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This paper is devoted to study patterns and dynamics in the diffusive minimal sediment model. Firstly, we analyze nonnegative constant equilibrium solutions and investigate patterns induced by diffusions. Then we study the properties and existence of nonconstant steady states. Moreover, we describe the local bifurcation structure and global bifurcation structure from two positive constant solutions, respectively. The main tools used here include the stability theory, degree theory, bifurcation theory. From extensive numerical simulations, the theoretical results are confirmed and complemented, and the influence of parameters δ 1 , δ 2 on these patterns is depicted. [ABSTRACT FROM AUTHOR]

In this work, we study a spatially heterogeneous predator–prey model where the interaction is governed by a Holling II functional response. The existence of coexistence states of the two species by using the bifurcation theory is discussed. As a result, it is shown that the degeneracy for the prey and the predator has distinctly different effects on the coexistence states of the two species when the intrinsic growth rate of the prey is above a certain critical value. We also study the asymptotic behavior of positive stationary solutions as the intrinsic growth rate of the predator tends to a certain critical value. [ABSTRACT FROM AUTHOR]

This paper is concerned with positive solutions of a predator–prey model with cross-diffusion. By virtue of the Leray–Schauder degree theory and the bifurcation theory, some sufficient conditions for the existence of positive solutions of the system are established. In particular, we derive the multiplicity results when some parameters are suitably large. Moreover, the existence and stability of nonconstant solutions are studied by the techniques of space decomposition and the implicit function theorem. Finally, the uniqueness of positive solution is studied when the spatial dimension is one. [ABSTRACT FROM AUTHOR]

In this paper, a two-species glycolysis model is investigated in which one species is substrate and the other is activator. A linear stability analysis shows that there is a critical value for the diffusion rate of the substrate above which the constant steady state solution is of Turing’s instability. Next, the steady state bifurcations are analyzed not only from a simple eigenvalue, but more difficultly, from a double one. The theoretical results are confirmed by numerical simulations. Our main methods are based on bifurcation theory, Lyapunov–Schmidt technique and singularity theory. [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with a -dimensional competition model between plasmid-bearing and plasmid-free organisms in the unstirred chemostat, which incorporates the Beddington–DeAngelis functional response and inhibitor. By the application of degree theory in cones, bifurcation theory and perturbation technique, we obtain the uniqueness, multiplicity and stability of the coexistence solutions of the steady-state system when the effects of the inhibitor is large enough. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider the dynamic behavior of an impulsive delay differential equation model with the effects of interspecific allelopathic interaction. A good understanding of the permanence, extinction and the existence of positive periodic solutions is gained. It turns out that the impulsive controls play a crucial role in shaping the above dynamics of the system. Numerical simulations are presented to substantiate the analytical results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a predator–prey model possessing a non-monotonic conversion rate. The main purpose is to determine the multiple existence and stability of positive steady-state solutions to this system. The results show that if the parameter is suitably large, then the system contains an -shaped global bifurcation curve with respect to a bifurcation parameter. That is, the system has two or three positive solutions for a suitable range of parameters. Moreover, the stability of positive solutions on this curve is also given. If is properly small, both uniqueness and non-uniqueness results can occur. The main tools used here include the bifurcation theory, the Lyapunov–Schmidt procedure, and the perturbation technique. [Copyright &y& Elsevier]

Abstract: This paper is purported to study a reaction diffusion system arising from a ratio-dependent predator–prey model with stage structure. The global asymptotic stability of the unique positive constant equilibrium is obtained under suitable conditions; existence and non-existence results of non-trivial solution are derived. [Copyright &y& Elsevier]

This paper is concerned with a diffusive predator–prey system with multiple Allee effects induced by fear factors subject to Neumann boundary conditions. Firstly, we introduce a priori estimates of non-negative and non-trivial solutions. Moreover, nonexistence of non-constant positive solutions are shown for certain parameters ranges. Secondly, we study the stability of non-negative constant solutions by the linearized theory. Then, by choosing the Allee threshold θ as the bifurcation parameter, we investigate the spatially homogeneous and non-homogeneous periodic solutions as well as non-constant steady-state solutions. Finally, some theoretical results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we study the stationary problem for the Lotka–Volterra competition system with cross-diffusion in a spatially heterogeneous environment. Although some sufficient conditions for the existence of positive solutions are obtained by using global bifurcation theory, the information for their structure is far from complete. In order to get better understanding of the competition system with cross-diffusion, we focus on the asymptotic behaviour of positive solutions and derive two shadow systems as the cross-diffusion coefficient tends to infinity, moreover, the structure of positive solutions of the limiting system is analysed. The result of asymptotic behaviour also reveals different phenomena from that studied in Wang and Li (2013). [ABSTRACT FROM AUTHOR]

In this paper, we study the coexistence of an extended Klausmeier model with cross-diffusion and nonlocal sustained grazing. First, we analyze a saddle–node bifurcation of spatially homogeneous system. Second, we focus on the reaction–diffusion system with nonlocal sustained grazing. Our main result is that nonlocal terms promote linear stability, and the system may produce pattern under the influences of self-diffusion and cross-diffusion. Moreover, both the grazing parameter and rainfall rate can induce transitions among bare soil state, vegetation pattern state and homogeneous vegetation state. Finally, we address the nonlocal reaction–diffusion system as a bifurcation problem, and analyze the existence and stability of bifurcation solutions. Furthermore, numerical simulations have been illustrated to verify our theoretical findings. [ABSTRACT FROM AUTHOR]

We study a reaction-diffusion system of the unstirred chemostat model with toxins in the N-dimensional case. Firstly, some sufficient conditions for the existence of positive steady-state solutions are obtained by means of the fixed point index theory. Secondly, the local structure of positive steady-state solutions and their stability are discussed by the bifurcation theory. Lastly, the effect of the toxins is investigated by the perturbation technique. The results show that when the parameter β, which measures the effect of toxins, is large enough, this model has no coexistence solution pHrovided that the maximal growth rate b of the species v is greater than σ ˆ 1 1 − k, and all positive solutions of this model are controlled by a limit system provided that b belongs to the interval (σ 1 1 − k, σ ˆ 1 1 − k). [ABSTRACT FROM AUTHOR]

Abstract: This paper deals with the unstirred chemostat model in the presence of an internal inhibitor. The main purpose of the paper is to determine the exact range of the maximal growth of two species where the system possesses positive solutions. It turns out that is a connected unbounded region in , whose boundary consists of two monotone nondecreasing curves and . For every inside the system has positive solutions and for outside there exists no positive solution. The functions and are constructed in terms of the limit of the corresponding time-dependent solution with a specific initial function. In particular, it is also shown that the system has at least two positive solutions in certain subregion of . [Copyright &y& Elsevier]

In this paper, we investigate a Lotka–Volterra competition model with Danckwerts boundary conditions in a one-dimensional habitat where one species assumes pure random diffusion while another one undergoes mixed movement (both random and directed movements). We focus on the joint influence of advection rate, intrinsic growth rate and interspecific competition coefficient on the competition outcomes. It turns out that there exist some critical curves which separate the stable region of the semitrivial steady states from the unstable one. The locations of these curves determine whether coexistence or bistability occurs. More precisely, there are various tradeoffs between advection rate, intrinsic growth rate and interspecific competition coefficient that allow the transition of competition outcomes including competition exclusion, coexistence and bistability. We illustrate our results in various parameter spaces. [ABSTRACT FROM AUTHOR]