Rosenzweig-MacArthur Predator-prey Model Research Articles

Dynamics of a discrete Rosenzweig–MacArthur predator–prey model with piecewise-constant arguments

This paper investigates the dynamics of a new discrete form of the classical continuous Rosenzweig–MacArthur predator–prey model and the biological implications of the dynamics. The discretization method used here is to modify the continuous model to another with piecewise-constant arguments and then to integrate the modified model, which is quite different from the traditional Euler discretization method. First, the existence and local stability of fixed points have been thoroughly discussed. Then, all codimension-1 bifurcations have been studied, including the transcritical bifurcations at the trivial fixed point and the boundary fixed point, and the Neimark–Sacker bifurcation at the unique positive fixed point. Furthermore, it is proved that there are no codimension-2 bifurcations. Finally, the control of the bifurcations is studied. Regarding the biological significance, we have considered the following four aspects: When no harvesting effort is applied to both predators and prey, it is observed that providing more resources to the prey species leads to predator extinction, causing the ecosystem to lose stability. This implies that the paradox of enrichment occurs. When predators are harvested, the system exhibits the hydra effect, i.e. increasing the harvesting effort on predators will lead to an increase in the mean population density of predators. When the prey is harvested, numerical examples indicate that increasing the harvesting effort initially reduces prey density. Upon reaching the bifurcation point, prey density stabilizes while the predator density continues to decline. When both prey and predators are harvested, the biological significance is similar to the previous case. The outcomes of this paper have significant theoretical meaning in the study of population ecology. By MATLAB, the bifurcation diagrams, maximal Lyapunov exponent diagrams, phase portraits and mean population density curves are plotted. Numerical simulations not only show the correctness of the theoretical findings but also reveal many new and interesting dynamics.

Stability of fronts in the diffusive Rosenzweig–MacArthur model

AbstractWe consider a diffusive Rosenzweig–MacArthur predator–prey model in the situation when the prey diffuses at a rate much smaller than that of the predator. In a certain parameter regime, the existence of fronts in the system is known: the underlying dynamical system in a singular limit is reduced to a scalar Fisher–KPP (Kolmogorov–Petrovski–Piskunov) equation and the fronts supported by the full system are small perturbations of the Fisher–KPP fronts. The existence proof is based on the application of the Geometric Singular Perturbation Theory with respect to two small parameters. This paper is focused on the stability of the fronts. We show that, for some parameter regime, the fronts are spectrally and asymptotically stable using energy estimates, exponential dichotomies, the Evans function calculation, and a technique that involves constructing unstable augmented bundles. The energy estimates provide bounds on the unstable spectrum which depend on the small parameters of the system; the bounds are inversely proportional to these parameters. We further improve these estimates by showing that the eigenvalue problem is a small perturbation of some limiting (as the modulus of the eigenvalue parameter goes to infinity) system and that the limiting system has exponential dichotomies. Persistence of the exponential dichotomies then leads to bounds uniform in the small parameters. The main novelty of this approach is related to the fact that the limit of the eigenvalue problem is not autonomous. We then use the concept of the unstable augmented bundles and by treating these as multiscale topological structures with respect to the same two small parameters consequently as in the existence proof, we show that the stability of the fronts is also governed by the scalar Fisher–KPP equation. Furthermore, we perform numerical computations of the Evans function to explicitly identify regions in the parameter space where the fronts are spectrally stable.

Dynamics of the generalized Rosenzweig–MacArthur model in a changing and patchy environment

In this paper, we propose a generalized Rosenzweig–MacArthur predator–prey model with two patches and Allee effects in predators. In a constant environment, for the single-patch model we show the existence of degenerate Bogdanov–Takens bifurcation with codimension up to 4 and degenerate Hopf bifurcation with codimension up to 3, which indicate that the nilpotent cusp of codimension 4 is the organizing center of the bifurcation set. Our results show that Allee effects in predators can not only induce much more complex bifurcations and dynamics (such as three limit cycles, multitype tristability), but also serve as a destabilizing factor to make predators extinction (deterministically) when the environment is sufficiently enriched, which provides strong support for the so-called “paradox of enrichment”. For the two-patch model, we numerically reveal the existence of eight positive steady states including two new coexisting attractors, which means that population dispersal can help species coexist or increase the population abundance. In a changing environment, for the single-patch model we find that the population can track unstable states, fast positive or slow negative environmental change can be beneficial to predators by delaying or even avoiding extinction; while for the two-patch model we observe some interesting phenomena: fast negative (or slow positive) environmental change can make predators in different patches trend to extinction synchronously, and rate-induced different regime shifts.

