2,996 research outputs found
Predators-prey models with competition Part I: existence, bifurcation and qualitative properties
We study a mathematical model of environments populated by both preys and
predators, with the possibility for predators to actively compete for the
territory. For this model we study existence and uniqueness of solutions, and
their asymptotic properties in time, showing that the solutions have different
behavior depending on the choice of the parameters. We also construct
heterogeneous stationary solutions and study the limits of strong competition
and abundant resources. We then use these information to study some properties
such as the existence of solutions that maximize the total population of
predators. We prove that in some regimes the optimal solution for the size of
the total population contains two or more groups of competing predators.Comment: 61 pages, no figure
Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction
We consider a system of differential equations with nonlinear Steklov
boundary conditions, related to the fractional problem where ,
, , and . When we develop a
quasi-optimal regularity theory in , uniformly w.r.t. ,
for every ; moreover we show that the
traces of the limiting profiles as are Lipschitz continuous
and segregated. Such results are extended to the case of densities,
with some restrictions on , and
Multidimensional entire solutions for an elliptic system modelling phase separation
For the system of semilinear elliptic equations we devise a new method to construct entire solutions. The
method extends the existence results already available in the literature, which
are concerned with the 2-dimensional case, also in higher dimensions .
In particular, we provide an explicit relation between orthogonal symmetry
subgroups, optimal partition problems of the sphere, the existence of solutions
and their asymptotic growth. This is achieved by means of new asymptotic
estimates for competing system and new sharp versions for monotonicity formulae
of Alt-Caffarelli-Friedman type.Comment: Final version: presentation of the results improved, and several
minor corrections with respect to the first versio
Entire solutions with exponential growth for an elliptic system modeling phase-separation
We prove the existence of entire solutions with exponential growth for the
semilinear elliptic system [\begin{cases} -\Delta u = -u v^2 & \text{in }
-\Delta v= -u^2 v & \text{in } u,v>0, \end{cases}] for every .
Our construction is based on an approximation procedure, whose convergence is
ensured by suitable Almgren-type monotonicity formulae. The construction of
\emph{some} solutions is extended to systems with components, for every
- …