Lotka-Volterra Research Articles

Dynamics of a non-local intraspecific competition predator–prey model with memory effect

In the paper, we investigate a diffusive predator–prey model with nonlocal intraspecific prey competition and spatial memory under Neumann boundary conditions. Through stability and bifurcation analysis, we find that the memory-based diffusion coefficient and the spatiotemporal diffusive delay have important effects on the dynamics of the model. By using the spatiotemporal diffusive delay as a bifurcation parameter, the critical values are determined for the stability of the positive constant steady state and the associated Hopf bifurcation. We find that the system may admit no stability switch, one stability switch and multiple stability switches.

Ecological and epidemiological ramification of fear: Exploring deterministic and stochastic dynamics in a predator–prey system with predator switching and harvesting

In an ecosystem, harvesting infected prey can assist in managing and containing the spread of the illness within the prey species. On the other hand, the harvesting of predators can be beneficial as it regulates their numbers, preventing them from over-consuming prey and subsequently preserving existence of the prey population. This study introduces a predator–prey model that encompasses prey infection, predator–prey interactions influenced by fear, switching and harvesting. We derive an analytic expression for the basic reproduction number, a critical determinant of disease spread. We investigate the global stability of disease-free and endemic equilibria contingent on the basic reproduction number’s value, highlighting the potential for disease eradication by maintaining it below unity. In-depth analysis of the deterministic model is undertaken, with a focus on Hopf bifurcations that delineate thresholds for disease-free and endemic states. Furthermore, the deterministic model is extended to incorporate environmental stochasticity. We obtain the conditions under which population extinction occurs. Our findings elucidate how the intensity of environmental noise influences population dynamics, providing valuable insights into extinction risks under varying noise levels.

Novel Framework for Exploring Human–Water Symbiosis Relationship: Analysis, Quantification, Discrimination, and Attribution

There is an interdependent symbiotic relationship between humans and water; scientific and effective assessment of the human–water symbiosis relationship is of great significance for the promotion of sustainable development. This study developed a novel framework of the human–water symbiosis relationship under an integrated perspective, which included theoretical interpretation, quantitative assessment, pattern discrimination, and an attribution analysis. Based on the symbiosis theory, the theoretical analysis of the human–water relationship was carried out to analyze the three basic elements of the human–water system, and then the evaluation index system of the human–water symbiosis system was constructed to quantitatively assess the development level of the human system and the water system. The Lotka–Volterra model was used to identify the symbiotic pattern, and the human–water symbiosis index was calculated to characterize the health state of the human–water symbiosis system. The main influencing factors of the human–water symbiosis system were further identified through an attribution analysis. Finally, a case study was carried out with 18 cities in Henan Province. Results reveal that (a) the proposed method can effectively realize the quantitative characterization of the human–water symbiosis relationship, with good applicability and obvious advantages; (b) the human–water symbiosis pattern of cities in Henan Province is dominated by the “human system parasitizes water system (H+W−)” pattern, and more attention should be paid to the water system in the subsequent development of it; and (c) the main factors influencing the human system, the water system, and the human–water symbiosis system are the research and development (R&D) personnel equivalent full-time (H7), per capita water resources (W1), and proportion of water conservancy and ecological water conservancy construction investment (W6), respectively. The findings can provide theoretical and methodological support for the study of the human–water symbiosis relationship and sustainable development in other regions.

Exploring the influence of refuges and additional foods on predator–prey interactions amidst environmental stochasticity and water level fluctuations

By considering that the predators have some additional foods apart from the focal prey and the prey take refuge, a predator–prey model with the effect of seasonality in a fluctuating environment is investigated in this study. Gaussian white noises, created by random fluctuation of water level, are introduced on the prey’s growth rate and predator’s mortality rate. Both the deterministic and stochastic systems are analyzed mathematically as well as numerically. Our analytical findings show that the intensity of environmental noise, additional foods and prey refuge play significant roles in survival as well as extinction of both prey and predator populations in the aquatic system. Our numerical results show that the effectual level of additional foods, depending on water level, has positive as well as negative effects on the predator and prey populations. We find that both species strongly persist at certain water level and for the low intensity of noise. However, for an increased value of noise intensity, they persist weakly and then go to extinction. Overall, our findings show that the variations of water level together with additional food and white noise plays crucial roles in persistence and extinction of species in the ecosystem.

