Existence Of Global Positive Solution Research Articles

Blowup estimates for a family of semilinear SPDEs with time-dependent coefficients

Abstract We investigate the blowup and stability of semilinear stochastic partial differential equations with time-dependent coefficients using stopping times of exponential functionals of Brownian martingales and a non-homogeneous heat semigroup. In particular we derive lower bounds for the probability of blowup in finite time, and we provide sufficient conditions for the existence of global positive solutions.

Analysis of a stochastic ratio-dependent predator–prey model driven by Lévy noise

In this paper we consider a non-autonomous ratio-dependent predator–prey system driven by Lévy noise. Firstly, we show the existence of global positive solution and stochastic boundedness. Secondly, the conditions of persistent in mean and extinction are established and we also give the asymptotic properties of the solution. Finally, we simulate the model to illustrate our main analytical results.

Dynamics Analysis of a Stochastic SIR Epidemic Model

We investigate an SIR epidemic model with stochastic perturbations. We assume that stochastic perturbations are of a white noise type which is directly proportional to the distances of three variables from the steady-state values, respectively. By constructing suitable Lyapunov functions and applying Itô’s formula, some qualitative properties are obtained, such as the existence of global positive solutions, stochastic boundedness, and permanence. A series of numerical simulations to illustrate these mathematical findings are presented.

Stochastic Logistic Systems with Jumps

This paper is concerned with a stochastic nonautonomous logistic model with jumps. In the model, the martingale and jump noise are taken into account. This model is new and more feasible and applicable. Sufficient criteria for the existence of global positive solutions are obtained; then asymptotic boundedness inpth moment, stochastically ultimate boundedness, and asymptotic pathwise behavior are to be considered.

Global solutions to norm-preserving non-local flows of porous media type

In this paper, we study the global existence of positive solutions to the norm-preserving non-local heat flow of the porous-media type equations on the compact Riemannian manifold (M, g) with the Cauchy data u0 > 0 on M, where r ≥ 1, p > 1 and λ(t) is chosen to make the L2-norm of the solution u (or a power of u) constant. We show that the limit is an eigenfunction for the Laplacian operator. We use some tricky estimates through the Sobolev imbedding theorem and the Moser iteration method.

Stochastic Delay Population Dynamics under Regime Switching: Global Solutions and Extinction

This paper is concerned with a delay Lotka-Volterra model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall inequality and Young’s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Finally, an example is given to illustrate the main results.

Dynamical Analysis of a Stochastic Predator-Prey Model with an Allee Effect

We present and analyze a modified Holling type-II predator-prey model that includes some important factors such as Allee effect, density-dependence, and environmental noise. By constructing suitable Lyapunov functions and applying Itô formula, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. A series of numerical simulations to illustrate these mathematical findings are presented.

QUALITATIVE ANALYSIS OF A STOCHASTIC PREDATOR–PREY SYSTEM WITH DISEASE IN THE PREDATOR

A stochastic predator–prey system with disease in the predator population is proposed, the existence of global positive solution is derived. When the white noise is small, there is a stationary distribution. In addition, conditions of global stability for the deterministic system are also established from the above result. By Lyapunov function, the long time behavior of solution around the disease-free equilibrium of deterministic system is derived. These results mean that stochastic system has the similar property with the corresponding deterministic system. When the white noise is small, however, large environmental noise makes the result different. Finally, numerical simulations are carried out to support our findings.

Positive solutions for a class of multi-parameter elliptic systems

In this paper, we deal with the global existence of positive solutions for a multi-parameter system of second-order elliptic equations with respect to parameters, by using the fixed point index theory in cones, the sub-supersolutions method, Leray–Schauder degree theory and a priori estimates technique. In addition, for the computation of Leray–Schauder degree on the Cartesian product of two ordered intervals, we also establish a result about Leray–Schauder degree on a strict ordered interval induced by a pair of strict lower and upper solutions for a system of second-order elliptic equations.

A multiplicity result of positive solutions for a class of multi-parameter ordinary differential systems

In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As applications, we prove the global existence of positive solutions for a multi-parameter system of second-order ordinary differential equations with respect to parameters. The discussion is based on the result of Leray-Schauder degree on the order interval and the fixed point index theory in cones.

DYNAMICAL PROPERTIES OF A STOCHASTIC TWO-SPECIES SCHOENER'S COMPETITIVE MODEL

A stochastic two-species Schoener's competitive model is proposed and investigated. Sufficient conditions for the existence of global positive solutions, boundedness, uniform continuity, global attractivity, stochastic permanence and extinction are obtained. Moreover, the upper-growth rate and the average in time of the sample paths of solutions are also estimated. Finally, some figures are introduced to illustrate the main results.

