Two-patch Model Research Articles

Modeling the impact of short-term displacement of domestic animals on the transmission dynamics of brucellosis

Brucellosis, a prevalent zoonotic disease caused by bacteria from the genus Brucella, presents a substantial public health and economic challenges worldwide. In this paper, we formulate a two-patch deterministic model to investigate the impact of short-term displacement of domestic animals on the transmission dynamics of brucellosis within and between patches with heterogeneous risk of infection. The model analysis is performed, and the basic reproduction number (R0) is computed and used in the stability analysis of equilibria. A global sensitivity analysis is performed using Latin Hypercube Sampling (LHS) approach to check the impact of model parameters over a period of time. Numerical simulation results indicate that the increase in time spent by domestic animals in high-risk areas, such as communal grazing areas or wildlife interfaces, significantly increases the spread of infections. This underscores the importance of considering short-term animal movements to communal grazing areas and water points in brucellosis management strategies. It equips policymakers and stakeholders with actionable insights to combat the burden of the disease.

Global mathematical analysis of a patchy epidemic model

The dissemination of a disease within a homogeneous population can typically be modeled and managed in a uniform fashion. Conversely, in non-homogeneous populations, it is essential to account for variations among subpopulations to achieve more precise predictive modeling and efficacious intervention strategies. In this study, we introduce and examine the comprehensive behavior of a deterministic two-patch epidemic model alongside its stochastic counterpart to assess disease dynamics between two heterogeneous populations inhabiting distinct regions. First, utilizing a specific Lyapunov function, we demonstrate that the disease-free equilibrium of the deterministic model is globally asymptotically stable. For the stochastic model, we establish that it is well-posed, meaning it possesses a unique positive solution with probability one. Subsequently, we ascertain the conditions necessary to ensure the total extinction of the disease across both regions. Furthermore, we explicitly determine a threshold condition under which the disease persists in both areas. Additionally, we discuss a scenario wherein the disease persists in one region while simultaneously becoming extinct in the other. The article concludes with a series of numerical simulations that corroborate the theoretical findings.

Sharp habitat shifts, evolutionary tipping points and rescue: Quantifying the perilous path of a specialist species towards a refugium in a changing environment

Specialist species thriving under specific environmental conditions in narrow geographic ranges are widely recognized as heavily threatened by climate deregulation. Many might rely on both their potential to adapt and to disperse towards a refugium to avoid extinction. It is thus crucial to understand the influence of environmental conditions on the unfolding process of adaptation. Here, I study the eco-evolutionary dynamics of a sexually reproducing specialist species in a two-patch quantitative genetic model with moving optima. Thanks to a separation of ecological and evolutionary time scales and the phase-line study of the selection gradient, I derive the critical environmental speed for persistence, which reflects how the existence of a refugium impacts extinction patterns and how it relates to the cost of dispersal. Moreover, the analysis provides key insights about the dynamics that arise on the path towards this refugium. I show that after an initial increase of population size, there exists a critical environmental speed above which the species crosses a tipping point, resulting into an abrupt habitat switch. In addition, when selection for local adaptation is strong, this habitat switch passes through an evolutionary “death valley”, leading to a phenomenon related to evolutionary rescue, which can promote extinction for lower environmental speeds than the critical one.

Ecological success of no-take marine protected areas: Using population dynamics theory to inform a global meta-analysis.

Adaptively managing marine protected areas (MPAs) requires accurately assessing whether established MPAs are achieving their goals of protecting and conserving biomass, especially for harvested populations. Ecological MPA assessments commonly compare inside of the MPA to a reference point outside of and/or before implementation (i.e., calculating "response ratios"). Yet, MPAs are not simple ecological experiments; by design, protected populations interact with those outside, and population dynamic responses can be nonlinear. This complicates assessment interpretations. Here, we used a two-patch population model to explore how MPA response ratios (outside-inside, before-after, and before-after-control-impact [BACI]) for fished populations behave under different conditions, like whether the population is receiving a sustainable larval supply or if it is declining despite protection from harvest. We then conducted a Bayesian evaluation of MPA effects on fish and invertebrate populations based on data collected from 82 published studies on 264 no-take MPAs worldwide, using the results of an earlier global meta-analysis as priors. We considered the effects of calculating different summary metrics on these results, drawing on the theoretical insights from our population model as a comparative framework. We demonstrate that not all response ratio comparison types provide the same information: For example, outside-inside and BACI comparisons can fail to detect population decline within MPAs, whereas before-after comparisons likely detect that pattern. Considering these limitations, we nonetheless found that MPAs globally are producing positive outcomes, with on average greater biomass, density, and organism size within their boundaries than reference sites. However, only a small portion of studies (18 of 82) provided the temporal data necessary to determine that protection, on average, has led to increased abundance of populations within MPAs over time. These findings demonstrate the importance of considering the underlying system dynamics when assessing MPA effects. Assuming that large outside-inside or BACI response ratios always reflect large and net positive conservation effects may lead to misleading conclusions, we recommend that: (1) when assessing specific MPA effects, empirical findings be considered alongside theoretical knowledge relevant to that MPA system, and (2) management should respond to the local conditions and outcomes, rather than a blanket expectation for positive MPA effects.

