Two-patch Environment Research Articles

Complex dynamics in prey-predator systems with cross-coupling: Exploring nonlinear interactions and population oscillations

This study investigates the problem of ecosystem dynamics in fragmented landscapes, specifically focusing on a two-patch environment with interacting prey and predators. The research examines the impact of cross-predation on these interactions. Using bifurcation analysis, we explored the structural arrangement of attractors and identified complex dynamics such as symmetric, asymmetric, and asynchronous attractors induced by varying cross-coupling levels. Notably, our study describes a novel mechanism for the formation of anti-phase synchrony in the patches. Unlike typical occurrences of a cycle following Hopf bifurcation, our model reveals that the anti-phase cycle stabilizes via Neimark-Sacker (NS) bifurcation of a two-period unstable cycle branch emanating from the synchronous cycle branch. Our findings also demonstrate that cross-feeding can lead to significant ecosystem asymmetry and branching, culminating in the dominance of a single cross-feeding chain. These results challenge traditional models and highlight the presence of multistability and the potential for ecosystem evolution towards distinct subsystem branches due to cross-predation. The study’s insights offer valuable contributions to population and evolutionary biology, enhancing our understanding of the intricate dynamics within fragmented ecosystems.

Analysis of an SIRS Model in Two-Patch Environment in Presence of Optimal Dispersal Strategy

Migration or dispersal of population plays an important role in disease transmission during an outbreak. In this work, we have proposed an SIRS compartmental epidemic model in order to analyze the system dynamics in a two-patch environment. Both the deterministic and fractional order systems have been considered in order to observe the impact of population dispersal. The following analysis has shown that we can have an infected system even if the basic reproduction number in one patch becomes less than unity. Moreover, higher dispersal towards a patch controls the infection level in the other patch to a greater extent. In the optimal control problem (both integer order and fractional), it is assumed that people’s dispersal rate will depend on the disease prevalence, and as such will be treated as a time-dependent control intervention. The numerical results reveal that there is a higher amount of recovery cases in both patches in the presence of optimal dispersal (both integer order and fractional). Not only that, implementation of people’s awareness reduces the infection level significantly even if people disperse at a comparatively higher rate. In a fractional system, it is observed that there will be a higher amount of recovery cases if the order of derivative is less than unity. The effect of fractional order is omnipotent in achieving a stable situation.

Population dynamics of two competing species in an advective heterogeneous two-patch environment

Population dynamics of two competing species in an advective heterogeneous two-patch environment

Consequences of Traceable Mobility in Populations Exhibiting Strong Allee Effect

In this research, we study the impacts of the traceable mobility in a two-patch environment when the population in each patch exhibits strong Allee effects. Traveling individuals are traced across patches by budgeting the average time spent in each patch while keeping their place of residency. Particularly, we focus on the impact that the effective population (residents and visitors) produces on regional dynamics.Our results show that low mobility across regions produces simple dynamics, where orbits converge to single or double extinction, or to a coexistence steady state. We derive mobility conditions under which an endangered population may benefit of the presence of a visitant one and avoid extinction -- the rescue effect. Nonetheless, increments in the visiting population would also lead the resident population to extinction -- the induced extinction effect.

Impacts of Seasonal and Spatial Variations on the Transmission of Typhoid Fever

Typhoid fever is a disease caused by the bacteria Salmonella Typhi through the ingestion of contaminated food or water, and it is still serious in developing countries. The infection routes include both human-to-human transmission and environment-to-human transmission. It was observed that higher incidence of typhoid fever occur during the rainy season and people living near water bodies may have a higher rate of typhoid infection. On the other hand, asymptomatically infected individuals also play a central role in the transmission of typhoid since they are not experiencing any symptoms but they are able to shed S. Typhi into the environment for years. Thus, a well-described model of the Typhoid transmission should include the asymptomatical compartment and the factors of spatial homogeneity and seasonality. This motivates us to develop a periodic two-patch system to investigate the spatial and seasonal effects on the transmission of Typhoid fever, in which the bacteria in the environment is included, and the population of human is divided into five classes, namely, susceptible individuals, infected individuals, carrier individuals, individuals under treatment and recovered individuals. We first introduce the basic reproduction number for the model, then we show that the extinction/persistence of Typhoid can be determined by <I>R</I>₀. Our numerical results indicate that an outbreak of Typhoid fever in a two-patch environment could be eliminated if migration between patches is prohibited. Finally, we also numerically observe that the infection risks of Typhoid may be underestimated if seasonal effects are ignored.

