Due to various imprecisions in nature, imprecise parameters in biological modeling should be taken into account. This paper studies the spatial dynamics of an imprecise prey‐predator model of the Leslie–Gower type by presenting imprecise parameters as interval parameters. First, conditions of Turing instability are obtained via bifurcation analysis and interval‐valued functions. Then, the effects of interval parameters on pattern selection are discussed via multiple‐scale analysis. We discover that when all the parameters of the model are interval parameters, the value of the controlled parameter increases, and the range of the pattern selection domain expands as the value of the interval variable increases, i.e., both the controlled parameter and boundary of the pattern selection domain are interval numbers. Finally, under the effects of the interval parameters of diffusion and the prey’s conversion rate into biomass for the predator, the density of the prey decreases or increases, respectively, and the structure or the microstructure of the pattern of the model changes with the growing value of the interval variable. This paper provides a new perspective on the study of the spatial predator‐prey model.

In order to study the dissemination mechanism of knowledge, a SIRI dynamics model with the learning rate, the forgetting rate and the recalling rate is constructed in this paper. Stability of equilibria and global dynamics of the SIRI model are analyzed. Two thresholds that determine whether knowledge is disseminated are given. We describe the stability of the equilibria for the SIRI model in which there are an equilibrium and a line of equilibria. In particular, we find the dividing curve function which is used to partition invariant set in order to discuss the local stability, and obtain the equation of the wave peak value or wave trough value in the process of knowledge dissemination. Numerical simulations are provided to support the theoretical results. The complicated dynamics properties exhibit that the model is very sensitive to variation of parameters, which play an important role on controlling and administering the knowledge dissemination.

One of the most efficient predator‐prey models with spatial effects is the one with ratio‐dependent functional response. However, there is a need to further explore the effects of spatial motion on the dynamic behavior of population. In this work, we study a ratio‐dependent predator‐prey model with diffusion terms. The aim of this work is to investigate the changes in predator’s distribution in space as the prey populations change their mobility. We observe that the frequency diffusion of the prey gives rise to the sparse density of the predator. Moreover, we also observe that the increasing rate of the conversion into predator biomass induces pattern transitions of the predator. Specifically speaking, Turing pattern of the predator populations goes gradually from a spotted pattern to a black‐eye pattern, with the intermediate state being the mixture of spot and stripe pattern. The simulation results and analysis of this work illustrate that the diffusion rate and the real intrinsic factor influence the persistence of the predator‐prey system mutually.

Spatial predator‐prey models have been studied by researchers for many years, because the exact distributions of the population can be well illustrated via pattern formation. In this paper, amplitude equations of a spatial Holling–Tanner predator‐prey model are studied via multiple scale analysis. First, by amplitude equations, we obtain the corresponding intervals in which different kinds of patterns will be onset. Additionally, we get the conclusion that pattern transitions of the predator are induced by the increasing rate of conversion into predator biomass. Specifically, pattern transitions of the predator between distinct Turing pattern structures vary in an orderly manner: from spotted patterns to stripe patterns, and finally to black‐eye patterns. Moreover, it is discovered that pattern transitions of prey can be induced by cross‐diffusion; that is, patterns of prey transmit from spotted patterns to stripe patterns and finally to a mixture of spot and stripe patterns. Meanwhile, it is found that both effects of cross‐diffusion and interaction between the prey and predator can lead to the complicated phenomenon of dynamics in the system of biology.

Due to the incomplete information caused by measurements, data collection, and initial value determination processes, imprecise parameters in epidemic modeling should be taken into account. This paper studies spatial dynamics of an imprecise epidemic model with the Beddington–DeAngelis incidence rate by presenting imprecise parameters as a form of interval parameters. First, Turing instability is analyzed via bifurcation analysis and an interval‐valued function. Then, the effects of interval parameters on pattern selection are discussed via multiple scale analysis. We find that an interval parameter does not change the type of patterns on the condition that the value of the interval number of the control parameter lies in the corresponding pattern selection domain. Moreover, we discover that when the incidence rate and diffusion rate are selected as interval numbers, respectively, the type of pattern or the spot pattern either transits orderly or becomes sparse; meanwhile, the spatial density of the infected decreases as the interval variable increases. Moreover, our results show that the effects of both different interval parameters and the same interval parameter with different interval value on the system are different. This paper provides a new perspective to study a spatial epidemic model.