of adaptation in ensembles of similar organisms under load of environmental factors and prove that violation of Liebig’s law follows from adaptation effects. If the fitness of an organism in a fixed environment satisfies the Law of the Minimum then adaptation equalizes the pressure of essential factors and, therefore, acts against the Liebig’s law. This is the the Law of the Minimum paradox: if for a randomly chosen pair “organism–environment” the Law of the Minimum typically holds, then in a well-adapted system, we have to expect violations of this law.

Salinas, Dino; Reyes, Juan; Fuente, Milton

doi: 10.1007/s11538-010-9602-8pmid: 21108012

Biological membranes contain many domains enriched in phospholipid lipids and there is not yet clear explanation about how these domains can control the activity of phospholipid metabolizing enzymes. Here we used the surface dilution kinetic theory to derive general equations describing how complex substrate distributions affect the activity of enzymes following either the phospholipid binding kinetic model (which assumes that the enzyme molecules directly bind the phospholipid substrate molecules), or the surface-binding kinetic model (which assumes that the enzyme molecules bind to the membrane before binding the phospholipid substrate). Our results strongly suggest that, if the enzyme follows the phospholipid binding kinetic model, any substrate redistribution would increase the enzyme activity over than observed for a homogeneous distribution of substrate. Besides, enzymes following the surface-binding model would be independent of the substrate distribution. Given that the distribution of substrate in a population of micelles (each of them a lipid domain) should follow a Poisson law, we demonstrate that the general equations give an excellent fit to experimental data of lipases acting on micelles, providing reasonable values for kinetic parameters—without invoking special effects such as cooperative phenomena. Our theory will allow a better understanding of the cellular-metabolism control in membranes, as well as a more simple analysis of the mechanisms of membrane acting enzymes.

Hilbe, Christian

doi: 10.1007/s11538-010-9608-2pmid: 21181502

Classical replicator dynamics assumes that individuals play their games and adopt new strategies on a global level: Each player interacts with a representative sample of the population and if a strategy yields a payoff above the average, then it is expected to spread. In this article, we connect evolutionary models for infinite and finite populations: While the population itself is infinite, interactions and reproduction occurs in random groups of size N. Surprisingly, the resulting dynamics simplifies to the traditional replicator system with a slightly modified payoff matrix. The qualitative results, however, mirror the findings for finite populations, in which strategies are selected according to a probabilistic Moran process. In particular, we derive a one-third law that holds for any population size. In this way, we show that the deterministic replicator equation in an infinite population can be used to study the Moran process in a finite population and vice versa. We apply the results to three examples to shed light on the evolution of cooperation in the iterated prisoner’s dilemma, on risk aversion in coordination games and on the maintenance of dominated strategies.

Bennett, Max; Farnell, Les; Gibson, William

doi: 10.1007/s11538-010-9614-4pmid: 21174230

Synaptic spines grow as a consequence of the formation of F-actin filaments at the spine head. The dynamics of F-actin in the spine head upon excitation of N-methy-D-aspartate (NMDA) receptors has recently been investigated experimentally, but there is no quantitative account of how these dynamic changes occur upon activation of these receptors; this we now supply. Dynamics of F-actin at the apex of lamellipodia have been investigated in detail, giving rise to the treadmilling theory of F-actin dynamics, involving catalysis by profilin, for which quantitative models are now available. Here, we adapt such a model to describe the dynamics of F-actin in the synaptic-spine head and show that it gives quantitative descriptions of this treadmilling phenomena which are well fitted by Monte Carlo simulations. Next, the means by which excitation of NMDA receptors enhances the activity of profilin through activity of the Rho small GTPase RhoA and the specific kinase ROCK is discussed. This is then used to model the NMDA receptor excitatory enhancement of profilin and so the treadmilling process of F-actin dynamics in spine growth. Such modelling provides a quantitative description of the synaptic-spine dynamics of the filamentous to globular actin ratio that is observed experimentally.

Cartwright, Reed

doi: 10.1007/s11538-010-9619-zpmid: 21191662

The ability of humans to communicate via language is a complex, adapted phenotype, which undoubtedly has a recently evolved genetic component. However, the evolutionary dynamics of language-associated alleles are poorly understood. To improve our knowledge of such systems, a population-genetics model for language-associated genes is developed. (The model is general and applicable to social interactions other than communication.) When an allele arises that potentially improves the ability of individuals to communicate, it will experience positive frequency-dependent selection because its fitness will depend on how many other individuals communicate the same way. Consequently, new and rare alleles are selected against, posing a problem for the evolutionary origin of language. However, the model shows that if individuals form language-based cliques, then novel language-associated alleles can sweep through a population. Thus, the origin of language ability can be sufficiently explained by Darwinian processes operating on genetic diversity in a finite population of human ancestors.