Frits Veerman

A Predator--2 Prey Fast--Slow Dynamical System for Rapid Predator Evolution

We consider adaptive change of diet of a predator population that switches its feeding between two prey populations. We develop a novel 1 fast--3 slow dynamical system to describe the dynamics of the three populations amidst continuous but rapid evolution of the predators diet choice. The two extremes at which the predators diet is composed solely of one prey correspond to two branches of the three-branch critical manifold of the fast--slow system. By calculating the points at which there is a fast transition between these two feeding choices (i.e., branches of the critical manifold), we prove that the system has a two-parameter family of periodic orbits for sufficiently large separation of the time scales between the evolutionary and ecological dynamics. Using numerical simulations, we show that these periodic orbits exist, and that their phase difference and oscillation patterns persist, when ecological and evolutionary interactions occur on comparable time scales. Our model also exhibits periodic orbits that agree qualitatively with oscillation patterns observed in experimental studies of the coupling between rapid evolution and ecological interactions.

Breathing pulses in singularly perturbed reaction-diffusion systems

The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures is presented. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm–Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.

Destabilization Mechanisms of Periodic Pulse Patterns Near a Homoclinic Limit

It has been observed in the Gierer--Meinhardt equations that destabilization mechanisms are rather complex when spatially periodic pulse patterns approach a homoclinic limit. In this paper we show ...

Parameter inference with analytical propagators for stochastic models of autoregulated gene expression

Stochastic gene expression in regulatory networks is conventionally modelled via the Chemical Master Equation (CME) (van Kampen 1981). As explicit solutions to the CME, in the form of so-called propagators, are not readily available, various approximations have been proposed (Zechner et al. 2013, Feigelman et al. 2016, Popović, Marr and Swain 2016). A recently developed analytical method (Veerman, Marr and Popović 2017) is based on a scale separation that assumes significant differences in the lifetimes of mRNA and protein in the network, allowing for the efficent approximation of propagators from asymptotic expansions for the corresponding generating functions. Here, we showcase the applicability of that method to a ‘telegraph’ model for gene expression that is extended with an autoregulatory mechanism. We demonstrate that the resulting approximate propagators can be successfully applied for Bayesian parameter inference in the non-regulated model with synthetic data; moreover, we show that in the extended autoregulated model, autoactivation or autorepression may be refuted under certain assumptions on the model parameters. Our results indicate that the method showcased here may allow for successful parameter inference and model identification from longitudinal single cell data.