Time delays are an important feature in many biological systems and are widely used in mathematical biology, with applications ranging from population dynamics, gene expression, virus dynamics, and cancer growth. In ecological and biological systems, interactions among population agents play a crucial role in the growth and persistence of populations. These agents, whether they are cells within an organism or animals in nature, can interact and exchange information, and based on this information, some action occurs. The effects of these interactions are not always immediate, and often, there is a time lag between the initial interaction and its observable impact on the population. Therefore, delay is a common phenomenon across different scales of biological organization. For instance, when a gene is triggered to produce a protein, this doesn’t happen immediately. Instead, the cell undergoes stages such as transcription (DNA to RNA copying) and translation (using RNA for protein synthesis). Each of these stages requires time, leading to a lag between the gene’s activation and the appearance of the protein product. These delays can also be observed in communication and evolutionary changes in populations. It has been observed that these time lags significantly influence the behavior of systems. They can induce changes like fluctuations in protein levels, enhance stability, or alter existing patterns. In cancer studies, researchers leverage time delays to gain insights into tumor growth mechanisms and immune responses against them. These delays reflect the duration needed for processes to unfold, such as cancer cell proliferation or immune cell identification and attack on tumors. For instance, there exists a delay from when an immune cell recognizes a cancer cell to when it becomes prepared to combat it.

These are just a few examples of delays observed in biological systems. Looking at past research, you can find many other biological processes where delays are important.

Until now, we have presented several examples of real-world phenomena exhibiting time delays. However, these examples represent only a fraction of the many situations in which time delay can be observed. The occurrence of time delays is widespread across various fields and systems. Now is the time to answer why we are interested in investigating time delays in EGT. In general, EGT offers a framework for modeling various social and biological processes as populations of interacting individuals. These populations can range from animals sharing food to people spreading news on social networks, or even cells in our bodies. A crucial aspect of EGT is the interaction between individuals within these systems. Agents in a population interact and, based on these interactions and other factors, receive points that influence their ability to persist, spread, compete, or potentially become extinct. Traditionally, EGT models assume that the interactions between individuals and the resulting responses are immediate. However, to create more realistic models, we should consider the possibility of time delays between actions and their consequences. Of course, the scale of these delays can vary depending on the type of population.

The replicator dynamic is usually the first and most famous model that new researchers learn about in EGT. It is a mathematical equation that shows how the fraction of different strategies in a population changes over time. Therefore, it is obvious that replicator dynamics are considered the earliest work of studying time delay in evolutionary games. Alboszta and Miękisz, in their research on discrete systems, identified two distinct categories of time delay in replicator dynamics: social delay and biological delay. Social delay exclusively impacts payoffs, with individuals reproducing based on past payoffs at each time step. In contrast, biological delay affects both payoffs and frequencies, representing scenarios where individuals born in the past reproduce according to historical payoffs. Their findings demonstrated that these different types of time delays exert varying influences on the stability of mixed evolutionarily stable strategies. They demonstrated that for small time delays in the social type of delay, the stable stationary state of replicator dynamics remains unchanged. However, as the delay increases beyond a critical threshold, a Hopf bifurcation occurs, leading to oscillatory behavior. This finding aligns with previous research on payoff delays, which Alboszta and Miękisz later termed “social type of delay.” In contrast, biological delays do not affect the stability of the stationary state, with one stable state persisting even under significant time delays. Interestingly, further studies revealed that when biological delay is strategy-dependent, it can shift the stationary state’s value, such that increasing the delay for a particular strategy disadvantages that strategy.

Subsequent researchers incorporated time delay into game theory models, exploring both fixed delays and those following distributions. They investigated delay in conjunction with other factors such as mutation, migration, punishment, and penalties, discovering that in certain scenarios, delay could significantly alter previously established results. More recently, studies have examined the impact of time delay in other frameworks, including stochastic processes affecting fixation and cooperative behavior. These investigations have yielded novel insights into the role of time delay factors in evolutionary game models.