Effect of dispersal in two-patch environment with Richards growth on population dynamics

In this paper, we consider a two-patch model coupled by migration terms, where each patch follows a Richards law. First, we prove the global stability of the model. Second, in the case when the migration rate tends to infinity, the total carrying capacity is given, which in general is different from the sum of the two carrying capacities and depends on the parameters of the growth rate and also on the migration terms. Using the theory of singular perturbations, we give an approximation of the solutions of the system in this case. Finally, we determine the conditions under which fragmentation and migration can lead to a total equilibrium population which might be greater or smaller than the sum of two carrying capacities and we give a complete classification for all possible cases. The total equilibrium population formula for a large migration rate plays an important role in this classification. We show that this choice of local dynamics has an influence on the effect of dispersal. Comparing the dynamics of the total equilibrium population as a function of the migration rate with that of the logistic model, we obtain the same behavior. In particular, we have only three situations that the total equilibrium population can occur: it is always greater than the sum of two carrying capacities, always smaller, and a third case, where the effect of dispersal is beneficial for lower values of the migration rate and detrimental for the higher values. We end by examining the two-patch model where one growth rate is much larger than the second one, we compare the total equilibrium population with the sum of the two carrying capacities.

The best of both worlds: Combining population genetic and quantitative genetic models

Numerous traits under migration–selection balance are shown to exhibit complex patterns of genetic architecture with large variance in effect sizes. However, the conditions under which such genetic architectures are stable have yet to be investigated, because studying the influence of a large number of small allelic effects on the maintenance of spatial polymorphism is mathematically challenging, due to the high complexity of the systems that arise. In particular, in the most simple case of a haploid population in a two-patch environment, while it is known from population genetics that polymorphism at a single major-effect locus is stable in the symmetric case, there exist no analytical predictions on how this polymorphism holds when a polygenic background also contributes to the trait. Here we propose to answer this question by introducing a new eco-evo methodology that allows us to take into account the combined contributions of a major-effect locus and of a quantitative background resulting from small-effect loci, where inheritance is encoded according to an extension to the infinitesimal model. In a regime of small variance contributed by the quantitative loci, we justify that traits are concentrated around the major alleles, according to a normal distribution, using new convex analysis arguments. This allows a reduction in the complexity of the system using a separation of time scales approach. We predict an undocumented phenomenon of loss of polymorphism at the major-effect locus despite strong selection for local adaptation, because the quantitative background slowly disrupts the rapidly established polymorphism at the major-effect locus, which is confirmed by individual-based simulations. Our study highlights how segregation of a quantitative background can greatly impact the dynamics of major-effect loci by provoking migrational meltdowns. We also provide a comprehensive toolbox designed to describe how to apply our method to more complex population genetic models.

Effects of Predator-Driven Prey Dispersal on Sustainable Harvesting Yield

Dispersal of organisms between patches is a common phenomenon in ecology and plays an important role in predator–prey population dynamics. We propose a nonsmooth Filippov predator–prey model in a two-patch environment characterized by a generalist predator-driven intermittent refuge protection of an apprehensive prey along with a balanced dispersal of the prey between refuge and nonrefuge areas. By employing qualitative techniques of nonsmooth dynamical systems, we see that the switching surface is a repeller whenever the interior equilibria are virtual, causing long-term population fluctuations. We find that the level of prey vigilance and the rate of prey dispersal play pivotal roles in the total harvesting yield. We observe that a sustainable high harvesting yield is possible when the prey is less vigilant and obtain the harvesting efforts for maximum sustainable total yield (MSTY). We further modify the model by considering a continuous threshold predator-driven prey dispersal and show that the model exhibits a Hopf bifurcation when the level of prey vigilance exceeds some critical threshold value. By comparing the dynamics of the two models we see that for a sustainable high harvesting yield of the system with continuous threshold dispersal, the prey needs to be highly vigilant compared to that of the system with intermittent dispersal of the prey. Further, we find numerically that the estimated MSTY from both models remains the same.

