Lotka-Volterra Research Articles

Dynamic analysis of an infected predator-prey system with prey herd behavior

ABSTRACT To protect themselves from predation, sometimes prey starts roaming in groups and shows herd behavior. This paper addresses the predator-prey system dynamics with prey herd behavior and infection in predators. We explore the conditions under which all equilibria exist and the stability of each equilibrium point of the system. The outcomes of this work suggest that prey herd behavior is beneficial for the prey population when the predators are diseased. Under suitable conditions, the three populations can coexist and maintain the ecological balance of nature.

Spatio-temporal dynamics in a delayed prey–predator model with nonlinear prey refuge and harvesting

In the current study, we introduce a model for the temporal and spatial interactions between prey and predator. The model incorporates a nonlinear refuge mechanism for prey, along with linear harvesting of prey and nonlinear harvesting for predators. Initially, we examined the well-posed nature of the model by analyzing the presence of all feasible equilibria and investigating the corresponding dynamics. Following that, we delve into the dynamics of the temporal model, focusing specifically on aspects such as uniform boundedness, permanence, and stability of viable equilibria. We demonstrate analytically that the proposed model experiences transcritical, saddle–node, Hopf, and Bogdanov–Takens bifurcations. It shows a variety of intricate dynamics involving Generalized Hopf and double Hopf. Then, discrete-time delay effects arising from the gestation of predator species have been incorporated into the temporal system. Hopf bifurcation for the delay parameter was detected in this investigation. Subsequently, we established conditions for self-diffusion instability and Turing instability in the spatiotemporal model, both with and without delay, employing the homogeneous Neumann boundary condition. Moreover, the discussion of sensitivity analysis (PRCC) serves to illustrate how crucial parameters influence the dynamics of the system. In addition, we conduct numerical simulations aiming to corroborate and validate the analytical results obtained.

The dynamics in an intraguild prey-predator model with Holling type III functional response

The dynamics in an intraguild prey-predator model with Holling type III functional response

Different patterns of coupled predator-prey dynamics when the same species interact in different locations.

Individual predator and prey species exhibit coupled population dynamics in simple laboratory systems and simple natural communities. It is unclear how often such pairwise coupling occurs in more complex communities, in which an individual predator species might feed on several prey species and an individual prey species might be attacked by several predators. To examine this problem, we applied multivariate autoregressive state-space (MARSS) models to 5-year time-series of monthly surveys of a predatory fish, the eastern mosquitofish (Gambusia holbrooki), and its littoral zone prey species, the least killifish (Heterandria formosa), in three locations in north Florida. The MARSS models were consistent with coupled predator-prey dynamics at two of the three locations. In one of these two locations, the estimated densities of the two species displayed classic predator-prey oscillations. In the third location, there was a positive effect of killifish density on mosquitofish density but no detectable effect of mosquitofish density on killifish density. In all three locations, increased submergent vegetation cover was associated with increased prey density but not increased predator density. Eigenvalues analyses for the joint predator-prey dynamics indicated that one of the cyclic locations had more stable dynamics than the other locations. The three different patterns demonstrate that the dynamics of a pairwise predator-prey interaction emerge not only from the characteristics of the prey and the predator, but also those of the habitat and trophic web in which the predator and prey are embedded.

Complex dynamic behaviour on fractional predator–prey model of mathematical ecology

Complex dynamic behaviour on fractional predator–prey model of mathematical ecology

Exploring chaos and bifurcation in a discrete prey–predator based on coupled logistic map

This research paper investigates discrete predator-prey dynamics with two logistic maps. The study extensively examines various aspects of the system’s behavior. Firstly, it thoroughly investigates the existence and stability of fixed points within the system. We explores the emergence of transcritical bifurcations, period-doubling bifurcations, and Neimark-Sacker bifurcations that arise from coexisting positive fixed points. By employing central bifurcation theory and bifurcation theory techniques. Chaotic behavior is analyzed using Marotto’s approach. The OGY feedback control method is implemented to control chaos. Theoretical findings are validated through numerical simulations.

