We will consider microscopic models of interacting objects (proteins in gene expression and players in evolutionary games). Time delays in our models correspond to particular mechanisms of interactions. We will show that stability of stationary states depends on the type of a time delay. In some situations time delays may change stationary states. We will discuss joint effects of random perturbations and time delays in certain dynamical systems. Our examples
will involve delayed degradation of proteins in gene expression [1,2],
strategy dependent delays in evolutionary games [3,4], and stochastic stability
in three-player games [5,6].
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References- J. Miękisz, J. Poleszczuk, M. Bodnar, U. Foryś, Stochastic model of gene expression with delayed degradation, Bull. Math. Biol. 73, 2011, 2231.
- J. Miękisz, Stochasticity and time delays in gene expression and evolutionary game theory, Probabilistic Engineering Mechanics 26, 2011, 33.
- J. Alboszta, J. Miękisz, Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay, J. Theor. Biol. 231, 2004, 175.
- J. Miękisz, , R. Jankowski, M. Matuszak, Replicator dynamics with strategy dependent time delays, preprint, 2014.
- J. Miękisz, S. Wesołowski, Stochasticity and time delays in evolutionary games, Dynamic Games and Applications 1, 2011, 440.
- J. Miękisz, M. Matuszak, J. Poleszczuk, Stochastic stability in three-player games with time delays, Dynamic Games and Applications, 2014.
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