Pattern formation and coexistence domains for a nonlocal population dynamics

Patterning of nonlocal transport models in biology: The impact of spatial dimension

Throughout developmental biology and ecology, transport can be driven by nonlocal interactions. Examples include cells that migrate based on contact with pseudopodia extended from other cells, and animals that move based on their awareness of other animals. Nonlocal integro-PDE models have been used to investigate contact attraction and repulsion in cell populations in 1D. In this paper, we generalise the analysis of pattern formation in such a model from 1D to higher spatial dimensions. Numerical simulations in 2D demonstrate complex behaviour in the model, including spatio-temporal patterns, multi-stability, and patterns with wavelength and shape that differ significantly depending on whether interactions are attractive or repulsive. Through linear stability analysis in N dimensions, we demonstrate how, unlike in local Turing reaction-diffusion models, the capacity for pattern formation fundamentally changes with dimensionality for this nonlocal model. Most notably, pattern formation is possible only in higher than one spatial dimension for both the single species system with repulsive interactions, and the two species system with 'run-and-chase' interactions. The latter case may be relevant to zebrafish stripe formation, which has been shown to be driven by run-and-chase dynamics between melanophore and xanthophore pigment cells.

Enhanced species coexistence in Lotka-Volterra competition models due to nonlocal interactions

We introduce and analyze a spatial Lotka-Volterra competition model with local and nonlocal interactions. We study two alternative classes of nonlocal competition that differ in how each species' characteristics determine the range of the nonlocal interactions. In both cases, nonlocal interactions can create spatial patterns of population densities in which highly populated clumps alternate with unpopulated regions. These non-populated regions provide spatial niches for a weaker competitor to establish in the community and persist in conditions in which local models predict competitive exclusion. Moreover, depending on the balance between local and nonlocal competition intensity, the clumps of the weaker competitor vary from M-like structures with higher densities of individuals accumulating at the edges of each clump to triangular structures with most individuals occupying their centers. These results suggest that long-range competition, through the creation of spatial patterns in population densities, might be a key driving force behind the rich diversity of species observed in natural ecological communities.

Nonlinear self-organized population dynamics induced by external selective nonlocal processes

Self-organization evolution of a population is studied considering generalized reaction-diffusion equations. We proposed a model based on non-local operators that has several of the equations traditionally used in research on population dynamics as particular cases. Then, employing a relatively simple functional form of the non-local kernel, we determined the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we established a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our proposal reproduces some of the experimental results that other approaches considered previously had not been able to obtain.

Nonlinear self-organized population dynamics induced by external selective nonlocal processes

Self-organization evolution of a population is studied considering generalized reaction-diffusion equations. We proposed a model based on non-local operators that has several of the equations traditionally used in research on population dynamics as particular cases. Then, employing a relatively simple functional form of the non-local kernel, we determined the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we established a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our proposal reproduces some of the experimental results that other approaches considered previously had not been able to obtain.

Single-species fragmentation: The role of density-dependent feedback

Internal feedback is commonly present in biological populations and can play a crucial role in the emergence of collective behavior. To describe the temporal evolution of the distribution of a single-species population, we consider a generalization of the Fisher-KPP equation. This equation includes the elementary processes of random motion, reproduction, and, importantly, nonlocal interspecific competition, which introduces a spatial scale of interaction. In addition, we take into account feedback mechanisms in diffusion and growth processes, mimicked by power-law density dependencies. This feedback includes, for instance, anomalous diffusion, reaction to overcrowding or to the rarefaction of the population, as well as Allee-like effects. We show that, depending on the kind of feedback that takes place, the population can self-organize splitting into disconnected subpopulations, in the absence of external constraints. Through extensive numerical simulations, we investigate the temporal evolution and the characteristics of the stationary population distribution in the one-dimensional case. We discuss the crucial role that density-dependence has on pattern formation, particularly on fragmentation, which can bring important consequences to processes such as epidemic spread and speciation.

Pattern Formation through Temporal Fractional Derivatives

It is well known that temporal first-derivative reaction-diffusion systems can produce various fascinating Turing patterns. However, it has been found that many physical, chemical and biological systems are well described by temporal fractional-derivative reaction-diffusion equations. Naturally arises an issue whether and how spatial patterns form for such a kind of systems. To address this issue clearly, we consider a classical prey-predator diffusive model with the Holling II functional response, where temporal fractional derivatives are introduced according to the memory character of prey’s and predator’s behaviors. In this paper, we show that this fractional-derivative system can form steadily spatial patterns even though its first-derivative counterpart can’t exhibit any steady pattern. This result implies that the temporal fractional derivatives can induce spatial patterns, which enriches the current mechanisms of pattern formation.

