Lotka-Volterra Predator-prey Model Research Articles

Dynamic Analysis of Stochastic Lotka–Volterra Predator-Prey Model with Discrete Delays and Feedback Control

In this paper, a stochastic Lotka–Volterra predator-prey model with discrete delays and feedback control is studied. Firstly, the existence and uniqueness of global positive solution are proved. Further, we investigate the asymptotic property of stochastic system at the positive equilibrium point of the corresponding deterministic model and establish sufficient conditions for the persistence and extinction of the model. Finally, the correctness of the theoretical derivation is verified by numerical simulations.

Conformable Fractional Order Lotka–Volterra Predator–Prey Model: Discretization, Stability and Bifurcation

Abstract The purpose of this study is to discuss dynamic behaviors of conformable fractional-order Lotka–Volterra predator–prey system. First of all, the piecewise constant approximation is used to obtain the discretize version of the model then, we investigate stability, existence, and direction of Neimark–Sacker bifurcation of the positive equilibrium point of the discrete system. It is observed that the discrete system shows much richer dynamic behaviors than its fractional-order form such as Neimark–Sacker bifurcation and chaos. Finally, numerical simulations are used to demonstrate the accuracy of analytical results.

The concept of force in population dynamics

The area of population dynamics has a rich history of the development and analysis of models of biological and social phenomena using ordinary differential equations. This paper describes a method for understanding the influence one variable exerts on another in such models as a force, with the relative effects of these forces providing a narrative explanation of the curvature in variable behaviour. Using the stock/flow form of a model, a symbolic notation is developed that identifies the forces with the causal pathways of the model’s feedback loops. A force is measured by its impact, defined as the ratio of acceleration to rate of change, computed by differentiation along its associated pathway between variables. Different phases of force dominance are determined to enhance the standard stability analysis of the models, providing an explanation of model behaviour in Newtonian mechanical terms. The concepts developed are applied to well-known models from mathematical biology: the Spruce Budworm model, where force dominance identifies scenarios that give clarity to intervention points; and the Lotka–Volterra predator–prey model where the analysis highlights the importance of dissipative forces in achieving stability. Conclusions are drawn on the explanatory power of this approach, with suggestions made for future work.

Comparing stochastic Lotka–Volterra predator-prey models

The phenomenology of population extinction is one of the central themes in population biology which it is an inherently stochastic event. In the present investigation, we study this problem for three different stochastic models built from a single Lotka–Volterra deterministic model. More concretely, we study their mean-extinction time which satisfies the backward Kolmogorov differential equation, a linear second-order partial differential equation with variable coefficients; hence, we can only compute numerical approximations. We suggest a finite element method using FreeFem++. Our analysis and numerical results allow us to conclude that there are important differences between the three models. These differences enable us to choose the most “natural way” to turn a the deterministic model into a stochastic model.

Persistence and extinction for stochastic ecological models with internal and external variables.

The dynamics of species' densities depend both on internal and external variables. Internal variables include frequencies of individuals exhibiting different phenotypes or living in different spatial locations. External variables include abiotic factors or non-focal species. These internal or external variables may fluctuate due to stochastic fluctuations in environmental conditions. The interplay between these variables and species densities can determine whether a particular population persists or goes extinct. To understand this interplay, we prove theorems for stochastic persistence and exclusion for stochastic ecological difference equations accounting for internal and external variables. Specifically, we use a stochastic analog of average Lyapunov functions to develop sufficient and necessary conditions for (i) all population densities spending little time at low densities i.e. stochastic persistence, and (ii) population trajectories asymptotically approaching the extinction set with positive probability. For (i) and (ii), respectively, we provide quantitative estimates on the fraction of time that the system is near the extinction set, and the probability of asymptotic extinction as a function of the initial state of the system. Furthermore, in the case of persistence, we provide lower bounds for the expected time to escape neighborhoods of the extinction set. To illustrate the applicability of our results, we analyze stochastic models of evolutionary games, Lotka-Volterra dynamics, trait evolution, and spatially structured disease dynamics. Our analysis of these models demonstrates environmental stochasticity facilitates coexistence of strategies in the hawk-dove game, but inhibits coexistence in the rock-paper-scissors game and a Lotka-Volterra predator-prey model. Furthermore, environmental fluctuations with positive auto-correlations can promote persistence of evolving populations and persistence of diseases in patchy landscapes. While our results help close the gap between the persistence theories for deterministic and stochastic systems, we highlight several challenges for future research.