Adaptation in a heterogeneous environment I: persistence versus extinction.

Understanding how a diversity of plants in agroecosystems affects the adaptation of pathogens is a key issue in agroecology. We analyze PDE systems describing the dynamics of adaptation of two phenotypically structured populations, under the effects of mutation, selection and migration in a two-patch environment, each patch being associated with a different phenotypic optimum. We consider two types of growth functions that depend on the n-dimensional phenotypic trait: either local and linear or nonlocal nonlinear. In both cases, we obtain existence and uniqueness results as well as a characterization of the large-time behaviour of the solution (persistence or extinction) based on the sign of a principal eigenvalue. We show that migration between the two environments decreases the chances of persistence, with in some cases a 'lethal migration threshold' above which persistence is not possible. Comparison with stochastic individual-based simulations shows that the PDE approach accurately captures this threshold. Our results illustrate the importance of cultivar mixtures for disease prevention and control.

On predation effort allocation strategy over two patches

In this paper, we formulate an ODE model to describe the population dynamics of one non-dispersing prey and two dispersing predators in a two-patch environment with spatial heterogeneity. The dispersals of the predators are implicitly reflected by the allocation of their presence (foraging time) in each patch. We analyze the dynamics of the model and discuss some biological implications of the theoretical results on the dynamics of the model. Particularly, we relate the results to the evolution of the allocation strategy and explore the impact of the spatial heterogeneity and the difference in fitness of the two predators on the allocation strategy. Under certain range of other parameters, we observe the existence of an evolutionarily stable strategy (ESS) while in some other ranges, the ESS disappears. We also discuss some possible extensions of the model. Particularly, when the model is modified to allow distinct preys in the two patches, we find that the heterogeneity in predation rates and biomass transfer rates in the two patches caused by such a modification may lead to otherwise impossible bi-stability for some pairs of equilibria.

Dynamics of an epidemic model with relapse over a two-patch environment

In this paper, with the assumption that infectious individuals, once recovered for a period of fixed length, will relapse back to the infectious class, we derive an epidemic model for a population living in a two-patch environment (cities, towns, or countries, etc.). The model is given by a system of delay differential equations with a fixed delay accounting for the fixed constant relapse time and a non-local term caused by the mobility of the individuals during the recovered period. We explore the dynamics of the model under two scenarios: (i) assuming irreducibility for three travel rate matrices; (ii) allowing reducibility in some of the three matrices. For (i), we establish the global threshold dynamics in terms of the principal eigenvalue of a 2×2 matrix. For (ii), we consider three special cases so that we can obtain some explicit results, which allow us to explicitly explore the impact of the travel rates. We find that the role that the travel rate of recovered and infectious individuals differs from that of susceptible individuals. There is also an important difference between case (i) and (ii): under (ii), a boundary equilibrium is possible while under (i) it is impossible.

Non-linear population discrete models with two time scales: re-scaling of part of the slow process

In this work we present a reduction result for discrete-time systems with two time scales. In order to be valid, previous results in the field require some strong hypotheses that are difficult to check in practical applications. Roughly speaking, the iterates of a map as well as their differentials must converge uniformly on compact sets. Here, we eliminate the hypothesis of uniform convergence of the differentials at no significant cost in the conclusions of the result. This new result is then used to extend to non-linear cases the reduction of some population discrete models involving processes acting at different time scales. In practical cases, some processes that occur at a fast time scale are often only measured at slow time intervals, notably mortality. For a general class of linear models that include such a kind of processes, it has been shown that a more realistic approach requires the re-scaling of those processes to be considered at the fast time scale. We develop the same type of re-scaling in some non-linear models and prove the corresponding reduction results. We also provide an application to a particular model of a structured population in a two-patch environment.

Loose social organisation of AB strain zebrafish groups in a two-patch environment.

We study the collective behaviour of zebrafish shoals of different numbers of individuals (1, 2, 3, 5, 7, 10 and 20 AB zebrafish Danio rerio) in a constraint environment composed of two identical square rooms connected by a corridor. This simple set-up is similar to a natural patchy environment. We track the positions and the identities of the fish and compute the metrics at the group and at the individual levels. First, we show that the number of fish affects the behaviour of each individual in a group, the cohesion of the groups, the preferential interactions and the transition dynamics between the two rooms. Second, during collective departures, we show that the rankings of exit correspond to the topological organisations of the fish prior to their collective departure. This spatial organisation appears in the group a few seconds before a collective departure. These results provide new evidences on the spatial organisation of the groups and the effect of the number of fish on individual and collective behaviours in a patchy environment.

