Lotka-Volterra Predator-prey Model Research Articles

A new numerical approach method to solve the Lotka–Volterra predator–prey models with discrete delays

This paper proposes a new approach called Extreme Learning Machine (ELM) to solve the Lotka–Volterra predator–prey models using a novel approximation form. Unlike the traditional methods that involve constructing complex algebraic systems and performing inverse matrix operations, ELM transforms the system of differential equations into a set of nonlinear equations and solves for the parameters through optimization iterations to obtain an approximate solution. Furthermore, we construct a solution form that satisfies the initial conditions, eliminating the need to handle initial conditions during the solving process, making this method more concise. Finally, by comparing with other numerical methods using two sets of models and parameters, ELM can produce high-precision results and further demonstrate the advantages of our method.

Codimension-two bifurcations of a two-dimensional discrete time Lotka-Volterra predator-prey model

Codimension-two bifurcations of a two-dimensional discrete time Lotka-Volterra predator-prey model

Genome-Wide Network Analysis of Above- and Below-Ground Co-growth in Populus euphratica.

Tree growth is the consequence of developmental interactions between above- and below-ground compartments. However, a comprehensive view of the genetic architecture of growth as a cohesive whole is poorly understood. We propose a systems biology approach for mapping growth trajectories in genome-wide association studies viewing growth as a complex (phenotypic) system in which above- and below-ground components (or traits) interact with each other to mediate systems behavior. We further assume that trait-trait interactions are controlled by a genetic system composed of many different interactive genes and integrate the Lotka-Volterra predator-prey model to dissect phenotypic and genetic systems into pleiotropic and epistatic interaction components by which the detailed genetic mechanism of above- and below-ground co-growth can be charted. We apply the approach to analyze linkage mapping data of Populus euphratica, which is the only tree species that can grow in the desert, and characterize several loci that govern how above- and below-ground growth is cooperated or competed over development. We reconstruct multilayer and multiplex genetic interactome networks for the developmental trajectories of each trait and their developmental covariation. Many significant loci and epistatic effects detected can be annotated to candidate genes for growth and developmental processes. The results from our model may potentially be useful for marker-assisted selection and genetic editing in applied tree breeding programs. The model provides a general tool to characterize a complete picture of pleiotropic and epistatic genetic architecture in growth traits in forest trees and any other organisms.

Spatial-Temporal Patterns Induced by Time Delay and Taxis in a Predator–Prey System

Time delays and taxi effects are important factors in the predator–prey interaction. This paper focuses on calculating the normal form on the center manifold near the Hopf bifurcation point for a general delayed diffusive predator–prey system with taxis under the Neumann boundary condition. A delayed diffusive Lotka–Volterra predator–prey model with prey-taxis is considered as an application. By numerical simulations, in the two-parameter plane, different types of spatial-temporal patterns are observed by varying delay or taxis.

System of Caputo Fractional Differential Equations with Applications to Predator and Prey Model

In this research article, we provide a methodology to solve the three systems of qth order linear Caputo fractional differential equations, where 0 < q < 1. Since the Caputo derivative is in the convolution form, we can apply the Laplace transform technique. The solution of the two linear system can be used as a tool to study the stability of the equilibrium solution of the Lotka-Volterra predator-prey model. We have referenced three system SIR model in this work. Due to the global nature of the Caputo derivative, the solution obtained is closer to the real data than the integer derivative.

Quantifying population dynamics via a geometric mean predator-prey model.

An integrable Hamiltonian variant of the two species Lotka-Volterra (LV) predator-prey model, shortly referred to as geometric mean (GM) predator-prey model, has been recently introduced. Here, we perform a systematic comparison of the dynamics underlying the GM and LV models. Though the two models share several common features, the geometric mean dynamics exhibits a few peculiarities of interest. The structure of the scaled-population variables reduces to the simple harmonic oscillator with dimensionless natural time TGM varying as ωGMt with ωGM=c12c21. We found that the natural timescales of the evolution dynamics are amplified in the GM model compared to the LV one. Since the GM dynamics is ruled by the inter-species rather than the intra-species coefficients, the proposed model might be of interest when the interactions among the species, rather than the individual demography, rule the evolution of the ecosystems.

Multi-species prey–predator dynamics during a multi-strain pandemic

Small and large scale pandemics are a natural phenomenon repeatably appearing throughout history, causing ecological and biological shifts in ecosystems and a wide range of their habitats. These pandemics usually start with a single strain but shortly become multi-strain due to a mutation process of the pathogen causing the epidemic. In this study, we propose a novel eco-epidemiological model that captures multi-species prey–predator dynamics with a multi-strain pandemic. The proposed model extends and combines the Lotka–Volterra prey–predator model and the Susceptible–Infectious–Recovered epidemiological model. We investigate the ecosystem’s sensitivity and stability during such a multi-strain pandemic through extensive simulation relying on both synthetic cases as well as two real-world configurations. Our results are aligned with known ecological and epidemiological findings, thus supporting the adequacy of the proposed model in realistically capturing the complex eco-epidemiological properties of the multi-species multi-strain pandemic dynamics.