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Relaxation Oscillation, Homoclinic Orbit and Limit Cycles in a Piecewise Smooth Predator–Prey Model

Existence of forced waves and gap formations for a Lotka-Volterra competition system with nonlocal dispersal in a time periodic shift habitat

This paper is devoted to the study of a class of time periodic Lotka-Volterra competition system with nonlocal dispersal and shifting habitats. By using some known results of the periodic KPP model and employing the iterative techniques, we prove that there exist two positive numbers $c_{0}(d_{1})$ and $c_{0}(d_{2})$ such that the system admits a forced wave provided that the forcing speed $c\in(-c_{0}(d_{2}),c_{0}(d_{1}))$. In addition, based on the theoretical results, we show that the gap formations exist for $c>c_{0}(d_{1})$ and $c<-c_{0}(d_{2})$.

Stable coexistence in indefinitely large systems of competing species

The Lotka–Volterra system is a set of ordinary differential equations describing growth of interacting ecological species. One of the debated questions is understanding how the number of species in the system influences the stability of the model. Robert May studied how large systems may become unstable when species–species interactions do not vanish. This outcome has frequently been interpreted as a universal phenomenon and summarized as ‘large systems are unstable’. By exploring general interaction networks, we show that the competitive Lotka–Volterra system may maintain stability even for large networks, despite non-vanishing interaction strength. We establish sufficient conditions for stability as a threshold on the interspecific interaction strength, formulated in terms of the maximum and minimum degrees (or weights) rather than the network’s size. For values below this threshold, coexistence of all species is attained, regardless of the system size. Our finding generalizes May’s result by showing that it is the outlier nodes with large degree that cause instability rather than the large number of species in the system.

A Stage Structure Prey Predator Model Using Pentagonal Fuzzy Numbers and Functional Response

في هذه الدراسة، قمنا بدراسة نموذج مفترس للفريسة مع هيكل مرحلة للفريسة. الهدف من الدراسة هو إيجاد سلوك النموذج باستخدام قيم المعلمات في وجود أرقام ضبابية خماسية.. يتم التفاعل بين الأنواع باستخدام الاستجابات الوظيفية، مثل تفاعل هولينج من النوع الأول للفريسة الناضجة واستجابة كرولي مارتن الوظيفية للفريسة الناضجة . فكرة المشكلة هي بناء نموذج رياضي في بيئة غامضة باستخدام المعلمات الغامضة والقيم الأولية.. يتم تنفيذ وجود نقاط التوازن. وباستخدام مفهوم قطع ألفا للمعلمات المستخدمة في نموذج الفريسة – المفترس تم التعامل مع الأعداد الغامضة الخماسية. يمكن اعتبار المعلمات التي استخدموها في الصياغة الرياضية قيمة واضحة من خلال تطبيق طريقة إزالة الضبابية. هنا يتم استخدام طريقة تقنية التصنيف القوية. تتم دراسة استقرار كل نقطة توازن عن طريق حساب مصفوفة جاكوبي وإيجاد القيم الذاتية التي تم تقييمها عند كل نقطة توازن. من خلال الاستفادة من تحليل استقرار هيكل مرحلة الفريسة يتم اكتشافه أيضًا. بالنسبة للنظام الديناميكي تم توفير عمليات محاكاة عددية باستخدام برنامج MATLAB للكمبيوتر حتى نتمكن من عرض سلوك النظام وتحديد ما إذا كان مستقرًا أم غير مستقر.

Dynamic behavior of a class of predator–prey model with two time delays

Dynamic behavior of a class of predator–prey model with two time delays

Limit cycle oscillations in the zonal-flow-catalyzed interactions of ion-temperature-gradient turbulence

Limit-cycle oscillations are studied for ion temperature gradient turbulence, which, in the absence of large diamagnetic (mean) shear flows, saturates through energy transfer from unstable modes to large-scale stable modes via zonal-flow intermediary modes. Oscillations of zonal flow and turbulence levels are strongly constrained by the reactive, largely non-dissipative character of the zonal flows. Since existing predator–prey models for observed oscillations in experiments do not include energy transfer through zonal flows to stable modes, low-order fluid models with this physics are constructed and investigated. A simple three-wave truncation produces low-amplitude zonal flows that slowly oscillate around a zero mean, with turbulence oscillations between coupled wavenumbers that exceed linear frequencies by orders of magnitude. This inconsistency with experimental observations is caused by the weak non-linear drive of zonal flows in three-wave systems and the lack of multiple-wavenumber turbulent interactions. A more comprehensive model that preserves multiple wavenumber interactions within the context of conservative zonal-flow-mediated energy transfer to stable modes accurately reflects observed dynamics when the phase between stable and unstable modes is occasionally randomized.

