Narrow Results

Brusselator model is a very typical autocatalytic reaction diffusion system. The bifurcation of steady-states of Brusselator model can be used to explain spot patterns of certain animals such as leopard and jaguar. Periodic patterns can be found throughout whole natural world, so it is very interesting to study patterns generated by the bifurcation of periodic solutions in extended Brusselator (EB) model, which extends Brusselator to T-periodic coefficients. In this paper, we study extended simplified Brusselator (ESB) model, which is EB model without diffusion terms. We find a unique T-periodic solution x₀(t) in the strictly positively invariant region and prove its stability. This result establishes a foundation to study the bifurcation of EB model from x₀(t). We also develop techniques of using degree theory and Floquet theory to analyze existence, uniqueness and stability of a periodic solution.

In this paper, we considered an eco-epidemic model with impulsive birth. By using the coincidence degree theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one strictly positive periodic solutions.

By employing a fixed point theorem in cones, this paper investigates the existence of almost periodic solutions for an impulsive logistic equation with infinite delay. A set of sufficient conditions on the existence of almost periodic solutions of the equation is obtained.

A new non-autonomous predator-prey system with the effect of viruses on the prey is investigated. By using the method of coincidence degree, some sufficient conditions are obtained for the existence of a positive periodic solution. Moreover, with the help of an appropriately chosen Lyapunov function, the global attractivity of the positive periodic solution is discussed. In the end, a numerical simulation is used to illustrate the feasibility of our results.

In this paper, a reaction–diffusion model describing temporal development of tumor tissue, normal tissue and excess H⁺ ion concentration is considered. Based on a combination of perturbation methods, the Fredholm theory and Banach fixed point theorem, we theoretically justify the existence of the traveling wave solution for this model.

The asymptotic behavior of an almost periodic competitive system is investigated. By using differential inequality, the module containment theorem and the Lyapunov function, a good understanding of the existence and global asymptotic stability of positive almost periodic solutions is obtained. Finally, an example and numerical simulations are performed for justifying the theoretical results.

In this paper, a class of nonautonomous Lotka–Volterra type multispecies competitive systems with delays is studied. By employing Lyapunov functional, some sufficient conditions to guarantee the existence of almost periodic solutions for the Lotka–Volterra system are obtained.

A mathematical model for avian influenza with optimal control strategies is presented as a system of discrete time delay differential equations (DDEs) and its important mathematical features are analyzed. In alignment to manage this, we develop an optimally controlled pandemic model of avian influenza and insert a time delay with exponential factor. Then we apply two controlled functions in the form of biosecurity of poultry and the education campaign against avian influenza to control the disperse of the disease. Our optimal control strategies will minimize the number of contaminated humans and contaminated birds. We also derive the basic reproduction number to examine the dynamical behavior of the model and demonstrate the existence of the controlled system. For the justification of our work, we present numerical simulations.

In this paper, a free boundary problem for a solid avascular tumor growth under the action of periodic external inhibitors with time delays in proliferation is studied. Sufficient conditions for the global stability of tumor-free equilibrium are given. Moreover, if external concentration of nutrients is large, we also prove that the tumor will not disappear and determine the conditions under which there exist periodic solutions to the model. The results show that the periodicity of the inhibitor may imply periodicity of the size of the tumor. More precisely, if ${\sigma }_{\infty }$ (the concentration of external nutrients) is greater than $\mu {\beta }^{\ast }+\nu$, where $\mu ,\nu$ are two constants; ${\beta }^{\ast }={max}_{0\le t\le \omega }\varphi (t)$; $\varphi (t)$ is a periodic function which can be interpreted as a treatment and $\omega$ is the period of $\varphi (t)$. Results are illustrated by computer simulations.

In this paper, we investigate the global existence of nonnegative solutions of a two-species Keller–Segel model with Lotka–Volterra competitive source terms. By raising the regularity of a solution from $L^1$ to $L^p(p>1)$, the existence and uniqueness of the classical global in time solution to this chemotaxis model is proved for any chemotactic coefficients ${\chi }_1,{\chi }_2>0$ when the space dimension is one. Furthermore, it is shown that the model has a unique classical global solution in two and three space dimensions if the chemotactic coefficients ${\chi }_1$ and ${\chi }_2$ are small as compared to the diffusion coefficient $d_3$ of the chemoattractant.

The purpose of this paper is to investigate asymptotic behaviors of the solutions for a competition system with random vs. nonlocal dispersal. We first prove the existence of invasion traveling waves via using the theory of asymptotic speeds of spread. Then we prove the invasion traveling waves are exponentially stable as perturbation in some exponentially weighted spaces by using the weighted energy and the squeezing technique.

This paper is concerned with the traveling wave solutions for a discrete SIR epidemic model with a saturated incidence rate. We show that the existence and non-existence of the traveling wave solutions are determined by the basic reproduction number $R_0$ of the corresponding ordinary differential system and the minimal wave speed $c^{\ast }$. More specifically, we first prove the existence of the traveling wave solutions for $R_0>1$ and $c>c^{\ast }$ via considering a truncated initial value problem and using the Schauder’s fixed point theorem. The existence of the traveling wave solutions with speed $c=c^{\ast }$ is then proved by using a limiting argument. The main difficulty is to show that the limit of a decreasing sequence of the traveling wave solutions with super-critical speeds is non-trivial. Finally, the non-existence of the traveling wave solutions for $R_0>1,$$0<c<c^{\ast }$ and $R_0\le 1,$$c>0$ is proved.

We investigate an optimal harvesting problem for age-structured species, in which elder individuals are more competitive than younger ones, and the population is modeled by a highly nonlinear integro-partial differential equation with a global feedback boundary condition. The existence of optimal strategies is established by means of compactness and maximizing sequences, and the maximum principle obtained via an adjoint system, tangent-normal cones and a new continuity result. In addition, some numerical experiments are presented to show the effects of the price function and younger’s weight on the optimal profits.

A diffusive SIS epidemic model with Holling II incidence rate is studied in this paper. We introduce the basic reproduction number ${{ℛ}}_0$ first. Then the existence of endemic equilibrium (EE) can be determined by the sizes of ${{ℛ}}_0$ as well as the diffusion rates of susceptible and infected individuals. We also investigate the effect of diffusion rates on asymptotic profile of EE. Our results conclude that the infected population will die out if the diffusion rate of susceptible individuals is small and the total population $N$ is below a certain level; while the two populations persist eventually if at least one of the diffusion rates of the susceptible and infected individuals is large.

In this paper, three competitive systems with different kinds of state-dependent control are presented and investigated. The existence of the order-1 homoclinic orbit and order-1 periodic solution of the two systems that incorporate just one kind of state-dependent control is obtained by applying differential equation geometry theory, and the stability of the order-1 periodic solution of each system is also given. Besides, sufficient conditions for the existence and stability of the order-2 periodic solution of the system that incorporate two kinds of state-dependent control are gained by successor function method and analogue of Poincaré criterion, respectively. Finally, numerical simulations are carried out to verify the theoretical results.