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We study the local and global existence and uniqueness of strict solution and the local well-posedness for a class of abstract "explicit" neutral integro-differential equations with state-dependent delay. In the last section some examples motivated by explicit neutral ordinary integro-differential equations arising in the theory of population dynamics are presented. [ABSTRACT FROM AUTHOR]

We argue that self-regulation in tick population dynamics in the natural world is not associated with the egg-production saturation, that was normally assumed in existing literature and modeled using the Ricker reproduction function. We argue that the nonlinearity and self-regulatory inhibition appear in the synergistic effect of tick cooperation and the crowding within groups, and an appropriate nonlinearity can be the sub-linear Gamma-Ricker function. We also noticed that the diapaused development leads to multiple time lags in a structured tick population dynamics model, leading to a scalar delay differential equation with two time lags appearing in the sub-linear Gamma-Ricker nonlinearity. This model has always one and only one positive equilibrium. Here we develop sharp conditions for this model to exhibit persistence and global convergence behaviors. [ABSTRACT FROM AUTHOR]

We consider a scalar delay differential equation x ˙ (t) = - d x (t) + f ((1 - α) ρ x (t - τ) + α ρ x (t - 2 τ)) with an instant mortality rate d > 0 , the nonlinear Rick reproductive function f, a survival rate during all development stages ρ , and a proportion constant α ∈ [ 0 , 1 ] with which population undergoes a diapause development. We consider global continuation of a branch of periodic solutions locally generated through the Hopf bifurcation mechanism, and we establish the existence of periodic solutions with periods within (3 τ , 6 τ) for a wide range of parameter values. We show this existence of periodic solutions not only for the delay τ near the first critical value τ ∗ when a local Hopf bifurcation takes place near the positive equilibrium, but for all τ > τ ∗ . We obtain this (global) existence of periodic solutions by using the equivalent-degree based global Hopf bifurcation theory, coupled with an application of the Li–Muldowney technique to rule out periodic solutions with period 3 τ . We conduct some numerical simulations to illustrate that this global continuation is completely due to the diapause-delay since solutions of the delay differential equation with only normal development delay in the given biologically realistic range all converge to the positive equilibrium. [ABSTRACT FROM AUTHOR]

We consider a coupled system of delay differential equations for a single-species tick population dynamics, assuming feeding adult ticks are distributed by their hosts in a spatially heterogeneous environment consisting of two patches where egg ticks produced will complete their life cycles with different, normal and diapause, developmental delays. We show that the mobility of adult tick host and the diapause developmental delay combined drive a synchronized oscillation in the total tick populations around a uniquely defined positive equilibrium, and this synchronization makes the oscillatory patterns much simpler in comparison with multi-peak oscillations exhibited in the absence of host mobility. [ABSTRACT FROM AUTHOR]

We study the approximate controllability of a general class of first order abstract control differential problems with state-dependent delay. To this end, we establish and prove several new results on local and global existence and uniqueness of mild and strict solutions for abstract differential equations with state-dependent delay. In the last section, some examples are presented. [ABSTRACT FROM AUTHOR]

• Vector co-occurrence in hosts impacts co-feeding vector-borne disease transmission. • Attachment and grooming behaviors determine vector-over-host distribution patterns. • A dynamic model provides insight on bistability and oscillation pattern formation. Towards gaining a mechanistic understanding of the co-feeding transmission dynamics of tick-borne diseases, we develop a delay differential equation model for vector-host population dynamics. In addition to the intrinsic demographic dynamics of both vector and host populations, the model has the distribution dynamics of vector individuals on hosts governed by vector attachment and host grooming behaviour. We introduce the concept of basic infestation number, derive analytic formulae for calculating it and use these formulae to characterize the distribution patterns. We also show how some of these patterns naturally lead to bi-stability and nonlinear oscillations in the vector and host populations. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of mild and strict solutions for abstract neutral differential equations with state-dependent delay. Some examples related to partial neutral differential equations are presented. [ABSTRACT FROM AUTHOR]

