Lotka-Volterra Predator-prey Model Research Articles

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.

Increase of maximum sustainable yield for fishery in two patches with fast migration

We present a model of a fish population present on two patches connected by migrations. Fish grow logistically on each patch and are caught. We assume that migrations between the two sites are fast relative to local growth and fishing. Taking advantage of the time scales, we use methods of aggregation of variables to obtain a reduced model governing the total biomass of the fish population at a slow time scale. We are looking for the maximum sustainable yield (MSY) for the system of the two connected patches. We show that although the total equilibrium population may be greater than the sum of the carrying capacities on each isolated site, the total catch is always less than or equal to the sum of the catches on the isolated fishing sites. We then consider a prey–predator community of fish in the same environment. We assume that only the predator is caught and not its prey, growing logistically on each site. We study the Lotka–Volterra prey–predator model as well as the model with a type II Holling functional response. We show that the total catch at MSY of the system of connected sites can be greater than the sum of the captures on each isolated site. This result is obtained when a fishing site with a large prey carrying capacity and an average growth rate is connected to a site with a small carrying capacity but a large growth rate. Finally, we discuss fishery management methods on two fishing sites for the Lotka–Volterra model as well as the Holling type II model in the case of a prey refuge.

Limit cycle oscillations, response time, and the time-dependent solution to the Lotka–Volterra predator–prey model

In this work, the time-dependent solution for the Lotka–Volterra predator–prey model is derived with the help of the Lambert W function. This allows an exact analytical expression for the period of the associated limit cycle oscillations and also for the response time between predator and prey population. These results are applied to the predator–prey interaction of zonal density corrugations and turbulent particle flux in gyrokinetic simulations of the collisionless trapped-electron mode turbulence. In the turbulence simulations, the response time is shown to increase when approaching the linear threshold, and the same trend is observed in the Lotka–Volterra model.

Stability analysis and Neimark-Sacker bifurcation of a nonstandard finite difference scheme for Lotka-Volterra prey-predator model

Stability analysis and Neimark-Sacker bifurcation of a nonstandard finite difference scheme for Lotka-Volterra prey-predator model

An operational matrix method to solve the Lotka–Volterra predator–prey models with discrete delays

In this paper, we consider the Lotka–Volterra predator–prey model with discrete delays. We apply the operational matrix method in [20] to this model which corresponds to a problem of nonlinear differential equations. The method is based on the operational matrices of the standard bases functions. In order to find the polynomial approximations of the known functions in the equation, the operational matrices are created by the least-squares method. The method converts the problem into a system of algebraic equations. By solving this system, the coefficients are determined. The approximate solutions are obtained by writing the coefficients in form of the approximate solutions. In addition, the accuracy of the approximate solution is checked and an error estimation technique is presented. The method is applied to some examples and the numerical results are reported. Numerical applications and comparisons with other methods show the efficiency of the current one.

On the Boundedness of the Numerical Solutions’ Mean Value in a Stochastic Lotka–Volterra Model and the Turnpike Property

In this paper, we study some properties of the solutions of a stochastic Lotka–Volterra predator-prey model, namely, the boundedness in the mean of numerical solutions, the strong convergence for this kind of solutions, and the turnpike property of solutions of an optimal control problem in a population modelled by a Lotka–Volterra system with stochastic environmental fluctuations. Even though there are numerous results in the deterministic case, there are few results for the behavior of numerical solutions in a population dynamic with random fluctuations. First, we show, using the Euler–Maruyama scheme, that the boundedness of numerical solutions and the convergence of the scheme are preserved in the stochastic case. Second, we analyze a property of the long-term behavior of a Lotka–Volterra system with stochastic environmental fluctuations known as turnpike property. In optimal control theory, the optimal solutions dwell mostly in the neighborhood of a balanced equilibrium path, corresponding to the optimal steady-state solution. Our study shows, by means of the Stochastic Maximum Principle, that this turnpike property is preserved, when the noise in the system is small. Numerical simulations are implemented to support our results.

