In an arid or semi-arid environment, precipitation plays a vital role in vegetation growth. Recent researches reveal that the response of vegetation growth to precipitation has a lag effect. To explore the mechanism behind the lag phenomenon, we propose and investigate a water-vegetation model with spatiotemporal nonlocal effects. It is shown that the temporal kernel function does not affect Turing bifurcation. For better understanding the influences of lag effect and nonlocal competition on the vegetation pattern formation, we choose some special kernel functions and obtain some insightful results: (i) Time delay does not trigger the vegetation pattern formation, but can postpone the evolution of vegetation. In addition, in the absence of diffusion, time delay can induce the occurrence of stability switches, while in the presence of diffusion, spatially nonhomogeneous time-periodic solutions may emerge, but there are no stability switches; (ii) The spatial nonlocal interaction may trigger the pattern onset for small diffusion ratio of water and vegetation, and can change the number and size of isolated vegetation patches for large diffusion ratio. (iii) The interaction between time delay and spatial nonlocal competition may induce the emergence of traveling wave patterns, so that the vegetation remains periodic in space, but is oscillating in time. These results demonstrate that precipitation can significantly affect the growth and spatial distribution of vegetation.

We study a phenomenon of stochastic generation of waveform patterns for reaction-diffusion systems in the Turing stability zone where the homogeneous equilibrium is a single attractor. In this analysis, we use a distributed variant of the Selkov glycolytic model with diffusion and random forcing. It is shown that in the Turing stability zone, random disturbances can induce a diversity of metastable spatial patterns with different waveforms. We carry out the parametric analysis of statistical characteristics of evolution of these patterns, and reveal the dominant patterns in the stochastic flow of mixed spatial structures.

Diffusion has been widely applied to model animal movement that follows Brownian motion. However, animals typically move in non-Brownian ways due to their perceptual judgment. Spatial memory and cognition recently have received much attention in characterizing complicated animal movement behaviours. Explicit spatial memory is modeled via a distributed delayed diffusion term in this paper. The distributed time represents the memory growth and decay over time, and the spatial nonlocality reflects the dependence of spatial memory on location. When the temporal delay kernel is weak under the assumption that animals can immediately acquire knowledge and memory decays over time, the equation is equivalent to a Keller-Segel chemotaxis model. For the strong kernel with learning and memory decay stages, rich spatiotemporal dynamics, such as Turing and checker-board patterns, appear via spatially non-homogeneous steady-state and Hopf bifurcations.

A subtle property of speech gestures is the fact that they are spatially and temporally extended, meaning that phonological contrasts are expressed using spatially extended constrictions, and have a finite duration. This paper shows how this spatiotemporal particulation of the vocal tract, for the purpose of linguistic signaling, comes about. It is argued that local uniform computations among topographically organized microscopic units that either constrict or relax individual points of the vocal tract yield the global spatiotemporal macroscopic structures we call constrictions, the locus of phonological contrast. The dynamical process is a morphogenetic one, based on the Turing and Hopf patterns of mathematical physics and biology. It is shown that reaction-diffusion equations, which are introduced in a tutorial mathematical style, with simultaneous Turing and Hopf patterns predict the spatiotemporal particulation, as well as concrete properties of speech gestures, namely the pivoting of constrictions, as well as the intermediate value of proportional time to peak velocity, which is well-studied and observed. The goal of the paper is to contribute to Bernstein\'s program of understanding motor processes as the emergence of low degree of freedom descriptions from high degree of freedom systems by actually pointing to specific, predictive, dynamics that yield speech gestures from a reaction-diffusion morphogenetic process.

Non-local interaction describes the effects of mobility of a population species in their spatial locations. Non-local interaction is incorporated into a prey-predator model by introducing an integral term with appropriate kernel function. The kernel function determines the nature and range of the accessibility of the resources. We consider a nonlocal spatio-temporal prey-predator model with additive weak Allee effect in prey growth and density-dependent predator mortality. The dynamics of the model is examined in presence of a parabolic and a triangular kernel functions. Comparisons are made between the resulting dynamics of the nonlocal model with these kernels for the same range of nonlocal interaction. A linear stability analysis is performed for both the kernels to derive Turing and spatial-Hopf bifurcation conditions. In general, Turing bifurcation curves are different for the two kernels and the same is true for spatial-Hopf bifurcation curves. This results in different dynamics for the two kernels for some parameter values. However, for a fixed range of nonlocal interaction, the dynamics of the model with these two kernels are almost similar for most of the parameter values explored. Increase in the range of nonlocal interaction up to a certain value stabilizes the system dynamics, and then, it destabilizes. Extensive numerical simulations are performed to illustrate the system dynamics.

Many ecological systems show striking non-homogeneous population distributions. Diffusion-driven instabilities are commonly studied as mechanisms of pattern formation in many fields of biology but only rarely in ecology, in part because some of the conditions seem quite restrictive for ecological systems. Seasonal variation is ubiquitous in temperate ecosystems, yet its effect on pattern formation has not yet been explored. We formulate and analyze an impulsive reaction-diffusion system for a resource and its consumer in a two-season environment. While the resource grows throughout the \'summer\' season, the consumer reproduces only once per year. We derive conditions for diffusion-driven instability in the system, and we show that pattern formation is possible with a Beddington-DeAngelis functional response. More importantly, we find that a low overwinter survival probability for the resource enhances the propensity for pattern formation: diffusion-driven instability occurs even when the diffusion rates of prey and predator are comparable (although not when they are equal).