Pair formation models with maturation period

The standard model for pair formation is generalized to include a maturation period. This model in the form of three coupled delay equations is a special case of the general age-structured model for a two-sex population. The exact conditions for the existence of an exponential (persistent) two-sex solution are derived. It is shown that this solution is unique and locally stable. In order to achieve these results the theory of homogeneous differential equations is extended to a class of homogeneous delay equations.     

Evolution of the pair rule gene network: Insights from a centipede

Comparative studies have examined the expression and function of homologues of the Drosophila melanogaster pair rule and segment polarity genes in a range of arthropods. The segment polarity gene homologues have a conserved role in the specification of the parasegment boundary, but the degree of conservation of the upstream patterning genes has proved more variable. Using genomic resources we identify a complete set of pair rule gene homologues from the centipede Strigamia maritima, and document a detailed time series of expression during trunk segmentation. We find supportive evidence for a conserved hierarchical organisation of the pair rule genes, with a division into early- and late-activated genes which parallels the functional division into primary and secondary pair rule genes described in insects. We confirm that the relative expression of sloppy-paired and paired with respect to wingless and engrailed at the parasegment boundary is conserved between myriapods and insects; suggesting that functional interactions between these genes might be an ancient feature of arthropod segment patterning. However, we find that the relative expression of a number of the primary pair rule genes is divergent between myriapods and insects. This corroborates suggestions that the evolution of upper tiers in the segmentation gene network is more flexible. Finally, we find that the expression of the Strigamia pair rule genes in periodic patterns is restricted to the ectoderm. This suggests that any direct role of these genes in segmentation is restricted to this germ layer, and that mesoderm segmentation is either dependent on the ectoderm, or occurs through an independent mechanism.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Models for contact-mediated pattern formation: cells that form parallel arrays

Kinetic continuum models are derived for cells that crawl over a 2D substrate, undergo random reorientation, and turn in response to contact with a neighbor. The integro-partial differential equations account for changes in the distribution of orientations in the population. It is found that behavior depends on parameters such as total mass, random motility, adherence, and sloughing rates, as well as on broad aspects of the contact response. Linear stability analysis, and numerical, and cellular automata simulations reveal that as parameters are varied, a bifurcation leads to loss of stability of a uniform (isotropic) steady state, in favor of an (anisotropic) patterned state in which cells are aligned in parallel arrays.     

Models for age structured populations with distributed maturation rates

In the use of age structured population models for agricultural applications such as the modeling of crop-pest interactions it is often essential that the model take into account the distribution in maturation rates present in some or all of the populations. The traditional method for incorporating distributed maturation rates into crop and pest models has been the so-called distributed delay method. In this paper we review the application of the distributed delay formalism to the McKendrick equation of an age structured population. We discuss the mathematical properties of the system of ordinary differential equations arising out of the distributed delay formalism. We then discuss an alternative method involving modification of the Leslie matrix.     

Sporophyte formation in experimentally-induced unisexual female and bisexual gametophytes ofLygodium japonicum

Unisexual female and male and bisexual gametophytes were experimentally induced inLygodium japonicum. A single bisexual gametophyte was isolated in a dish and a female gametophyte was paired with a male one to allow intragametophytic selfing and intergametophytic mating, respectively. About 30% of the females formed sporophytes but no bisexual gametophytes formed them.     

Optimal rates of bisexual reproduction in cyclical parthenogens with density-dependent growth

This work explores theoretical patterns of reproduction that maximize the production of resting eggs and the long-term fitness of genotypes in cyclical parthenogens. Our focus is on density-dependent reproduction as it influences the consequences of a trade-off between producing amictic daughters – which reproduce parthenogenetically and subitaneously – and producing mictic daughters – which undergo meiosis and bisexual reproduction. Amictic females increase competitive ability and allow the population to achieve a larger size; mictic females directly contribute to population survival through harsh periods by producing resting eggs. Although morphologically indistinguishable, the two types of females differ greatly in their ecological and reproductive roles. What factors underlie the differential allocation of resources to produce amictic and mictic females? Using a demographic model based on readily accessible parameters we demonstrate the existence of a frequency of mictic females that will maximize the population's long-term fitness. This frequency, termed the optimal mictic ratio, m_o, is 1 ? (q/b)^1/2, where q is the mortality rate and b is the maximum birth rate. Using computer simulation we compared the fitness of a population with this constant mictic ratio with populations having multiple switches from complete parthenogenetic growth to complete allocation in mixis (mictic ratio either 0 or 1). Two important conclusions for optimal mixis in density-dependent growth conditions are: (1) intermediate mictic ratios are optimal, and (2) optimal mictic ratios are higher when habitat conditions are better. Physiological cues responding to differences in birth and death rates are common so that it is possible that populations may adjust their relative rates of mictic and amictic female production in response to environmentally induced changes to the optimum mictic ratio. Our analysis demonstrates that different patterns of mixis are expected in different type of habitats. Since the optimal mictic ratio is sensitive to the effects of a variety of environmental challenges, our model makes possible a new means to evaluate life history evolution in cyclical parthenogens.     