New insights into the combined effect of dispersal and local dynamics in a two-patch population model

Understanding the effect of dispersal on fragmented populations has drawn the attention of ecologists and managers in recent years, and great efforts have been made to understand the impact of dispersal on the total population size. All previous numerical and theoretical findings determined that the possible response scenarios of the overall population size to increasing dispersal are monotonic or hump-shaped, which has become a common assumption in ecology. Against this, we show in this paper that many other response scenarios are possible by using a simple two-patch discrete-time model. This fact evidences the interplay of local dynamics and dispersal and has significant consequences from a management perspective that will be discussed.

Stability and bifurcation in a two-patch commensal symbiosis model with nonlinear dispersal and additive Allee effect

In this paper, a two-patch model with additive Allee effect, nonlinear dispersal and commensalism is proposed and studied. The stability of equilibria and the existence of saddle-node bifurcation, transcritical bifurcation are discussed. Through qualitative analysis of the model, we know that the persistence and the extinction of population are influenced by the Allee effect, dispersal and commensalism. Combining with numerical simulation, the study shows that the total population density will increase when the Allee effect constant [Formula: see text] increases or [Formula: see text] decreases. In addition to suppress the Allee effect, nonlinear dispersal and commensalism are crucial to the survival of the species in the two patches.

KPP transition fronts in a one-dimensional two-patch habitat.

This paper is concerned with the existence of transition fronts for a one-dimensional two patch model with KPP reaction terms. Density and flux conditions are imposed at the interface between the two patches. We first construct a pair of suitable super- and sub solutions by making full use of information of the leading edges of two KPP fronts and gluing them through the interface conditions. Then, an entire solution obtained thanks to a limiting argument is shown to be a transition front moving from one patch to the other one. This propagating solution admits asymptotic past and future speeds, and it connects two different fronts, each associated with one of the two patches. The paper thus provides the first example of a transition front for a KPP-type two-patch model with interface conditions.

Impacts of pleiotropy and migration on repeated genetic adaptation.

Observations of genetically repeated evolution (repeatability) in complex organisms are incongruent with the Fisher-Orr model, which implies that repeated use of the same gene should be rare when mutations are pleiotropic (i.e. affect multiple traits). When spatially divergent selection occurs in the presence of migration, mutations of large effect are more strongly favored, and hence, repeatability is more likely, but it is unclear whether this observation is limited by pleiotropy. Here, we explore this question using individual-based simulations of a two-patch model incorporating multiple quantitative traits governed by mutations with pleiotropic effects. We explore the relationship between fitness trade-offs and repeatability by varying the alignment between mutation effect and spatial variation in trait optima. While repeatability decreases with increasing trait dimensionality, trade-offs in mutation effects on traits do not strongly limit the contribution of a locus of large effect to repeated adaptation, particularly under increased migration. These results suggest that repeatability will be more pronounced for local rather than global adaptation. Whereas pleiotropy limits repeatability in a single-population model, when there is local adaptation with gene flow, repeatability can occur if some loci are able to produce alleles of large effect, even when there are pleiotropic trade-offs.

Dynamics of a two-patch logistic model with diffusion and time delay

In this paper, we proposed a two-patch logistic model connected by diffusion, where one patch includes the Gamma type distribution time delay while the other patch does not include the time delay. In general, Routh–Hurwitz criterion is applied to the derivation for the conditions of Hopf bifurcation, but the more the order of the time delay increases the more the difficulty rises. Hence we adopt the polar form method for the characteristic equation to study the stability of coexistence equilibrium. Our findings show that the diffusion prevents the instabilization of the coexistence equilibrium. Besides, we found that the coexistence equilibrium is stable when time delay is small, and becomes unstable as the delay increases. But it can be restabilized for further increasing of time delay and continues to be stable afterwards. In other words, the diffusion and the time delay are beneficial to the stability of the coexistence equilibrium.

A reaction-diffusion model for population dynamics in patchy landscapes

We address the linear eigenvalue problem associated with reaction-diffusion models for patchy linear domains with any finite number of unique patches. Interface conditions between adjacent patches may be chosen independently to accurately reflect organism behavior. We show that there exists a real principal eigenvalue with a corresponding non-negative eigenfunction that is non-zero interior to the domain. We demonstrate that our model is a generalization of several existing two-patch models. We consider simple three- and four-patch landscapes representing fragmented reserves to illustrate how our model can be used to analyze population persistence, spatial distribution, and migration dynamics. Fragmented landscapes with increased heterogeneity capture migration dynamics not found in landscapes with only two patch types, such as favorable habitat patches that experience net immigration. Conditions for the existence and non-existence of zero-flux conditions interior to three-patch systems are shown, which provide insight to the location of sources and direction(s) of migration.