Phase separation driven by density-dependent movement: A novel mechanism for ecological patterns

Many ecosystems develop strikingly regular spatial patterns because of small-scale interactions between organisms, a process generally referred to as spatial self-organization. Self-organized spatial patterns are important determinants of the functioning of ecosystems, promoting the growth and survival of the involved organisms, and affecting the capacity of the organisms to cope with changing environmental conditions. The predominant explanation for self-organized pattern formation is spatial heterogeneity in establishment, growth and mortality, resulting from the self-organization processes. A number of recent studies, however, have revealed that movement of organisms can be an important driving process creating extensive spatial patterning in many ecosystems. Here, we review studies that detail movement-based pattern formation in contrasting ecological settings. Our review highlights that a common principle, where movement of organisms is density-dependent, explains observed spatial regular patterns in all of these studies. This principle, well known to physics as the Cahn-Hilliard principle of phase separation, has so-far remained unrecognized as a general mechanism for self-organized complexity in ecology. Using the examples presented in this paper, we explain how this movement principle can be discerned in ecological settings, and clarify how to test this mechanism experimentally. Our study highlights that animal movement, both in isolation and in unison with other processes, is an important mechanism for regular pattern formation in ecosystems.

Effect of environment fluctuations on pattern formation of single species

System-environment interactions are intrinsically nonlinear and dependent on the interplay between many degrees of freedom. The complexity may be even more pronounced when one aims to describe biologically motivated systems. In that case, it is useful to resort to simplified models relying on effective stochastic equations. A natural consideration is to assume that there is a noisy contribution from the environment, such that the parameters that characterize it are not constant but instead fluctuate around their characteristic values. From this perspective, we propose a stochastic generalization of the nonlocal Fisher-KPP equation where, as a first step, environmental fluctuations are Gaussian white noises, both in space and time. We apply analytical and numerical techniques to study how noise affects stability and pattern formation in this context. Particularly, we investigate noise-induced coherence by means of the complementary information provided by the dispersion relation and the structure function.

Extinction, coexistence, and localized patterns of a bacterial population with contact-dependent inhibition

BACKGROUND

Contact-dependent inhibition (CDI) has been recently revealed as an intriguing but ubiquitous mechanism for bacterial competition in which a species injects toxins into its competitors through direct physical contact for growth suppression. Although the molecular and genetic aspects of CDI systems are being increasingly explored, a quantitative and systematic picture of how CDI systems benefit population competition and hence alter corresponding competition outcomes is not well elucidated.

RESULTS

By constructing a mathematical model for a population consisting of CDI+ and CDI- species, we have systematically investigated the dynamics and possible outcomes of population competition. In the well-mixed case, we found that the two species are mutually exclusive: Competition always results in extinction for one of the two species, with the winner determined by the tradeoff between the competitive benefit of the CDI+ species and its growth disadvantage from increased metabolic burden. Initial conditions in certain circumstances can also alter the outcome of competition. In the spatial case, in addition to exclusive extinction, coexistence and localized patterns may emerge from population competition. For spatial coexistence, population diffusion is also important in influencing the outcome. Using a set of illustrative examples, we further showed that our results hold true when the competition of the population is extended from one to two dimensional space.

CONCLUSIONS

We have revealed that the competition of a population with CDI can produce diverse patterns, including extinction, coexistence, and localized aggregation. The emergence, relative abundance, and characteristic features of these patterns are collectively determined by the competitive benefit of CDI and its growth disadvantage for a given rate of population diffusion. Thus, this study provides a systematic and statistical view of CDI-based bacterial population competition, expanding the spectrum of our knowledge about CDI systems and possibly facilitating new experimental tests for a deeper understanding of bacterial interactions.

Nonlinear diffusion effects on biological population spatial patterns

Motivated by the observation that anomalous diffusion is a realistic feature in the dynamics of biological populations, we investigate its implications in a paradigmatic model for the evolution of a single species density u(x,t). The standard model includes growth and competition in a logistic expression, and spreading is modeled through normal diffusion. Moreover, the competition term is nonlocal, which has been shown to give rise to spatial patterns. We generalize the diffusion term through the nonlinear form ∂tu(x,t)=D∂xxu(x,t)ν (with D,ν>0), encompassing the cases where the state-dependent diffusion coefficient either increases (ν>1) or decreases (ν<1) with the density, yielding subdiffusion or superdiffusion, respectively. By means of numerical simulations and analytical considerations, we display how that nonlinearity alters the phase diagram. The type of diffusion imposes critical values of the model parameters for the onset of patterns and strongly influences their shape, inducing fragmentation in the subdiffusive case. The detection of the main persistent mode allows analytical prediction of the critical thresholds.