On β-extinction and Stability of a Stochastic Lotka-Volterra System with Infinite Delay

On β-extinction and Stability of a Stochastic Lotka-Volterra System with Infinite Delay

Research on the group effect of new energy vehicles from different policy perspectives

Abstract Policies play a pivotal role in fostering the growth of new energy vehicles. This paper delves into the benefits and risks associated with the development of new energy vehicles under varying policy landscapes. Firstly, the intricate relationship between the competition and growth of new energy vehicles and internal combustion engine vehicles is captured through the application of the extended Lotka-Volterra model. Secondly, a simulation flow diagram is constructed, taking into account the actual mediums of policy impact and drawing insights from diverse research methodologies. This flow diagram enables a comprehensive evaluation of environmental benefits, health risks, and traffic pressure. Finally, a critical evaluation of the actual impact of policies on new energy vehicles is conducted, along with an exploration of the root causes of potential policy paradoxes. The findings reveal that a standalone infrastructure support policy, research and development support policy, or carbon tax policy can, in certain periods, lead to an increase in carbon emissions and traffic pressure. Although the driving restriction policy does not exert negative impacts on the environment, health, and traffic at the national level, it can result in a surge in the total vehicle population. Therefore, it is imperative to adopt a combination of policies to mitigate the potential risks associated with any single policy.

A novel binary data classification algorithm based on the modified reaction-diffusion predator-prey system with Holling-II function.

In this study, we propose a modified reaction-diffusion prey-predator model with a Holling-II function for binary data classification. In the model, we use u and v to represent the densities of prey and predators, respectively. We modify the original equation by substituting the term v with f-v to obtain a stable and clear nonlinear decision surface. By employing a finite difference method for numerical solution of the original model, we conduct various experiments in two-dimensional and three-dimensional spaces to validate the feasibility of the classifier. Additionally, with consideration for wide real applications, we perform classification experiments on electroencephalogram signals, demonstrating the effectiveness and robustness of the classifier in binary data classification.

Dynamics of a slow-fast Leslie-Gower predator-prey model with prey harvesting.

For the Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, the known results are on the saddle-node bifurcation and the Hopf bifurcation of codimensions 1, the Bogdanov-Takens bifurcations of codimensions 2 and 3, and on the cyclicity of singular slow-fast cycles. Here, we focus on the global dynamics of the model in the slow-fast setting and obtain much richer dynamical phenomena than the existing ones, such as global stability of an equilibrium; an unstable canard cycle exploding to a homoclinic loop; coexistence of a stable canard cycle and an inner unstable homoclinic loop; and, consequently, coexistence of two canard cycles: a canard explosion via canard cycles without a head, canard cycles with a short head and a beard and a relaxation oscillation with a short beard. This last one should be a new dynamical phenomenon. Numerical simulations are provided to illustrate these theoretical results.

Population dynamics in a Leslie-Gower predator-prey model with proportional prey refuge at low densities

In this paper we propose a mathematical Leslie-Gower predator-prey model, in which the prey takes refuge from the predator when its population size is below a critical threshold, the functional response of the predator is represented by a Holling II function, and the growth of the prey in the absence of the predator is subject to a semi-saturation parameter that affects its birth curve. Since the model is composed of two vector fields, its qualitative analysis includes, in addition to the determination of the number and stability of the equilibria for each vector field and belonging to the biological sense set, the study of the dynamics in the trajectories close to the dividing curve of the two vector fields in order to determine possible pseudo-equilibria. As a result, if the proposed model has a single inner equilibrium, then there is the possibility of having between one or at least two limit cycles, coexisting or not in both vector fields and around the inner equilibrium. Likewise, the model has a stable pseudo-equilibrium which may be surrounded by at least two limit cycle.

COMPLEX DYNAMICAL BEHAVIORS OF A DISCRETE MODIFIED LESLIE–GOWER PREDATOR–PREY MODEL WITH PREY HARVESTING

Recently, the research on the modified Leslie–Gower model has become an appealing topic. Due to economic efficiency and the complexity of discrete models, we investigate a discrete modified Leslie–Gower predator–prey model with prey harvesting in this paper. The stability of fixed points and bifurcations of the interior fixed points are studied. According to bifurcation theory and normal forms, we derived the conditions of codimension 2 bifurcations occurred, including 1:1 strong resonance bifurcation and fold-flip bifurcation. These two bifurcations are unusual in bifurcation analysis on discrete systems. In addition, the continuation curves, bifurcation diagrams, and phase diagrams are used to demonstrate theoretical results. Our study shows the interesting dynamics of this model that are very different from the continuous one.

