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### Open Access Journals

DCDS

This special issue of

Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.

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

*Discrete and Continuous Dynamical Systems-A*is dedicated to Peter W. Bates on the occasion of his 60th birthday, and in recognition of his outstanding contributions to infinite dimensional dynamical systems and the mathematical theory of phase transitions.Peter Bates was born in Manchester, England on December 27, 1947. He graduated from the University of London in mathematics in 1969 after which he moved to United States with his family. Later, he attended the University of Utah and received his Ph.D. in 1976. Following his graduation, Peter moved to Texas and taught at University of Texas at Pan American and Texas A&M University. He returned to Utah in 1984 and taught at Brigham Young University until 2004. He is currently a professor of mathematics at Michigan State University.

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

keywords:

DCDS

The precise dynamics of a reaction-diffusion model of autocatalytic
chemical reaction is described. It is shown that exactly either one, two,
or three steady states exists, and the solution of dynamical problem always approaches
to one of steady states in the long run. Moreover it is shown that a global codimension one
manifold separates the basins of attraction of the two stable steady states.
Analytic ingredients include exact multiplicity of semilinear elliptic equation,
the theory of monotone dynamical systems and the
theory of asymptotically autonomous dynamical systems.

DCDS-B

A reaction-diffusion logistic population model with spatially nonhomogeneous harvesting is considered. It is shown that when the intrinsic growth rate is larger than the principal eigenvalue of the protection zone, then the population is always sustainable; while in the opposite case, there exists a maximum allowable catch to avoid the population extinction. The existence of steady state solutions is also studied for both cases. The existence of an optimal harvesting pattern is also shown, and theoretical results are complemented by some numerical simulations for one-dimensional domains.

PROC

We consider a class of singular systems of
Lane-Emden type
\begin{equation} \nonumber
\begin{cases}
\Delta u + \la u^{p_1} v^{q_1}=0, & x\in D,\\
\Delta v + \la u^{p_2} v^{q_2}=0, & x\in D,\\
u=v=0, & x\in \partial D,
\end{cases}
\end{equation}
with $p_1\le 0, \; p_2> 0, \; q_1> 0, \; q_2\le 0$, and $D$ a
smooth domain in $\R^n$. In case the system is sublinear we prove
existence of a positive solution. If $D$ is a ball in $\R^n$, we
prove both existence and uniqueness of positive radially symmetric
solution.

DCDS

The existences of an asymptotic spreading speed and traveling wave
solutions for a diffusive model which describes the interaction of
mistletoe and bird populations with nonlocal diffusion and delay
effect are proved by using monotone semiflow theory. The effects of
different dispersal kernels on the asymptotic spreading speeds are
investigated through concrete examples and simulations.

DCDS

[]

CPAA

In this paper, we investigate the existence and uniqueness of crossing
limit cycle for a planar nonlinear Liénard system which is discontinuous along a straight line (called a discontinuity line).
By using the Poincaré mapping method and
some analysis techniques, a criterion for the existence, uniqueness and stability of a crossing limit cycle in the discontinuous differential system is established. An application to Schnakenberg model of an autocatalytic chemical reaction is given to illustrate the effectiveness of our result. We also consider a class of discontinuous piecewise linear differential systems and give a necessary condition of the existence of crossing limit cycle, which can be used to prove the non-existence of crossing limit cycle.

keywords:
Liénard system
,
uniqueness.
,
crossing limit cycle
,
(non) existence
,
Discontinuous system

DCDS-B

The effects of a higher vorticity moment on a variational problem
for barotropic vorticity on a rotating sphere is examined rigorously
in the framework of the Direct Method. This variational model
differs from previous work on the Barotropic Vorticity Equation
(BVE) in relaxing the angular momentum constraint, which then allows
us to state and prove theorems that give necessary and sufficient
conditions for the existence and stability of constrained energy
extremals in the form of super and sub-rotating solid-body steady
flows. Relaxation of angular momentum is a necessary step in the
modeling of the important tilt instability where the rotational axis
of the barotropic atmosphere tilts away from the fixed north-south
axis of planetary spin. These conditions on a minimal set of
parameters consisting of the planetary spin, relative enstrophy and
the fourth vorticity moment, extend the results of
previous work and clarify the role of the higher vorticity moments in models of geophysical flows.

DCDS-B

It is known that some predator-prey system can possess a unique
limit cycle which is globally asymptotically stable. For a
prototypical predator-prey system, we show that the solution curve
of the limit cycle exhibits temporal patterns of a relaxation
oscillator, or a Heaviside function, when certain parameter is
small.

PROC

A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained
and in certain cases, exact multiplicity is proved.

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