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Z. Miao, N. Popovic, P. Szmolyan:
"Oscillations in a cAMP signaling model for cell aggregation - a geometric analysis";
Journal of Mathematical Analysis and Applications, 483 (2020), 1.

We study a singularly perturbed model for a cyclic adenosine monophosphate (cAMP) signalling system that controls aggregation of the amoeboid microorganism Dictyostelium discoideum. The model, which is based on a classical model due to Martiel and Goldbeter [16], takes the form of a planar system of ordinary differential equations with two singular perturbation parameters which manifest very differently: while one parameter encodes the separation of scales between the slow and fast variables, the other induces a non-uniformity in the corresponding vector field in the singular limit. We apply geometric singular perturbation theory and the desingularisation technique known as "blow-up" to construct a family of attracting, periodic (relaxation-type) orbits; in the process, we elucidate the novel singular structure of the model, and we describe in detail the resulting oscillatory dynamics.

cAMP signalling; Singular perturbation; Geometric singular perturbation theory; Blow-up technique; Relaxation oscillation; Periodic orbit

http://dx.doi.org/10.1016/j.jmaa.2019.123577

https://doi.org/10.1016/j.jmaa.2019.123577