Lotka-Volterra predator-prey model with periodically varying carrying capacity.

We study the stochastic spatial Lotka-Volterra model for predator-prey interaction subject to a periodically varying carrying capacity. The Lotka-Volterra model with on-site lattice occupation restrictions (i.e., finite local carrying capacity) that represent finite food resources for the prey population exhibits a continuous active-to-absorbing phase transition. The active phase is sustained by the existence of spatiotemporal patterns in the form of pursuit and evasion waves. Monte Carlo simulations on a two-dimensional lattice are utilized to investigate the effect of seasonal variations of the environment on species coexistence. The results of our simulations are also compared to a mean-field analysis in order to specifically delineate the impact of stochastic fluctuations and spatial correlations. We find that the parameter region of predator and prey coexistence is enlarged relative to the stationary situation when the carrying capacity varies periodically. The (quasi-)stationary regime of our periodically varying Lotka-Volterra predator-prey system shows qualitative agreement between the stochastic model and the mean-field approximation. However, under periodic carrying capacity-switching environments, the mean-field rate equationspredict period-doubling scenarios that are washed out by internal reaction noise in the stochastic lattice model. Utilizing visual representations of the lattice simulations and dynamical correlation functions, we study how the pursuit and evasion waves are affected by ensuing resonance effects. Correlation function measurements indicate a time delay in the response of the system to sudden changes in the environment. Resonance features are observed in our simulations that cause prolonged persistent spatial correlations. Different effective static environments are explored in the extreme limits of fast and slow periodic switching. The analysis of the mean-field equationsin the fast-switching regime enables a semiquantitative description of the (quasi-)stationary state.

A note on stability criteria in the periodic Lotka–Volterra predator-prey model

We present a stability result for T-periodic solutions of the periodic predator–prey Lotka–Volterra model. In 2021, R. Ortega gave a stability criteria in terms of the L1 norm of the coefficients of a planar linear system associated to the model. Previously, in 1994, Z. Amine and R. Ortega proved another stability criteria formulated in terms of the L∞ norm. The present work gives a Lp criterion, building a bridge between the two previous results.

Complex dynamics of COVID-19 mathematical model on Erdős–Rényi network

In this study, a conformable fractional order Lotka–Volterra predator-prey model that describes the COVID-19 dynamics is considered. By using a piecewise constant approximation, a discretization method, which transforms the conformable fractional-order differential equation into a difference equation, is introduced. Algebraic conditions for ensuring the stability of the equilibrium points of the discrete system are determined by using Schur–Cohn criterion. Bifurcation analysis shows that the discrete system exhibits Neimark–Sacker bifurcation around the positive equilibrium point with respect to changing the parameter d and e. Maximum Lyapunov exponents show the complex dynamics of the discrete model. In addition, the COVID-19 mathematical model consisting of healthy and infected populations is also studied on the Erdős Rényi network. If the coupling strength reaches the critical value, then transition from nonchaotic to chaotic state is observed in complex dynamical networks. Finally, it has been observed that the dynamical network tends to exhibit chaotic behavior earlier when the number of nodes and edges increases. All these theoretical results are interpreted biologically and supported by numerical simulations.

The effects of diffusion on the dynamics of a Lotka-Volterra predator-prey model with a protection zone

The effects of diffusion on the dynamics of a Lotka-Volterra predator-prey model with a protection zone

Novel predator-prey model admitting exact analytical solution.

The Lotka-Volterra predator-prey model still represents the paradigm for the description of the competition in population dynamics. Despite its extreme simplicity, it does not admit an analytical solution, and for this reason, numerical integration methods are usually adopted to apply it to various fields of science. The aim of the present work is to investigate the existence of new predator-prey models sharing the broad features of the standard Lotka-Volterra model and, at the same time, offer the advantage of possessing exact analytical solutions. To this purpose, a general Hamiltonian formalism, which is suitable for treating a large class of predator-prey models in population dynamics within the same framework, has been developed as a first step. The only existing model having the property of admitting a simple exact analytical solution, is identified within the above class of models. The solution of this special predator-prey model is obtained explicitly, in terms of known elementary functions, and its main properties are studied. Finally, the generalization of this model, based on the concept of power-law competition, as well as its extension to the case of N-component competition systems, are considered.

Global existence and stabilization in a diffusive predator–prey model with population flux by attractive transition

The diffusive Lotka–Volterra predator–prey model ut=∇⋅d1∇u+χv2∇(uv)+u(m1−u+av),x∈Ω,t>0,vt=d2Δv+v(m2−bu−v),x∈Ω,t>0,is considered in a bounded domain Ω⊂Rn, n∈{2,3}, under Neumann boundary condition, where d1,d2,m1,χ,a,b are positive constants and m2 is a real constant. The purpose of this paper is to establish global existence and boundedness of classical solutions in the case n=2 and global existence of weak solutions in the case n=3 as well as show long-time stabilization. More precisely, we prove that the solutions (u(⋅,t),v(⋅,t)) converge to the constant steady state (u∗,v∗) as t→∞, where u∗,v∗ solves u∗(m1−u∗+av∗)=v∗(m2−bu∗−v∗)=0 with u∗>0 (covering both coexistence as well as prey-extinction cases).

