Traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity

In this paper, we consider the existence of traveling wave solutions in delayed higher dimensional lattice differential systems with partial monotonicity. By relaxing the monotonicity of the upper solutions and allowing it greater than positive equilibrium point, we establish the existence of traveling wave solutions by means of Schauder's fixed point theorem. And then, we apply our results to delayed competition‐cooperation systems on higher dimensional lattices.     

Travelling waves of a predator–prey model with a spatiotemporal delay

This paper is concerned with the existence of traveling waves to a predator–prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper–lower solutions and by using the cross‐iteration method as well as the Schauder's fixed‐point theorem, the existence of a traveling wave solution connecting the semi‐trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.     

Traveling wave solutions in a delayed lattice competition-cooperation system

In this paper, we first reduce the existence of traveling wave solutions in a delayed lattice competition-cooperation system to the existence of a pair of upper and lower solutions by means of Schauder’s fixed point theorem and the cross iteration scheme, and then we construct a pair of upper and lower solutions to obtain the existence and nonexistence of traveling wave solutions. We also consider the asymptotic behaviour of any nonnegative traveling wave solutions at negative infinity.     

Global bifurcation of solutions for a predator–prey model with prey‐taxis

We study pattern formations in a predator–prey model with prey‐taxis. It is proved that a branch of nonconstant solutions can bifurcate from the positive equilibrium only when the chemotactic is repulsive. Furthermore, we find the stable bifurcating solutions near the bifurcation point under suitable conditions. Copyright © 2014 John Wiley & Sons, Ltd.     

Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity. Based on two different mixed-quasi monotonicity reaction terms, we propose new conditions on the reaction terms and new definitions of upper and lower solutions. By using Schauder’s fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive diffusive Lotka-Volterra systems.     

Global existence of solutions to reaction–diffusion systems without conditions on the nonlinearities growth

It is well known that, for reaction–diffusion systems, if the nonlinearities grow faster than a polynomial, nothing seems to be known for instance. The purpose of this paper is to give sufficient conditions guaranteeing global existence, uniqueness and uniform boundedness of solutions for coupled reaction–diffusion equations without condition growth on the reactions terms f and g in case f + g ≠ 0. These systems possess many and various applications in physics as the diffusion of the Phosphorus in the Silicone or some models describing some nuclear reactions; there have also been other applications in chemistry and biology. Our techniques are based on the Lyapunov functional methods. Copyright © 2010 John Wiley & Sons, Ltd.     

Traveling wave solutions for a delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies

In this paper, we study the traveling wave solutions of a delayed diffusive SIR epidemic model with nonlinear incidence rate and constant external supplies. We find that the existence of traveling wave solutions is determined by the basic reproduction number of the corresponding spatial‐homogenous delay differential system and the minimal wave speed. The existence is proved by applying Schauder's fixed point theorem and Lyapunov functional method. The non‐existence of traveling waves is obtained by two‐sided Laplace transform. Copyright © 2016 John Wiley & Sons, Ltd.     

Traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity

This paper deals with the existence of traveling wave solutions in delayed nonlocal diffusion systems with mixed monotonicity. Based on two different mixed-quasimonotonicity reaction terms, we propose new definitions of upper and lower solutions. By using Schauder's fixed point theorem and a new cross-iteration scheme, we reduce the existence of traveling wave solutions to the existence of a pair of upper and lower solutions. The general results obtained have been applied to type-K monotone and type-K competitive nonlocal diffusive Lotka-Volterra systems.     

Multiplicity of positive almost periodic solutions in a delayed Hassell–Varley‐type predator–prey model with harvesting on prey

In this paper, we consider a delayed Hassell–Varley‐type predator–prey model with harvesting on prey. By means of Mawhin's continuation theorem of coincidence degree theory, some new sufficient conditions are obtained for the existence of at least two positive almost periodic solutions for the aforementioned model. To the best of the author's knowledge, so far, the result of this paper is completely new. An example is employed to illustrate the result of this paper. Copyright © 2013 John Wiley & Sons, Ltd.     

Traveling wave solutions for the combustion model of a shear flow in a cylinder

We establish traveling wave solutions for the combustion model of a shear flow in a cylinder. We study two cases: the infinite Lewis number and an arbitrary Lewis number. For the infinite Lewis number, we establish the existence of traveling wave fronts for both non‐minimal and minimal speeds. For an arbitrary Lewis number, we establish the uniform bounds and exponential decay rates. Copyright © 2015 John Wiley & Sons, Ltd.     