Coexistence and extinction for two competing species in patchy environments.

A system of two competing species μ and ν that diffuse over a two-patch environment is investigated. When u-species has smaller birth rate in the first patch and larger birth rate in the second patch than v-species, and the average birth rate for u-species is larger than or equal to v-species, it was shown in a previous publication that two species coexist in a slow diffusion environment, whereas u-species drives v-species into extinction in a fast diffusion environment. In this paper, we analyze global dynamics and bifurcations for the same model with identical order of birth rates, but with opposite order of average birth rates, i.e., the average birth rate of u-species is less than that of v-species. We observe richer dynamics with two scenarios, depending on the relative difference between the variation in the birth rates of v-species on two patches and the variation in the average birth rates of two species. When the variation in average birth rates is relatively large, there is no stability switch for the semitrivial equilibria. On the other hand, such a stability switch takes place when the variation in average birth rates is relatively mild. In both cases, v-species, with larger average birth rate, prevails in a fast diffusion environment, whereas in a slow diffusion environment, the two species can coexist or u-species that has the greatest birth rate among both species and patches will persist and drive v-species to extinction.

Stability analysis of a two-patch predator\u2013prey model with two dispersal delays

This paper deals with a predator–prey model with both species in the delayed-dispersal case in a two-patch environment. The purpose of this paper is to study the effect of two dispersal delays on the stability of three equilibria. It turns out that the stability of the trivial equilibrium and the boundary equilibrium is delay-independent. However, the stability of the coexistence equilibrium is delay-dependent. Numerical simulations are performed to demonstrate the obtained results.

食饵庇护所对斑块环境下Leslie-Gower捕食系统的影响

PDF HTML阅读 XML下载 导出引用 引用提醒 食饵庇护所对斑块环境下Leslie-Gower捕食系统的影响 DOI: 10.5846/stxb201704130654 作者: 作者单位: 河南大学 数学与统计学院 应用数学研究所 作者简介: 通讯作者: 中图分类号: Q141 基金项目: 中国博士后科学基金项目（2017M612388）；国家自然科学基金项目（31200312）；河南大学科学基金项目（yqpy20140045） The effect of prey refuge in a patchy Leslie-Gower predation system Author: Affiliation: Henan UniversityInstitute of Applied Mathematics, School of Mathematics and Information Sciences, Fund Project: 摘要 | 图/表 | 访问统计 | 参考文献 | 相似文献 | 引证文献 | 资源附件 | 文章评论 摘要:建立了一个显式含有空间庇护所的两斑块Leslie-Gower捕食者-食饵系统。假设只有食饵种群在斑块间以常数迁移率迁移，且在每个斑块上食饵间的迁移比局部捕食者-食饵相互作用发生的时间尺度要快。利用两个时间尺度，可以构建用来描述所有斑块总的食饵和捕食者密度的综合系统。数学分析表明，在一定条件下，存在唯一的正平衡点，并且此平衡点全局稳定。进一步，捕食者的数量随着食饵庇护所数量增加而降低；在一定条件下，食饵的数量随着食饵庇护所数量增加先增加后降低，在足够强的庇护所强度下，两物种出现灭绝。对比以往研究，利用显式含有和隐含空间庇护所的数学模型所得结论不一致，这意味着在研究庇护所对捕食系统种群动态影响时，空间结构可能起着重要作用。 Abstract:Predation is a common and important species interaction in many ecological systems. Mathematical models are important and very useful tools to understand and analyze the dynamic behavior of the predator-prey systems. Classical predator-prey models such as the Lotka-Volterra model and Leslie-Gower model can be used in a homogeneous environment. However, in general, the environment is heterogeneous and this can be represented using a set of discrete patches connected by migration. In the simplest situation, the two patches predator-prey model is used, which is composed of the local population process and the dispersal from patch to patch. Prey escaping predation in space or time is widely observed. Spatial or temporal refuges are well-known examples of this class of mechanism. Most theoretical studies have focused on how refuges add stability to the system with predator-prey interactions. The role of spatial and temporal refuges on species coexistence in communities with intraguild predation has also been investigated. These studies have two characteristics. First, most of the research is based on the Lotka-Volterra framework, only a few on the Leslie-Gower framework. Second, the traditional way to model this is to modify the functional response of predators and consider prey refuges, implicitly. The prey refuge can positively affect the growth of prey and negatively that of predators, because the decrease of predation success can lead to the reduction of prey mortality. On the other hand, the hiding behavior of prey could be either advantageous or detrimental for the involved populations. For example, the prey population in the refuges has a low or even negative growth rate because they are rarely offered feeding or mating opportunities. That is, there is a different population structure between the prey refuge and the normal habitat patch. Therefore, it is more reasonable to consider prey refuge explicitly. Based on the above consideration, this paper proposes a Leslie-Gower predator-prey model incorporating the effect of prey refuge explicitly in a two-patch environment. We assume that the prey dispersal is at a constant migration rate and is faster than the local predator-prey interaction, while the predator cannot disperse between patches. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. The mathematical analysis of the aggregated model shows that there exists a unique and globally stable positive equilibrium under a certain condition. Simple mathematical analysis shows that increasing the amount of refuge can increase predator densities. As far as the prey species is concerned, under a certain condition, there exists a threshold, such that, for the prey refuge smaller than this threshold, increasing the amount of prey refuge can increase the prey densities, and if the prey refuge is larger than this threshold, increasing the amount of prey refuge can decrease the prey densities. Furthermore, prey and predator will become extinct when the effect of prey refuge is strong enough. In contrast to previous research, the effects of refuges used by the prey are different in spatially implicit and spatially explicit models. The discrepancy suggests that the spatial structure is important when refuges are included. 参考文献 相似文献 引证文献