Positivity-preserving symplectic methods for the stochastic Lotka–Volterra predator-prey model

In this paper, positivity-preserving symplectic numerical approximations are investigated for the 2d-dimensional stochastic Lotka–Volterra predator-prey model driven by multiplicative noises, which plays an important role in ecosystem. The model is shown to possess both a unique positive solution and a stochastic symplectic geometric structure, and hence can be interpreted as a stochastic Hamiltonian system. To inherit the intrinsic biological characteristic of the original system, a class of stochastic Runge–Kutta methods is presented, which is proved to preserve positivity of the numerical solution and possess the discrete stochastic symplectic geometric structure as well. Uniform boundedness of both the exact solution and the numerical one are obtained, which are crucial to derive the conditions for convergence order one in the \(\mathbb {L}^1(\varOmega )\)-norm. Numerical examples illustrate the stability and structure-preserving property of the proposed methods over long time.

Stochastic dynamics of predator-prey interactions.

The interaction between a consumer (such as, a predator or a parasitoid) and a resource (such as, a prey or a host) forms an integral motif in ecological food webs, and has been modeled since the early 20th century starting from the seminal work of Lotka and Volterra. While the Lotka-Volterra predator-prey model predicts a neutrally stable equilibrium with oscillating population densities, a density-dependent predator attack rate is known to stabilize the equilibrium. Here, we consider a stochastic formulation of the Lotka-Volterra model where the prey’s reproduction rate is a random process, and the predator’s attack rate depends on both the prey and predator population densities. Analysis shows that increasing the sensitivity of the attack rate to the prey density attenuates the magnitude of stochastic fluctuations in the population densities. In contrast, these fluctuations vary non-monotonically with the sensitivity of the attack rate to the predator density with an optimal level of sensitivity minimizing the magnitude of fluctuations. Interestingly, our systematic study of the predator-prey correlations reveals distinct signatures depending on the form of the density-dependent attack rate. In summary, stochastic dynamics of nonlinear Lotka-Volterra models can be harnessed to infer density-dependent mechanisms regulating predator-prey interactions. Moreover, these mechanisms can have contrasting consequences on population density fluctuations, with predator-dependent attack rates amplifying stochasticity, while prey-dependent attack rates countering to buffer fluctuations.

Attraction Region for the Classical Lotka−Volterra Predator−Prey model Caused by impulsive Effects

Discontinuous phenomena appear in various research fields. Impulsive differential equations are often used to model such discontinuous dynamics. This study deals with the Lotka–Volterra predator–prey model dominated by impulsive effects. In the modeling, impulses are added considering the ratios of the interior equilibrium to the current populations of the prey and predator. In this case, there is no restriction of the time interval between the impulse and the next impulse being the same. By focusing on the time interval between impulses adjacent to one another as well as the impulsive effect amount, a simple sufficient condition for the interior equilibrium to become globally asymptotically stable and sufficient conditions that are useful for estimating an attraction region are provided. Our results reveal that impulse control for the Lotka–Volterra predator–prey model can reduce the variation in the population of the prey and predator and enable the stable coexistence of both.

Estimating Equations for Density Dependent Markov Jump Processes

Reaction networks are important tools for modeling a variety of biological phenomena across a wide range of scales, for example as models of gene regulation within a cell or infectious disease outbreaks in a population. Hence, calibrating these models to observed data is useful for predicting future system behavior. However, the statistical estimation of the parameters of reaction networks is often challenging due to intractable likelihoods. Here we explore estimating equations to estimate the reaction rate parameters of density dependent Markov jump processes (DDMJP). The variance–covariance weights we propose to use in the estimating equations are obtained from an approximating process, derived from the Fokker–Planck approximation of the chemical master equation for stochastic reaction networks. We investigate the performance of the proposed methodology in a simulation study of the Lotka–Volterra predator–prey model and by fitting a susceptible, infectious, removed (SIR) model to real data from the historical plague outbreak in Eyam, England.

Existence and Uniqueness of the Positive Steady State Solution for a Lotka-Volterra Predator-Prey Model with a Crowding Term

This paper deals with a Lotka-Volterra predator-prey model with a crowding term in the predator equation. We obtain a critical value λ 1 (Ω0), and demonstrate that the existence of the predator in $${\overline \Omega _0}$$ only depends on the relationship of the growth rate μ of the predator and λ 1 (Ω0), not on the prey. Furthermore, when μ < λ 1 (Ω0), we obtain the existence and uniqueness of its positive steady state solution, while when μ ≥ λ 1 (Ω0), the predator and the prey cannot coexist in $${\overline \Omega _0}$$ . Our results show that the coexistence of the prey and the predator is sensitive to the size of the crowding region $${\overline \Omega _0}$$ , which is different from that of the classical Lotka-Volterra predator-prey model.