Influence of dispersal on the dynamics of an eco-epidemiological two-patch predator-prey model in deterministic and stochastic environment with ratio-dependent functional responses

Influence of dispersal on the dynamics of an eco-epidemiological two-patch predator-prey model in deterministic and stochastic environment with ratio-dependent functional responses

Dynamics of the generalized Rosenzweig–MacArthur model in a changing and patchy environment

In this paper, we propose a generalized Rosenzweig–MacArthur predator–prey model with two patches and Allee effects in predators. In a constant environment, for the single-patch model we show the existence of degenerate Bogdanov–Takens bifurcation with codimension up to 4 and degenerate Hopf bifurcation with codimension up to 3, which indicate that the nilpotent cusp of codimension 4 is the organizing center of the bifurcation set. Our results show that Allee effects in predators can not only induce much more complex bifurcations and dynamics (such as three limit cycles, multitype tristability), but also serve as a destabilizing factor to make predators extinction (deterministically) when the environment is sufficiently enriched, which provides strong support for the so-called “paradox of enrichment”. For the two-patch model, we numerically reveal the existence of eight positive steady states including two new coexisting attractors, which means that population dispersal can help species coexist or increase the population abundance. In a changing environment, for the single-patch model we find that the population can track unstable states, fast positive or slow negative environmental change can be beneficial to predators by delaying or even avoiding extinction; while for the two-patch model we observe some interesting phenomena: fast negative (or slow positive) environmental change can make predators in different patches trend to extinction synchronously, and rate-induced different regime shifts.

One way or another: Combined effect of dispersal and asymmetry on total realized asymptotic population abundance

Understanding the consequences on population dynamics of the variability in dispersal over a fragmented habitat remains a major focus of ecological and environmental inquiry. Dispersal is often asymmetric: wind, marine currents, rivers, or human activities produce a preferential direction of dispersal between connected patches. Here, we study how this asymmetry affects population dynamics by considering a discrete-time two-patch model with asymmetric dispersal. We conduct a rigorous analysis of the model and describe all the possible response scenarios of the total realized asymptotic population abundance to a change in the dispersal rate for a fixed symmetry level. In addition, we discuss which of these scenarios can be achieved just by restricting mobility in one specific direction. Moreover, we also report that changing the order of events does not alter the population dynamics in our model, contrary to other situations discussed in the literature.

Destabilization of synchronous periodic solutions for patch models: A criterion by period functions

In this paper, we study the destabilization of synchronous periodic solutions for patch models. By applying perturbation theory for matrices, we derive asymptotic expressions of the Floquet spectra and provide a destabilization criterion for synchronous periodic solutions arising from closed orbits or degenerate Hopf bifurcations in terms of period functions. Finally, we apply the main results to the well-known two-patch Holling-Tanner model.

Dynamics of a two-patch SIR model with disease surveillance mediated infection force

Travel between patches and behavior changes of susceptible individuals are two crucial factors affecting the disease dynamics for any emerging infectious disease. Therefore studies of their impacts on disease dynamics are of great importance, especially to public health policy-makers. In this paper, we propose an SIR model that incorporates these two factors into consideration. For our two-patch SIR model with infection force mediated by (past) disease surveillance, we identify the basic reproduction number and show how it is related to travel modified basic reproduction numbers. We also carry out stability and Hopf bifurcation analysis and obtain conditions under which the disease persists in one patch or both patches. Numerical simulations are carried out to illustrate how travel between two patches and disease surveillance mediated infection force affect the disease dynamics and thus provide us some helpful insights on proper control of emerging infectious diseases.

Optimal dispersal and diffusion-enhanced robustness in two-patch metapopulations: origin’s saddle-source nature matters

A two-patch logistic metapopulation model is investigated both analytically and numerically focusing on the impact of dispersal on population dynamics. First, the dependence of the global dynamics on the stability type of the full extinction equilibrium point is tackled. Then, the behaviour of the total population with respect to the dispersal is studied analytically. Our findings demonstrate that diffusion plays a crucial role in the preservation of both subpopulations and the full metapopulation under the presence of stochastic perturbations. At low diffusion, the origin is a repulsor, causing the orbits to flow nearly parallel to the axes, risking stochastic extinctions. Higher diffusion turns the repeller into a saddle point. Orbits then quickly converge to the saddle’s unstable manifold, reducing extinction chances. This change in the vector field enhances metapopulation robustness. On the other hand, the well-known fact that asymmetric conditions on the patches is beneficial for the total population is further investigated. This phenomenon has been studied in previous works for large enough or small enough values of the dispersal. In this work, we complete the theory for all values of the dispersal. In particular, we derive analytically a formula for the optimal value of the dispersal that maximizes the total population.

