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In this paper, the dynamic behavior of a diffusive Rosenzweig-MacArthur (R-M) predatorprey model with hyperbolic tangent functional response and fear effect was investigated. For the local system, we gave a detailed classification of equilibria and performed bifurcation analysis. It was shown by numerical simulation that both the capture rate and fear factor have a stabilizing effect. Furthermore, the existence of limit cycles was discussed when the prey was in low fear or carrying capacity was sufficiently large. For the reaction-diffusion system, we considered the local stability of a unique positive equilibrium, Turing instability of both positive equilibrium and homogeneous periodic orbits under weak fear effect or strong carrying capacity, the direction of Hopf bifurcation and the stability of bifurcating periodic solutions under small fear cost and large diffusion coefficients, as well as the existence of positive nonconstant steady states. However, in the absence of fear effect, Turing instability of both positive equilibrium and homogeneous periodic orbits did not occur. Meanwhile, numerical examples were given to illustrate the corresponding analytic results. [ABSTRACT FROM AUTHOR]

This paper examines the discrete-time Rosenzweig-MacArthur predator–prey model featuring a general predator functional response. We determine the closed and positively invariant set D, where the system's dynamics are analyzed. The set contains extinction, boundary, and interior equilibrium points, which we explore in detail. Our research shows that extinction and boundary equilibrium points can be globally asymptotically stable. We observe transcritical and period-doubling at the boundary equilibrium point with stable two-cycle bifurcation. During period-doubling bifurcation at the boundary equilibrium, one Lyapunov exponent is zero, while the other is negative. The interior equilibrium point exhibits Neimark-Sacker and period-doubling bifurcation with unstable two-cycle. For the Neimark-Sacker bifurcation, we provide an exact form of the first Lyapunov exponent. Our research also shows that the system is permanent in set D. Finally, we apply our theoretical findings to various well-known predator functional responses. [ABSTRACT FROM AUTHOR]

This paper examines the stability of the Rosenzweig–MacArthur model distributed to identical discrete habitat patches. Migration between patches is assumed to follow the non-diffusive rule that individuals have a fixed rate of leaving their local habitat patch and migrating to another. Under this non-diffusive migration rule, we found that population dispersal on a non-regular and connected habitat network can both stabilize and destabilize the Rosenzweig–MacArthur model. It is also shown that our non-diffusive migration rule apparently becomes diffusive if the habitat network is regular. [ABSTRACT FROM AUTHOR]

Understanding the transport mechanism is important key to interpret behavioral and physiological responses of organisms or to predict the success by which they find their resources. In this paper, the effect of transport mechanism in the prey-predator populations is investigated. The models suggested are based on classical Lotka-Volterra (LV) and Rosenzweig-MacArthur (RM) systems in which the passive dispersal (diffusion) and active dispersal (directed movement per se or taxis) are considered. The numerical simulation results obtained from this coupling model confirms two important features: 1) Active dispersal playing the convection role and considered as a simple mechanism to stabilize the prey-predator system; and 2) the coupling effect between active and passive dispersals generates heterogeneous distributions of both species of prey or of predator even in the homogenous environment. [ABSTRACT FROM AUTHOR]

The Comparative Functional Response Approach (CFRA) was developed to provide a practical methodology by which short-term experiments can be used to forecast the longer-term impacts of a potential invading consumer. The CFRA makes inferences about potential invader impact based on comparisons of the functional responses of invader and native consumers on native resources in a common experimental venue. Application of the CFRA and derivative approaches have proliferated since it was introduced in 2014. Here we examine the conceptual foundations of the CFRA within the context of basic Lotka–Volterra consumer-resource theory. Our goals are to assess whether core predictions of the CFRA hold within this framework, to consider the relative importance of background mortality and consumer assimilation efficiency in determining predator impact, and to leverage this conceptual framework to expand the discussion regarding stability and long term consumer and resource dynamics. The CFRA assertion that consumers with a higher functional response will have larger impacts on resources only holds as long as all other parameters are equal, but basic theory indicates that predator impacts on prey abundance and stability will depend more on variation in conversion efficiency and background mortality. While examination of the CFRA within this framework highlights limitations about its current application, it also points to potential strengths that are only revealed when a theoretical context is identified, in this case the implications for stability and conceptual links to competition theory. [ABSTRACT FROM AUTHOR]

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.