Influence of technological progress and renewability on the sustainability of ecosystem engineers populations.

Overpopulation and environmental degradation due to inadequate resource-use are outcomes of human's ecosystem engineering that has profoundly modified the world's landscape. Despite the age-old concern that unchecked population and economic growth may be unsustainable, the prospect of societal collapse remains contentious today. Contrasting with the usual approach to modeling human-nature interactions, which are based on the Lotka-Volterra predator-prey model with humans as the predators and nature as the prey, here we address this issue using a discrete-time population dynamics model of ecosystem engineers. The growth of the population of engineers is modeled by the Beverton-Holt equation with a density-dependent carrying capacity that is proportional to the number of usable habitats. These habitats (e.g., farms) are the products of the work of the individuals on the virgin habitats (e.g., native forests), hence the denomination engineers of ecosystems to those agents. The human-made habitats decay into degraded habitats, which eventually regenerate into virgin habitats. For slow regeneration resources, we find that the dynamics is dominated by rounds of prosperity and collapse, in which the population reaches vanishing small densities. However, increase of the efficiency of the engineers to explore the resources eliminates the dangerous oscillatory patterns of feast and famine and leads to a stable equilibrium that balances population growth and resource availability. This finding supports the viewpoint of growth optimists that technological progress may avoid collapse.

Global existence and boundedness of solutions in a Lotka-Volterra reaction-diffusion system of predator-prey model with nonlinear prey-taxis

In this paper, we investigate a diffusive Lotka-Volterra predator-prey model with nonlinear prey-taxis under Neumann boundary conditions. This system describes a prey-taxis mechanism that is an immediate movement of the predator u in response to a change of the prey v (which lead to the collection of u). We apply some methods to overcome the substantial difficulty of the existence of nonlinear prey-taxis term and prove that the unique global classical solutions of Lotka-Volterra predator-prey model are globally bounded.

Existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models

This paper is concerned with the existence and non-monotonicity of traveling wave solutions for general diffusive predator-prey models. By using Schauder's fixed point theorem and the existence of contracting rectangles, we obtain the existence result. Then we investigate the asymptotic behavior of positive monotone traveling wave solutions by using the modified Ikehara's Theorem. With the help of their asymptotic behavior, we provide a sufficient condition which guarantee that all positive traveling wave solutions of the system are non-monotone. Furthermore, to illustrate our main results, the existence and non-monotonicity of traveling wave solutions of Lotka-Volterra predator-prey model and modified Leslie-Gower predator-prey models with different kinds of functional responses are also discussed.

Review of a predator-prey model with two limit cycles

ABSTRACTIt is well-known that the Lotka–Volterra predator-prey model has a family of periodic orbits, but does not possess limit cycles and therefore the model is said to be structurally unstable. The Lotka–Volterra model is a special case of a much larger group namely the quadratic population models and it can be shown that none of them can produce limit cycles. The surprising finding is that by combining two quadratic models a quadratic population model with two limit cycles is uncovered. Although the model looks simple at first glance it provides a rich source of dynamics and deserves attention. In this paper, we revisit a model that has its origin in the work of Dubois and Closset. A set of two quadratic population models interact as piecewise defined differential equations. The model has been discussed by Ren Yongtai and Han Li, cryptically written and showing some linguistic and typographical errors, but providing an excellent vehicle for developing skills in mathematical modelling, differential equations and technology for the young researcher. We explore the model in clearer detail and supplement the theory with rich graphical illustration. The paper has the purpose of providing an example of how a young researcher, such as a postgraduate student in biomathematics, can expand on an existing model by making use of current technology.