Dynamical analysis of a discrete two-patch model with the Allee effect and nonlinear dispersal.

The dynamic behavior of a discrete-time two-patch model with the Allee effect and nonlinear dispersal is studied in this paper. The model consists of two patches connected by the dispersal of individuals. Each patch has its own carrying capacity and intraspecific competition, and the growth rate of one patch exhibits the Allee effect. The existence and stability of the fixed points for the model are explored. Then, utilizing the central manifold theorem and bifurcation theory, fold and flip bifurcations are investigated. Finally, numerical simulations are conducted to explore how the Allee effect and nonlinear dispersal affect the dynamics of the system.

Multi-scale effects of habitat loss and the role of trait evolution.

Habitat loss (HL) is a major cause of species extinctions. Although the effects of HL beyond the directly impacted area have been previously observed, they have not been modelled explicitly, especially in an eco-evolutionary context. To start filling this gap, we study a two-patch deterministic consumer-resource model, with one of the patches experiencing loss of resources as a special case of HL. Our model allows foraging and mating within a patch as well as between patches. We then introduce heritable variation in consumer traits related to resource utilization and patch use to investigate eco-evolutionary dynamics and compare results with constant and no trait variation scenarios. Our results show that HL in one patch can indeed reduce consumer densities in the neighbouring patch but can also increase consumer densities in the neighbouring patch when the resources are overexploited. Yet at the landscape scale, the effect of HL on consumer densities is consistently negative. Patch isolation increases consumer density in the patch experiencing HL but has generally negative effects on the neighbouring patch, with context-dependent results at the landscape scale. With high cross-patch dependence and coupled foraging and mating preferences, local HL can sometimes even lead to landscape-level consumer extinction. Eco-evolutionary dynamics can rescue consumers from such extinction in some cases if their death rates are sufficiently small. More generally, trait evolution had positive or negative effects on equilibrium consumer densities after HL, depending on the evolving trait and the spatial scale considered. In summary, our findings show that HL at a local scale can affect the neighbouring patch and the landscape as a whole, where heritable trait variation can, in some cases, alleviate the impact of HL. We thus suggest joint consideration of multiple spatial scales and trait variation when assessing and predicting the impacts of HL.

Animal-vectored nutrient flows across resource gradients influence the nature of local and meta-ecosystem functioning

Organisms move across landscapes mediating nutrient, energy, and biomass exchanges. Meta-ecosystem ecology offers a framework to study how these flows affect ecosystem functions in space and time. However, meta-ecosystem models often represent consumer movement as diffusion along gradients of resources. Crucially, this assumes that consumer movement connects the same trophic compartments among patches of the same ecosystem. Yet, empirical evidence shows that organisms move across different ecosystems and connect diverse trophic compartments in diffusive and non-diffusive ways. Here, we derive a two-patch meta-ecosystem model that accounts for both types of organismal movement, and we investigate their influences on local and meta-ecosystem functions. We apply a novel approach, modelling consumer movement through a dispersers’ pool that captures the fraction of moving organisms and allows us to partition local and regional dynamics. We show that non-diffusive consumer movement increases landscape heterogeneity while diffusive consumer movement enhances source–sink dynamics. Local ecosystem differences driven by consumer movement type are less prevalent at meta-ecosystem extents. Thus, movement type is essential for predicting local ecosystem dynamics. Our results support recent calls to consider explicitly the role of consumers in shaping and maintaining ecosystem functions across spatial extents.

Effect of avian influenza scare on transmission of zoonotic avian influenza: A case study of influenza A (H7N9)

Avian influenza scare is a human psychological factor that asserts both positive and negative effects on the transmission of zoonotic avian influenza. In order to study the dichotomous effect of avian influenza scare on disease transmission, taking H7N9 avian influenza as a typical case, a two-patch epidemic model is proposed. The global dynamics and the threshold criteria are established by LaSalle invariant principle and the theory of asymptotic autonomous system. To mitigate the negative effects and curb illegal poultry trade, a game-theoretic model is adopted to explore the optimal policy of culling subsidies to reasonably compensate stakeholders for their economic losses resulting from the scare. The optimal policy of culling subsidy is found to heavily depend on the penalty of illegal poultry trade, the stakeholders’ income, the intensity of control measures, and the prevalence level of the disease. The negative effect of avian influenza scare on disease transmission is considerably more significant than the positive effect. In order to avoid a widespread outbreak of zoonotic avian influenza across the region, a comprehensive national global control strategy is essential and effective, even in the presence of the negative effect of the avian influenza scare.