117 research outputs found
Selected topics on reaction-diffusion-advection models from spatial ecology
We discuss the effects of movement and spatial heterogeneity on population
dynamics via reaction-diffusion-advection models, focusing on the persistence,
competition, and evolution of organisms in spatially heterogeneous
environments. Topics include Lokta-Volterra competition models, river models,
evolution of biased movement, phytoplankton growth, and spatial spread of
epidemic disease. Open problems and conjectures are presented
Patchiness and Demographic Noise in Three Ecological Examples
Understanding the causes and effects of spatial aggregation is one of the
most fundamental problems in ecology. Aggregation is an emergent phenomenon
arising from the interactions between the individuals of the population, able
to sense only -at most- local densities of their cohorts. Thus, taking into
account the individual-level interactions and fluctuations is essential to
reach a correct description of the population. Classic deterministic equations
are suitable to describe some aspects of the population, but leave out features
related to the stochasticity inherent to the discreteness of the individuals.
Stochastic equations for the population do account for these
fluctuation-generated effects by means of demographic noise terms but, owing to
their complexity, they can be difficult (or, at times, impossible) to deal
with. Even when they can be written in a simple form, they are still difficult
to numerically integrate due to the presence of the "square-root" intrinsic
noise. In this paper, we discuss a simple way to add the effect of demographic
stochasticity to three classic, deterministic ecological examples where
aggregation plays an important role. We study the resulting equations using a
recently-introduced integration scheme especially devised to integrate
numerically stochastic equations with demographic noise. Aimed at scrutinizing
the ability of these stochastic examples to show aggregation, we find that the
three systems not only show patchy configurations, but also undergo a phase
transition belonging to the directed percolation universality class.Comment: 20 pages, 5 figures. To appear in J. Stat. Phy
Mathematical models for cell migration: A non-local perspective
We provide a review of recent advancements in non-local continuous models for migration, mainly from the perspective of its involvement in embryonal development and cancer invasion. Particular emphasis is placed on spatial non-locality occurring in advection terms, used to characterize a cell's motility bias according to its interactions with other cellular and acellular components in its vicinity (e.g. cell-cell and cell-tissue adhesions, non-local chemotaxis), but we also briefly address spatially non-local source terms. Following a short introduction and description of applications, we give a systematic classification of available PDE models with respect to the type of featured non-localities and review some of the mathematical challenges arising from such models, with a focus on analytical aspects. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'
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Differential Equations arising from Organising Principles in Biology
This workshop brought together experts in modeling and analysis of organising principles of multiscale biological systems such as cell assemblies, tissues and populations. We focused on questions arising in systems biology and medicine which are related to emergence, function and control of spatial and inter-individual heterogeneity in population dynamics. There were three main areas represented of differential equation models in mathematical biology. The first area involved the mathematical description of structured populations. The second area concerned invasion, pattern formation and collective dynamics. The third area treated the evolution and adaptation of populations, following the Darwinian paradigm. These problems led to differential equations, which frequently are non-trivial extensions of classical problems. The examples included but were not limited to transport-type equations with nonlocal boundary conditions, mixed ODE-reaction-diffusion models, nonlocal diffusion and cross-diffusion problems or kinetic equations
Analytical solution of the nitracline with the evolution of subsurface chlorophyll maximum in stratified water columns
In a stratified water column, the nitracline is a layer where the nitrate concentration increases below the nutrient-depleted upper layer, exhibiting a strong vertical gradient in the euphotic zone. The subsurface chlorophyll maximum layer (SCML) forms near the bottom of the euphotic zone, acting as a trap to diminish the upward nutrient supply. Depth and steepness of the nitracline are important measurable parameters related to the vertical transport of nitrate into the euphotic zone. The correlation between the SCML and the nitracline has been widely reported in the literature, but the analytic solution for the relationship between them is not well established. By incorporating a piecewise function for the approximate Gaussian vertical profile of chlorophyll, we derive analytical solutions of a specified nutrient-phytoplankton model. The model is well suited to explain basic dependencies between a nitracline and an SCML. The analytical solution shows that the nitracline depth is deeper than the depth of the SCML, shoaling with an increase in the light attenuation coefficient and with a decrease in surface light intensity. The inverse proportional relationship between the light level at the nitracline depth and the maximum rate of new primary production is derived. Analytic solutions also show that a thinner SCML corresponds to a steeper nitracline. The nitracline steepness is positively related to the light attenuation coefficient but independent of surface light intensity. The derived equations of the nitracline in relation to the SCML provide further insight into the important role of the nitracline in marine pelagic ecosystems
Spatial patterns of competing random walkers
We review recent results obtained from simple individual-based models of
biological competition in which birth and death rates of an organism depend on
the presence of other competing organisms close to it. In addition the
individuals perform random walks of different types (Gaussian diffusion and
L\'{e}vy flights). We focus on how competition and random motions affect each
other, from which spatial instabilities and extinctions arise. Under suitable
conditions, competitive interactions lead to clustering of individuals and
periodic pattern formation. Random motion has a homogenizing effect and then
delays this clustering instability. When individuals from species differing in
their random walk characteristics are allowed to compete together, the ones
with a tendency to form narrower clusters get a competitive advantage over the
others. Mean-field deterministic equations are analyzed and compared with the
outcome of the individual-based simulations.Comment: 38 pages, including 6 figure
Effect of spatial configuration of an extended nonlinear Kierstead-Slobodkin reaction-transport model with adaptive numerical scheme
In this paper, we consider the numerical simulations of an extended nonlinear form
of Kierstead-Slobodkin reaction-transport system in one and two dimensions. We
employ the popular fourth-order exponential time differencing Runge-Kutta (ETDRK4)
schemes proposed by Cox and Matthew (J Comput Phys 176:430-455,
2002), that was modified by Kassam and Trefethen (SIAM J Sci Comput 26:1214-1233,
2005), for the time integration of spatially discretized partial differential equations. We demonstrate
the supremacy of ETDRK4 over the existing exponential time differencing integrators
that are of standard approaches and provide timings and error comparison. Numerical
results obtained in this paper have granted further insight to the question "What is the
minimal size of the spatial domain so that the population persists?" posed by Kierstead
and Slobodkin (J Mar Res 12:141-147,
1953
), with a conclusive remark that the popula-
tion size increases with the size of the domain. In attempt to examine the biological
wave phenomena of the solutions, we present the numerical results in both one- and
two-dimensional space, which have interesting ecological implications. Initial data and
parameter values were chosen to mimic some existing patternsScopus 201
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