Complex Dynamics of a Discrete Prey–Predator Model Exposing to Harvesting and Allee Effect on the Prey Species with Chaos Control

Complex Dynamics of a Discrete Prey–Predator Model Exposing to Harvesting and Allee Effect on the Prey Species with Chaos Control

Discretization of the Lotka–Volterra System and Asymptotic Focal and Prefocal Sets

Discretization of the Lotka–Volterra System and Asymptotic Focal and Prefocal Sets

Global stability of three trophic levels predator–prey model with alarm-taxis

Global stability of three trophic levels predator–prey model with alarm-taxis

Role of prey refuge and fear level in fractional prey–predator model with anti-predator

Ecological modeling is an effective tool for studying the interactions between predator and prey species by considering various functional responses and ecological effects. Employing computational or mathematical models allows us to determine the impact of specific human interventions or animal behaviors on the evolution of these species. The fear experienced by prey due to the presence of predators plays a crucial role in shaping the dynamics of their interactions. The manuscript focuses on developing and examining a novel prey–predator model that considers predation fear, prey refuge, and anti-predator effects. Caputo fractional derivative is utilized in the construction and analysis of the model, which integrates ecological principles like memory effects to improve our comprehension of species relationship. The study investigates aspects such as stability, well-posedness, and solution uniqueness for the proposed model. We carried out extensive numerical simulation to support theoretical results. Graphical results are provided for the model encompassing a broad spectrum of fractional order values. The effect of the fear level, growth rate of prey, saturation rate and prey refuse on the behavior of the solution are discussed.

A Prey-Predator Mathematical Model With Diffusion and Home Ranges

A prey-predator model with diffusion and home ranges is considered. The model consists of partial differential and integral equations. The model incorporates complex mathematical expressions, which make it hard to analyze mathematically. Therefore, a numerical solution is provided in two cases. The first case considers the prey population growing logistically, while we consider the exponential growth of the prey population in the second case. We study the dynamic behavior of the two species for both cases. Special attention goes to the impact of home ranges and diffusion coefficients on the dynamics of prey and predator populations.

A Prey-Predator Mathematical Model With Diffusion and Home Ranges

A prey-predator model with diffusion and home ranges is considered. The model consists of partial differential and integral equations. The model incorporates complex mathematical expressions, which make it hard to analyze mathematically. Therefore, a numerical solution is provided in two cases. The first case considers the prey population growing logistically, while we consider the exponential growth of the prey population in the second case. We study the dynamic behavior of the two species for both cases. Special attention goes to the impact of home ranges and diffusion coefficients on the dynamics of prey and predator populations.

Impact of hunting cooperation in predator and anti‐predator behaviors in prey in a predator–prey model

In this paper, we propose a predator–prey model with hunting cooperation in predator and anti‐predator behaviors in prey. The conditions for the existence and the stability of the unique positive constant equilibrium are given. It is found that with the increasing of the birth rate of the prey, the trivial solution loses its stability, and the semi‐trivial solution emerges and also loses its stability. For the positive constant solution, we find that as the hunting cooperation in predator increases or the fear decreases, the positive constant equilibrium loses its stability, and Hopf bifurcation occurs. We also derive the existence of limit cycles by Poincaré‐Bendixson theorem. We also study a diffusive model and derive that self‐diffusion can induce Turing instability. Finally, we conduct numerical simulations to present our conclusions.

Impact of hunting cooperation in predator and anti‐predator behaviors in prey in a predator–prey model

In this paper, we propose a predator–prey model with hunting cooperation in predator and anti‐predator behaviors in prey. The conditions for the existence and the stability of the unique positive constant equilibrium are given. It is found that with the increasing of the birth rate of the prey, the trivial solution loses its stability, and the semi‐trivial solution emerges and also loses its stability. For the positive constant solution, we find that as the hunting cooperation in predator increases or the fear decreases, the positive constant equilibrium loses its stability, and Hopf bifurcation occurs. We also derive the existence of limit cycles by Poincaré‐Bendixson theorem. We also study a diffusive model and derive that self‐diffusion can induce Turing instability. Finally, we conduct numerical simulations to present our conclusions.

Revealing endogenous conditions for Peto’s paradox via an ordinary differential equation model

Cancer, a disease intimately linked to cellular mutations, is commonly believed to exhibit a positive association with the cell count and lifespan of a species. Despite this assumption, the observed uniformity in cancer rates across species, referred to as the Peto’s paradox, presents a conundrum. Recognizing that tumour progression is not solely dependent on cancer cells but involves intricate interactions among various cell types, this study employed a Lotka-Volterra (LV) ordinary differential equation model to analyze the evolution of cancerous cells and the cancer incidence in an immune environment. As a result, this study uncovered the sufficient conditions underlying the absence of correlation in Peto’s paradox and provide insights into the reasons for the equitable distribution of cancer incidence across diverse species by applying nondimensionalization and drawing an analogy between the characteristic time interval for the variation of cell populations in the ODE model and that of cell cycles of a species.