Regime shift in Rosenzweig–Macarthur predator–prey model in presence of strong Allee effect in prey

Regime shift in Rosenzweig–Macarthur predator–prey model in presence of strong Allee effect in prey

A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model

In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens’ method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.

No sensitivity to functional forms in the Rosenzweig–MacArthur model with strong environmental stochasticity

The classic Rosenzweig–MacArthur predator–prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig–MacArthur model with the three functional responses considered by Fussmann & Blasius (2005), I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator–prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues’ real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator–prey or other ecological systems.

Stability scenarios and period‐doubling onset of chaos in a population model with delayed harvesting

In this work, we investigate complete stability behaviors of the Rosenzweig–MacArthur predator–prey model with delayed harvesting of the prey. The unharvested system exhibits either a steady‐state or an oscillatory dynamics for the coexistence of the species. We explore how the delayed harvesting affects the dynamics of these two modes by analyzing the system stability in effort‐delay bi‐parameter plane. Some novel dynamical scenarios and intricate dynamics are obtained. Analytical conditions for different stability scenarios are derived by examining the associated quasi‐polynomial eigenvalue equation. For invariant harvesting effort, the time delay induces four stability scenarios: stability invariance, instability invariance, stability change, and stability switching. On the other hand, the effort instigates five stability scenarios: stability invariance, instability invariance, stability change, instability change, and instability switching, when the delay strength is fixed. Majority of literatures on harvesting reported that harvesting stabilizes predator–prey interactions. However, we will show that the delayed harvesting can destabilize the system. One of the novelties of the study is to unveil the occurrence of effort‐induced chaos via period‐doubling mechanism. Interestingly, the effort‐induced switching phenomena and chaos do not occur for non‐delayed harvesting.

Bifurcation and overexploitation in Rosenzweig-MacArthur model

<p style='text-indent:20px;'>In this paper, we propose a Rosenzweig–MacArthur predator-prey model with strong Allee effect and trigonometric functional response. The local and global stability of equilibria is studied, and the existence of bifurcation is determined in terms of the carrying capacity of the prey, the death rate of the predator and the Allee effect. An analytic expression is employed to determine the criticality and codimension of Hopf bifurcation. The existence of supercritical Hopf bifurcation and the non-existence of Bogdanov–Takens bifurcation at the positive equilibrium are proved. A point-to-point heteroclinic cycle is also found. Biologically speaking, such a heteroclinic cycle always indicates the collapse of the system after the invasion of the predator, i.e., overexploitation occurs. It is worth pointing out that heteroclinic relaxation cycles are driven by either the strong Allee effect or the high per capita death rate. In addition, numerical simulations are given to demonstrate the theoretical results.</p>

Modelling Holling type II functional response in deterministic and stochastic food chain models with mass conservation

The Rosenzweig-MacArthur predator-prey model is the building block in modeling food chain, food webs and ecosystems. There are a number of hidden assumptions involved in the derivation. For instance the prey population growth is logistic without predation but also with predation. In order to reveal these we will start with modelling a resource-predator-prey system in a closed spatially homogeneous environment. This allows us to keep track of the nutrient flow. With an instantaneous remineralisation of the products excreted in the environment by the populations and dead body mass there is conservation of mass. This allows for a model dimension reduction and yields the mass balance predator-prey model. When furthermore the searching and handling processes are much faster that the population changing rates, the trophic interaction is described by a Holling type II functional response, also assumed in the Rosenzweig-MacArthur model. The derivation uses an extended deterministic model with number of searching and handling predators as model variables where the ratio of the predator/prey body masses is used as a mechanistic time-scale parameter. This extended model is also used as a starting point for the derivation of a stochastic model. We will investigate the stochastic effects of random switching between searching and handling of the predators and predator dying. Prey growth by consumption of ambient resources is still deterministic and therefore the stochastic model is hybrid. The transient dynamics is studied by numerical Monte Carlo simulations and also the quasi-equilibrium distribution for the population quantities is calculated. The body mass of the prey individual is the scaling parameter in the stochastic model formulation. This allows for a quantification of the mean-field approximation criterion for the justification of replacement of the stochastic by a deterministic model.

