Integrodifference Equations Research Articles

Separation of scales in an integrodifference equation model of metapopulation dispersal and homing predicts changes in equilibria

Separation of scales in an integrodifference equation model of metapopulation dispersal and homing predicts changes in equilibria

How to quantify the regional effects of ocean temperature rise due to climate change: implications of Octopus maya ecophysiology on food security of the Yucatan shelf artisanal fishermen

The vital survival, maturation, and reproduction rates of Octopus maya were formulated according to the thermal preferences in each stage (juvenile and adult) and the bottom temperature of the Yucatan shelf projected from different shared socioeconomic pathways (SSPs): SSP1-5, to describe the species population growth in shallow waters. The dispersion of each individual in the population and the spread of the offshore population were incorporated with an integrodifference equation. For each SSP, the food security of the artisanal fishermen in the Western, Central, and Eastern Yucatan shelf region was analyzed based on the proposed availability, access, and utilization indices of Octopus maya as food. The analysis was complemented with an average monthly protein and income poverty indicators that this species’ shallow water fishery would have the capacity to supply by the year 2100. Although the proportion of the legally O. maya catchable population may be favored with the gradual increase in temperature in the coming years, this increase may not translate into greater food security for the artisanal fishermen who catch this species in shallow waters. Moreover, this fishery alone may not have the capacity to supply the average annual intake of 10.6 kg of protein per person recommended and maintain fishermen and their families above the income poverty line by 2100.

Nyström discretization of integrodifference equations: numerical continuation of periodic solutions and Floquet multipliers

AbstractIntegrodifference equations are discrete-time counterparts to reaction-diffusion equations and have various applications in, e.g., theoretical ecology. Their behavior is often illustrated using numerical simulations, which require a spatial discretization. In this paper, we establish that periodic solutions to time-periodic integrodifference equations, their stability and their Floquet spectrum persist under discretization of Nyström-type, which replaces integrals by quadrature or cubature rules. Moreover, it is shown that the convergence rates of the particular integration rules are preserved. By means of a typical model from theoretical ecology, these results are demonstrated in terms of a numerical continuation for periodic solutions and their Floquet multipliers.

Unraveling trends in schistosomiasis: deep learning insights into national control programs in China.

To achieve the ambitious goal of eliminating schistosome infections, the Chinese government has implemented diverse control strategies. This study explored the progress of the 2 most recent national schistosomiasis control programs in an endemic area along the Yangtze River in China. We obtained village-level parasitological data from cross-sectional surveys combined with environmental data in Anhui Province, China from 1997 to 2015. A convolutional neural network (CNN) based on a hierarchical integro-difference equation (IDE) framework (i.e., CNN-IDE) was used to model spatio-temporal variations in schistosomiasis. Two traditional models were also constructed for comparison with 2 evaluation indicators: the mean-squared prediction error (MSPE) and continuous ranked probability score (CRPS). The CNN-IDE model was the optimal model, with the lowest overall average MSPE of 0.04 and the CRPS of 0.19. From 1997 to 2011, the prevalence exhibited a notable trend: it increased steadily until peaking at 1.6 per 1,000 in 2005, then gradually declined, stabilizing at a lower rate of approximately 0.6 per 1,000 in 2006, and approaching zero by 2011. During this period, noticeable geographic disparities in schistosomiasis prevalence were observed; high-risk areas were initially dispersed, followed by contraction. Predictions for the period 2012 to 2015 demonstrated a consistent and uniform decrease. The proposed CNN-IDE model captured the intricate and evolving dynamics of schistosomiasis prevalence, offering a promising alternative for future risk modeling of the disease. The comprehensive strategy is expected to help diminish schistosomiasis infection, emphasizing the necessity to continue implementing this strategy.

Spread and persistence for integro-difference equations with shifting habitat and strong Allee effect.

