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### Open Access Journals

DCDS-B

Some models in population dynamics with intra-specific
competition lead to integro-differential equations where the
integral term corresponds to nonlocal consumption of resources
[8][9]. The principal difference of such
equations in comparison with traditional reaction-diffusion
equation is that homogeneous in space solutions can lose their
stability resulting in emergence of spatial or spatio-temporal
structures [4]. We study the existence and global bifurcations of
such structures. In the case of unbounded domains, transition
between stationary solutions can be observed resulting in
propagation of generalized travelling waves (GTW).
GTWs are introduced in [18] for reaction-diffusion systems as
global in time propagating solutions. In this work their
existence and properties are studied for the integro-differential
equation. Similar to the reaction-diffusion equation in
the monostable case, we prove the existence of generalized
travelling waves for all values of the speed greater or equal to the minimal one.
We illustrate these results by numerical
simulations in one and two space dimensions and observe a variety
of structures of GTWs.

NHM

A reaction-diffusion equation with nonlinear boundary condition is
considered in a two-dimensional infinite strip. Existence of
waves in the bistable case is proved by the Leray-Schauder
method.

CPAA

A linear second order elliptic equation describing heat or mass
diffusion and convection on a given velocity field is considered
in $R^3$. The corresponding operator $L$ may not satisfy the
Fredholm property. In this case, solvability conditions for the
equation $L u = f$ are not known. In this work, we
derive solvability conditions in $H^2(R^3)$ for
the non self-adjoint problem by relating it to a self-adjoint
Schrödinger type operator, for which solvability conditions are
obtained in our previous work [13].

DCDS-B

The paper is devoted to integro-differential equations arising in population
dynamics. The integral term describes the nonlocal consumption of resources.
We study the Fredholm property of the corresponding linear operators and
use it to prove the existence of travelling waves when the support of the
integral is sufficiently small. In this case, the integro-differential
operator is close to the differential operator and we can use the implicit
function theorem. We carry out numerical simulations in order to study the
case where the support of the integral is not small. We observe various
regimes of wave propagation. Some of them, in particular periodic waves do
not exist for the usual reaction-diffusion equation.

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