Dynamical Behaviours of Prey-predator Fishery Model with Harvesting Affected by Toxic Substances

In this paper we consider a harvesting model of predator-prey fishery in which the prey is directly infected by some external toxic substances. The toxic infection is indirectly transmitted to the predator during the feeding process. The model is a modied version from the classic Lotka-Volterra predator-prey model. The stability and bifurcation analyses are addressed. Numerical simulations of the model are performed and bifurcation diagrams are studied to investigate the dynamical behaviours between the predator and the prey. The eects of toxicity and harvesting on the stability of steady states found in the model are discussed.

Ergodic property of a Lotka–Volterra predator–prey model with white noise higher order perturbation under regime switching

In this paper, we investigate a classical Lotka–Volterra predator–prey model with telephone noise and a higher order perturbation of white noise. The existence of a unique positive solution is discussed and sufficient conditions for the existence of an ergodic stationary distribution is established. Some simulation figures are presented to illustrate the analytical findings.

Effects of degeneracy and response function in a diffusion predator–prey model

In this paper, we consider positive solutions of a diffusion predator–prey model with a degeneracy under the Dirichlet boundary conditions. We first obtain sufficient conditions of the existence of positive solutions by the Leray–Schauder degree theory, and then analyze the limiting behavior of positive solutions as the growth rate of the predator goes to infinity and the conversion rates of the predator goes to zero, respectively. It is shown that these results for Holling II response function (i.e. m > 0) reveal interesting contrast with that for the classical Lotka–Volterra predator–prey model (i.e. m = 0).

A numerical method for solutions of Lotka–Volterra predator–prey model with time-delay

In this paper, we present a numerical scheme to obtain polynomial approximations for the solutions of continuous time-delayed population models for two interacting species. The method includes taking inner product of a set of monomials with a vector obtained from the problem under consideration. Doing this, the problem is transformed to a nonlinear system of algebraic equations. This system is then solved, yielding coefficients of the approximate polynomial solutions. In addition, the technique of residual correction, which aims to increase the accuracy of the approximate solution by estimating its error, is discussed in some detail. The method and the residual correction technique are illustrated with two examples.

The Impacts of Harvesting Activities on Prey-Predator Fishery Model in the Presence of Toxin

The present paper discusses a prey-predator fishery model where both species are subjected to harvesting effort. Both species release some toxic substances to each other. The prey species obeys the law of logistic growth. The model is a modified version from the classic Lotka-Volterra predator-prey model. The equilibria of the model are obtained and the dynamical behaviors of the proposed system are examined. Simulations of the model are performed and bifurcation diagrams are studied. The effects of harvesting and toxin on the stability of steady states are examined. In the present research, we found that the impact of harvesting activities is more influential on the dynamical behaviors in the fishery model than that of the toxin.

Hybrid control of Hopf bifurcation in a Lotka-Volterra predator-prey model with two delays

In this paper, the Hopf bifurcation control for a Lotka-Volterra predator-prey model with two delays is studied by using a hybrid control strategy. By analyzing the associated characteristic equation, its local stability and the existence of Hopf bifurcation with respect to both delays are established. In addition, the onset of an inherent bifurcation is delayed. Based on the normal form theory and the center manifold theorem, explicit formulas are derived to determine the direction of Hopf bifurcation and stability of the bifurcating periodic solution. Numerical simulation results confirm that the hybrid controller is efficient in controlling Hopf bifurcation.

