48 research outputs found
A Nonlinear Splitting Algorithm for Systems of Partial Differential Equations with self-Diffusion
Systems of reaction-diffusion equations are commonly used in biological
models of food chains. The populations and their complicated interactions
present numerous challenges in theory and in numerical approximation. In
particular, self-diffusion is a nonlinear term that models overcrowding of a
particular species. The nonlinearity complicates attempts to construct
efficient and accurate numerical approximations of the underlying systems of
equations. In this paper, a new nonlinear splitting algorithm is designed for a
partial differential equation that incorporates self-diffusion. We present a
general model that incorporates self-diffusion and develop a numerical
approximation. The numerical analysis of the approximation provides criteria
for stability and convergence. Numerical examples are used to illustrate the
theoretical results
A remark on "Study of a Leslie-Gower-type tritrophic population model" [Chaos, Solitons and Fractals 14 (2002) 1275-1293]
In [Aziz-Alaoui, 2002] a three species ODE model, based on a modified
Leslie-Gower scheme is investigated. It is shown that under certain
restrictions on the parameter space, the model has bounded solutions for all
positive initial conditions, which eventually enter an invariant attracting
set. We show that this is not true. To the contrary, solutions to the model can
blow up in finite time, even under the restrictions derived in [Aziz-Alaoui,
2002], if the initial data is large enough. We also prove similar results for
the spatially extended system. We validate all of our results via numerical
simulations.Comment: 10 pages, 4 figure
Biological control via "ecological" damping: An approach that attenuates non-target effects
In this work we develop and analyze a mathematical model of biological
control to prevent or attenuate the explosive increase of an invasive species
population in a three-species food chain. We allow for finite time blow-up in
the model as a mathematical construct to mimic the explosive increase in
population, enabling the species to reach "disastrous" levels, in a finite
time. We next propose various controls to drive down the invasive population
growth and, in certain cases, eliminate blow-up. The controls avoid chemical
treatments and/or natural enemy introduction, thus eliminating various
non-target effects associated with such classical methods. We refer to these
new controls as "ecological damping", as their inclusion dampens the invasive
species population growth. Further, we improve prior results on the regularity
and Turing instability of the three-species model that were derived in earlier
work. Lastly, we confirm the existence of spatio-temporal chaos
What is India speaking: The "Hinglish" invasion
While language competition models of diachronic language shift are
increasingly sophisticated, drawing on sociolinguistic components like variable
language prestige, distance from language centers and intermediate bilingual
transitionary populations, in one significant way they fall short. They fail to
consider contact-based outcomes resulting in mixed language practices, e.g.
outcome scenarios such as creoles or unmarked code switching as an emergent
communicative norm. On these lines something very interesting is uncovered in
India, where traditionally there have been monolingual Hindi speakers and
Hindi/English bilinguals, but virtually no monolingual English speakers. While
the Indian census data reports a sharp increase in the proportion of
Hindi/English bilinguals, we argue that the number of Hindi/English bilinguals
in India is inaccurate, given a new class of urban individuals speaking a mixed
lect of Hindi and English, popularly known as "Hinglish". Based on
predator-prey, sociolinguistic theories, salient local ecological factors and
the rural-urban divide in India, we propose a new mathematical model of
interacting monolingual Hindi speakers, Hindi/English bilinguals and Hinglish
speakers. The model yields globally asymptotic stable states of coexistence, as
well as bilingual extinction. To validate our model, sociolinguistic data from
different Indian classes are contrasted with census reports: We see that
purported urban Hindi/English bilinguals are unable to maintain fluent Hindi
speech and instead produce Hinglish, whereas rural speakers evidence
monolingual Hindi. Thus we present evidence for the first time where an
unrecognized mixed lect involving English but not "English", has possibly taken
over a sizeable faction of a large global population.Comment: This paper has been withdrawan as the model has now been modified and
the existing model has some error