428 research outputs found
Depleting the signal: Analysis of chemotaxis-consumption models -- A survey
We give an overview of analytical results concerned with chemotaxis systems
where the signal is absorbed. We recall results on existence and properties of
solutions for the prototypical chemotaxis-consumption model and various
variants and review more recent findings on its ability to support the
emergence of spatial structures
Depleting the signal: Analysis of chemotaxis-consumption models—A survey
We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures
Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis
This paper proves the global existence and boundedness of solutions to a general reaction diffusion predator prey system with prey-taxis defined on a smooth bounded domain with no-flux boundary condition. The result holds for domains in arbitrary spatial dimension and small prey-taxis sensitivity coefficient. This paper also proves the existence of a global attractor and the uniform persistence of the system under some additional conditions. Applications to models from ecology and chemotaxis are discussed. (C) 2016 Elsevier Inc. All rights reserved
Dynamics of a Leslie-Gower predator-prey system with cross-diffusion
A Leslie–Gower predator–prey system with cross-diffusion subject to Neumann boundary conditions is considered. The global existence and boundedness of solutions are shown. Some sufficient conditions ensuring the existence of nonconstant solutions are obtained by means of the Leray–Schauder degree theory. The local and global stability of the positive constant steady-state solution are investigated via eigenvalue analysis and Lyapunov procedure. Based on center manifold reduction and normal form theory, Hopf bifurcation direction and the stability of bifurcating timeperiodic solutions are investigated and a normal form of Bogdanov–Takens bifurcation is determined as well
Global solvability and stability of an alarm-taxis system
This paper is concerned with the global boundedness and stability of
classical solutions to an alarm-taxis system describing the burglar alarm
hypothesis as an important mechanism of anti-predation behavior when species
are threaten by predators. Compared to the existing prey-taxis systems, the
alarm-taxis system has more complicated coupling structure and additionally
requires the gradient estimate of the primary predator density to attain the
global boundedness of solutions. By the sophisticated coupling energy estimates
based on the Neumann semigroup smoothing properties, we establish the existence
of globally bounded solutions in two dimensions with Neumann boundary
conditions and furthermore prove the global stability of co-existence
homogeneous steady states under certain conditions on the system parameters
Mathematical models for chemotaxis and their applications in self-organisation phenomena
Chemotaxis is a fundamental guidance mechanism of cells and organisms,
responsible for attracting microbes to food, embryonic cells into developing
tissues, immune cells to infection sites, animals towards potential mates, and
mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of
the bedrock of mathematical biology, a go-to-choice for modellers and analysts
alike. For the former it is simple yet recapitulates numerous phenomena; the
latter are attracted to these rich dynamics. Here I review the adoption of PKS
systems when explaining self-organisation processes. I consider their
foundation, returning to the initial efforts of Patlak and Keller and Segel,
and briefly describe their patterning properties. Applications of PKS systems
are considered in their diverse areas, including microbiology, development,
immunology, cancer, ecology and crime. In each case a historical perspective is
provided on the evidence for chemotactic behaviour, followed by a review of
modelling efforts; a compendium of the models is included as an Appendix.
Finally, a half-serious/half-tongue-in-cheek model is developed to explain how
cliques form in academia. Assumptions in which scholars alter their research
line according to available problems leads to clustering of academics and the
formation of "hot" research topics.Comment: 35 pages, 8 figures, Submitted to Journal of Theoretical Biolog
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