Behavioural Analysis of the Modified Lotka-Volterra Predator-Prey Model Using a Computer Simulation

In a real-world scenario, problems are identified, assumptions are made, data are collected and models are proposed. The underlying mathematical structure of the models is analyzed to appraise the sensitivity of the conclusions when the assumptions are not precisely met. In this work, we present the formulation of a modified Lotka-Volterra Predator-Prey Model to include terms which represent the degrees of internal competition of both predators and prey for their limited resources and a simulation of the stability behaviour of the resulting system. A computer simulation of the behaviour - stability of the solution of the model - of the system under different conditions is also presented. The critical points, eigenvalues and phase portrait of the system in terms of the model parameters are also obtained. The model interpretation and applicability are also discussed. The trajectories of the proposed model are not periodic and tend to equilibrium level with time, thus, making the system to be asymptotically stable. This behaviour is in line with the observable natural predator-prey system.

About the minimal time crisis problem and applications

We study the optimal control problem where the cost functional to be minimized represents the so-called time of crisis, i.e. the time spent by a trajectory solution of a control system outside a given set K. Such a problematic finds applications in population dynamics, such as in prey-predator models, which require to find a control strategy that may leave and enter the crisis domain K a number of time that increases with the time interval. One important feature of the time crisis function is that it can be expressed using the characteristic function of K that is discontinuous preventing the use of the standard Maximum Principle. We provide an approximation scheme of the problem based on the Moreau-Yosida approximation of the indicator function of K and prove the convergence of an optimal sequence for the approximated problem to an optimal solution of the original problem when the regularization parameter goes to zero. We illustrate this approach on a simple example and then on the Lotka-Volterra prey-predator model.

Ciclos límites en modelos depredador-presa

The classic Lotka-Volterra model belongs to a family of differential equations known as “Generalized Lotka-Volterra”, which is part of a classification of four models of quadratic fields with center. These models have been studied to address the Hilbert infinitesimal problem, which consists in determine the number of limit cycles of a perturbed hamiltonian system with center. In this work, we first present an alternative proof of the existence of centers in Lotka-Volterra predator-prey models. This new approach is based in algebraic equations given by Kapteyn, which arose to answer Poincare’s problem for quadratic fields. In addition, using Hopf Bifurcation theorem, we proof that more realistic models, obtained by a non-linear perturbation of a classic Lotka-Volterra model, also possess limit cycles.

Ecological monitoring in a discrete-time prey–predator model

The paper is aimed at the methodological development of ecological monitoring in discrete-time dynamic models. In earlier papers, in the framework of continuous-time models, we have shown how a systems-theoretical methodology can be applied to the monitoring of the state process of a system of interacting populations, also estimating certain abiotic environmental changes such as pollution, climatic or seasonal changes.In practice, however, there may be good reasons to use discrete-time models. (For instance, there may be discrete cycles in the development of the populations, or observations can be made only at discrete time steps.) Therefore the present paper is devoted to the development of the monitoring methodology in the framework of discrete-time models of population ecology. By monitoring we mean that, observing only certain component(s) of the system, we reconstruct the whole state process. This may be necessary, e.g., when in a complex ecosystem the observation of the densities of certain species is impossible, or too expensive. For the first presentation of the offered methodology, we have chosen a discrete-time version of the classical Lotka–Volterra prey–predator model. This is a minimal but not trivial system where the methodology can still be presented.We also show how this methodology can be applied to estimate the effect of an abiotic environmental change, using a component of the population system as an environmental indicator. Although this approach is illustrated in a simplest possible case, it can be easily extended to larger ecosystems with several interacting populations and different types of abiotic environmental effects.

Steady-state solutions of a diffusive prey-predator model with finitely many protection zones

This paper is concerned with a diffusive Lotka-Volterra prey-predator model with finitely many protection zones for the prey species. We discuss the stability of trivial and semi-trivial steady-state solutions, and we also study the existence and non-existence of positive steady-state solutions. It is proved that there exists a certain critical growth rate of the prey for survival. Moreover, it is shown that when cross-diffusion is present, under certain conditions, the critical value decreases as the number of protection zones increases. On the other hand, it is also shown that when cross-diffusion is absent, the critical value does not always decrease even if the number of protection zones increases.