Cover page and Preface
Shouchuan Hu Xin Lu
The Tenth AIMS International Conference on Dynamical Systems, Di eren- tial Equations and Applications took place in the magni cent Madrid, Spain, July 7 - 11, 2014. The present volume is the Proceedings, consisting of some carefully selected submissions after a rigorous refereeing process.

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Global periodicity in a class of reaction-diffusion systems with time delays
Wei Feng Xin Lu
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).
keywords: Reaction-diffusion systems time-delay system.
Global stability in a class of reaction-diffusion systems with time-varying delays
Wei Feng Xin Lu
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keywords: stability and asymptotic behavior. time-varying delay effects Diffusive logistic equations
Mathematical analysis on an extended Rosenzweig-MacArthur model of tri-trophic food chain
Wei Feng Nicole Rocco Michael Freeze Xin Lu
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.
keywords: stability and asymptotic behavior extinction or coexistence numerical simulations. limiting nutrient response Tri-trophic food-chain models
Optimal controls for a 3-compartment model for cancer chemotherapy with quadratic objective
Wei Feng Shuhua Hu Xin Lu
keywords: Optimal control cancer chemotherapy. three compartment model field of extremals
Global attractors of reaction-diffusion systems modeling food chain populations with delays
Wei Feng C. V. Pao Xin Lu
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.
keywords: coexistence and permanence Population dynamics numerical simulations. reaction diffusion systems positive steady state and global attractor 4-species food chain
Population dynamics in a model for territory acquisition
Wei Feng Xin Lu Richard John Donovan Jr.
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Coexistence and asymptotic stability in stage-structured predator-prey models
Wei Feng Michael T. Cowen Xin Lu
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.
keywords: stability analysis stage-structured models Predator-prey interactions coexistence and simulations.
On existence of wavefront solutions in mixed monotone reaction-diffusion systems
Wei Feng Weihua Ruan Xin Lu
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.
keywords: mixed monotone functions numerical simulations. coexistence in ecological models Reaction-diffusion systems existence of wavefront solutions
Regularity of densities in relaxed and penalized average distance problem
Xin Yang Lu
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.
keywords: average-distance regularity. optimal transport Kantorovich potential Nonlocal variational problem

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