The dynamics of the solution flow of a two-species Lotka-Volterra
competition model with an extra equation for simple inhibitor
dynamics is investigated. The model fits into the abstract
framework of two-species competition systems (or $K$-monotone
systems), but the equilibrium representing the extinction of both
species is not a repeller. This feature distinguishes our problem
from the case of classical two-species competition without
inhibitor (classical case for short), where a basic assumption
requires that equilibrium to be a repeller. Nevertheless, several
results similar to those in the classical case, such as
competitive exclusion and the existence of a "thin" separatrix,
are obtained, but differently from the classical case, coexistence
of the two species or extinction of one of them may depend on the
initial conditions. As in almost all two species competition
models, the strong monotonicity of the flow (with respect to a
certain order on $\mathbb R^3$) is a key ingredient for establishing the
main results of the paper.
The paper extends the Faber--Krahn inequality for
elliptic and periodic parabolic problems to random and general
nonautonomous parabolic problems. Under proper assumptions, it
also provides necessary and sufficient conditions for the
Faber-Krahn inequality being equality.
The present paper is devoted to the study of transition fronts of nonlocal Fisher-KPP equations in time heterogeneous media. We first construct transition fronts with exact decaying rates as the space variable tends to infinity and with prescribed interface location functions, which are natural generalizations of front location functions in homogeneous media. Then, by the general results on space regularity of transition fronts of nonlocal evolution equations proven in the authors' earlier work (), these transition fronts are continuously differentiable in space. We show that their space partial derivatives have exact decaying rates as the space variable tends to infinity. Finally, we study the asymptotic stability of transition fronts. It is shown that transition fronts attract those solutions whose initial data decays as fast as transition fronts near infinity and essentially above zero near negative infinity.
An integral formula is given representing the generalized principal Lyapunov exponent for random linear parabolic PDEs. As an application, an upper estimate of the exponent is obtained.
The present paper is devoted to the study of transition fronts in nonlocal reaction-diffusion equations with time heterogeneous nonlinearity of ignition type. It is proven that such an equation admits space monotone transition fronts with finite speed and space regularity in the sense of uniform Lipschitz continuity. Our approach is first constructing a sequence of approximating front-like solutions and then proving that the approximating solutions converge to a transition front. We take advantage of the idea of modified interface location, which allows us to characterize the finite speed of approximating solutions in the absence of space regularity, and leads directly to uniform exponential decaying estimates.
The current paper is concerned with the global
dynamics of a class of nonlinear oscillators driven by real or
white noises, of which a typical example is a shunted
Josephson junction exposed to some random medium.
Applying random dynamical systems theory,
it is shown that a driven oscillator in the class under consideration
with a tempered real noise
has a one-dimensional global random attractor
provided that the damping is not too small. Moreover, restricted
to the global attractor, the oscillator induces a random dynamical
system on $S^1$. It is then shown that there is a rotation number
associated to the oscillator which characterizes the speed at
the solutions of the oscillator move around the global attractor. The results
extend the existing ones for time periodic and quasi-periodic
Josephson junctions and can be applied to Josephson junctions
driven by white noises.
In this paper, we study the spectral theory for nonlocal dispersal operators with time periodic indefinite weight functions subject to Dirichlet type, Neumann type and spatial periodic type boundary conditions. We first obtain necessary and sufficient conditions for the existence of a unique positive principal spectrum point for such operators. We then investigate upper bounds of principal spectrum points and sufficient conditions for the principal spectrum points to be principal eigenvalues. Finally we discuss the applications to nonlinear mathematical models from biology.
Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary conditions are allowed to depend on time. The theory of the principal spectrum/principal Lyapunov exponents is shown to apply to those equations. In the nonautonomous case, the main result states that the principal eigenvalue of any time-averaged equation is not larger than the supremum of the principal spectrum and that there is a time-averaged equation whose principal eigenvalue is not larger than the infimum of the principal spectrum. In the random case, the main result states that the principal eigenvalue of the time-averaged equation is not larger than the principal Lyapunov exponent.
The current paper is devoted to the study of pullback attractors for
general nonautonomous and random parabolic equations on non-smooth
domains $D$. Mild solutions are considered for such equations. We
first extend various fundamental properties for solutions of smooth
parabolic equations on smooth domains to solutions of general
parabolic equations on non-smooth domains, including continuous
dependence on parameters, monotonicity, and compactness, which are
of great importance in their own. Under certain dissipative
conditions on the nonlinear terms, we prove that mild solutions
with initial conditions in $L_q(D)$ exist globally for $q$ » $1$. We
then show that pullback attractors for nonautonomous and random
parabolic equations on non-smooth domains exist in $L_q(D)$ for
$1$ « $q$ < $\infty$.
The strong interest in infinite dimensional dissipative systems originated from the observation that the dynamics of large classes of partial differential equations and systems resembles the behavior known from the modern theory of finite-dimensional dynamical systems. Reaction-diffusion problems are typical examples in this context. In biological applications linear diffusion represents random dispersal of a species, but in many cases other dispersal strategies occur, which has led to models with cross diffusion and nonlocal dispersal.
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