Multiplicity of positive solutions for a [formula omitted]-Laplacian system and its applications

In this paper, we establish the product formula for the fixed point index on product cone, and the relation between Leray–Schauder degree and a pair of strict lower and upper solutions for a (p1,p2)-Laplacian system. Based on the product formula of the fixed point index and Leray–Schauder degree theory, we deal with the multiplicity of positive solutions for a class of (p1,p2)-Laplacian systems. As applications, we prove the global existence of positive solutions for a multi-parameter system of (p1,p2)-Laplacian equations with respect to parameters.

Stochastic Delay Logistic Model under Regime Switching

This paper is concerned with a delay logistical model under regime switching diffusion in random environment. By using generalized Itô formula, Gronwall′s inequality, and Young′s inequality, some sufficient conditions for existence of global positive solutions and stochastically ultimate boundedness are obtained, respectively. Also, the relationships between the stochastic permanence and extinction as well as asymptotic estimations of solutions are investigated by virtue of V‐function technique, M‐matrix method, and Chebyshev′s inequality. Finally, an example is given to illustrate the main results.

The Complex Dynamics of a Stochastic Predator‐Prey Model

A modified stochastic ratio‐dependent Leslie‐Gower predator‐prey model is formulated and analyzed. For the deterministic model, we focus on the existence of equilibria, local, and global stability; for the stochastic model, by applying Itô formula and constructing Lyapunov functions, some qualitative properties are given, such as the existence of global positive solutions, stochastic boundedness, and the global asymptotic stability. Based on these results, we perform a series of numerical simulations and make a comparative analysis of the stability of the model system within deterministic and stochastic environments.

Heat flow method for Lichnerowicz type equations on closed manifolds

In this paper, we establish existence results for positive solutions to the Lichnerowicz equation of the following type in closed manifolds -\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M, where $p>1, q>0$, and $A(x)>0$, $B(x)\geq0$ are given smooth functions. Our analysis is based on the global existence of positive solutions to the following heat equation {ll} u_t-\Delta u=A(x)u^{-p}-B(x)u^{q},\quad in\quad M\times\mathbb{R}^{+}, u(x,0)=u_0,\quad in\quad M with the positive smooth initial data $u_0$.

A degenerate parabolic equation with a nonlocal source and an absorption term

This article deals with a class of nonlocal and degenerate quasilinear parabolic equation u t = f(u)(Δu + a∫Ω u(x, t)dx − u) with homogeneous Dirichlet boundary conditions. The local existence of positive classical solutions is proved by using the method of regularization. The global existence of positive solutions and blow-up criteria are also obtained. Furthermore, it is shown that, under certain conditions, the solutions have global blow-up property. When f(s) = s p , 0 < p ≤ 1, the blow-up rate estimates are also obtained.

Existence and uniqueness of global positive solutions to the stochastic functional Kolmogorov-type system

In general, any population systems are often subject to various environmental noise. To examine whether the presence of such noise affects these systems significantly, we examine a stochastic functional Kolmogorov-type population system with the general stochastic perturbation. In this paper, we show that the environmental noise may suppress the potential population explosion and guarantee global existence of positive solutions when the noise intensity is strongly dependent on the population size. However, when the noise intensity is weakly dependent on the population size, the stochastic population system behaves similarly to the corresponding deterministic system. To illustrate our idea clearly, we also discuss some stochastic Lotka-Volterra systems as special cases.

Finite-time blowup and existence of global positive solutions of a semi-linear SPDE

We consider stochastic equations of the prototype d u ( t , x ) = ( Δ u ( t , x ) + u ( t , x ) 1 + β ) d t + κ u ( t , x ) d W t on a smooth domain D ⊂ R d , with Dirichlet boundary condition, where β , κ are positive constants and { W t , t ≥ 0 } is a one-dimensional standard Wiener process. We estimate the probability of finite-time blowup of positive solutions, as well as the probability of existence of non-trivial positive global solutions.

Population dynamical behavior of Lotka–Volterra system under regime switching

In this paper, we investigate a Lotka–Volterra system under regime switching d x ( t ) = diag ( x 1 ( t ) , … , x n ( t ) ) [ ( b ( r ( t ) ) + A ( r ( t ) ) x ( t ) ) d t + σ ( r ( t ) ) d B ( t ) ] , where B ( t ) is a standard Brownian motion. The aim here is to find out what happens under regime switching. We first obtain the sufficient conditions for the existence of global positive solutions, stochastic permanence and extinction. We find out that both stochastic permanence and extinction have close relationships with the stationary probability distribution of the Markov chain. The limit of the average in time of the sample path of the solution is then estimated by two constants related to the stationary distribution and the coefficients. Finally, the main results are illustrated by several examples.

Existence of global positive solutions of semilinear elliptic equations

Using a very simple approach, we prove the existence of asymptotically vanishing global positive solutions of the semilinear elliptic equation Δ u + f ( x , u ) + g ( | x | ) x ⋅ ∇ u = 0 in all of R n , n ≥ 3 . In particular, earlier conditions on the negative part of the function f are dropped.