A dynamic surge pricing model throughout product lifecycle

The shorter lifecycle and faster upgrading of the product make the potential demand change rapidly, thereby pricing based on market changes is critical to increasing profits. In response to changes in the potential demand, we present a dynamic surge pricing model that characterizes sales trajectories and gradient pricing. We introduce the Lotka–Volterra system to construct the sale-forecast system and prove its effectiveness in predicting the product lifecycle curve that reflects the change of the potential demand. In contrast to the linear demand function, we propose a gradient pricing mechanism based on marginal sales and total sales to describe the relationship between price and potential demand throughout the life cycle. Particularly, the dynamic surge pricing model degrades to Cournot model in the maturity phase of the market. We characterize the dynamic equilibrium and conduct the sensitivity analysis of parameters, showing that the dynamic surge pricing model outperforms Cournot model in terms of profit. A numerical example illustrates that the profit of the dynamic surge pricing model is nearly 21.82% higher than that of Cournot model.

Development of a method to assess synergy and competition for water use among water-energy-food nexus in the Yellow River basin: Water quantity-quality dimensions

Water, as an indispensable component of the water-energy-food (WEF) nexus, plays a pivotal role in shaping its stability and safety. The competition for water between the food and energy systems is accentuated in the concept of the nexus. However, existing quantitative frameworks and assessment mechanisms have limitations as they often neglect to consider water quality in water competition within the WEF nexus. Therefore, a WEF nexus synergy and competition assessment method was proposed by coupling water footprint theory and the Lotka-Volterra model in this paper. This method establishes two scenarios based on the water footprint perspective, namely, water quantity and water quality-quantity. It effectively addresses the deficiencies in quantitative analysis of WEF system trade-offs and synergies. To illustrate the application of the method, this study conducted a case study in the Yellow River Basin (YRB) of China, evaluating the complex competitive and synergistic mechanisms of water use in the energy and food industries. Compared to the water quantity scenario, the results reveals that the inclusion of gray water footprint intensifies the competition for water in the food and energy industries in the YRB from 2000 to 2020, particularly in resource-based and economically developed cities. The spatial distribution characteristics of water resource competition and synergy in the YRB’s WEF system align with WEF endowment and economic technology levels. To address water conflicts in the WEF nexus and enhance nexus security, this study proposes a regulatory pathway for WEF system synergistic security based on two aspects (water competition and synergy characteristics and resource endowment). It contributes to a more comprehensive understanding of water competition and synergy in the WEF nexus and provides valuable insights for resource management.

Complex Dynamics in a Predator–Prey Model with Fear Affected Transmission

Complex Dynamics in a Predator–Prey Model with Fear Affected Transmission

Global dynamics of a diffusive competitive Lotka–Volterra model with advection term and more general nonlinear boundary condition

We investigate a competitive diffusion–advection Lotka–Volterra model with more general nonlinear boundary condition. Based on some new ideas, techniques, and the theory of the principal spectral and monotone dynamical systems, we establish the influence of the following parameters on the dynamical behavior of system (1.2): advection rates αu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{u}$\\end{document} and αv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\alpha _{v}$\\end{document}, interspecific competition intensities cu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{u}$\\end{document} and cv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$c_{v}$\\end{document}, the resources functions ru\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r_{u}$\\end{document} and rv\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$r_{v}$\\end{document} of the two competitive species, and nonlinear boundary functions g1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{1}$\\end{document} and g2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{2}$\\end{document}. The models of (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018) are particular cases of our results when gi≡const\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$g_{i}\\equiv const$\\end{document} for i=1,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$i=1,2$\\end{document}, and hence this paper extends some of the conclusions from (Tang and Chen in J. Differ. Equ. 269(2):1465–1483, 2020; Zhou and Zhao in J. Differ. Equ. 264:4176–4198, 2018).

The persistence of bipartite ecological communities with Lotka–Volterra dynamics

The assembly and persistence of ecological communities can be understood as the result of the interaction and migration of species. Here we study a single community subject to migration from a species pool in which inter-specific interactions are organised according to a bipartite network. Considering the dynamics of species abundances to be governed by generalised Lotka–Volterra equations, we extend work on unipartite networks to we derive exact results for the phase diagram of this model. Focusing on antagonistic interactions, we describe factors that influence the persistence of the two guilds, locate transitions to multiple-attractor and unbounded phases, as well as identifying a region of parameter space in which consumers are essentially absent in the local community.