We consider a stage-structure Rosenzweig-MacArthur model describing the predator-prey interaction. Here, the prey population is divided into two sub-populations namely immature prey and mature prey. We assume that predator only consumes immature prey, where the predation follows the Holling type Ⅱ functional response. We perform dynamical analysis including existence and uniqueness, the positivity and the boundedness of the solutions of the proposed model, as well as the existence and the local stability of equilibrium points. It is shown that the model has three equilibrium points. Our analysis shows that the predator extinction equilibrium exists if the intrinsic growth rate of immature prey is greater than the death rate of mature prey. Furthermore, if the predation rate is larger than the death rate of predator, then the coexistence equilibrium exists. It means that the predation process on the prey determines the growing effects of the predator population. Furthermore, we also show the existence of forward and Hopf bifurcations. The dynamics of our system are confirmed by our numerical simulations.

The article shows the possibility of solving the problem of the transition between periodic and point attractors in the bistable Rosenzweig–MacArthur model with modifications for the dynamics of root hemiparasitic plants and their hosts. [ABSTRACT FROM AUTHOR]

While it is somewhat well known that spatial PDE extensions of the Rosenzweig–MacArthur predator–prey model do not admit spatial pattern formation through the Turing mechanism, in this paper we demonstrate that the addition of allochthonous resources into the system can result in spatial patterning and colony formation. We study pattern formation, through Turing and Turing–Hopf mechanisms, in two distinct spatial Rosenzweig–MacArthur models generalized to include allochthonous resources. Both models have previously been shown to admit heterogeneous spatial solutions when prey and allochthonous resources are confined to different regions of the domain, with the predator able to move between the regions. However, pattern formation in such cases is not due to the Turing mechanism, but rather due to the spatial separation between the two resources for the predator. On the other hand, for a variety of applications, a predator can forage over a region where more than one food source is present, and this is the case we study in the present paper. We first consider a three PDE model, consisting of equations for each of a predator, a prey, and an allochthonous resource or subsidy, with all three present over the spatial domain. The second model we consider arises in the study of two independent predator–prey systems in which a portion of the prey in the first system becomes an allochthonous resource for the second system; this is referred to as a predator–prey–quarry–resource–scavenger model. We show that there exist parameter regimes for which these systems admit Turing and Turing–Hopf bifurcations, again resulting in spatial or spatiotemporal patterning and hence colony formation. This is interesting from a modeling standpoint, as the standard spatially extended Rosenzweig–MacArthur predator–prey equations do not permit the Turing instability, and hence, the inclusion of allochthonous resources is one route to realizing colony formation under Rosenzweig–MacArthur kinetics. Concerning the ecological application, we find that spatial patterning occurs when the predator is far more mobile than the prey (reflected in the relative difference between their diffusion parameters), with the prey forming colonies and the predators more uniformly dispersed throughout the domain. We discuss how this spatially heterogeneous patterning, particularly of prey populations, may constitute one way in which the paradox of enrichment is resolved in spatial systems by way of introducing allochthonous resource subsidies in conjunction with spatial diffusion of predator and prey populations. [ABSTRACT FROM AUTHOR]

• We study the Rosenzweig-MacArthur model with time-dependent carrying capacity. • The three―dimensional model exhibits a rate-induced critical transition (RICT). • The RICT only occurs when the carrying capacity changes beyond a critical rate. • This critical rate is related to the position of the maximal canard trajectory. • Which can be approximated using methods from singular perturbation theory. Critical transitions or regime shifts are sudden and unexpected changes in the state of an ecosystem, that are usually associated with dangerous levels of environmental change. However, recent studies show that critical transitions can also be triggered by dangerous rates of environmental change. In contrast to classical regime shifts, such rate-induced critical transitions do not involve any obvious loss of stability, or a bifurcation, and thus cannot be explained by the linear stability analysis. In this work, we demonstrate that the well-known Rosenzweig-MacArthur predator-prey model can undergo a rate-induced critical transition in response to a continuous decline in the habitat quality, resulting in a collapse of the predator and prey populations. Rather surprisingly, the collapse occurs even if the environmental change is slower than the slowest process in the model. To explain this counterintuitive phenomenon, we combine methods from geometric singular perturbation theory with the concept of a moving equilibrium , and study critical rates of environmental change with dependence on the initial state and the system parameters. Moreover, for a fixed rate of environmental change, we determine the set of initial states that undergo a rate-induced population collapse. Our results suggest that ecosystems may be more sensitive to how fast environmental conditions change than previously assumed. In particular, unexpected critical transitions with dramatic ecological consequences can be triggered by environmental changes that (i) do not exceed any dangerous levels, and (ii) are slower than the natural timescales of the ecosystem. This poses an interesting research question whether regime shifts observed in the natural world are predominantly rate-induced or bifurcation-induced. [ABSTRACT FROM AUTHOR]

