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### Open Access Journals

DCDS-B

In this paper, we investigate the inside dynamics of the positive solutions of integro-differential equations
\begin{equation*}
\partial_t u(t,x)= (J\star u)(t,x) -u(t,x) + f(u(t,x)), \ t>0 \hbox{ and } x\in\mathbb{R},
\end{equation*}
with both

*thin-tailed*and*fat-tailed*dispersal kernels $J$ and a monostable reaction term $f.$ The notion of inside dynamics has been introduced to characterize traveling waves of some reaction-diffusion equations [23]. Assuming that the solution is made of several fractions $\upsilon^i\ge 0$ ($i\in I \subset \mathbb{N}$), its inside dynamics is given by the spatio-temporal evolution of $\upsilon^i$. According to this dynamics, the traveling waves can be classified in two categories: pushed and pulled waves. For thin-tailed kernels, we observe no qualitative differences between the traveling waves of the above integro-differential equations and the traveling waves of the classical reaction-diffusion equations. In particular, in the KPP case ($f(u)\leq f'(0)u$ for all $u\in(0,1)$) we prove that all the traveling waves are pulled. On the other hand for fat-tailed kernels, the integro-differential equations do not admit any traveling waves. Therefore, to analyse the inside dynamics of a solution in this case, we introduce new notions of pulled and pushed solutions. Within this new framework, we provide analytical and numerical results showing that the solutions of integro-differential equations involving a fat-tailed dispersal kernel are pushed. Our results have applications in population genetics. They show that the existence of long distance dispersal events during a colonization tend to preserve the genetic diversity.
DCDS-B

In this paper, we construct radially symmetric solutions of a nonlinear non-cooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.

DCDS

In this paper, we analyse the structure of the set of positive solutions of an heterogeneous nonlocal equation of the form:
$$ \int_{\Omega} K(x, y)u(y)\,dy -\int_ {\Omega}K(y, x)u(x)\, dy + a_0u+\lambda a_1(x)u -\beta(x)u^p=0 \quad \text{in}\quad \Omega$$
where $\Omega\subset \mathbb{R}^n$ is a bounded domain, $K\in C(\mathbb{R}^n\times \mathbb{R}^n) $ is non-negative, $a_i,\beta \in C(\Omega)$ and $\lambda\in\mathbb{R}$.
Such type of equation appears in some studies of population dynamics where the population evolves in a partially controlled heterogeneous landscape and disperses on long ranges.
Under some fairly general assumptions on $K,a_i$ and $\beta$, we first establish a necessary and sufficient condition for the existence of a unique positive solution. Then, we analyse the structure of the set of positive solutions $(\lambda,u_\lambda)$ depending on the presence or absence of a refuge zone (i.e $\omega$ so that $\beta_{|\omega}\equiv 0$).

DCDS-B

We are interested in the long time behaviour of the positive solutions of the Cauchy problem involving the integro-differential equation

$\partial_t u(t,x)\\=\int_{\Omega }m(x,y)\left(u(t,y)-u(t,x)\right)\,dy+\left(a(x)-\int_{\Omega }k(x,y)u(t,y)\,dy\right)u(t,x),$ |

supplemented by the initial condition

in

, where the domain

is a, the functions

and

are non-negative kernels satisfying integrability conditions and the function

is continuous. Such a problem is used in population dynamics models to capture the evolution of a clonal population structured with respect to a phenotypic trait. In this context, the function

represents the density of individuals characterized by the trait, the domain of trait values

is a bounded subset of

, the kernels

and

respectively account for the competition between individuals and the mutations occurring in every generation, and the function

represents a growth rate. When the competition is independent of the trait, that is, the kernel

is independent of

, (

), we construct a positive stationary solution which belongs to

inthe space of Radon measures on

.

.Moreover, in the case where this measure

is regular and bounded, we prove its uniqueness and show that, for any non-negative initial datum in

, the solution of the Cauchy problem converges to this limit measure in

. We also exhibit an example for which the measure is singular and non-unique, and investigate numerically the long time behaviour of the solution in such a situation. The numerical simulations seem to reveal a dependence of the limit measure with respect to the initial datum.

$u(0,\cdot)=u_0$ |

$\Omega $ |

$\Omega $ |

$k$ |

$m$ |

$a$ |

$u$ |

$\Omega $ |

$\mathbb{R}^N$ |

$k$ |

$m$ |

$a$ |

$k$ |

$x$ |

$k(x,y)=k(y)$ |

$d\mu$ |

$\Omega $ |

$\mathbb{M}(\Omega )$ |

$d\mu$ |

$L^1(\Omega )\cap L^{\infty}(\Omega )$ |

$L^2(\Omega )$ |

NHM

We consider a so-called random
obstacle model for the motion of a hypersurface through a field of
random obstacles,
driven by a constant driving field.
The resulting semi-linear parabolic PDE with random coefficients does not
admit a global nonnegative stationary solution,
which implies that an interface that was flat originally cannot get
stationary.
The absence of global stationary solutions is shown by
proving lower bounds on the growth of stationary solutions on
large domains with Dirichlet boundary conditions.
Difficulties arise because the random
lower order part of the equation cannot be bounded uniformly.

DCDS

We are concerned with travelling wave solutions
arising in a reaction diffusion equation with bistable and
nonlocal nonlinearity, for which the comparison principle does not
hold. Stability of the equilibrium $u\equiv 1$ is not assumed. We
construct a travelling wave solution connecting 0 to an unknown
steady state, which is "above and away", from the
intermediate equilibrium. For focusing kernels we prove that, as
expected, the wave connects 0 to 1. Our results also apply readily
to the nonlocal ignition case.

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