704 research outputs found
Spatiotemporal dynamics in a spatial plankton system
In this paper, we investigate the complex dynamics of a spatial plankton-fish
system with Holling type III functional responses. We have carried out the
analytical study for both one and two dimensional system in details and found
out a condition for diffusive instability of a locally stable equilibrium.
Furthermore, we present a theoretical analysis of processes of pattern
formation that involves organism distribution and their interaction of
spatially distributed population with local diffusion. The results of numerical
simulations reveal that, on increasing the value of the fish predation rates,
the sequences spots spot-stripe mixtures
stripes hole-stripe mixtures holes wave pattern is
observed. Our study shows that the spatially extended model system has not only
more complex dynamic patterns in the space, but also has spiral waves.Comment: Published Pape
On some Free Boundary Problems of the Prey-predator Model
In this paper we investigate some free boundary problems for the
Lotka-Volterra type prey-predator model in one space dimension. The main
objective is to understand the asymptotic behavior of the two species (prey and
predator) spreading via a free boundary. We prove a spreading-vanishing
dichotomy, namely the two species either successfully spread to the entire
space as time goes to infinity and survive in the new environment, or they
fail to establish and die out in the long run. The long time behavior of
solution and criteria for spreading and vanishing are also obtained. Finally,
when spreading successfully, we provide an estimate to show that the spreading
speed (if exists) cannot be faster than the minimal speed of traveling
wavefront solutions for the prey-predator model on the whole real line without
a free boundary.Comment: 28 page
Modelling chemotaxis of microswimmers: from individual to collective behavior
We discuss recent progress in the theoretical description of chemotaxis by
coupling the diffusion equation of a chemical species to equations describing
the motion of sensing microorganisms. In particular, we discuss models for
autochemotaxis of a single microorganism which senses its own secretion leading
to phenomena such as self-localization and self-avoidance. For two
heterogeneous particles, chemotactic coupling can lead to predator-prey
behavior including chase and escape phenomena, and to the formation of active
molecules, where motility spontaneously emerges when the particles approach
each other. We close this review with some remarks on the collective behavior
of many particles where chemotactic coupling induces patterns involving
clusters, spirals or traveling waves.Comment: to appear as a contribution to the book "Chemical kinetics beyond the
textbook
Dynamics of a diffusive predator-prey system with fear effect in advective environments
We explore a diffusive predator-prey system that incorporates the fear effect
in advective environments. Firstly, we analyze the eigenvalue problem and the
adjoint operator, considering Constant-Flux and Dirichlet (CF/D) boundary
conditions, as well as Free-Flow (FF) boundary conditions. Our investigation
focuses on determining the direction and stability of spatial Hopf bifurcation,
with the generation delay serving as the bifurcation parameter.
Additionally, we examine the influence of both linear and Holling-II functional
responses on the dynamics of the model. Through these analyses, we aim to gain
a better understanding of the intricate relationship between advection,
predation, and prey response in this system
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
Multiple wave solutions in a diffusive predator-prey model with strong Allee effect on prey and ratio-dependent functional response
A thorough analysis is performed in a predator-prey reaction-diffusion model
which includes three relevant complex dynamical ingredients: (a) a strong Allee
effect; (b) ratio-dependent functional responses; and (c) transport attributes
given by a diffusion process. As is well-known in the specialized literature,
these aspects capture adverse survival conditions for the prey, predation
search features and non-homogeneous spatial dynamical distribution of both
populations. We look for traveling-wave solutions and provide rigorous results
coming from a standard local analysis, numerical bifurcation analysis, and
relevant computations of invariant manifolds to exhibit homoclinic and
heteroclinic connections and periodic orbits in the associated dynamical system
in . In so doing, we present and describe a diverse zoo of traveling wave
solutions; and we relate their occurrence to the Allee effect, the spreading
rates and propagation speed. In addition, homoclinic chaos is manifested via
both saddle-focus and focus-focus bifurcations as well as a Belyakov point. An
actual computation of global invariant manifolds near a focus-focus homoclinic
bifurcation is also presented to enravel a multiplicity of wave solutions in
the model. A deep understanding of such ecological dynamics is therefore
highlighted.Comment: 35 pages, 22 figure
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