Stability Analysis of a Biological Model of a Marine Resources Allowing Density Dependent Migration

Biology of a marine resources is a descriptive science. The description is the first step towards understanding a system. However, the main objective is to present a rigorous mathematical analysis and numerical simulation of these spatio temporal models. In the present paper, we consider a two species food chain, i.e. a prey and predator populations modeled in a two-patch environment, one of which is a free fishing zone and the other one is protected zone. We study the qualitative analysis of solutions and we establish sufficient conditions under which the endemic and trivial equilibria are asymptotically stable.The asymptotic stability corresponding to the equilibria is graphically shown.

Dynamics of the predator-prey models on the two-patch fragmented habitat with dispersal

In this work, we consider the population-dispersal dynamics for predator-prey interactions in a two-patch environment. On each fragmented patch, there is a two-species predator-prey ecological system. It is assumed that the predator species are mobile. The existence and local dynamics of boundary equilibria and interior equilibria with respect to parameters are completely classified. Moreover, global extinction results are established analytically. In particular, the phenomenon of over-exploitation is also found in these discrete patches models. Finally, some biological interpretations are discussed.

Natural enemies deployment in patchy environments for augmentative biological control

Biological control is an important tool for ecologically friendly crop protection against pests, that consists in using a biological organism (predator, parasitoid, pathogen) to reduce the population density of the targeted pest. We examine the effectiveness of periodic impulsive releases of biocontrol agents (beneficial species) into a two-patch environment through mathematical modeling. In this paper, we consider a spatio-temporal Lotka–Volterra pest–predator system defined over two patches. We show that the threshold predator release rate guaranteeing the stability of the pest-free solution is actually independent of the release period when predators in both patches follow balanced dynamics or pests do not disperse. Otherwise, the stability threshold becomes period-dependent and more specifically it is an increasing function of the release period. This implies that the deployment of biological control agents at a given release rate can possibly succeed if releases are frequent and small and fail otherwise. In the various cases we also show what the optimal strategy is that minimizes the total release rate or how to spread a given release rate between the two patches.

Periodic solutions for an impulsive semi-ratio-dependent predator-prey model with patches and time delays

In this paper, we study the existence of a positive periodic solution for a two-species semi-ratio-dependent predator-prey system with time delays and impulses in a two-patch environment. By using the method of coincidence degree theorem, a set of easily verifiable conditions are obtained for the existence of at least one strictly positive periodic solution for the system. In particular, our result generalizes some known criteria.