Dynamics of a spatially coupled model with delayed prey dispersal

ABSTRACT Dispersal of species from one region to another one is a common occurrence in ecology. Several studies have been conducted on predator–prey interactions subjected to population dispersal between patches. In this paper, we consider a two-patch Rosenzweig-MacArthur predator-prey model with prey dispersal. In absence of predator, the movement of prey is density-independent. Predator-influenced prey dispersal is also taken into account because predators have a potential to control prey movement. Travelling time (time delay) linked with the movement mechanism among the prey community is incorporated. The positivity and boundedness of the solutions in the spatially coupled system are established. Stability behaviours of the coexisting equilibrium are explored by considering delay as the bifurcating parameter. It is found that, delayed prey dispersal can potentially alter the stability (resp. instability), and even causes stability switching (resp. instability switching) around the interior equilibrium. However, after some consecutive changes in stability, the equilibrium undergoes instability for larger delay. Analysis of the stability is performed by estimating the distance between critical values of the time delay. In addition, numerical examples are provided to illustrate the findings.

Analysis of The Rosenzweig-MacArthur Predator-Prey Model with Anti-Predator Behavior

This paper discusses the analysis of the Rosenzweig-MacArthur predator-prey model with anti-predator behavior. The analysis is started by determining the equilibrium points, existence, and conditions of the stability. Identifying the type of Hopf bifurcation by using the divergence criterion. It has shown that the model has three equilibrium points, i.e., the extinction of population equilibrium point (E0), the non-predatory equilibrium point (E1), and the co-existence equilibrium point (E2). The existence and stability of each equilibrium point can be shown by satisfying several conditions of parameters. The divergence criterion indicates the existence of the supercritical Hopf-bifurcation around the equilibrium point E2. Finally, our model's dynamics population is confirmed by our numerical simulations by using the 4th-order Runge-Kutta methods.

Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

Effect of Spatial Average on the Spatiotemporal Pattern Formation of Reaction-Diffusion Systems

Some quantities in reaction-diffusion models from cellular biology or ecology depend on the spatial average of density functions instead of local density functions. We show that such nonlocal spatial average can induce instability of constant steady state, which is different from classical Turing instability. For a general scalar equation with spatial average, the occurrence of the steady state bifurcation is rigorously proved, and the formula to determine the bifurcation direction and the stability of the bifurcating steady state is given. For the two-species model, spatially non-homogeneous time-periodic orbits could arise due to spatially non-homogeneous Hopf bifurcation from the constant equilibrium. Examples from a nonlocal cooperative Lotka-Volterra model and a nonlocal Rosenzweig-MacArthur predator-prey model are used to demonstrate the bifurcation of spatially non-homogeneous patterns.

Building mean field ODE models using the generalized linear chain trick & Markov chain theory

ABSTRACT The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

Evolutionarily stable strategies in stable and periodically fluctuating populations: The Rosenzweig–MacArthur predator–prey model

An evolutionarily stable strategy (ESS) is an evolutionary strategy that, if adapted by a population, cannot be invaded by any deviating (mutant) strategy. The concept of ESS has been extensively studied and widely applied in ecology and evolutionary biology [M. Smith, On Evolution (1972)] but typically on the assumption that the system is ecologically stable. With reference to a Rosenzweig-MacArthur predator-prey model [M. Rosenzweig, R. MacArthur, Am. Nat. 97, 209-223 (1963)], we derive the mathematical conditions for the existence of an ESS when the ecological dynamics have asymptotically stable limit points as well as limit cycles. By extending the framework of Reed and Stenseth [J. Reed, N. C. Stenseth, J. Theoret. Biol. 108, 491-508 (1984)], we find that ESSs occur at values of the evolutionary strategies that are local optima of certain functions of the model parameters. These functions are identified and shown to have a similar form for both stable and fluctuating populations. We illustrate these results with a concrete example.