We study an integro-difference equation model that describes the spatial dynamics of a species with a strong Allee effect in a shifting habitat. We examine the case of a shifting semi-infinite bad habitat connected to a semi-infinite good habitat. In this case we rigorously establish species persistence (non-persistence) if the habitat shift speed is less (greater) than the asymptotic spreading speed of the species in the good habitat. We also examine the case of a finite shifting patch of hospitable habitat, and find that the habitat shift speed must be less than the asymptotic spreading speed associated with the habitat and there is a critical patch size for species persistence. Spreading speeds and traveling waves are established to address species persistence. Our numerical simulations demonstrate the theoretical results and show the dependence of the critical patch size on the shift speed.

Dynamical behavior of a temporally discrete non-local reaction-diffusion equation on bounded domain

This paper focuses on the study of global dynamics of a class of temporally discrete non-local reaction-diffusion equations on bounded domains. Similar to classical reaction diffusion equations and integro-difference equations, temporally discrete reaction-diffusion equations can also be used to describe the dispersal phenomena in population dynamics. In this paper, we first derived a temporally discrete reaction diffusion equation model with time delay and nonlocal effects to model the evolution of a single species population with age-structured located in a bounded domain. By establishing a new maximum principle and applying the monotone iteration method, the global stabilities of the trivial solution and the positive steady state solution are obtained respectively under some appropriate assumptions.

Propagation dynamics for a class of integro-difference equations in a shifting environment

In this paper, we study the propagation dynamics for a class of integro-difference equations with a shifting habitat. We first use the moving coordinates to transform such an equation to an integro-difference equation with a new kernel function containing the shifting speed c. In two directions of the spatial variable, the resulting equation has two limiting equations with spatial translation invariance. Under the hypothesis that each of these two limiting equations has both leftward and rightward spreading speeds, we establish the spreading properties of solutions and the existence of nontrivial forced waves for the original equation by appealing to the abstract theory of nonmonotone semiflows with asymptotic translation invariance. Further, we prove the stability and uniqueness of forced waves under appropriate conditions.

Introduction for special issue: integro-difference equation models in ecology

Introduction for special issue: integro-difference equation models in ecology

Modelling and bifurcation analysis of spatiotemporal hormetic effects on pest control

A discrete hormetic Ricker model (HRM) of a population with instant pulse perturbation between two consecutive generations can generate hormetic effects. For example, analysis of complex three-parameter spaces with the intrinsic growth rate, intervention strength, and dose timing as parameters revealed hormetic biphasic dose and dose timing responses. These responses exhibited either a J-shaped or an inverted U-shaped pattern, yielding a homeostatic change or a catastrophic shift and hormetic effects in many parameter regions. However, whether it is pest control or the effectiveness of treatments of tumors with radiotherapy and/or chemotherapy, the phenomenon is linked to the size of the space in which the parameters are located. Thus, the occurrence of hormetic effects is associated with spatiotemporal heterogeneity. To show this, we have developed an integro-difference equation based on the HRM that can describe the complex dynamics and hormetic effects in pest populations in two-dimensional space. Our findings indicate that factors such as the spatial domain, spatial grid, the dosage, and timing of control applications, and the intrinsic growth rate of the pest can significantly affect the spatial and temporal characteristics of pest populations and diverse biological phenomena in two-dimensional space. In particular, under the same size of spatial domains (or the same number of spatial grids), the smaller the spatial grids (or the larger the spatial domain), the stronger the hormetic effects of low dose stimulation and high dose inhibition will be. These factors include larger maximum responses and higher toxic thresholds. Therefore, optimal pest control measures should not only rely on the efficiency of a pesticide and its application time but should also be based on the spatial domain and the designing of reasonable grid-based control strategies to avoid hormetic effects.

