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PROC

The Tenth AIMS International Conference on Dynamical Systems, Dieren-
tial Equations and Applications took place in the magnicent Madrid, Spain,
July 7 - 11, 2014. The present volume is the Proceedings, consisting of some
carefully selected submissions after a rigorous refereeing process.

For more information please click the “Full Text” above.

For more information please click the “Full Text” above.

keywords:

DCDS-B

In this paper we study a class of
reaction-diffusion systems modelling the dynamics of
"food-limited" populations with periodic environmental data and
time delays. The existence of a global attracting positive
periodic solution is first established in the model without time
delay. It is further shown that as long as the magnitude of the
instantaneous self-limitation effects is larger than that of the
time-delay effects, the positive periodic solution is also the
global attractor in the time-delay system. Numerical simulations
for both cases (with or without time delays) demonstrate the same
asymptotic behavior (extinction or converging to the positive
$T$-periodic solution, depending on the growth rate of the
species).

PROC

Please refer to Full Text.

DCDS-S

In this paper, we study a new model as an extension of the
Rosenzweig-MacArthur tritrophic food chain model in which the
super-predator consumes both the predator and the prey. We first obtain
the ultimate bounds and conditions for exponential convergence for these
populations. We also find all possible equilibria and investigate their stability or
instability in relation with all the ecological parameters. Our main focus is on the
conditions for the existence, uniqueness and stability of a coexistence equilibrium.
The complexity of the dynamics in this model is theoretically discussed and graphically
demonstrated through various examples and numerical simulations.

PROC

n/a

CPAA

In this paper, we study a reaction-diffusion system modeling the
population dynamics of a four-species food chain with time delays.
Under Dirichlet and Neumann boundary conditions, we discuss the
existence of a positive global attractor which demonstrates the
presence of a positive steady state and the permanence effect in the
ecological system. Sufficient conditions on the interaction rates
are given to ensure the persistence of all species in the food
chain. For the case of Neumann boundary condition, we further obtain
the uniqueness of a positive steady state, and in such case the
density functions converge uniformly to a constant solution.
Numerical simulations of the food-chain models are also given to
demonstrate and compare the asymptotic behavior of the
time-dependent density functions.

MBE

In this paper we analyze the effects of a stage-structured
predator-prey system where the prey has two stages, juvenile and
adult. Three different models (where the juvenile or adult prey
populations are vulnerable) are studied to evaluate the impacts of
this structure to the stability of the system and coexistence of the
species. We assess how various ecological parameters, including
predator mortality rate and handling times on prey, prey growth rate
and death rate, prey capture rate and nutritional values
in two stages, affect the existence and stability of all possible
equilibria in each of the models, as well as the ultimate bounds and the dynamics of the populations.
The main focus of this paper is to find general conditions to ensure the presence and stability of the coexistence equilibrium
where both the predator and prey can co-exist
Through specific examples, we demonstrate the
stability of the trivial and co-existence equilibrium as well as the dynamics in each
system.

DCDS-B

In this article, we give an existence-comparison
theorem for wavefront solutions in a general class of
reaction-diffusion systems. With mixed quasi-monotonicity and
Lipschitz condition on the set bounded by coupled upper-lower
solutions, the existence of wavefront solution is proven by applying
the Schauder Fixed Point Theorem on a compact invariant set. Our
main result is then applied to well-known examples: a
ratio-dependent predator-prey model, a three-species food chain model of
Lotka-Volterra type and a three-species competition model of
Lotka-Volterra type. For each model, we establish conditions on the
ecological parameters for the presence of wavefront solutions
flowing towards the coexistent states through suitably constructed
upper and lower solutions. Numerical simulations on those models are
also demonstrated to illustrate our theoretical results.

NHM

The average distance problem finds application in data parameterization, which involves
``representing'' the data using lower dimensional objects. From
a computational point of view it is often convenient to restrict
the unknown to the family of parameterized curves.
The original formulation of the average distance problem exhibits several undesirable properties.
In this paper we propose an alternative variant:
we minimize the functional
\begin{equation*}
\int_{{\mathbb{R}}^d\times \Gamma_\gamma} |x-y|^p {\,{d}}\Pi(x,y)+\lambda L_\gamma +\varepsilon\alpha(\nu)
+\varepsilon' \eta(\gamma)+\varepsilon''\|\gamma'\|_{TV},
\end{equation*}
where $\gamma$ varies among the family of parametrized curves, $\nu$ among probability measures on $\gamma$,
and $\Pi$ among transport plans between $\mu$ and $\nu$. Here $\lambda,\varepsilon,\varepsilon',\varepsilon''$ are given parameters,
$\alpha$ is a penalization term on $\mu$, $\Gamma_\gamma$ (resp. $L_\gamma$) denotes the graph (resp. length)
of $\gamma$, and $\|\cdot\|_{TV}$ denotes the total variation semi-norm.
We will use techniques from
optimal transport theory and calculus of variations.
The main aim is to prove essential boundedness,
and Lipschitz continuity for Radon-Nikodym derivative of $\nu$, when $(\gamma,\nu,\Pi)$
is a minimizer.

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