Class Of Reaction-diffusion Models Research Articles

Does reaction-diffusion support the duality of fragmentation effect?

There is a gap between single-species model predictions, and empirical studies, regarding the effect of habitat fragmentation per se, i.e., a process involving the breaking apart of habitat without loss of habitat. Empirical works indicate that fragmentation can have positive as well as negative effects, whereas, traditionally, single-species models predict a negative effect of fragmentation. Within the class of reaction-diffusion models, studies almost unanimously predict such a detrimental effect. In this paper, considering a single-species reaction-diffusion model with a removal – or similarly harvesting – term, in two dimensions, we find both positive and negative effects of fragmentation of the reserves, i.e., the protected regions where no removal occurs. Fragmented reserves lead to higher population sizes for time-constant removal terms. On the other hand, when the removal term is proportional to the population density, higher population sizes are obtained on aggregated reserves, but maximum yields are attained on fragmented configurations, and for intermediate harvesting intensities.

New exact travelling wave solutions for some famous nonlinear partial differential equations using the improved tanh-function method

The improved tanh-function method is a powerful tool for obtaining exact travelling wave solutions. The method is used for constructing exact travelling wave solutions and new kinds of solutions for the modified dispersive water wave equation, the Abrahams–Tsuneto reaction diffusion system, and for a class of reaction diffusion models. The solutions obtained are different from those reported in the literature.

Formation and deformation of patterns through diffusion

Simulating various patterns exhibited on biological forms with mathematical models has become an important supplement to theoretical biology. Models based on a certain mechanism are intended to provide explanations to the formation of a basic pattern. However, in real phenomena, among a basic pattern there always exist some difference between any two individuals. Such differences are consequences of environmental factors posed during the developmental processes. These factors, such as temperature, affect the diffusion rates of corresponding morphogenes which, in turn, alter a basic pattern to certain extent. We provide, in this paper, a quantitative characterization of this effect for a class of reaction-diffusion models. Mathematically, we study the emergence of stationary patterns and their dependence on diffusion rates for this class of models (RD-equations) with no-flux boundary conditions. The results are generalized to systems with homogeneous Dirichlet boundary conditions when the kinetic terms are odd functions. Through an analysis of the phase dynamics, we show that the deformation of stationary patterns, as the diffusion rates change, is governed by the variation of certain plane curves in the phase space. A constructive proof is given which shows explicitly how to obtain such curves. Applications of this study are illustrated with three model examples. We use these models to explain the biological implications of the mathematical features we investigated. Results from computer simulations are presented and compared with physical patterns.

Simple population models with diffusion

A class of reaction-diffusion models is discussed. These models contain internal reactions (kinetics) and motion in one space dimension (diffusion). Two kinds of diffusion, random and directed, are considered. Simple models in population and epidemic theory are presented to illustrate the flexibility of the directed diffusion concept.