We discuss effects of stochasticity and time delays in simple models of population dynamics. In social-type models, where individuals react to the information concerning the state of the population at some earlier time, sufficiently large time delays may cause oscillations. In biological-type models, where some changes already take place in the population at an earlier time, oscillations might not be present for any time delay. We illustrate this idea in models of delayed random walks, gene expression, and population dynamics of evolutionary game theory.
Many socio-economic and biological processes can be modeled as systems of interacting objects. One may then try to derive their global behavior from individual interactions between their basic entities such as animals in ecological and evolutionary models, RNA and protein molecules in biochemical reactions of gene expression and regulation, and people in social processes. If the number of interacting objects is small, to describe and analyze the time evolution of such systems we should use stochastic modeling.
It was usually assumed that reactions take place instantaneously and effects of individual interactions are immediate. In reality, all biochemical processes take a certain time and there is a substantial time delay between the beginning of a reaction and the appearance of new products in the system. Similarly, in ecological models results of interactions between individuals may appear in the future, and in social models, individuals or players may act, that is choose appropriate strategies, on the basis of the information concerning events in the past.
It is well known that time delays may cause oscillations in solutions of ordinary differential equations [1], [2], [3], [4] and [5]. The main goal of this paper is to show that the presence of oscillations depends on particular causes of a time delay. We divide models with time delays into two families. In social-type models, where individuals react to the information concerning the state of the population at some earlier time, we should expect oscillations. On the other hand, in biological-type models, where some changes already take place in the population at an earlier time, oscillations might not be present for any time delay. We illustrate our idea with two examples: gene expression with a delayed degradation and an evolutionary game with the stable coexistence of two strategies.
It was argued recently in [6] that combined effects of the time delay of protein degradation and stochasticity may cause an oscillatory behavior in simple models of gene expression. It was shown in [7] that if one assumes that a process of degradation is consuming, that is molecules which started to degrade cannot take part in other processes, then oscillatory behavior is no longer present in such systems. The key point here is that although protein molecules will completely degrade at some time in the future, they have already changed the state of the system at an earlier time. We say that such models are of biological type. However, if we change the model and allow protein molecules to be chosen many times for degradation, that is we do not see any change at an earlier time, then we obtain formally a delayed random walk of a social type [8] and [9]. In such a random walk, oscillations are present for sufficiently large delays. We compare here these two models and show that they are equivalent in the limit of small time delays. We derive an analytical expression for the variance of the number of protein molecules in a simple model of gene expression with a time-delay degradation.
We will also discuss two evolutionary game theory models with stationary coexistence of two strategies in the replicator dynamics [10] and [11]. In the social-type model, players imitate opponents taking into account average payoffs of games played some time ago. In the biological-type model, new players are born from parents who played in the past. We show that in the first type of dynamics, the stationary point is asymptotically stable for small time delays and becomes unstable for big ones. In the second type of dynamics, however, the stationary point is asymptotically stable for any time delay.
It is well known that time delays may cause oscillations in solutions of ordinary differential equations. Usually a unique globally asymptotically stable stationary point loses the stability for large time delays. More precisely, there exists a critical time delay at which the system undergoes the Hopf bifurcation and a stable limit cycle appears. Here we demonstrated that the presence of oscillations depends on particular causes of a time delay. In particular, in social-type models, where individuals react to the information concerning the state of the population at some earlier time, we should expect oscillations. On the other hand, in biological-type models, where some physical change already takes place in the population at an earlier time, oscillations might not be present for any time delay.
We compared a delayed random walk model (a social-type model with oscillations) to a corresponding production–degradation model (biological-type model without oscillations). We derived an analytical expressions for the variance of the number of protein molecules in a simple model of gene expression with a small time delay degradation. We also presented two population dynamics models–evolutionary games–with and without oscillations.
It is important to study more complex systems with time delays, especially combined effects of time delays and stochasticity, and in particular the possibility of stable oscillations in such systems.
