Oscillations in a cAMP signalling model for cell aggregation -- a geometric analysis
In this article, the authors perform a geometric analysis of a singularly perturbed model for a cyclic adenosine monophosphate (cAMP) signalling system that controls aggregation of the amoeboid microorganism \textit{Dictyostelium discoideum} and which can be rewritten into the form of a planar system of ordinary differential equations with two small parameters \(\kappa\) and \(\varepsilon,\) \[ R'=\kappa(U+\mathcal{P}\varepsilon)\frac{(U+\varepsilon)^2+\Lambda R^2 U^2}{(U+\varepsilon)(U+\frac{\varepsilon}{c})}\left[\tilde\mu\varepsilon^{\frac12}(U+\varepsilon)-(U+\tilde d\varepsilon^{\frac32})R \right], \] \[ U'=\tilde b\varepsilon^2(U+\varepsilon)^2+\Theta R^2 U^2 -U\left[(U+\varepsilon)^2 +\Lambda R^2 U^2\right], \] where the state variables \(R\) and \(U\) correspond to the total proportion of receptors in the active state and the concentration of extracellular cAMP, respectively. Using the geometric singular perturbation theory and blow-up techniques, the authors prove that for \(\varepsilon\in(0,\varepsilon_0]\) there exists \(\kappa_0=\kappa_0(\varepsilon_0)\) such that for \(\kappa\in(0,\kappa_0]\) the above system admits a unique family of attracting periodic orbits \(\Gamma_{\kappa\varepsilon}\) which tends to a singular cycle \(\Gamma_{0\varepsilon}\) as \(\kappa\to0\) uniformly for \(\varepsilon\in(0,\varepsilon_0],\) and to a singular cycle \(\Gamma_{00}\) as \((\kappa,\varepsilon)\to(0,0).\)
scientific article; zbMATH DE number 7138437