Enhanced ionic interaction in narrow pores and ion pair formation

The same physical phenomenon that gives rise to the increase in the electrostatic self-energy of an ion within a narrow water-filled pore is shown to result in interionic electrical interactions within the pore that are much stronger and of longer range than those between the same ions in the same solution in bulk. Because of the much enhanced attraction between ions of opposite charge within the pore the formation of ion pairs becomes likely, even for strong electrolytes that are fully dissociated in the same solution when not spatially confined. Some predicted consequences of ion pair formation in narrow pores that may be experimentally detected are discussed. It is shown that, in a simple passive pore, due to ion pair formation, an Ussing unidirectional flux ratio exponent of less than 1 is predicted. This is usually thought to characterize a carrier rather than a pore.     

Exploring stable pattern formation in models of tussock moth populations

1. The western tussock moth ( Orgyia vetusta ) at the University of California Bodega Marine Reserve (Sonoma County, California, USA) exhibits dense, localized populations in the midst of extensive habitats where variation in host plant quality or predator abundance is unable to explain the restricted extent of the outbreaks.
2. Two primary features suggest that the host patterning is intrinsically generated: (i) female tussock moths are wingless, producing a low effective dispersal distance for the hosts; and (ii) the tussock moth population is attacked by several species of widely dispersing wasp and fly parasitoids.
3. We consider a set of spatially explicit host–parasitoid models to examine whether intrinsically generated patterns are possible within this system. These models include a spatially extended Nicholson–Bailey model to examine general features of pattern formation in host–parasitoid systems, and two system-specific models, an individual-based simulation and a population-level analytic model, to examine the details of this empirical system.
4. Both stable patterning and rapid extinction of the host population are initial-condition dependent outcomes of the general and specific models, implying that an intrinsically generated stable host pattern is feasible within the tussock moth system.
5. Stable patterning is enhanced by a large parasitoid-to-host dispersal ratio, local host resource limitation, and increased parasitism at the host patch's edge.     

A master equation for a spatial population model with pair interactions

We derive a closed master equation for an individual-based population model in continuous space and time. The model and master equation include Brownian motion, reproduction via binary fission, and an interaction-dependent death rate moderated by a competition kernel. Using simulations we compare this individual-based model with the simplest approximation, the spatial logistic equation. In the limit of strong diffusion the spatial logistic equation is a good approximation to the model. However, in the limit of weak diffusion the spatial logistic equation is inaccurate because of spontaneous clustering driven by reproduction. The weak-diffusion limit can be partially analyzed using an exact solution of the master equation applicable to a competition kernel with infinite range. This analysis shows that in the case of a top-hat kernel, reducing the diffusion can increase the total population. For a Gaussian kernel, reduced diffusion invariably reduces the total population. These theoretical results are confirmed by simulation.     

Evidence for Wg-independent tergite boundary formation in the millipede Glomeris marginata

The correlation between dorsal and ventral segmental units in diplopod myriapods is complex and disputed. Recent results with engrailed (en), hedgehog (hh), wingless (wg), and cubitus-interruptus (ci) have shown that the dorsal segments are patterned differently from the ventral segments. Ventrally, gene expression is compatible with the classical autoregulatory loop known from Drosophila to specify the parasegment boundary. In the dorsal segments, however, this Wg/Hh autoregulatory loop cannot be present because the observed gene expression patterns argue against the involvement of Wg signalling. In this paper, we present further evidence against an involvement of Wg signalling in dorsal segmentation and propose a hypothesis about how dorsal segmental boundaries may be controlled in a wg-independent way. We find that (1) the Notum gene, a modulator of the Wg gradient in Drosophila, is not expressed in the dorsal segments. (2) The H15/midline gene, a repressor of Wg action in Drosophila, is not expressed in the dorsal segments, except for future heart tissue. (3) The patched (ptc) gene, which encodes a Hh receptor, is strongly expressed in the dorsal segments, which is incompatible with Wg-Hh autoregulation. The available data suggest that anterior-posterior (AP) boundary formation in dorsal segments could instead rely on Dpp signalling rather than Wg signalling. We present a hypothesis that relies on Hh-mediated activation of Dpp signalling and optomotor-blind (omb) expression to establish the dorsal AP boundary (the future tergite boundary). The proposed mechanism is similar to the mechanism used to establish the AP boundary in Drosophila wings and ventral pleura.     

