Here, we study the dynamics of the singularly perturbed logistic difference equation with two different continuous arguments. First of all, local stability of the fixed points is investigated by analyzing the corresponding characteristic equations of the linearized equations. Secondly, we illustrate that the considered system exhibits Hopf bifurcation. A discretized analogue of the original system is obtained using the method of steps. Local stability and bifurcation analysis of the discretized system are investigated. Explicit conditions for the occurrence of a variety of complex dynamics such as fold and Neimark-Sacker bifurcations are reached. We compare the results with those of the associated difference equation with continuous argument when the perturbation parameter $\epsilon \longrightarrow 0$ and with those of the logistic delay differential equation with two different delays when $\epsilon \longrightarrow 1$. Finally, numerical simulations including Lyapunov exponent, bifurcation diagrams and phase portraits are carried out to confirm the theoretical analysis obtained and to illustrate more complex dynamics of the system.

A malaria transmission disease model with host selectivity and Insecticide treated bed nets (ITNs), as an intervention for controlling the disease, is formulated. Since the vector is an insect, the vector time scale is much more expeditious than the host time scale. This leads to a singularly perturbed model with two distinctive intrinsic time scales, two-slow for the host and one-fast for the vector. The basic reproduction number R0 is calculated and the local stability analysis is performed at equilibria of the model when the perturbation parameter ɛ > 0. The model is analyzed when ɛ → 0 using asymptotic expansions technique. Merging bed-net control, vector-bias, and singular perturbation have a notable effect on the model dynamics. It is shown that if over %30 of humans use ITNs, malaria disease burden can be reduced. The dynamics on the slow surface indicate that the infected vectors decays very fast when ɛ = 0.001 according to the numerical simulations.