## Journals

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DCDS

Random dispersal is essentially a local
behavior which describes the movement of organisms between
adjacent spatial locations. However, the movements and
interactions of some organisms can occur between non-adjacent
spatial locations. To address the question about which dispersal
strategy can convey some competitive advantage, we consider a
mathematical model consisting of one reaction-diffusion equation
and one integro-differential equation, in which two competing
species have the same population dynamics but different dispersal
strategies: the movement of one species is purely by random walk
while the other species adopts a non-local dispersal strategy.
For spatially periodic and heterogeneous environments we show
that (i) for fixed random dispersal rate,
if the nonlocal dispersal distance is sufficiently small, then
the non-local disperser can invade the random disperser but not vice versa;
(ii) for fixed nonlocal dispersal distance,
if the random dispersal rate becomes sufficiently small,
then
the random disperser can invade the nonlocal disperser but not
vice versa.
These results suggest
that for spatially periodic heterogeneous environments,
the competitive advantage may belong to the species with much lower
effective rate of dispersal. This is in agreement with previous results
for the evolution of random dispersal [9, 13] that the slower disperser
has an advantage. Nevertheless, if random dispersal strategy with either zero Dirichlet or zero Neumann boundary condition is compared with non-local dispersal strategy with hostile
surroundings, the species with much lower effective rate of dispersal may not have the competitive advantage. Numerical results will be presented to shed light on the global dynamics of the system for general values of non-local interaction distance and also to point to future research directions.

keywords:
Reaction-diffusion
,
Competition
,
Non-local dispersal
,
Random dispersal
,
Integral kernel.

DCDS-B

We study an initial-boundary value problem for a fourth-order
parabolic partial differential equation with an unknown velocity.
The equation originated from the linearization of a two-dimensional
Couette flow model, that was recently proposed by Benilov and
Vynnycky. In the case of a $180^{\circ}$-- contact angle between
liquid and a moving plate Benilov and Vynnycky conjectured that the
speed of the contact line blows up to infinity in finite time. In
this paper we present numerical simulations and qualitative analysis
of the model. We show that depending on the initial data and
parameter values different long time behaviors of velocity can be observed. The speed of
the contact line may blow up to infinity or converge to a constant.

keywords:
contact line singularity
,
contact angle
,
fourth-order parabolic equation.
,
blow-up
,
Couette flow

MBE

This paper is concerned with an indefinite weight linear eigenvalue
problem in cylindrical domains. We investigate the minimization of the positive
principal eigenvalue under the constraint that the weight is bounded by
a positive and a negative constant and the total weight is a fixed negative
constant. Biologically, this minimization problem is motivated by the question
of determining the optimal spatial arrangement of favorable and unfavorable
regions for a species to survive. Both our analysis and numerical simulations
for rectangular domains indicate that there exists a threshold value such that
if the total weight is below this threshold value, then the optimal favorable
region is a circular-type domain at one of the four corners, and a strip at the
one end with shorter edge otherwise.

DCDS-B

Random dispersal describes the movement of organisms between
adjacent spatial locations. However, the movement of some organisms
such as seeds of plants can occur between non-adjacent
spatial locations and is thus non-local.
We propose to study a mixed dispersal strategy, which is a combination
of random dispersal and non-local dispersal. More specifically, we assume that a
fraction of individuals in the population adopt random dispersal, while the remaining fraction assumes
non-local dispersal. We investigate how such mixed dispersal affects
the invasion of a single species and also how mixed dispersal strategy will evolve
in spatially heterogeneous but temporally constant environment.

keywords:
reaction-diffusion
,
integral kernel.
,
random dispersal
,
competition
,
Non-local dispersal

## Year of publication

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