Lotka-Volterra Predator-prey Model Research Articles

KAN-ODEs: Kolmogorov–Arnold network ordinary differential equations for learning dynamical systems and hidden physics

Kolmogorov–Arnold networks (KANs) as an alternative to multi-layer perceptrons (MLPs) are a recent development demonstrating strong potential for data-driven modeling. This work applies KANs as the backbone of a neural ordinary differential equation (ODE) framework, generalizing their use to the time-dependent and temporal grid-sensitive cases often seen in dynamical systems and scientific machine learning applications. The proposed KAN-ODEs retain the flexible dynamical system modeling framework of Neural ODEs while leveraging the many benefits of KANs compared to MLPs, including higher accuracy and faster neural scaling, stronger interpretability and generalizability, and lower parameter counts. First, we quantitatively demonstrated these improvements in a comprehensive study of the classical Lotka–Volterra predator–prey model. We then showcased the KAN-ODE framework’s ability to learn symbolic source terms and complete solution profiles in higher-complexity and data-lean scenarios including wave propagation and shock formation, the complex Schrödinger equation, and the Allen–Cahn phase separation equation. The successful training of KAN-ODEs, and their improved performance compared to traditional Neural ODEs, implies significant potential in leveraging this novel network architecture in myriad scientific machine learning applications for discovering hidden physics and predicting dynamic evolution.

Optimized Parameter Estimation and Integrating Neural Network Forecasting of Dynamic Plant-Livestock Model for Early Warning in Agro-Environment Control Systems

The research utilizes the Lotka-Volterra prey-predator model to study Plant-Herbivore dynamics, focusing on the relationship between traditional livestock farming and vegetation conditions. Advanced methods are developed to improve the precision and efficiency of parameter estimation in these models. Neural networks are incorporated to enhance forecasting abilities, and an extension of the Plant-Herbivore models includes Botswana's climate and livestock variables. Efficient parameter space exploration is achieved using the Runge-Kutta method along with Multistart and the local solver $fmincon$ in MATLAB. This method improves parameter estimation accuracy. To address the impact of homogeneity assumptions in the data, estimate aggregation through weighting and time conversion is applied. Furthermore, the study investigates the use of nonlinear least squares to further refine the process, allowing for the identification of parameters that best fit observed livestock data, even with non-linearity. By using optimized parameter estimation techniques along with normalized nonlinear least squares, the cumulative error was reduced from an initial 1563.4521 to a final value of 0.0038, well within the specified thresholds (1.0, 0.1, and 0.01). Comparisons between Autoregressive Integrated Moving Average (ARIMA) and Neural Network Auto-Regressive (NNAR) models showed that NNAR models outperformed ARIMA models, with lower variance estimates (0.000004 - 0.000562) compared to ARIMA (0.103 - 0.155). NNAR models displayed Mean Error (ME) values ranging from -0.0012 to 0.0140, indicating a close match between forecasts and actual values with minor deviations. As a result, NNAR forecasting was used for predicting soil moisture, death, and harvest rates, which were integrated into the extended Plant-Herbivore model. This integration enabled the estimation of livestock production trajectories for 2021-2022, along with corresponding interpretations. The study also assessed the uncertainty propagation from NNAR forecasts onto the Plant-Herbivore dynamic model, revealing an increase in uncertainty with longer lead times.

Persistence and Stochastic Extinction in a Lotka–Volterra Predator–Prey Stochastically Perturbed Model

The classical Lotka–Volterra predator–prey model is globally stable and uniformly persistent. However, in real-life biosystems, the extinction of species due to stochastic effects is possible and may occur if the magnitudes of the stochastic effects are large enough. In this paper, we consider the classical Lotka–Volterra predator–prey model under stochastic perturbations. For this model, using an analytical technique based on the direct Lyapunov method and a development of the ideas of R.Z. Khasminskii, we find the precise sufficient conditions for the stochastic extinction of one and both species and, thus, the precise necessary conditions for the stochastic system’s persistence. The stochastic extinction occurs via a process known as the stabilization by noise of the Khasminskii type. Therefore, in order to establish the sufficient conditions for extinction, we found the conditions for this stabilization. The analytical results are illustrated by numerical simulations.

ФРАКЦИОНИРОВАНИЕ ПЛАНКТОНА С ПОМОЩЬЮ ПОСЛЕДОВАТЕЛЬНОЙ ФИЛЬТРАЦИИ И ПОСТРОЕНИЕ ПИРАМИД БИОРАЗНООБРАЗИЯ

Lotka-Volterra predator-prey models are used to study community ecology, but their ability to generate ecological pyramids compared to field data has not been investigated in detail. In this paper, agent-based modeling (ABM) was used instead of systems of ordinary differential equations (ODE). It was shown that the two-component producer-consumer system is unstable, whereas the three-component system with consumers of the 1st and 2nd order is stable under prolonged simulation. Time slices as the program progresses can generate both ecological pyramids and cascades. Simulation results are consistent with experiments on separation of the Black Sea plankton from the area of Cape Fiolent (Crimea) into fractions ranging in size from 2 mm to 2 microns. Although biodiversity in individual samples at different points in time as well as abundance vary widely, both predictably decline with rising trophic levels in cases where the number of tests increases over time.

Modeling the Impact of Lamprey Population Dynamics on Aquatic Food Webs: A Leslie and Lotka-Volterra Approach

In recent years, the widely distributed invasive parasitic species of marine lampreys have had a profound impact on host populations and the entire aquatic ecosystem through their unique survival and reproduction strategies, highlighting the importance of species interactions and ecological balance in ecosystems. In response to this, this study establishes a species growth model for lampreys based on the Leslie model from the perspective of gender ratio changes. The idea of the Lotka-Volterra predator-prey model is used to establish the food web centered on lamprey and the fluctuation mode model of various groups in it. Based on the model results, we conclude that the growth trend of the lamprey itself is difficult to be affected by other populations, but the lamprey population will have serious adverse effects on other species. The results of this model are in good agreement with the actual situation, and have profound significance in laying the foundation for the study of the community characteristics of lampreys and their related ecosystems.

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.