In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions.

A linear growth-diffusion equation is studied in a time-dependent interval whose location and length both vary. We prove conditions on the boundary motion for which the solution can be found in exact form and derive the explicit expression in each case. Next, we prove the precise behaviour near the boundary in a ‘critical’ case: when the endpoints of the interval move in such a way that near the boundary there is neither exponential growth nor decay, but the solution behaves like a power law with respect to time. The proof uses a subsolution based on the Airy function with argument depending on both space and time. Interesting links are observed between this result and Bramson's logarithmic term in the nonlinear FKPP equation on the real line. Each of the main theorems is extended to higher dimensions, with a corresponding result on a ball with a time-dependent radius.

Reaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.

We present a full classification of the short-time behaviour of the interfaces and local solutions to the nonlinear parabolic p-Laplacian type reaction–diffusion equation of non-Newtonian elastic filtration:

$$\begin{equation*} u_t-\Big(|u_x|^{p-2}u_x\Big)_x+bu^{\beta}=0, \ p>2, \beta >0. \end{equation*}$$

The paper is devoted to a reaction-diffusion equation in an infinite two-dimensionalstrip with nonlinear boundary conditions. The existence of travelling waves is proved inthe bistable case by the Leray-Schauder method. It is based on a topological degree forelliptic problems in unbounded domains and on a priori estimates of solutions.

We improve several recent results in the asymptotic integration theory of nonlinear ordinary differential equations via a variant of the method devised by J. K. Hale and N. Onuchic The results are used for investigating the existence of positive solutions to certain reaction-diffusion equations.

This paper proposes a quantitative model of the reaction-diffusion type to examine thedistribution of interferon-α (IFNα) in a lymph node(LN). The numerical treatment of the model is based on using an original unstructured meshgeneration software Ani3D and nonlinear finite volume method for diffusion equations. Thestudy results in suggestion that due to the variations in hydraulic conductivity ofvarious zones of the secondary lymphoid organs the spatial stationary distribution ofIFNα is essentially heterogeneous across the organs. Highly protecteddomains such as sinuses, conduits, co-exist with the regions in which where the stationaryconcentration of IFNα is lower by about 100-fold. This is the first studywhere the spatial distribution of soluble immune factors in secondary lymphoid organs ismodelled for a realistic three-dimensional geometry.

The G-function formalism has beenwidely used in the context of evolutionary games for identifyingevolutionarily stable strategies (ESS). This formalism wasdeveloped for and applied to point processes. Here, we examine the G-functionformalism in the settings of spatial evolutionarygames and strategy dynamics, based on reaction-diffusion models. We startbyextending the point process maximum principle to reaction-diffusion modelswith homogeneous, locally stable surfaces. We then develop the strategy dynamics forsuch surfaces. When the surfaces are locally stable, but nothomogenous, the standard definitions of ESS and the maximumprinciple fall apart. Yet, we show by examples that strategy dynamics leads toconvergent stable inhomogeneous strategies that are possibly ESS, in the sensethat for many scenarios which we simulated,invaders could not coexist with the exisiting strategies.

Conditions are given under which a space-time jump Markov process describing the stochastic model of non-linear chemical reactions with diffusion converges to the homogeneous state solution of the corresponding reaction-diffusion equation. The deviation is measured by a central limit theorem. This limit is a distribution-valued Ornstein–Uhlenbeck process and can be represented as the mild solution of a certain stochastic partial differential equation.