1,008 research outputs found
Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes
Spatial variation in population densities across a landscape is a feature of many ecological systems, from
self-organised patterns on mussel beds to spatially restricted insect outbreaks. It occurs as a result of
environmental variation in abiotic factors and/or biotic factors structuring the spatial distribution of
populations. However the ways in which abiotic and biotic factors interact to determine the existence
and nature of spatial patterns in population density remain poorly understood. Here we present a new
approach to studying this question by analysing a predator–prey patch-model in a heterogenous
landscape. We use analytical and numerical methods originally developed for studying nearest-
neighbour (juxtacrine) signalling in epithelia to explore whether and under which conditions patterns
emerge. We find that abiotic and biotic factors interact to promote pattern formation. In fact, we find a
rich and highly complex array of coexisting stable patterns, located within an enormous number of
unstable patterns. Our simulation results indicate that many of the stable patterns have appreciable
basins of attraction, making them significant in applications. We are able to identify mechanisms for
these patterns based on the classical ideas of long-range inhibition and short-range activation, whereby
landscape heterogeneity can modulate the spatial scales at which these processes operate to structure
the populations
Existence of Periodic Solutions for a Delayed Ratio-Dependent Three-Species Predator-Prey Diffusion System on Time Scales
This paper investigates the existence of periodic solutions of a ratio-dependent predator-prey diffusion system with Michaelis-Menten functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain suffcient criteria for the existence of periodic solutions for the system. Moreover, when the time scale 𝕋 is chosen as ℝ or ℤ, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations
Partial Differential Equations in Ecology
Partial differential equations (PDEs) have been used in theoretical ecology research for more than eighty years. Nowadays, along with a variety of different mathematical techniques, they remain as an efficient, widely used modelling framework; as a matter of fact, the range of PDE applications has even become broader. This volume presents a collection of case studies where applications range from bacterial systems to population dynamics of human riots
Efficient numerical methods to solve some reaction-diffusion problems arising in biology
Philosophiae Doctor - PhDIn this thesis, we solve some time-dependent partial differential equations, and systems of such equations, that governs reaction-diffusion models in biology. we design and implement some novel exponential time differencing schemes to integrate stiff systems of ordinary differential equations which arise from semi-discretization of the associated partial differential equations. We split the semi-linear PDE(s) into a linear, which contains the highly stiff part of the problem, and a nonlinear part, that is expected to vary more slowly than the linear part. Then we introduce higher-order finite difference approximations for the spatial discretization. Resulting systems of stiff ODEs are then solved by using exponential time differencing methods. We present stability properties of these methods along with extensive numerical simulations for a number of different reaction-diffusion models, including single and multi-species models. When the diffusivity is small many of the models considered in this work are found to exhibit a form of localized spatiotemporal patterns. Such patterns are correctly captured by our proposed numerical schemes. Hence, the schemes that we have designed in this thesis are dynamically consistent. Finally, in many cases, we have compared our results with
those obtained by other researchers
Dispersal-induced instability in complex ecosystems
In his seminal work in the 1970s Robert May suggested that there was an upper
limit to the number of species that could be sustained in stable equilibrium by
an ecosystem. This deduction was at odds with both intuition and the observed
complexity of many natural ecosystems. The so-called stability-diversity debate
ensued, and the discussion about the factors making an ecosystem stable or
unstable continues to this day. We show in this work that dispersal can be a
destabilising influence. To do this, we combine ideas from Alan Turing's work
on pattern formation with May's random-matrix approach. We demonstrate how a
stable equilibrium in a complex ecosystem with two trophic levels can become
unstable with the introduction of dispersal in space. Conversely, we show that
Turing instabilities can occur more easily in complex ecosystems with many
species than in the case of only a few species. Our work shows that adding more
details to the model of May gives rise to more ways in which an equilibrium can
become unstable. Making May's simple model more realistic is therefore unlikely
to remove the upper bound on complexity.Comment: 30 pages, 6+1 figure
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