To explore the impact of pest‐control strategy through a fractional derivative, we consider three predator‐prey systems by simple modification of Rosenzweig‐MacArthur model. First, we consider fractional‐order Rosenzweig‐MacArthur model. Allee threshold phenomena into pest population is considered for the second case. Finally, we consider additional food to the predator and harvesting in prey population. The main objective of the present investigation is to observe which model is most suitable for the pest control. To achieve this goal, we perform the local stability analysis of the equilibrium points and observe the basic dynamical properties of all the systems. We observe fractional‐order system has the ability to stabilize Rosenzweig‐MacArthur model with low pest density from oscillatory state. In the numerical simulations, we focus on the bistable regions of the second and third model, and we also observe the effect of the fractional order α throughout the stability region of the system. For the third model, we observe a saddle‐node bifurcation due to the additional food and Allee effect to the pest densities. Also, we numerically plot two parameter bifurcation diagram with respect to the harvesting parameter and fractional order of the system. We finally conclude that fractional‐order Rosenzweig‐MacArthur model and the modified Rosenzweig‐MacArthur model with additional food for the predator and harvested pest population are more suitable models for the pest management. [ABSTRACT FROM AUTHOR]

The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator.

We propose novel positive numerical integrators for approximating predator-prey models. The schemes are based on suitable symplectic procedures applied to the dynamical system written in terms of the log transformation of the original variables. Even if this approach is not new when dealing with Hamiltonian systems, it is of particular interest in population dynamics since the positivity of the approximation is ensured without any restriction on the temporal step size. When applied to separable M-systems, the resulting schemes are proved to be explicit, positive, Poisson maps. The approach is generalized to predator-prey dynamics which do not exhibit an M-system structure and successively to reaction-diffusion equations describing spatially extended dynamics. A classical polynomial Krylov approximation for the diffusive term joint with the proposed schemes for the reaction, allows us to propose numerical schemes which are explicit when applied to well established ecological models for predator-prey dynamics. Numerical simulations show that the considered approach provides results which outperform the numerical approximations found in recent literature. [ABSTRACT FROM AUTHOR]

This article investigates the oscillatory patterns of the following discrete-time Rosenzweig–MacArthur model x1[n+1]=(1+λ)x1[n]−ax12[n]−bx1[n]x2[n]1+mx1[n],x2[n+1]=(1+μ)x2[n]+cx1[n]x2[n]1+mx1[n]. The system describes the evolution and interaction of the populations of two associated species (prey and predator) from generation to generation. We show that this system can exhibit co-dimension-1 bifurcations (flip and Neimark–Sacker bifurcations) as λ crosses some critical values and codimension-2 bifurcations (1:2, 1:3, and 1:4 resonances) for certain critical values of (λ,μ) at the positive equilibrium point. The normal form theory and the center manifold theorem are used to obtain the normal forms. For codimension-2 bifurcations, the bifurcation diagrams are established by using these normal forms along the orbits of differential equations. Numerical simulations are presented to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

One of the simplest population biological models displaying a Hopf bifurcation is the Rosenzweig–MacArthur model with Holling type II response function as essential ingredient. In seasonally forced versions the fixed point on one side of the Hopf bifurcation becomes a limit cycle and the Hopf limit cycle on the other hand becomes a torus, hence the Hopf bifurcation becomes a torus bifurcation, and via torus destruction by further increasing relevant parameters can follow deterministic chaos. We investigate this route to chaos also in view of stochastic versions, since in real world systems only such stochastic processes would be observed. However, the Holling type II response function is not directly related to a transition from one to another population class which would allow a stochastic version straight away. Instead, a time scale separation argument leads from a more complex model to the simple 2 dimensional Rosenzweig–MacArthur model, via additional classes of food handling and predators searching for prey. This extended model allows a stochastic generalization with the stochastic version of a Hopf bifurcation, and ultimately also with additional seasonality allowing a torus bifurcation under stochasticity. Our study shows that the torus destruction into chaos with positive Lyapunov exponents can occur in parameter regions where also the time scale separation and hence stochastic versions of the model are possible. The chaotic motion is observed inside Arnol’d tongues of rational ratio of the forcing frequency and the eigenfrequency of the unforced Hopf limit cycle. Such torus bifurcations and torus destruction into chaos are also observed in other population biological systems, and were for example found in extended multi-strain epidemiological models on dengue fever. To understand such dynamical scenarios better also under noise the present low dimensional system can serve as a good study case. [ABSTRACT FROM AUTHOR]