Complexity and chaos control in a discrete-time Lotka–Volterra predator–prey system

ABSTRACT In this paper, we investigate a discrete-time Lotka–Volterra predator–prey model that can undergo flip bifurcations and Hopf bifurcations by using the centre manifold theorem and bifurcation theory. Bifurcation diagrams, maximum Lyapunov exponents, and phase portraits verify the theoretical analysis, which exhibits complex dynamical behaviours such as period-7, 13, 14, 21, 27, 32 orbits, a cascade of period-doubling bifurcation in period-2, 4, 8, quasi-periodic orbits and the chaotic sets. Furthermore, the feedback control method is used to stabilize the chaotic orbits at an unstable fixed point. This result further reveals richer dynamics of the discrete model compared with the continuous model. It is found out that the appropriately carrying capacity can be helpful to stabilize the model. Instead, the higher or lower carrying capacity may destabilize the system producing more complex dynamical behaviours.

Bifurkasi Hopf pada Model Lotka-Volterra Orde-Fraksional dengan Efek Allee Aditif pada Predator

This article aims to study the dynamics of a Lotka-Volterra predator-prey model with Allee effect in predator. According to the biological condition, the Caputo fractional-order derivative is chosen as its operator. The analysis is started by identifying the existence, uniqueness, and non-negativity of the solution. Furthermore, the existence of equilibrium points and their stability is investigated. It has shown that the model has two equilibrium points namely both populations extinction point which is always a saddle point, and a conditionally stable co-existence point, both locally and globally. One of the interesting phenomena is the occurrence of Hopf bifurcation driven by the order of derivative. Finally, the numerical simulations are given to validate previous theoretical results.

Dynamics of Two Preys – One Predator System with Competition between Preys

We study the dynamics of two preys – one predator interaction with competition between prey populations. The proposed model is developed from the Lotka-Volterra predator-prey model. We first discuss some fundamental issues such as equilibrium points and the stability of each equilibrium point. The analysis shows that there are seven equilibrium points and some of them are conditionally locally asymptotically stable. The dynamical behavior of the proposed model is then verified numerically. We also observe numerically the occurrence of bistability, as well as a Hopf bifurcation which is driven by the conversion rate of predation into the predator growth rate.

An Improved Collocation Approach of Euler Polynomials Connected with Bernoulli Ones for Solving Predator-Prey Models with Time Lag

This paper deals with an approach to obtaining the numerical solution of the Lotka–Volterra predator-prey models with discrete delay using Euler polynomials connected with Bernoulli ones. By using the Euler polynomials connected with Bernoulli ones and collocation points, this method transforms the predator-prey model into a matrix equation. The main characteristic of this approach is that it reduces the predator-prey model to a system of algebraic equations, which greatly simplifies the problem. For these models, the explicit formula determining the stability and the direction is given. Numerical examples illustrate the reliability and efficiency of the proposed scheme.

Bifurcations and chaos control in a discrete-time biological model

In this paper, bifurcations and chaos control in a discrete-time Lotka–Volterra predator–prey model have been studied in quadrant-[Formula: see text]. It is shown that for all parametric values, model has boundary equilibria: [Formula: see text], and the unique positive equilibrium point: [Formula: see text] if [Formula: see text]. By Linearization method, we explored the local dynamics along with different topological classifications about equilibria. We also explored the boundedness of positive solution, global dynamics, and existence of prime-period and periodic points of the model. It is explored that flip bifurcation occurs about boundary equilibria: [Formula: see text], and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of [Formula: see text]. Further, it is also explored that about [Formula: see text] the model undergoes a N–S bifurcation, and meanwhile a stable close invariant curves appears. From the perspective of biology, these curves imply that between predator and prey populations, there exist periodic or quasi-periodic oscillations. Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23. The Maximum Lyapunov exponents as well as fractal dimension are computed numerically to justify the chaotic behaviors in the model. Finally, feedback control method is applied to stabilize chaos existing in the model.

Stability of synchronized steady state solution of diffusive Lotka–Volterra predator–prey model

We study the reaction–diffusion Lotka–Volterra predator–prey model with Dirichlet boundary condition. In the case of equal diffusion rates and equal growth rates, the synchronized steady state solution is proved to be locally asymptotically stable.