Numerical Study of the Reaction Diffusion Prey–Predator Model Having Holling II Increasing Function in the Predator Under Noisy Environment

The current study deals with the stochastic diffusive prey–predator model with Holling II increasing function in the predator. As it is a population dynamics its population can be affected by various factors such as disease in prey–predator species, environmental fluctuation, etc. So, the underlying model is considered under the impact of a temporal multiplicative noise. The numerical solution of the governing model is obtained by finite difference schemes. To check the accuracy of the numerical solution, the stability and consistency of the schemes in the mean square sense are derived. All the schemes are consistent with the model and von Neumann stable. Several simulations are drawn to check the efficacy of the scheme. A test problem is investigated with numerical schemes and for the different values of the parameters, 3D and 2D plots are drawn. A graphical representation of the given system is drawn which shows the behavior of both species and the efficiency of the novel schemes. Hope so, these results will motivate the researcher to consider the stochastic prey–predator model for a better understanding of the ecosystem.

An empirical predator‒prey model: Examining the interaction of Florida panthers, white‐tailed deer, and cattle in Florida

AbstractIn this article, we present a predator‒prey model for Florida panthers (FL panthers), white‐tailed deer (deer) and cattle in Florida. The model is used to construct the following four policy scenarios: (a) joint management of all three species (Scenario 1); (b) a target population size of FL panthers of 240 individuals (Scenario 2); (c) a target population size of FL panthers equal to the carrying capacity (Scenario 3); and (d) a target population size of FL panthers of zero (Scenario 4) (Scenario 4). We parameterized the model for FL panthers, deer and cattle in Florida, which enables us to obtain empirical results for the population, harvest and net benefit of all three species under each policy scenario. When comparing Scenarios 1 and 2, the total net benefit of all three species is only slightly higher in the former case. Under Scenarios 3 and 4, the net benefit of deer changes significantly compared to Scenario 1, while the effect on the net benefit of cattle is minor. When comparing Scenario 1 and an actual situation, we find that the optimal population of FL panthers is only slightly higher than the actual population.

Role of eggplant trichome in whitefly oviposition and its relevance to biological control under greenhouse conditions.

The combined release of the predatory mite Amblyseius swirskii (Athias-Henriot) and the mirid Nesidiocoris tenuis (Reuter) provides effective biological control of the tobacco whitefly (Bemisia tabaci (Gennadius)) in greenhouse eggplant. However, knowing how plants' trichomes affect pest-predator interactions could improve whitefly management. Here, the effect of two varieties with either the presence or absence of trichomes was assessed on naturally occurring whitefly populations and predator abundance in a first experiment under field conditions. Predator-prey models were developed to assess the effect of trichomes on pest and predator population dynamics under field conditions. In a second semi-field experiment, the occurrence and oviposition preferences of B. tabaci and A. swirskii in the same eggplant varieties were compared. Significantly higher numbers of whitefly and mite, adults and eggs, were found on the hairy variety in both experiments. However, no differences were found in N. tenuis abundance between varieties under field conditions. Predator-prey models showed that whitefly growth rate increased in the hairy variety. N. tenuis and A. swirskii showed different fitness parameters according to the variety, with the former displaying better performance in the hairless variety and the latter in the hairy variety. Both predators effectively controlled the increase in whitefly populations in both varieties. Overall, the findings suggest that the hairless variety is more effective in deterring whiteflies. Additionally, the higher population of A. swirskii on the hairy variety indicates that this predator benefited from both the presence of trichomes and the prey.

Traveling front in a diffusive predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a singular diffusive predator–prey model with Beddington–DeAngelis functional response, employing geometric singular perturbation theory and Bendixson's criteria. Our investigation revolves around transforming the reaction–diffusion equation into a multi‐scale four‐dimensional slow–fast system with two different orders of small parameters. Through once singular perturbation analysis, our focus shifts towards exploring the existence of heteroclinic orbits in a three‐dimensional system. We analyze these dynamics through the perspective of the Fisher–KPP equation in two limit cases. In the first case, only the normal to the two‐dimensional slow manifold is unstable. This allows for the deduction of existence of heteroclinic orbits in the three‐dimensional system through investigating the dynamics on the two‐dimensional slow manifold. Consequently, we obtain both monotonic traveling fronts and non‐monotonic fronts with oscillatory tails. In the second case, the normal to the one‐dimensional slow manifold exhibits both stable and unstable directions, then it is impossible to restrict the dynamics of the three‐dimensional system entirely to the slow manifold. Instead, we integrate the slow orbits of the reduced system with the fast orbits of the layer system to construct a singular heteroclinic orbit. According to Fenichel's theorem, we discover the existence of exact heteroclinic orbits of three‐dimensional system and derive the monotonic traveling fronts under weaker parameter conditions. Additionally, we also discuss the nonexistence of traveling fronts. Finally, we demonstrate our theoretical results with numerical simulations.