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# A typical phase portrait of the system (12). In this case, there is a single coexistence equilibrium that appears to be asymptotically stable. Nullclines are indicated by bold lines. The backward orbit from (M (0), 0), denoted γ (see Lemma 4), is present in the figure. The dashed line indicates the "upper" boundary of the region (32) of Theorem 4. Equilibrium points are indicated by dots, and arrow directions indicate approximate directions of solution velocity in forward time.

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There is an urgent need for more understanding of the effects of surveillance on malaria control. Indoor residual spraying has had beneficial effects on global malaria reduction, but resistance to the insecticide poses a threat to eradication. We develop a model of impulsive differential equations to account for a resistant strain of mosquitoes tha...

## Citations

... The theory of impulsive differential equations and inclusions has been systematically developed (see e.g., the monographs [1][2][3][4], and the references therein), among other things, especially because of many practical applications (see e.g., References [1,[4][5][6][7][8][9][10][11]). These applications concern fluctuations of pendulum systems under impulsive effects, remittent oscillators, population dynamics, oxygen-driven self-cycling fermentation process, nutrient-driven self-cycling fermentation process, various impulsive drug effects, optimal impulsive vaccination for an SIR control model, an SEIRS epidemic model, malaria vector model, impulsive insecticide spraying, HIV induction-maintenance therapy, and so forth. ...

Ordinary differential equations with n-valued impulses are examined via the associated Poincaré translation operators from three perspectives: (i) the lower estimate of the number of periodic solutions on the compact subsets of Euclidean spaces and, in particular, on tori; (ii) weakly locally stable (i.e., non-ejective in the sense of Browder) invariant sets; (iii) fractal attractors determined implicitly by the generating vector fields, jointly with Devaney’s chaos on these attractors of the related shift dynamical systems. For (i), the multiplicity criteria can be effectively expressed in terms of the Nielsen numbers of the impulsive maps. For (ii) and (iii), the invariant sets and attractors can be obtained as the fixed points of topologically conjugated operators to induced impulsive maps in the hyperspaces of the compact subsets of the original basic spaces, endowed with the Hausdorff metric. Five illustrative examples of the main theorems are supplied about multiple periodic solutions (Examples 1–3) and fractal attractors (Examples 4 and 5).

... We will call such a system an impulsive system with autonomous right-hand side; see [Bachar, Raimann & Kotanko, 2016;Church & Smith?, 2016;Rozins & Day, 2017;Xie et. al, 2017] for a few recent examples in mathematical biology. ...

... The partial derivatives of S at a given fixed point can be computed up to a given order by sequentially solving systems of linear inhomogeneous impulsive differential equations. This is the approach that is taken in the majority of the literature; see [Church & Smith?, 2016;Pang, Shen & Zhao, 2016;Xie et. al, 2017] for a few recent applications of these techniques to biological systems. ...

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov–Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point.

In this article, we examine nonautonomous bifurcation patterns in nonlinear systems of impulsive differential equations. The approach is based on Lyapunov-Schmidt reduction applied to the linearization of a particular nonlinear integral operator whose zeroes coincide with bounded solutions of the impulsive differential equation in question. This leads to sufficient conditions for the presence of fold, transcritical and pitchfork bifurcations. Additionally, we provide a computable necessary condition for bifurcation in nonlinear scalar impulsive differential equations. Several examples are provided illustrating the results.