Hopf Bifurcation of a Rosenzweig-Macarthur Hyperbolic Tangent-Type Predator-Prey Model

In this paper, the Rosenzweig-MacArthur predator-prey model with the hyperbolic tangent functional response is investigated. We choose capturing efficiency of predator as the bifurcation parameter and mainly discuss the existence, direction and stability of Hopf bifurcation by Poincaré-Andronov-Hopf bifurcation theorem.

A discrete-time fractional-order Rosenzweig-Macarthur predator-prey model involving prey refuge

In this article, the dynamical behaviors of a discrete-time fractional-order Rosenzweig-MacArthur model with prey refuge are studied. The piecewise constant arguments scheme is applied to obtain the discrete-time model. All possible fixed points and their existence conditions are investigated as well as the local behavior of nearby solutions in various contingencies. Numerical simulations such as the time series, phase portraits, and bifurcation diagrams are portrayed. Three types of bifurcations are shown numerically namely the forward, the period-doubling, and Neimark-Sacker bifurcations. Some phase portraits are depicted to justify the occurrence of those bifurcations.

Keep, break and breakout in food chains with two and three species.

In this paper, through Rosenzweig-MacArthur predator-prey model we study the cyclic coexistence and stationary coexistence and discuss temporal keep and break in the food chain with two species. Then species' diffusion is considered and its effect on oscillation and stability of the ODE system is studied concerning the two different states of coexistence. We find in cyclic coexistence temporal oscillation of population is translated into spatial oscillation although there is fluctuation at the beginning of population waves and finally more stable population evolution is observed. Furthermore, the presence of spatial diffusion of the species can lead to steady wavefront propagation and alter the population distribution in the food chain with two and three species. We show that lower-level species with slow propagation will limit higher-level species and help to keep food chain in space, but through fast propagation lower-level species can survive in a new space without predation and realize a breakout in the linear food chain.

A Predator-Prey model in the chemostat with Holling Type II response function

A model of predator-prey interaction in a chemostat with Holling Type II functional and numerical response functions of the Monod or Michaelis-Menten form is considered. It is proved that local asymptotic stability of the coexistence equilibrium implies that it is globally asymptotically stable. It is also shown that when the coexistence equilibrium exists but is unstable, solutions converge to a unique, orbitally asymptotically stable periodic orbit. Thus the range of the dynamics of the chemostat predator-prey model is the same as for the analogous classical Rosenzweig-MacArthur predator-prey model with Holling Type II functional response. An extension that applies to other functional rsponses is also given.

Self-diffusion-driven pattern formation in prey–predator system with complex habitat under fear effect

In the present work, we explore the influence of habitat complexity on the activities of prey and predator of a spatio-temporal system by incorporating self-diffusion. First, we modify the Rosenzweig–MacArthur predator–prey model by incorporating the effects of habitat complexity on the carrying capacity and fear effect of prey and predator functional response. We establish conditions for the existence and stability of all feasible equilibrium points of the non-spatial model, and later we prove the existence of Hopf and transcritical bifurcations in different parametric phase-planes analytically and numerically. The stability of the spatial system is studied, and we discuss the conditions for Turing instability. Selecting suitable control parameter from the Turing space, the existence conditions for stable patterns are derived using the amplitude equations. Results obtained from theoretical analysis of the amplitude equations are justified by numerical simulation results near the critical parameter value. Further, from numerical simulation, we illustrate the effect of diffusion of the dynamical system in the spatial domain by different pattern formations. Thus, our model clearly shows that the fear effect of prey and predator’s functional response makes an anti-predator behaviour including habitat complexity which helps the prey to survive in the spatio-temporal domain through diffusive process.