Positive almost periodic solution for a class of Nicholson’s blowflies model with multiple time-varying delays

In this paper, we study the existence and exponential convergence of positive almost periodic solutions for the generalized Nicholson’s blowflies model with multiple time-varying delays. Under proper conditions, we establish some criteria to ensure that the solutions of this model converge locally exponentially to a positive almost periodic solution. Moreover, we give some examples to illustrate our main results.     

A note on positive non-constant steady-state solutions to a reaction-diffusion model

In the present paper, we investigate a reaction-diffusion system with feedback effect subject to the homogeneous Neumann boundary condition and study the positive steady-state solutions. We establish a priori estimates for positive steady-state solutions and derive some results for non-existence of positive non-constant steady-state solutions. Our analysis complements the existing results on this model.     

Positive solutions of a diffusive prey–predator model in a heterogeneous environment

In this paper, we investigate a diffusive prey–predator model in a spatially degenerate heterogeneous environment. We are concerned with the positive solutions of the model, and obtain some results for the existence and non-existence of positive solutions. Moreover, the multiplicity, stability and asymptotical behaviors of positive solutions with respect to the parameters are also studied.     

Almost periodic solution for Nicholson’s blowflies model with patch structure and linear harvesting terms

In this paper, we study the existence and exponential convergence of positive almost periodic solutions for a class of Nicholson’s blowflies model with patch structure and multiple linear harvesting terms. Under appropriate conditions, we establish some criteria to ensure that the solutions of this system converge locally exponentially to a positive almost periodic solution. Moreover, we give some examples and numerical simulations to illustrate our main results.     

Positive almost periodic solution for a class of Nicholson’s blowflies model with a linear harvesting term

In this paper, we study the problem of positive almost periodic solutions for the generalized Nicholson’s blowflies model with a linear harvesting term and multiple time-varying delays. By applying the fixed point theorem and the Lyapunov functional method, we establish some criteria to ensure that the solutions of this model converge locally exponentially to a positive almost periodic solution. Moreover, we give an example to illustrate our main results.     

Positive solutions for a diffusive Bazykin model in spatially heterogeneous environment

In this paper, we investigate a diffusive Bazykin model in a spatially heterogeneous environment. We obtain some results on nonexistence and existence of positive solutions of the model. Moreover, the asymptotic behavior of positive solutions with respect to certain parameters is also studied.     

Positive solutions of impulsive boundary value problems for first-order nonlinear functional differential equations

In this paper, we study the problem on the existence of positive solutions for a class of impulsive periodic boundary value problems of first-order nonlinear functional differential equations. By using the fixed point theorem in cones and some analysis techniques, we present some sufficient conditions which guarantee the existence of one and multiple positive solutions for the impulsive periodic boundary value problems. Our results generalize and improve some previous results. Moreover, our results show that positive solutions for the impulsive periodic boundary value problems may be yielded completely by some proper impulsive conditions (see Example 4.1 and Remark 4.2 in Sect. 4), and also implies that proper impulsive conditions are of great significance to simulate processes, optimal control, population model and so on.     

Global asymptotic behaviour for a Nicholson model with patch structure and multiple delays

For a Nicholson’s blowflies model with patch structure and multiple discrete delays, we study some aspects of its global dynamics. Conditions for the absolute global asymptotic stability of both the trivial equilibrium and a positive equilibrium (when it exists) are given. The existence of positive heteroclinic solutions connecting the two equilibria is also addressed. We further consider a diffusive Nicholson-type model with patch structure, and establish a criterion for the existence of positive travelling wave solutions, for large wave speeds. Several applications illustrate the results, improving some criteria in the recent literature.     

Some nonexistence results for nonconstant stationary solutions to the Gray–Scott model in a bounded domain

In the present paper, we are concerned with a reaction–diffusion system well-known as the Gray–Scott model in a bounded domain and study the corresponding steady-state problem. We establish some results for the nonexistence of nonconstant positive stationary solutions.     

New results of existence and stability of periodic solution for a delay multispecies Logarithmic population model

In this Letter, for a general class of delayed periodic multispecies Logarithmic population model, we prove some new results on the existence of positive periodic solutions by contraction principle. The global exponential stability of positive periodic solutions is discussed further, and conditions for exponential convergence are given. The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases.     

Properties of positive solutions to an elliptic equation with negative exponent

In this paper, we study some quantitative properties of positive solutions to a singular elliptic equation with negative power on the bounded smooth domain or in the whole Euclidean space. Our model arises in the study of the steady states of thin films and other applied physics as well as differential geometry. We can get some useful local gradient estimate and L¹ lower bound for positive solutions of the elliptic equation. A uniform positive lower bound for convex positive solutions is also obtained. We show that in lower dimensions, there is no stable positive solutions in the whole space. In the whole space of dimension two, we can show that there is no positive smooth solution with finite Morse index. Symmetry properties of related integral equations are also given.     