Invasion and adaptive evolution for individual-based spatially structured populations

The interplay between space and evolution is an important issue in population dynamics, that is particularly crucial in the emergence of polymorphism and spatial patterns. Recently, biological studies suggest that invasion and evolution are closely related. Here, we model the interplay between space and evolution starting with an individual-based approach and show the important role of parameter scalings on clustering and invasion. We consider a stochastic discrete model with birth, death, competition, mutation and spatial diffusion, where all the parameters may depend both on the position and on the phenotypic trait of individuals. The spatial motion is driven by a reflected diffusion in a bounded domain. The interaction is modelled as a trait competition between individuals within a given spatial interaction range. First, we give an algorithmic construction of the process. Next, we obtain large population approximations, as weak solutions of nonlinear reaction–diffusion equations. As the spatial interaction range is fixed, the nonlinearity is nonlocal. Then, we make the interaction range decrease to zero and prove the convergence to spatially localized nonlinear reaction–diffusion equations. Finally, a discussion of three concrete examples is proposed, based on simulations of the microscopic individual-based model. These examples illustrate the strong effects of the spatial interaction range on the emergence of spatial and phenotypic diversity (clustering and polymorphism) and on the interplay between invasion and evolution. The simulations focus on the qualitative differences between local and nonlocal interactions.      

Processes of pair formation leading to assortative mating in biological populations: dynamic interaction model

A model of pair formation that treats mating between two individuals as a dynamic process rather than an instant event is suggested. A number of cases corresponding to polygamous and monogamous matings of individuals are considered. The individual mating preferences are represented in the model by a set of "compatibility probabilities." It is shown that the mating pattern (the distribution of characters among mating pairs) established in a population as a result of the process of pair formation does not uniquely reflect the pattern of mating preferences. Quite different mating patterns may occur in populations with very similar mating preferences, as well as similar mating patterns may occur in populations with different individual mating preferences. The model demonstrates that not only mating preferences of individuals but also specific mechanisms of pair formation play an important role in determining the mating pattern of a population.     

Derivation of stochastic partial differential equations for size- and age-structured populations

Stochastic partial differential equations (SPDEs) for size-structured and age- and size-structured populations are derived from basic principles, i.e. from the changes that occur in a small time interval. Discrete stochastic models of size-structured and age-structured populations are constructed, carefully taking into account the inherent randomness in births, deaths, and size changes. As the time interval decreases, the discrete stochastic models lead to systems of Itô stochastic differential equations. As the size and age intervals decrease, SPDEs are derived for size-structured and age- and size-structured populations. Comparisons between numerical solutions of the SPDEs and independently formulated Monte Carlo calculations support the accuracy of the derivations.     

Self-organization of mobile populations in cyclic competition

The formation of out-of-equilibrium patterns is a characteristic feature of spatially extended, biodiverse, ecological systems. Intriguing examples are provided by cyclic competition of species, as metaphorically described by the ‘rock-paper-scissors’ game. Both experimentally and theoretically, such non-transitive interactions have been found to induce self-organization of static individuals into noisy, irregular clusters. However, a profound understanding and characterization of such patterns is still lacking. Here, we theoretically investigate the influence of individuals’ mobility on the spatial structures emerging in rock-paper-scissors games. We devise a quantitative approach to analyze the spatial patterns self-forming in the course of the stochastic time evolution. For a paradigmatic model originally introduced by May and Leonard, within an interacting particle approach, we demonstrate that the system's behavior—in the proper continuum limit—is aptly captured by a set of stochastic partial differential equations. The system's stochastic dynamics is shown to lead to the emergence of entangled rotating spiral waves. While the spirals’ wavelength and spreading velocity is demonstrated to be accurately predicted by a (deterministic) complex Ginzburg-Landau equation, their entanglement results from the inherent stochastic nature of the system. These findings and our methods have important applications for understanding the formation of noisy patterns, e.g. in ecological and evolutionary contexts, and are also of relevance for the kinetics of (bio)-chemical reactions.