Interval neutrosophic stochastic dynamical systems driven by Brownian motion

Stochastic and neutrosophic theory are effective tools for modeling real-world phenomena and natural dynamical systems, where inputs are often affected by stochastic noises and outputs often contain both randomness and indeterminacy. In this work, we present a new type of stochastic differential equations (SDE) driven by an one-dimensional Brownian motion that can be considered as an efficient tool to describe the uncertain behavior of dynamical systems operating in interval neutrosophic environments with stochastic noises. After introducing some basic foundations on neutrosophic arithmetic, neutrosophic calculus and neutrosophic stochastic process, we define the new form of interval neutrosophic stochastic differential equations taking values in neutrosophic environment. Under some suitable conditions, the unique existence result of stochastic solution is obtained based on the use of Picard successive approximation. We also introduce an efficient numerical algorithm, namely Euler–Maruyama method, to solve the numerical solution of proposed problem and further demonstrate the effectiveness of the numerical method by solving some examples in stochastic biological systems such as stochastic logistic growth model, stochastic Lotka–Volterra predator–prey model, and stochastic SARS model, respectively.

Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients

Discarded plastic is subjected to weather effects from different ecosystems and becomes microplastic particles. Due to their small size, they have spread across the planet. Their presence in living organisms can have several harmful consequences, such as altering the interaction between prey and predator. Huang et al. successfully modeled this system presenting numerical results of ecological relevance. Here, we have rewritten their equations and solved a set of them analytically, confirming that microplastic particles accumulate faster in predators than in prey and calculating the time values from which it happens. Using these analytical solutions, we have retrieved the Lotka–Volterra predator–prey model with time-varying intraspecific coefficients, allowing us to interpret ecological quantities referring to microplastics dispersion. After validating our equations, we solved analytically particular situations of ecological interest, characterized by extreme effects on predatory performance, and proposed a second-order differential equation as a possible next step to address this model. Our results open space for further refinement in the study of predator–prey models under the effects of microplastic particles, either exploring the second-order equation that we propose or modify the Huang et al. model to reduce the number of parameters, embedding in the time-varying intraspecies coefficients all the adverse effects caused by microplastic particles.

Implementasi algoritma genetika dalam mengestimasi kepadatan populasi jackrabbit dan coyote

This article studies about the parameter estimation using genetic algorithm for a Lotka-Volterra prey-predator model. The secondary data consist of the density of jackrabbit as prey and coyote as predator in Southwest Presscott–Arizona are used. As results, the Mean Absolute Percentage Error (MAPE) are computed to compare the results of parameter estimation and the real data. We have shown that MAPE for jackrabbit and coyote respectively given by 7.75424% and 7.95283%. This results show that the parameter estimation with genetic algorithm using Lotka-Volterra model is passably. Furthermore, some numerical simulations are portrayed to show each population density for the next 100 years.

Dynamics for nonlocal diffusion problems with a free boundary

In this paper, we first consider two scalar nonlocal diffusion problems with a free boundary and a fixed boundary. We obtain global existence, uniqueness and longtime behavior of solution of these two problems. The spreading-vanishing dichotomy is established, and some sufficient conditions for spreading and vanishing are also derived. For scalar models, we prove that accelerated spreading could happen if and only if a threshold condition is violated by kernel function. Then we discuss a classical Lotka-Volterra predator-prey model with nonlocal diffusions and a free boundary which can be seen as nonlocal diffusion counterpart of the model in the work of Wang (2014) [24].

Harmonic Oscillators of Mathematical Biology: Many Faces of a Predator-Prey Model

Summary We show that a number of models in virus dynamics, epidemiology and plant biology can be presented as “damped” versions of the Lotka-Volterra predator-prey model, by analogy to the damped harmonic oscillator. The analogy deepens with the use of Lyapunov functions, which allow us to characterize their dynamics and even make some estimates.

Dynamics of stochastic Lotka-Volterra predator-prey models driven by three independent Brownian motions

Dynamics of stochastic Lotka-Volterra predator-prey models driven by three independent Brownian motions

Stability and Boundedness Analysis of Lotka-Volterra Prey-Predator Model with Prey Refuge and Predator Cannibalism

Abstract In this paper, a prey-predator system incorporating prey refuge and predator cannibalism is studied. The stability and ultimate boundedness of the analyzed state parameters (y_1,y_2) defining the system are obtained using the Lyapunov’s second or direct method. We construct a suitable complete Lyapunov function for the nonlinear system and demonstrate its efficacy. The method is built upon applying various theoretical Lyapunov functions. By constructing a Lyapunov function which possesses a functional relationship to the the original model system, we give sufficient conditions which ensure the stability and ultimate boundedness of the state parameters describing the nonlinear prey-predator system. We give a numeric example to illustrate the result obtained. JEL classification numbers: 34C27, 34D23, 34C25. Keywords: Prey-predator system, Prey refuge, Predator cannibalism, Stability, Boundedness, Lyapunov’s method.