We planned to develop predator-prey models using Paramecium and yeast, but they have not been empirically examined since work by Gause in the 1930s. Therefore, we evaluated if Paramecium aurelia ingests and grows on eight yeasts. Recognising that it ingested yeasts but could not grow, we assessed if it might grow on other yeasts, by empirically parameterising a predator-prey model that relies on ingestion, not growth. Simulations were compared to P. aurelia-yeast time-series data, from Gause. We hypothesised that if the model simulated predator-prey dynamics that mimicked the original data, then possibly P. aurelia could grow on yeast; simulations did not mimic the original data. Reviewing works by Gause exposed two issues: experiments were undoubtedly contaminated with bacteria, allowing growth on bacteria, not yeast; and the population cycle data cannot be considered a self-sustaining time series, as they were manipulated by adding yeast and ciliates. We conclude that past and future work should not rely on this system, for either empirical or theoretical evaluations. Finally, although we show that P. aurelia, P. caudatum, Euplotes patella, and Blepharisma sp. cannot grow on yeast, Tetrahymena pyriformis and Colpidium striatum can; these may provide models to explore predator-prey dynamics. [ABSTRACT FROM AUTHOR]

In the current study differences were evaluated between a complex 3D multistage population model (SINMOD) and a simpler consumer-resource population model for estimating the effects of crude oil on the marine copepod Calanus finmarchicus . The SINTEF OSCAR model was used to simulate hypothetical oil spills in the Lofoten area in 1995, 1997, and 2001. Both population models simulated a negligible effect of crude oil on the Calanus’ biomass when assuming low species sensitivity. The simple model estimated a larger effect on the biomass (up to a 100% decline) compared to the complex model (maximum decline of 60–80%) at high species sensitivity to crude oil. These differences may be related to the inclusion of copepod advection in the complex model. Our study showed that if little data is available to parameterize a model, or if computational resources are scarce, the simple model could be used for risk screening. Nevertheless, the possibility of including a dilution factor for time-varying biomass should be examined to improve the estimations of the simple model. The complex model should be used for a more in depth risk analysis, as it includes physical processes such as the drift of organisms and differentiation between developmental stages. [ABSTRACT FROM AUTHOR]

Predator and prey isoclines are estimated from data on yeast–protist population dynamics ( Gause et al., 1936 ). Regression analysis shows that the prey isocline is best fitted by an L-shaped function that has a vertical and a horizontal part. The predator isocline is vertical. This shape of isoclines corresponds with the Lotka–Volterra and the Rosenzweig–MacArthur predator–prey models that assume a prey refuge. These results further support the idea that a prey refuge changes the prey isocline of predator–prey models from a horizontal to an L-shaped curve. Such a shape of the prey isocline effectively bounds amplitude of predator–prey oscillations, thus promotes species coexistence. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of a predator-prey system with the assumption that both prey and predators use game theory-based strategies to maximize their per capita population growth rates. The predators adjust their strategies in order to catch more prey per unit time, while the prey, on the other hand, adjust their reactions to minimize the chances of being caught. We assume each individual is either mobile or sessile and investigate the evolution of mobility for each species in the predator-prey system. When the underlying population dynamics is of the Lotka-Volterra type, we show that strategies evolve to the equilibrium predicted by evolutionary game theory and that population sizes approach their corresponding stable equilibrium (i.e. strategy and population effects can be analyzed separately). This is no longer the case when population dynamics is based on the Holling II functional response, although the strategic analysis still provides a valuable intuition into the long term outcome. Numerical simulation results indicate that, for some parameter values, the system has chaotic behavior. Our investigation reveals the relationship between the game theory-based reactions of prey and predators, and their population changes. [ABSTRACT FROM AUTHOR]

A general framework for age-structured predator-prey systems is introduced. Individuals are distinguished into two classes, juveniles and adults, and several possible interactions are considered. The initial system of partial differential equations is reduced to a system of (neutral) delay differential equations with one or two delays. Thanks to this approach, physically correct models for predator-prey with delay are provided. Previous models are considered and analysed in view of the above results. A Rosenzweig-MacArthur model with delay is presented as an example. [ABSTRACT FROM AUTHOR]