Multiple periodic solutions in delayed Gause-type ratio-dependent predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple periodic solutions in delayed Gause-type ratio-dependent predator–prey systems with numerical responses. As corollaries, some applications are listed.     

Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity

This paper is concerned with the existence of traveling wave fronts for delayed non-local diffusion systems without quasimonotonicity, which can not be answered by the known results. By using exponential order, upper-lower solutions and Schauder's fixed point theorem, we reduce the existence of monotone traveling wave fronts to the existence of upper-lower solutions without the requirement of monotonicity. To illustrate our results, we establish the existence of traveling wave fronts for two examples which are the delayed non-local diffusion version of the Nicholson's blowflies equation and the Belousov-Zhabotinskii model. These results imply that the traveling wave fronts of the delayed non-local diffusion systems without quasimonotonicity are persistent if the delay is small.     

Higher-order compact finite difference method for systems of reaction–diffusion equations

This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction–diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.     

Well‐posedness,positivity, and time asymptotics properties for a reaction–diffusion model of plankton communities,involving a rational nonlinearity with singularity

In this work, we consider a reaction–diffusion system, modeling the interaction between nutrients, phytoplankton, and zooplankton. Using a semigroup approach in , we prove global existence, uniqueness, and positivity of the solutions. The nonlinearity is handled by providing estimates in , allowing to deal with most of the functional responses that describe predator/prey interactions (Holling I, II, III, Ivlev) in ecology. The paper finally exhibits some time asymptotic properties of the solutions.     

Dynamics of a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects

This paper is concerned with a non‐selective harvesting predator–prey model with Hassell–Varley type functional response and impulsive effects. By using the fixed point theory based on monotone operator, some simple conditions are obtained for the existence of at least one positive periodic solution of the model. The existence result of this paper implies that the functional response on prey does not influence the existence of positive periodic solution of the model, which completes some results given in recent years. Further, by applying the comparison theorem in impulsive differential equations and constructing a suitable Lyapunov functional, the permanence and global attractivity of the model are also investigated. The main results in this paper extend, complement, and improve the previously known result. And some examples and numerical simulations are given to illustrate the feasibility and effectiveness of the main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Existence of periodic solutions in predator–prey and competition dynamic systems

In this paper, we systematically explore the periodicity of some dynamic equations on time scales, which incorporate as special cases many population models (e.g., predator–prey systems and competition systems) in mathematical biology governed by differential equations and difference equations. Easily verifiable sufficient criteria are established for the existence of periodic solutions of such dynamic equations, which generalize many known results for continuous and discrete population models when the time scale is chosen as or , respectively. The main approach is based on a continuation theorem in coincidence degree theory, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in dynamic equations on time scales. This study shows that it is unnecessary to explore the existence of periodic solutions of continuous and discrete population models in separate ways. One can unify such studies in the sense of dynamic equations on general time scales.     

Stability and Hopf bifurcation of a delayed reaction–diffusion neural network

In this paper, a delayed reaction–diffusion neural network with Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed. The existence of Hopf bifurcation at the trivial steady state is established. Using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae are derived to determine the direction and stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2011 John Wiley & Sons, Ltd.     

A predator–prey system with two impulses on the diseased prey and a Beddington–DeAngelis response

A predator–prey system with two impulses on the diseased prey is formulated and analyzed for the purpose of integrated pest management. The local and global stability of the susceptible pest‐eradication periodic solution, as well as the permanence of the system, are obtained under the sufficient conditions by means of Floquet's theory for impulsive differential equations. Finally, we interpret our mathematical results. Copyright © 2009 John Wiley & Sons, Ltd.     

Travelling wave solutions for a nonlocal dispersal HIV infection dynamical model

This paper is devoted to developing a nonlocal dispersal HIV infection dynamical model. The existence of travelling wave solutions is investigated by employing Schauder's fixed point theorem. That is, we study the existence of travelling wave solutions for $R_0>1$ and each wave speed $c>c^{?}$. In addition, the boundary asymptotic behaviour of travelling wave solutions at +∞ is obtained by constructing suitable Lyapunov functions and employing Lebesgue dominated convergence theorem. By employing a limiting argument, we investigate the existence of travelling wave solutions for $R_0>1$ and $c=c^{?}$. The main difficulties are that the semiflow generated by the model does not have the order-preserving property and the solutions lack of regularity.