A qualitative study on general Gause-type predator–prey models with non-monotonic functional response

In this paper, we study a diffusive predator–prey model with general growth rates and non-monotonic functional response under homogeneous Neumann boundary condition. A local existence of periodic solutions and the asymptotic behavior of spatially inhomogeneous solutions are investigated. Moreover, we show the existence and non-existence of non-constant positive steady-state solutions. Especially, to show the existence of non-constant positive steady-states, the fixed point index theory is used without estimating the lower bounds of positive solutions. More precisely, calculating the indexes at the trivial, semi-trivial and positive constant solutions, some sufficient conditions for the existence of non-constant positive steady-state solutions are studied. This is in contrast to the works in previous papers. Furthermore, on obtaining these results, we can observe that the monotonicity of a prey isocline at the positive constant solution plays an important role.     

OPTIMAL EXISTENCE THEORY FOR SINGLE AND MULTIPLE POSITIVE PERIODIC SOLUTIONS TO FUNCTIONAL DIFFERENCE EQUATIONS WITH FEEDBACK CONTROL

Using a fixed point theorem in a cone, we obtain some optimal existence results for single and multiple positive periodic solutions to a functional difference system with feedback control. Moreover, we apply our results to a population model.     

Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis

In this work, we study the existence and global exponential convergence of positive almost periodic solutions for the generalized model of hematopoiesis. Under appropriate conditions, we employ a novel proof to establish some criteria for ensuring that all solutions of this model converge exponentially to the positive almost periodic solution.     

QUALITATIVE ANALYSIS OF BOBWHITE QUAIL POPULATION MODEL

In this paper, the qualitative behavior of solutions of the bobwhite quail pop-ulation modelwhere 0   

Unbounded connected component of the positive solution set of nonlinear operator equation and its applications

In this paper, by using global bifurcation theories we obtain some results for structure of positive solution set of some nonlinear equations with parameters. As a result, we obtain some existence results for positive solutions of the nonlinear operator equation. The main result can be applied to various of differential boundary value problems to obtain the existence results for positive solutions without the assumption that the nonlinearities are positone.     

Positive Solutions for a Lotka–Volterra Prey–Predator Model with Cross-Diffusion of Fractional Type

In this paper we consider a Lotka–Volterra prey–predator model with cross-diffusion of fractional type. The main purpose is to discuss the existence and nonexistence of positive steady state solutions of such a model. Here a positive solution corresponds to a coexistence state of the model. Firstly we study the stability of the trivial and semi-trivial solutions by analyzing the principal eigenvalue of the corresponding linearized system. Secondly we derive some necessary conditions to ensure the existence of positive solutions, which demonstrate that if the intrinsic growth rate of the prey is too small or the death rate (or the birth rate) of the predator is too large, the model does not possess positive solutions. Thirdly we study the sufficient conditions to ensure the existence of positive solutions by using degree theory. Finally we characterize the stable/unstable regions of semi-trivial solutions and coexistence regions in parameter plane.     

On a new fractional-order Logistic model with feedback control

In this paper, we formulate and analyze a new fractional-order Logistic model with feedback control, which is different from a recognized mathematical model proposed in our very recent work. Asymptotic stability of the proposed model and its numerical solutions are studied rigorously. By using the Lyapunov direct method for fractional dynamical systems and a suitable Lyapunov function, we show that a unique positive equilibrium point of the new model is asymptotically stable. As an important consequence of this, we obtain a new mathematical model in which the feedback control variables only change the position of the unique positive equilibrium point of the original model but retain its asymptotic stability. Furthermore, we construct unconditionally positive nonstandard finite difference(NSFD) schemes for the proposed model using the Mickens' methodology. It is worth noting that the constructed NSFD schemes not only preserve the positivity but also provide reliable numerical solutions that correctly reflect the dynamics of the new fractional-order model. Finally, we report some numerical examples to support and illustrate the theoretical results. The results indicate that there is a good agreement between the theoretical results and numerical ones.     

二阶时滞微分方程边值问题的上下解方法

主要利用上下解方法和Schauder不动点定理,研究一类二阶时滞微分方程边值问题的正解存在性.为了验证结论的正确性,本文在结尾部分给出了两个例子.     

Positive solutions for ratio-dependent predator-prey interaction systems

In this paper, we study the dynamics of predator-prey interaction systems between two species with ratio-dependent functional responses. First we provide sufficient and necessary conditions for positive steady-state solutions, and then we investigate the relationships between positive equilibria and positive solutions of the system over a large domain. Furthermore, we deal with the uniqueness and the stability of positive steady-states solutions with some assumptions. In addition, we discuss the extinction and the persistence results of time-dependent positive solutions to the system.