Habitat fragmentation is an important area of concern in species conservation. Habitat fragmentation can affect population distributions through reductions in suitable habitat, and through organism responses to different habitat types and the transitions between them. In earlier work, the effect of habitat fragmentation on cyclic populations was investigated in the context of populations that show no behavioural response to the interface between habitat types. In this paper, we extend the earlier work by adding edge-mediated behaviour to the models. That is, we investigate the dynamics that result when oscillatory predator and prey species also exhibit behavioural responses to habitat interfaces. Our results show generally that habitat loss decreases the amplitude and the average density of the prey and predator populations, but that most of the reponses observed in the two models exhibit marked differences. This work highlights the complexity of the interplay between population cycles, habitat fragmentation, and edge-mediated behaviour, and the need to study such systems in greater detail. [ABSTRACT FROM AUTHOR]

Abstract: Predator–prey systems of two species are considered, in which the predator consumes the prey and the prey has a harmful effect on the predator. Our aim is to show that the negative effect of the prey may lead to reduction and/or extinction of its predator, and to show that how and when the reduction/extinction happens. The specific system is a simple extension of the Rosenzweig–MacArthur model by introducing a negative interaction term in the predator equation. By phase-portrait analysis and numerical simulations, we demonstrate that interaction outcomes in the system may transition among predation, amensalism, competition, neutralism and commensalism. First, varying initial densities of species population alone can lead to the transition. In addition, critical regions of the initial densities are exhibited, in which small fluctuation of the densities results in both transition of outcomes and extinction of the predator. Second, varying one or more parameters (factors) may also lead to the transition. Moreover, specific regions of parameters are exhibited, in which small variation of the parameters leads to reduction and/or extinction of the predator. [Copyright &y& Elsevier]

The present study combines short-term experiments with food chain modeling to explore the long-term effects of the herbicide atrazine on consumer-resource dynamics in a marine intertidal ecosystem. Short-term (28 d) lab experiments indicated that the intrinsic rate of increase ( r) and carrying capacity ( K) of the marine diatom Seminavis robusta decreased with increasing atrazine exposure. This decrease did not show the concave shape expected from the lifetime productivity for nonexposed diatoms and from single-species toxicity data in the literature but instead was described best by a linear model. These experimentally observed atrazine-induced decreases of r and K were used to parameterize a Rosenzweig-MacArthur model representing a simple food chain including the tested diatom and its grazer, the harpacticoid copepod Delavalia palustris var. palustris. Stable oscillation zoo-phytobenthos systems were produced at diatom exposures of 0, 100, and 150 µg/L atrazine. An atrazine concentration of 150 µg/L contributed to a 15% increase of the oscillation periods of both diatoms and copepods as well as a 52% reduction of oscillation amplitudes compared with the control situation. Although the amplitudes of copepods increased only 7% at 150 µg/L atrazine, the maximum and minimum copepod densities at that concentration were reduced 61 and 63%, respectively. The effects of atrazine on periodicity and amplitudes were robust to 20% changes in the food-chain model parameters that represented allometric relationships. The simulations in the present study suggest food chain-mediated indirect effects on zoobenthos populations, indicating a reduced diatom and copepod availability throughout the year. Environ. Toxicol. Chem. 2013;32:1088-1095. © 2013 SETAC [ABSTRACT FROM AUTHOR]

Root hemiparasitic plants interact with their host plants through parasitism and competition. The interactions can be divided into aboveground and belowground interactions. Because both groups of plants are autotrophic, they compete for light aboveground. Belowground interactions are more complex. The host plants compete for resources in the soil and the hemiparasitic plants prey on the host plants through haustoria, using the hosts as the main source of water and nutrients. In this paper, we modeled the relationship between these two plant types, extending the well-known Rosenzweig-MacArthur predator-prey model to cover both light competition and intra-specific parasitism among hemiparasites. We included a realistic relationship of carrying capacity to environmental productivity and followed model behavior on a productivity gradient. The model shows that, at very low productivities, there are only a few poor hosts and hemiparasites have no chance to persist. As productivity increases, there is a range of productivity where both plant types coexist. A further increase in productivity gets the system out of the coexistence range, and only host plants survive. This final prediction successfully explains patterns observed in empirical data, contrary to the results of an earlier, oversimplified model of the explored interaction. Comparison of various models demonstrates that the model is able to reproduce the decline of hemiparasites with increasing productivity only when competition for light is included. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study a spatial (two-dimensional) Rosenzweig-MacArthur model under the following assumptions: (1) prey spread follows a nonlinear diffusion rule, (2) preys have a refuge zone (sometimes called "protection zone") where predators cannot enter, (3) predators move following linear diffusion. We present a bifurcation analysis for the system that shows the existence of positive solutions at the steady state. We complement the theoretical results with numerical computations and compare our results with those obtained in the case of having linear diffusion for the prey movement. Our results show that both models, with linear and nonlinear diffusion for the prey, have the same bifurcation point and the positive solution curves are virtually the same in a neighborhood of this point, but they get drastically different as the bifurcation parameter approaches zero. [ABSTRACT FROM AUTHOR]