Evolution of dispersal by memory and learning in integrodifference equation models

In this paper, we develop an integrodifference equation model that incorporates spatial memory and learning so that each year, a fraction of the population use the same dispersal kernel as the previous year, and the remaining individuals return to where they bred or were born. In temporally static environments, the equilibrium of the system corresponds to an ideal free dispersal strategy, which is evolutionarily stable. We prove local stability of this equilibrium in a special case, and we observe convergence towards this equilibrium in numerical computations. When there are periodic or stochastic temporal changes in the environment, the population is less able to match the environment, but is able to do so to some extent depending on the parameters. Overall, the mechanism proposed in this model shows a possible way for the dispersal kernel of a population to evolve towards an ideal free dispersal kernel.

Attractive sets of periodic integrodifference equations under discretization

We consider periodic difference equations in infinite-dimensional Banach spaces possessing compact asymptotically stable set A . It is established that such A persists under spatial discretizations by means of projection methods as nearby closed and bounded uniformly asymptotically stable sets, which moreover converge to A in the Hausdorff metric for increasingly more accurate schemes. The proof is based on a Lyapunov function guaranteed by an ambient converse theorem and a pullback construction. As application serves integrodifference equations on spaces of continuous and p-integrable functions over a compact habitat.

Local interactions affect spread of resource in a consumer-resource system with group defense

Integrodifference equations are a discrete-time spatially explicit model that describes the dispersal of ecological populations through space. This framework is useful to study spread dynamics of organisms and how ecological interactions can affect their spread. When studying interactions such as consumption, dispersal rates might vary with life cycle stage, such as in cases with dispersive juveniles and sessile adults. In the non-dispersive stage, resources may engage in group defense to protect themselves from consumption. These local nondispersive interactions may limit the number of dispersing recruits that are produced and therefore affect how fast populations can spread. We present a spatial consumer-resource system using an integrodifference framework with limited movement of their adult stages and group defense mechanisms in the resource population. We model group defense using a Type IV Holling functional response, which limits the survival of adult resource population and enhances juvenile consumer production. We find that high mortality levels for sessile adults can destabilize resource at carrying capacity. Furthermore, we find that at high resource densities, group defense leads to a slower local growth of resource in newly invaded regions due to intraspecific competition outweighing the effect of consumption on resource growth.

Block-pulse integrodifference equations

We present a hybrid method for calculating the equilibrium population-distributions of integrodifference equations (IDEs) with strictly increasing growth, for populations that are confined to a finite habitat-patch. This method is based on approximating the growth function of the IDE with a piecewise-constant function, and we call the resulting model a block-pulse IDE. We explicitly write out analytic expressions for the iterates and equilibria of the block-pulse IDEs as sums of cumulative distribution functions. We characterize the dynamics of one-, two-, and three-step block-pulse IDEs, including formal stability analyses, and we explore the bifurcation structure of these models. These simple models display rich dynamics, with numerous fold bifurcations. We then use three-, five-, and ten-step block-pulse IDEs, with a numerical root finder, to approximate models with compensatory Beverton–Holt growth and depensatory, or Allee-effect, growth. Our method provides a good approximation for the equilibrium distributions for compensatory and depensatory growth and offers numerical and analytical advantages over the original growth models.

Take me for a ride: Herbivores can facilitate plant reinvasions.

Herbivores shape plant invasions through impacts on demography and dispersal, yet only demographic mechanisms are well understood. Although herbivores negatively impact demography by definition, they can affect dispersal either negatively (e.g., seed consumption), or positively (e.g., caching). Exploring the nuances of how herbivores influence spatial spread will improve the forecasting of plant movement on the landscape. Here, we aim to understand how herbivores impact how fast plant populations spread through varying impacts on plant demography and dispersal. We strive to determine whether, and under what conditions, we see net positive effects of herbivores, in order to find scenarios where herbivores can help to promote spread. We draw on classic invasion theory to develop a stage-structured integrodifference equation model that incorporates herbivore impacts on plant demography and dispersal. We simulate seven herbivore "syndromes" (combinations of demographic and/or dispersal effects) drawn from the literature to understand how increasing herbivore pressure alters plant spreading speed. We find that herbivores with solely negative effects on plant demography or dispersal always slow plant spreading speed, and that the speed slows monotonically as herbivore pressure increases. However, we also find that plant spreading speed can be hump shaped with respect to herbivore pressure: plants spread faster in the presence of herbivores (for low herbivore pressure) and then slower (for high herbivore pressure). This result is robust, occurring across all syndromes in which herbivores have a positive effect on plant dispersal, and is a sign that the positive effects of herbivores on dispersal can outweigh their negative effects on demography. For all syndromes we find that sufficiently high herbivore pressure results in population collapse. Thus, our findings show that herbivores can speed up or slow down plant spread. These insights allow for a greater understanding of how to slow invasions, facilitate native species recolonization, and shape range shifts with global change.

