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A reaction-diffusion equation with memory kernel of Jeffreys type and with a balanced bistable reaction term is considered in a bounded interval of the real line. Taking advantage of the fact that in this case the integro-differential equation can be transformed into a local partial differential equation, it is proved that there exist solutions which evolve very slowly in time and maintain a transition layer structure for an exponentially long time Tε ≥ c1 exp(c2/ε) as ε → 0 +, where ε 2 is the diffusion coefficient. Hence, we extend to reaction-diffusion equations with memory kernel of Jeffreys type the well-known results valid for the classic Allen–Cahn equation.

Nowadays, there is an increasing focus on knowledge input and output measurement rather than assessing the characteristics of the knowledge itself. This phenomenon happens since knowledge becomes central into capitalistic production processes: the competitive advantage of countries is the principal engine for the production of high-value, non-ubiquitous and complex knowledge (see Dicken, 2007). In this paper, we present the reaction-diffusion equations and their application to a particular economic phenomenon: in which way knowledge complexity diffusion might affect or not affect the labour productivity across the Eastern European countries. More profoundly, the aim followed by the present research work is to provide Eastern European countries with a substitute approach: a strategy. This strategy has the objective of enabling these countries to avoid acting as knowledge predators towards other countries (i.e. those such as Sweden, Finland and Denmark, referred to as prey countries, which invest their resources in knowledge and therefore in technology). We simulate the knowledge competition and the spill-over productivity effects across the Eastern and some European countries using specific reaction-diffusion equations types: the Lotka-Volterra equations which are able to describe a complex dynamic system of the prey-predator phenomenon. Our approach is based on the two steps of analysis. In essence, the first step, we compute the Knowledge Complexity Index of patents, for quantifying the European knowledge complexity; describing the possible spatial patterns and transformation of the European knowledge. Furthermore, we use the canonical Lotka-Volterra models of patents from the European Patent Office and the labour productivity data from the Eurostat database from 2000 and 2017 for Eastern European countries and some European countries, such as Finland, Sweden, Austria and Germany. More specifically, we have identified the presence of lagged knowledge between the Eastern European countries and some Northern European, such as Finland, Sweden and Denmark. In conclusion, the implementation of specific knowledge diffusion policy strategies could be helpful for future applications. It may avoid intra-guild cannibalism phenomenon from being engaged in some Eastern European countries.