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Fluid-mechanics studies are applied in mechanical engineering, biomedical engineering, oceanography, meteorology and astrophysics. In this paper, we investigate a (2+1)-dimensional extended Kadomtsev–Petviashvili II equation in fluid mechanics. Based on the Hirota bilinear method, we give a bilinear Bäcklund transformation. Via the extended homoclinic test technique, we construct the breather-wave solutions under certain constraints. We obtain the velocities of the breather waves, which depend on the coefficients in that equation. Besides, we derive the lump solutions with the periods of the breather-wave solutions tending to the infinity. Based on the polynomial-expansion method, travelling-wave solutions are constructed. We observe that the shapes of a breather wave and a lump remain unchanged during the propagation. We graphically discuss the effects of those coefficients on the breather wave and lump.

This work is devoted to the study of semi-linear elliptic systems in unbounded cylinders with linear dependence of the components of the nonlinearity vector. We reduce the study of such a problem with non-Fredholm operator to the study of a perturbation of some reaction-diffusion operator which satisfies the Fredholm property. Then sufficient conditions that ensure the structural stability of particular solutions are given. These conditions are applied to derive some existence results for some combustion model with complex chemistry and for some KPP like system.

This work is devoted to the study of a singular reaction–diffusion system arising in modelling the introduction of a lethal pathogen within an invading host population. In the absence of the pathogen, the host population exhibits a bistable dynamics (or Allee effect). Earlier numerical simulations of the singular SI model under consideration have exhibited stable travelling waves and also, under some circumstances, a reversal of the wave front speed due to the introduction of the pathogen. Here we prove the existence of such travelling wave solutions, study their linear stability and give analytical conditions yielding a reversal of the wave front speed, i.e. the invading host population may eventually retreat following the introduction of the lethal pathogen.

The existence and structural stability of travelling waves of systems of the free cytosolic calcium concentration in the presence of immobile buffers are studied. The proof is carried out by passing to zero with the diffusion coefficients of buffers. Thus, its method is different from Ref. 13 where the existence is proved straightforwardly.

In this paper we consider a mathematical model of cancer cell invasion of tissue (extracellular matrix). Two crucial components of tissue invasion are (i) cancer cell proliferation, and (ii) over-expression and secretion of proteolytic enzymes by the cancer cells. The proteolytic enzymes are responsible for the degradation of the tissue, enabling the proliferating cancer cells to actively invade and migrate into the degraded tissue. Our model focuses on the role of nonlocal kinetic terms modelling competition for space and degradation. The model consists of a system of reaction-diffusion-taxis partial differential equations, with nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. First of all we prove results concerning the local existence, uniqueness and regularity of solutions. We then prove global existence. Using Green's functions, we transform our original nonlocal equations into a coupled system of parabolic and elliptic equations and we undertake a numerical analysis of this equivalent system, presenting computational simulation results from our model showing the effect of the nonlocal terms (travelling waves we observed have the shape closely linked to the nonlocal terms). Finally, in the discussion section, concluding remarks are made and open problems are indicated.

This paper deals with problems of stability and travelling waves for a class of recurrent neural networks with arbitrary exponents k and m. A novel model which is not only nonlinear but also coupled is proposed. This paper makes the following contributions: (1) Conditions for local stablility of 1-D networks and 2-D networks are established with a series of mathematical arguments. (2) Completely convergence of 1-D neural networks is proved by constructing a suitable energy function. (3) The nonuniform solution of the networks is obtained when the connectivity is Gaussian profile. (4) Travelling waves of the networks are analyzed with the connectivity profile. Finally, simulation examples are employed to illustrate the obtained results.

Individual decision making is described as a bistable dynamical system. It can be influenced by the environment represented by other individuals, public opinion, all kinds of visual, oral and other information. We will study how the interaction of the individual decision making with the environment results in various patterns of decision making in the society.

We propose a reaction-diffusion model of the mechanisms involved in the healing of corneal surface wounds. The model focuses on the stimulus for increased mitotic and migratory activity, specifically the role of epidermal growth factor. We determine an analytic approximation for the speed of travelling wave solutions of the model in terms of the parameters and verify the results numerically. By comparing the predicted speed with experimentally measured healing rates, we conclude that serum-derived factors can alone account for the overall features of the healing process, but that the supply of growth factors by the tear film, in the absence of serum-derived factors, is not sufficient to give the observed healing rate. Numerical solutions of the model equations also confirm the importance of both migration and mitosis for effective wound healing. By modifying the model, we obtain an analytic prediction for the healing rate of corneal surface wounds when epidermal growth factor is applied topically to the wound.

In this work we study a spatial model for the West Nile Virus (WNV) propagation across the USA from the east to the west. WNV is an arthropod-borne flavivirus that appeared at first time in New York city in the summer of 1999 and then spread prolifically within birds. Mammals, as human and horse, do not develop sufficiently high bloodstream titers to play a significant role in transmission, which is the reason to consider the mosquito-bird cycle. The proposed model aims to study this propagation in a system of partial differential reaction-diffusion equations considering the mosquito and the avian populations. The diffusion is allowed to both populations, being greater in avian than in the mosquito. When a threshold value R₀, depending on the model's parameters, is greater than one, the disease remains endemic and could propagate to regions previously free of disease. The travelling wave solutions of the model are studied to determine the speed of the disease propagation. This wave speed is obtained as a function of the model's parameters, for instance, vertical transmission rate and avian diffusion coefficient.

In this paper, a food-chain model in a mangrove ecosystem with detritus recycling is analyzed. From the stability analysis of the delayed homogeneous system, an interval for the parameter representing detritus-detritivores interaction rate is obtained that imparts stability to the system around the coexistent state. Next, we have studied the model in a nonhomogeneous environment. The analysis revealed the existence of a subinterval of the above mentioned interval such that when the above interaction-rate lies within this interval, the system will undergo diffusion driven instability. Finally, we show the existence of travelling wave solutions for the said ecosystem. Numerical simulations are carried out to augment analytical results.

The reduction method is used to obtain some optimal stability results in biomathematics: an epidemic model with diffusion and the May-Leonard system for competition between three species with diffusion. The existence of travelling waves solutions is proved for the epidemic model.