Numerical dynamics of integrodifference equations

Integrodifference equations are versatile models in theoretical ecology for the spatial dispersal of species evolving in non-overlapping generations. The dynamics of these infinite-dimensional discrete dynamical systems is often illustrated using computational simulations. This paper studies the effect of Nyström discretization to the local dynamics of periodic integrodifference equations having Hölder continuous functions over a compact domain as state space. We prove persistence and convergence for hyperbolic periodic solutions and their associated stable and unstable manifolds respecting the convergence order of the quadrature/cubature method.

Generalized ρ-Almost Periodic Sequences and Applications

In this paper, we analyze the Bohr ρ-almost periodic type sequences and the generalized ρ-almost periodic type sequences of the form F:I×X→Y, where ∅≠I⊆Zn, X and Y are complex Banach spaces and ρ is a general binary relation on Y. We provide many structural results, observations and open problems about the introduced classes of ρ-almost periodic sequences. Certain applications of the established theoretical results to the abstract Volterra integro-difference equations are also given.

Investigation of hybrid fractionalq-integro-difference equations supplemented with nonlocalq-integral boundary conditions

AbstractIn this article, we introduce and study a new class of hybrid fractionalqq-integro-difference equations involving Riemann-Liouvilleqq-derivatives, supplemented with nonlocal boundary conditions containing Riemann-Liouvilleqq-integrals of different orders. The existence of a unique solution to the given problem is shown by applying Banach’s fixed point theorem. We also present the existing criteria for solutions to the problem at hand by applying Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Illustrative examples are given to demonstrate the application of the obtained results. Some new results follow as special cases of this work.

Numerical Dynamics of Integrodifference Equations: Hierarchies of Invariant Bundles in Lp (Ω)

We study how the “full hierarchy” of invariant manifolds for nonautonomous integrodifference equations on the Banach spaces of p-integrable functions behaves under spatial discretization of Galerkin type. These manifolds include the classical stable, center-stable, center, center-unstable and unstable ones, and can be represented as graphs of -functions. For kernels with a smoothing property, our main result establishes closeness of these graphs in the -topology under numerical discretizations preserving the convergence order of the method. It is formulated in a quantitative fashion and technically results from a general perturbation theorem on the existence of invariant bundles (i.e. nonautonomous invariant manifolds).

Asymptotic Regimes of an Integro-Difference Equation with Discontinuous Kernel

This paper is concerned with an integral equation that models discrete time dynamics of a population in a patchy landscape. The patches in the domain are reflected through the discontinuity of the kernel of the integral operator at a finite number of points in the whole domain. We prove the existence and uniqueness of a stationary state under certain assumptions on the principal eigenvalue of the linearized integral operator and the growth term as well. We also derive criteria under which the population undergoes extinction (in which case the stationary solution is 0 everywhere).

Multiparameter bifurcation in periodic integrodifference equations

Integrodifference equations appear as popular discrete‐time growth‐dispersal models in theoretical ecology. For such applications, seasonal influences are well‐motivated but lead to a large number of system parameters. This paper provides a flexible and easy to verify sufficient criterion for multiparameter bifurcation of periodic solutions to periodic integrodifference equations. In terms of a “bunch theorem,” a generic condition to identify those directions is given, in which solution branches do bifurcate. Additionally, we show that symmetry properties of integrodifference equations extend to the bifurcating solution branches.