Narrow Results

In this paper, we obtain a lower bound for the maximum number of crossing limit cycles for cubic planar switching systems with two regions separated by a straight line. By pseudo Hopf bifurcation and $𝜖$-order Lyapunov constants, we prove that there exist cubic near-integrable switching systems with at least 26 small-amplitude limit cycles bifurcating from an elementary center after suitable perturbations. To the best of our knowledge, this is the best lower bound so far for the cubic class by using a first order analysis.

This paper investigates the dynamical behavior of the modified May–Holling–Tanner model in the presence of dynamic alternative resources. We study the role of dynamic alternative resources on the survival of the species when there is prey rarity. Detailed mathematical analysis and numerical evaluations, including the situation of ecosystem collapsing, have been presented to discuss the coexistence of species’, stability, occurrence of different bifurcations (saddle-node, transcritical, and Hopf) in three cases in the presence of prey and alternative resources, in the absence of prey and in the absence of alternative resources. It has been obtained that the multiple coexisting states and their stability are outcomes of variations in predation rate for alternative resources. Also, the occurrence of Hopf bifurcation, saddle-node bifurcation, and transcritical bifurcation are due to variations in the parameters of dynamic alternative resources. The impact of dynamic alternative resources on species’ density reveals the fact that if the predation rate for alternative resources increases, then the prey biomass increases (under some restrictions), and variations in the predator’s biomass widely depend upon the quality of food items. This study also points out that the survival of predators is possible in the absence of prey. In the theme of ecological balance, this study suggests some theoretical points of view for the eco-managers.