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This paper is concerned with a diffusive Holling–Tanner prey–predator model in a bounded domain with Dirichlet boundary condition and a free boundary. The global existence of the unique solution is proved. Moreover, the criteria governing spreading–vanishing are derived by mainly using the comparison principle. The results show that if the length of the occupying line is bigger than a threshold value (spreading barrier), then the spreading of predators will make an achievement, and, if the length of the occupying line is smaller than this spreading barrier and the spreading coefficient is relatively small depending on initial size of predators, then the predators will fail in establishing themselves and eventually die out. [ABSTRACT FROM AUTHOR]

This paper is concerned with a fractional diffusion predator-prey system with Holling type-II functional response in a bounded domain with no flux boundary condition. The local and global stabilities are investigated and sufficient conditions of stabilities are obtained. The existence of the non-constant steady states is considered and the sufficient conditions of the existence for the non-constant steady states are also obtained. The results show that the predator and the prey can coexist under some suitable conditions with fractional diffusion. [ABSTRACT FROM AUTHOR]

This paper is purported to investigate a ratio-dependent prey–predator system with cross-diffusion in a bounded domain under no flux boundary condition. The asymptotical stabilities of nonnegative constant solutions are investigated to this system. Moreover, without estimating the lower bounds of positive solutions, the existence, multiplicity of positive steady states are considered by using fixed points index theory, bifurcation theory, energy estimates and the differential method of implicit function and inverse function. [ABSTRACT FROM AUTHOR]

In this paper we consider a family of singularly perturbed delay differential equation of convection diffusion type. When the perturbation parameter is very small, the solution of the problem exhibits layer behavior. In the layer region the solution changes rapidly, while away from this region the change in the solution is moderate. This simultaneous presence of two different scales phenomena makes the problem stiff. In this work, the problem is solved by applying a new Liouville-Green transform and the asymptotic solutions are obtained. Application to multi-point boundary value problem is also illustrated. Several test examples are taken into account so as to test the efficiency of the proposed method. The method presented is compared with other existing numerical or asymptotic methods. It is observed that the method presented is very easy to implement and is capable of reducing the size of calculations significantly while still maintaining high accuracy of the solution. [ABSTRACT FROM AUTHOR]

Abstract: This paper is purported to investigate an acid-mediated invasion model for tumor–stromal interactions under no flux boundary condition. By applying the contraction mapping principle, the parabolic Schauder estimates, parabolic estimates and Sobolev embedding theorems, we prove that there exists a unique global classical solution of this model. [Copyright &y& Elsevier]