The harvesting management is developed to protect the biological resources from over-exploitation such as harvesting and trapping. In this article, we consider a predator–prey interaction that follows the fractional-order Rosenzweig–MacArthur model where the predator is harvested obeying a threshold harvesting policy (THP). The THP is applied to maintain the existence of the population in the prey–predator mechanism. We first consider the Rosenzweig–MacArthur model using the Caputo fractional-order derivative (that is, the operator with the power-law kernel) and perform some dynamical analysis such as the existence and uniqueness, non-negativity, boundedness, local stability, global stability, and the existence of Hopf bifurcation. We then reconsider the same model involving the Atangana–Baleanu fractional derivative with the Mittag–Leffler kernel in the Caputo sense (ABC). The existence and uniqueness of the solution of the model with ABC operator are established. We also explore the dynamics of the model with both fractional derivative operators numerically and confirm the theoretical findings. In particular, it is shown that models with both Caputo operator and ABC operator undergo a Hopf bifurcation that can be controlled by the conversion rate of consumed prey into the predator birth rate or by the order of fractional derivative. However, the bifurcation point of the model with the Caputo operator is different from that of the model with the ABC operator. [ABSTRACT FROM AUTHOR]

• Seasonal forcing of predator-prey dynamics is predominantly modelled as a sine wave. • There is no evidence that abrupt seasonal forcing results in the same dynamics. • We study the response of predator-prey dynamics to the shape of the forcing signal. • An abrupt forcing triggers bifurcations and chaos at lower forcing strengths. • These dynamical changes are not a mere consequence of the forcing signal's variance. Coupled dynamical systems in ecology are known to respond to the seasonal forcing of their parameters with multiple dynamical behaviours, ranging from seasonal cycles to chaos. Seasonal forcing is predominantly modelled as a sine wave. However, the transition between seasons is often more sudden as illustrated by the effect of snow cover on predation success. A handful of studies have mentioned the robustness of their results to the shape of the forcing signal but did not report any detailed analyses. Therefore, whether and how the shape of seasonal forcing could affect the dynamics of coupled dynamical systems remains unclear, while abrupt seasonal transitions are widespread in ecological systems. To provide some answers, we conduct a numerical analysis of the dynamical response of predator–prey communities to the shape of the forcing signal by exploring the joint effect of two features of seasonal forcing: the magnitude of the signal, which is classically the only one studied, and the shape of the signal, abrupt or sinusoidal. We consider both linear and saturating functional responses, and focus on seasonal forcing of the predator's discovery rate, which fluctuates with changing environmental conditions and prey's ability to escape predation. Our numerical results highlight that a more abrupt seasonal forcing mostly alters the magnitude of population fluctuations and triggers period–doubling bifurcations, as well as the emergence of chaos, at lower forcing strength than for sine waves. Controlling the variance of the forcing signal mitigates this trend but does not fully suppress it, which suggests that the variance is not the only feature of the shape of seasonal forcing that acts on community dynamics. Although theoretical studies may predict correctly the sequence of bifurcations using sine waves as a representation of seasonality, there is a rationale for applied studies to implement as realistic seasonal forcing as possible to make precise predictions of community dynamics. [ABSTRACT FROM AUTHOR]

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.

We consider a stage-structure Rosenzweig-MacArthur model describing the predator-prey interaction. Here, the prey population is divided into two sub-populations namely immature prey and mature prey. We assume that predator only consumes immature prey, where the predation follows the Holling type II functional response. We perform dynamical analysis including existence and uniqueness, the positivity and the boundedness of the solutions of the proposed model, as well as the existence and the local stability of equilibrium points. It is shown that the model has three equilibrium points. Our analysis shows that the predator extinction equilibrium exists if the intrinsic growth rate of immature prey is greater than the death rate of mature prey. Furthermore, if the predation rate is larger than the death rate of predator, then the coexistence equilibrium exists. It means that the predation process on the prey determines the growing effects of the predator population. Furthermore, we also show the existence of forward and Hopf bifurcations. The dynamics of our system are confirmed by our numerical simulations.