Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems

This paper studies the dynamics of the generalist predator–prey systems modeled in [E. Alexandra, F. Lutscher and G. Seo, Bistability and limit cycles in generalist predator–prey dynamics, Ecol. Complex. 14 (2013) 48–55]. When prey reproduces much faster than predator, by combining the normal form theory of slow-fast systems, the geometric singular perturbation theory and the results near non-hyperbolic points developed by Krupa and Szmolyan [Relaxation oscillation and canard explosion, J. Differential Equations 174(2) (2001) 312–368; Extending geometric singular perturbation theory to nonhyperbolic points—fold and canard points in two dimensions, SIAM J. Math. Anal. 33(2) (2001) 286–314], we provide a detailed mathematical analysis to show the existence of homoclinic orbits, heteroclinic orbits and canard limit cycles and relaxation oscillations bifurcating from the singular homoclinic cycles. Moreover, on global stability of the unique positive equilibrium, we provide some new results. Numerical simulations are also carried out to support the theoretical results.

Firing patterns of the CA1 pyramidal neuron with geometric singular perturbation: a model study

An investigation of CA1 pyramidal model is an important issue for applications, which is intimately related to the composition of ions in the extracellular environment and external stimulation. In this paper, it is demonstrated that the effects of different electrophysiological parameters such as muscarinic-sensitive potassium current activation variable and sustained sodium current inactivation variable on the firing sequence of model by numerical simulations. Furthermore, the paper also discusses that the temperature affects the firing of the CA1 model from direct current (DC) and alternating current (AC) stimuli. It is found that the model exhibits excellent spiking and bursting patterns, even chaotic patterns occur. Meanwhile, generalized mixed oscillations emerge in the model. Additionally, the firing modes are depicted by providing the response curve (RC), inter-spike interval curve (ISI), phase diagram curve (PDC) and the number of spikes per burst curve (NC). Mathematically, the paper elaborates the results which are presented to obtain two lower dimensional subsystems, which govern the fast and slow dynamics for giving insight into the dynamic behaviors of the full 5D system based on the geometric singular perturbation theory (GSPT). Particularly, we analyse the phase diagrams of the CA1 model to understand the properties better. The present results may contribute to further understand the information processing of the CA1 pyramidal neurons.