An important adaptive mechanism for ticks in respond to variable climate is diapause. Incorporating this physiological mechanism into a tick population dynamics model results in a delay differential system with multiple delays. Here, we consider a mechanistic model that takes into consideration of the development diapause by both larvae and nymph ticks, which share a common set of hosts. We introduce the concept of parametric trigonometric functions (convex combinations of two trigonometric functions with different oscillation frequencies) and explore their qualitative properties to derive an explicit formula of the critical diapause portion for the Hopf bifurcation to take place. Our work shows analytically that diapause can generate complex oscillations even though seasonality is not included. [ABSTRACT FROM AUTHOR]

Abstract: We develop a global Hopf bifurcation theory for a system of functional differential equations with state-dependent delay. The theory is based on an application of the homotopy invariance of -equivariant degree using the formal linearization of the system at a stationary state. Our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local Hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state-dependent delay. [Copyright &y& Elsevier]

We apply our recently developed global Hopf bifurcation theory to examine global continuation with respect to the parameter for periodic solutions of functional differential equations with state-dependent delay. We give sufficient geometric conditions to ensure the uniform boundedness of periodic solutions, obtain an upper bound of the period of non-constant periodic solutions in a connected component of Hopf bifurcation, and establish the existence of rapidly oscillating periodic solutions. [ABSTRACT FROM AUTHOR]

For a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function, we obtain the existence, stability and limiting profile of a phase-locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory. [Copyright &y& Elsevier]

We study the local and global existence and uniqueness of solution, the well-posedness and the existence of global attractor for abstract functional differential equations with state-dependent delay. Some examples concerning partial differential equations with state dependent delay are presented. [ABSTRACT FROM AUTHOR]

Abstract: The Vector Integration to Endpoint (VITE) circuit describes a real-time neural network model simulating behavioral and neurobiological properties of planned arm and hand movements by the interaction of two populations of neurons. We analyze the speed-accuracy trade-off generated by this circuit, generalized to include delayed feedback. With delay, two important new properties of the circuit emerge: a breakdown of Fitts'' law when the movement time is small relative to the delay; and a positive Fitts'' law Y-intercept. This breakdown of Fitts'' law for tasks with small Index of Difficulty has been previously observed experimentally, and we suggest it may be attributed at least in part to delay effects in the nervous system elaborated by the model. Additionally, this gives a theoretical explanation for why positive Fitts'' law Y-intercept should occur, and that it is related to the delay within the movement circuit. [Copyright &y& Elsevier]

We consider a delay differential equation for tick population with diapause, derived from an age-structured population model, with two time lags due to normal and diapause mediated development. We derive threshold conditions for the global asymptotic stability of biologically important equilibria, and give a general geometric criterion for the appearance of Hopf bifurcations in the delay differential system with delay-dependent parameters. By choosing the normal development time delay as a bifurcation parameter, we analyze the stability switches of the positive equilibrium, and examine the onset and termination of Hopf bifurcations of periodic solutions from the positive equilibrium. Under some technical conditions, we show that global Hopf branches are bounded and connected by a pair of Hopf bifurcation values. This allows us to show that diapause can lead to the occurrence of multiple stability switches, coexistence of two stable limit cycles, among other rich dynamical behaviours. [ABSTRACT FROM AUTHOR]

We examine the impact of delays on option pricing and obtain sufficient conditions for the robustness of delays for certain two-asset options, where the underlying assets involved are modeled by stochastic delay differential equations. We prove the robustness of delay parameters on prices of underlying asset and prices of three types of options, which differ in their payoff functions. We provide numerical experiments which validate our theoretical results and find that when the underlying assets exist delayed response, the corresponding impact on option prices is robust with respect to the small changes in delay parameters. • We studied the robustness of delays on some two-asset options. • We obtained sufficient conditions for the robustness of delays. • We proved the robustness result in detail with three types of two-asset options as examples. • We used the Euler-Maruyama scheme and Monte Carlo methods in numerical simulations. [ABSTRACT FROM AUTHOR]