Nonlocal Neumann Boundary Value Problem for Fractional Symmetric Hahn Integrodifference Equations

In this article, we present a nonlocal Neumann boundary value problems for separate sequential fractional symmetric Hahn integrodifference equation. The problem contains five fractional symmetric Hahn difference operators and one fractional symmetric Hahn integral of different orders. We employ Banach fixed point theorem and Schauder’s fixed point theorem to study the existence results of the problem.

Host--Parasitoid Dynamics and Climate-Driven Range Shifts

Climate change has created new and evolving environmental conditions, impacting all species, including hosts and parasitoids. I therefore present integrodifference equation (IDE) models of host--parasitoid systems to model population dynamics in the context of climate-driven shifts in habitats. I describe and analyze two IDE models of host--parasitoid systems to determine criteria for coexistence of the host and parasitoid. Specifically, I determine the critical habitat speed, beyond which the parasitoid cannot survive. By comparing the results from two IDE models, I investigate the impacts of assumptions that reduce the system to a single-species model. I also compare critical speeds predicted by a spatially-implicit difference-equation model with critical speeds determined from numerical simulations of the IDE system. The spatially-implicit model uses approximations for the dominant eigenvalue of an integral operator. The classic methods to approximate the dominant eigenvalue for IDE systems do not perform well for asymmetric kernels, including those that are present in shifting-habitat IDE models. Therefore, I compare several methods for approximating dominant eigenvalues and ultimately conclude that geometric symmetrization and iterated geometric symmetrization give the best estimates of the parasitoid critical speed.

On Periodic Fractional (p, q)-Integral Boundary Value Problems for Sequential Fractional (p, q)-Integrodifference Equations

We study the existence results of a fractional (p, q)-integrodifference equation with periodic fractional (p, q)-integral boundary condition by using Banach and Schauder’s fixed point theorems. Some properties of (p, q)-integral are also presented in this paper as a tool for our calculations.

Biphasic range expansions with short- and long-distance dispersal

Long-distance dispersal (LDD) has long been recognized as a key factor in determining rates of spread in biological invasions. Two approaches for incorporating LDD in mathematical models of spread are mixed dispersal and heavy-tailed dispersal. In this paper, I analyze integrodifference equation (IDE) models with mixed-dispersal kernels and fat-tailed (a subset of the heavy-tailed class) dispersal kernels to study how short- and long-distance dispersal contribute to the spread of invasive species. I show that both approaches can lead to biphasic range expansions, where an invasion has two distinct phases of spread. In the initial phase of spread, the invasion is controlled by short-distance dispersal. Long-distance dispersal boosts the speed of spread during the ultimate phase, and can have significant effects even when the probability of LDD is vanishingly small. For fat-tailed kernels, I introduce a method of characterizing the “shoulder” of a dispersal kernel, which separates the peak and tail.

On sequential fractional Caputo $ (p, q) $-integrodifference equations via three-point fractional Riemann-Liouville $ (p, q) $-difference boundary condition

<abstract><p>In this paper, we aim to study the problem of a sequential fractional Caputo $ (p, q) $-integrodifference equation with three-point fractional Riemann-Liouville $ (p, q) $-difference boundary condition. We use some properties of $ (p, q) $-integral in this study and employ Banach fixed point theorems and Schauder's fixed point theorems to prove existence results of this problem.</p></abstract>

Reliability optimization for non-repairable series-parallel systems with a choice of redundancy strategies and heterogeneous components: Erlang time-to-failure distribution

The redundancy allocation problem (RAP) of non-repairable series-parallel systems considering cold standby components and imperfect switching mechanism has been traditionally formulated with the objective of maximizing a lower bound on system reliability instead of exact system reliability. This objective function has been considered due to the difficulty of determining a closed-form expression for the system reliability equation. But, the solution that maximizes the lower bound for system reliability does not necessarily maximize exact system reliability and thus, the obtained system reliability may be far from the optimal reliability. This article attempts to overcome the mentioned drawback. Under the assumption that component time-to-failure is distributed according to an Erlang distribution and switch time-to-failure is exponentially distributed, a closed-form expression for the subsystem cold standby reliability equation is derived by solving an integrodifference equation. A semi-analytical expression is also derived for the reliability equation of a subsystem with mixed redundancy strategy. The accuracy and the correctness of the derived equations are validated analytically. Using these equations, the RAP of non-repairable series-parallel systems with a choice of redundancy strategies is formulated. The proposed mathematical model maximizes exact system reliability at mission time given system design constraints. Unlike most of the previous formulations, the possibility of using heterogeneous components in each subsystem is provided so that the active components can be of one type and the standby ones of the other. The results of an illustrative example demonstrate the high performance of the proposed model in determining optimal design configuration and increasing system reliability.

On Sequential Fractional q-Hahn Integrodifference Equations

In this paper, we prove existence and uniqueness results for a fractional sequential fractional q-Hahn integrodifference equation with nonlocal mixed fractional q and fractional Hahn integral boundary condition, which is a new idea that studies q and Hahn calculus simultaneously.

Skewed temperature dependence affects range and abundance in a warming world

Population growth metrics such as R 0 are usually asymmetric functions of temperature, with cold-skewed curves arising when the positive effects of a temperature increase outweigh the negative effects, and warm-skewed curves arising in the opposite case. Classically, cold-skewed curves are interpreted as more beneficial to a species under climate warming, because cold-skewness implies increased population growth over a larger proportion of the species's fundamental thermal niche than warm-skewness. However, inference based on the shape of the fitness curve alone, and without considering the synergistic effects of net reproduction, density and dispersal, may yield an incomplete understanding of climate change impacts. We formulate a moving-habitat integrodifference equation model to evaluate how fitness curve skewness affects species’ range size and abundance during climate warming. In contrast to classic interpretations, we find that climate warming adversely affects populations with cold-skewed fitness curves, positively affects populations with warm-skewed curves and has relatively little or mixed effects on populations with symmetric curves. Our results highlight the synergistic effects of fitness curve skewness, spatially heterogeneous densities and dispersal in climate change impact analyses, and that the common approach of mapping changes only in R 0 may be misleading.