Lotka-Volterra Research Articles

An explicit solution to harvesting behaviors in a predator–prey system

AbstractThis paper derives closed‐form solutions for a strategic, simultaneous harvesting in a predator–prey system. Using a parametric constraint, it establishes the existence and uniqueness of a linear feedback‐Nash equilibrium involving two specialized fleets and allows for continuous time results for a class of payoffs that have constant elasticity of the marginal utility. These results contribute to the scarce literature on analytically tractable predator–prey models with endogenous harvesting. A discussion based on industry size effects is provided to highlight the role played by biological versus strategic interactions in the multispecies context.Recommendations for Resource Managers This model presents a thorough examination of the economic inefficiencies inherent in the exploitation dynamics of two interdependent species, elucidating the complex interplay between ecological interactions and economic outcomes. The size of the fishing industries constitutes a significant variable that must be integrated into the formulation of pertinent policy recommendations. This constitutes an advancement towards a more time‐consistent approach to Ecosystem‐Based Fishery Management (EBFM).

Retraction Note: Coopetition analysis in industry upgrade and urban expansion based on fractional derivative gray Lotka–Volterra model

Retraction Note: Coopetition analysis in industry upgrade and urban expansion based on fractional derivative gray Lotka–Volterra model

INNOVATIVE MATHEMATICAL MODELLING IN PREDATOR-PREY DYNAMICS: A SYSTEMATIC REVIEW

Advancements in mathematical ecology have greatly expanded predator-prey models beyond the classical Lotka-Volterra framework, incorporating complex factors such as role reversal, fractional calculus, and the Allee effect. This review highlights innovative models and advanced mathematical techniques that enhance our understanding of predator-prey dynamics and their applications across disciplines. It analysed recent studies introducing new ecological factors, including maturation delays, disease dynamics, and spatial heterogeneity. Techniques like fractional calculus and network theory were examined for their effectiveness in capturing complex behaviours. Models addressing role reversal between adult prey and juvenile predators or incorporating generalized fractional derivatives reveal significant impacts on population stability. Additionally, maturation delays, handling time, and gestation periods markedly influence oscillatory dynamics. The reviewed models demonstrate versatility in guiding pest control, understanding disease spread, and optimizing biotechnological processes. This review shows that modern predator-prey models, enriched by complex ecological factors and advanced mathematics, provide profound insights into system dynamics, with practical applications across various fields. As global challenges grow, these models offer crucial guidance for developing more resilient and sustainable systems, underscoring their potential to addressissues like food security, disease outbreaks, and ecosystem degradation. Future research should integrate emerging factors, such as anthropogenic noise and industrial pollutants, to further enhance the models’ real-world relevance.

Bifurcations in a discrete-time Beddington–DeAngelis prey–predator model with fear effect, prey refuge and harvesting

Bifurcations in a discrete-time Beddington–DeAngelis prey–predator model with fear effect, prey refuge and harvesting

Bifurcation of a Leslie–Gower Predator–Prey Model with Nonlinear Harvesting and a Generalist Predator

A Leslie–Gower predator–prey model with nonlinear harvesting and a generalist predator is considered in this paper. It is shown that the degenerate positive equilibrium of the system is a cusp of codimension up to 4, and the system admits the cusp-type degenerate Bogdanov–Takens bifurcation of codimension 4. Moreover, the system has a weak focus of at least order 3 and can undergo degenerate Hopf bifurcation of codimension 3. We verify, through numerical simulations, that the system admits three different stable states, such as a stable fixed point and three limit cycles (the middle one is unstable), or two stable fixed points and two limit cycles. Our results reveal that nonlinear harvesting and a generalist predator can lead to richer dynamics and bifurcations (such as three limit cycles or tristability); specifically, harvesting can cause the extinction of prey, but a generalist predator provides some protection for the predator in the absence of prey.

The Volterra Integrable case. Novel analytical and numerical results

In the present paper we reconsider the integrable case of the Hamiltonian $N$-species Volterra system, as it has been introduced by Vito Volterra in 1937 and significantly enrich the results already published in the ArXiv in 2019 by two of the present authors (M. Scalia and O. Ragnisco). In fact, we present a new approach to the construction of conserved quantities and comment about the solutions of the equations of motion; we display mostly new analytical and numerical results, starting from the classical predator-prey model and arriving at the general $N$-species model

Note on the diffusive prey-predator model with variable coefficients and degenerate diffusion

It is of interest to understand effects of variable coefficients and degenerate diffusion on the longtime behaviors of solutions of reaction diffusion equations. Recently, Yang and Yao (2024) studied a classical prey-predator model and proved that the degradation of the diffusion coefficient of the prey and variable coefficients satisfying the appropriate conditions will not affect dynamical properties. In this note we shall simplify the proof of Yang and Yao (2024) and delete the condition (4).

Predator incited fear weakened by infertility of prey in a prey-predator model with memory effect

Predator incited fear weakened by infertility of prey in a prey-predator model with memory effect

Coexistence states for a Lotka–Volterra competitive model with aggressive movement strategy

This paper is concerned with a model of reaction–diffusion–advection equations arising in ecology. The equations model a situation in which the foreign species and the native species are both assumed to be competing for space in the same ecological niche, and the foreign species is just random diffusion while the native species is “braver” — a combination of random diffusion and a directed movement up the gradient of the foreign species. Our aim is to give the necessary and sufficient conditions for the coexistence of the foreign species and native species. For this, we first establish an a priori estimate result. Then we apply eigenvalue theory and homogenization principle to derive the necessary conditions for the coexistence. Finally, the sufficient conditions for the coexistence are given by using a global bifurcation method.

Propagation dynamics of a three-species predator–prey model with non-local dispersal in a shifting habitat

This paper is concerned with the propagation phenomenon of a non-local dispersal predator–prey system involving two preys and one predator in a shifting habitat. Assuming that the growth rate of each prey is moving to the right at a speed of [Formula: see text] and the habitat is suitable for each prey to survive before the shifting edge, while the habitat behind the boundary is not suitable for survival. For any given speed of the shifting habitat edge, we establish four types of forced extinction waves, which describe the conversion from the state of a saturated aboriginal prey with a pair of invading alien predator–prey, two aboriginal co-existent preys with an invading alien predator, a pair of aboriginal co-existent predator–prey and an invading alien prey, and the co-existence of three species to the extinction state, respectively. Meanwhile, we show the non-existence of some forced waves.

Effect of discontinuous harvesting on a diffusive predator-prey model

Effect of discontinuous harvesting on a diffusive predator-prey model

Global dynamics of a prey–predator model with component Allee effect for predator reproduction and Crowley–Martin-type functional response: the role of strong Allee effect

Global dynamics of a prey–predator model with component Allee effect for predator reproduction and Crowley–Martin-type functional response: the role of strong Allee effect

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.