Three-species Predator-prey Model Research Articles

Boundedness and stability of a quasilinear three-species predator–prey model with competition mechanism

Boundedness and stability of a quasilinear three-species predator–prey model with competition mechanism

Persistence and extinction of a modified LG-Holling type II predator-prey model with two competitive predators and Lévy jumps

In this paper, we study a three-species predator-prey model with modified LG-Holling type II with Lévy jumps, and we take the competition among predators into consideration. First, We use an Ornstein-Uhlenbeck process to describe the environmental stochasticity and prove that there is a unique positive solution to the system by mathematical analysis skills such as comparison theorem. Futhermore, the extinction or persistence in the mean of each species under different conditions is obtained. Finally, some numerical simulations are carried out to support our main results.

Global boundedness of a three-species predator-prey model with prey-taxis and competition

Global boundedness of a three-species predator-prey model with prey-taxis and competition

Analysis of the onset of a regime shift and detecting early warning signs of major population changes in a two-trophic three-species predator-prey model with long-term transients.

Identifying early warning signs of sudden population changes and mechanisms leading to regime shifts are highly desirable in population biology. In this paper, a two-trophic ecosystem comprising of two species of predators, competing for their common prey, with explicit interference competition is considered. With proper rescaling, the model is portrayed as a singularly perturbed system with fast prey dynamics and slow dynamics of the predators. In a parameter regime near singular Hopf bifurcation, chaotic mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations are observed as long-lasting transients before the system approaches its asymptotic state. To analyze the dynamical cause that initiates a large amplitude oscillation in an MMO orbit, the model is reduced to a suitable normal form near the singular-Hopf point. The normal form possesses a separatrix surface that separates two different types of oscillations. A large amplitude oscillation is initiated if a trajectory moves from the "inner" to the "outer side" of this surface. A set of conditions on the normal form variables are obtained to determine whether a trajectory would exhibit another cycle of MMO dynamics before experiencing a regime shift (i.e. approaching its asymptotic state). These conditions serve as early warning signs for a sudden population shift as well as detect the onset of a regime shift in this ecological model.

DYNAMICS ANALYSIS OF THREE-SPECIES REACTION-DIFFUSION SYSTEM VIA THE MULTIPLE SCALE PERTURBATION METHOD

In this paper, the general analysis of spatiotemporal dynamics of three-species reaction-diffusion system induced by Turing bifurcation is given. Firstly, by employing the Routh-Hurwitz criterion the conditions for Turing bifurcation in three-species reaction-diffusion equations are obtained. Secondly, through the tool of the multiple scale perturbation method the amplitude equations of Turing patterns are also given. Finally, we take a three-species predator-prey model as an example to illustrate the application of these general theoretical results, and meanwhile carry out many numerical simulations to depict spots pattern, stripes pattern and labyrinthine pattern and demonstrate the validity of these theories.

Correction to: Dynamics for a Three-Species Predator-Prey Model with Density-Dependent Motilities

Correction to: Dynamics for a Three-Species Predator-Prey Model with Density-Dependent Motilities

Dynamics for a Three-Species Predator-Prey Model with Density-Dependent Motilities

Dynamics for a Three-Species Predator-Prey Model with Density-Dependent Motilities

Appearance of Temporal and Spatial Chaos in an Ecological System: A Mathematical Modeling Study.

The ecological theory of species interactions rests largely on the competition, interference, and predator–prey models. In this paper, we propose and investigate a three-species predator–prey model to inspect the mutual interference between predators. We analyze boundedness and Kolmogorov conditions for the non-spatial model. The dynamical behavior of the system is analyzed by stability and Hopf bifurcation analysis. The Turing instability criteria for the Spatio-temporal system is estimated. In the numerical simulation, phase portrait with time evolution diagrams shows periodic and chaotic oscillations. Bifurcation diagrams show the very rich and complex dynamical behavior of the non-spatial model. We calculate the Lyapunov exponent to justify the dynamics of the non-spatial model. A variety of patterns like interference, spot, and stripe are observed with special emphasis on Beddington–DeAngelis function response. These complex patterns explore the beauty of the spatio-temporal model and it can be easily related to real-world biological systems.

Noise-induced control of environmental fluctuations in a three-species predator–prey model

A three-species ecological model is proposed that comprises one prey and two predators (low predator and top predator) to understand the persistence and/or extinction of prey and predators under a suitable environment. The realistic ecological system is typically inhomogeneous in terms of spatiotemporal variations of population density. With this motivation, white noise is introduced to the system to conceive the impact of environmental heterogeneity on the interaction of prey and predators. The boundedness of solutions of the system is presented. The condition for the local stability of the deterministic model is established. The stochastic analysis of the system suggests the impression of the environmental noise on the dynamical behaviour of solutions. Numerical results reveal that the proposed system exhibits an oscillatory pattern of solutions. It is shown that the white noise in the low predator has a dominating role in comparison with other noises to keep the oscillation in order. Moreover, to investigate the spatial dependency of the population density, diffusion term is introduced to the system that also displays oscillatory behaviour of the solutions. The results also confirm that the population density of prey, low predator, and top predator have a biological control in the oscillatory behaviour. In the presence of diffusion and white noise, numerical simulation illustrates the sensitivity of the diffusion coefficients and the amplitude of noises. Numerical results lead to a reasonable agreement with the analytical findings in both non-spatial and spatial systems.

Traveling wave solutions for a three-species predator–prey model with two aborigine preys

In this paper, we study the invading phenomenon of an alien predator to the habitat of two aborigine preys by traveling waves connecting the predator-free state to the co-existence state. We characterize the minimal wave speed of this invading process based on an application of Schauder’s fixed point theorem with the help of (generalized) upper-lower-solutions. New form of upper-lower-solutions are constructed to derive the existence of traveling waves for all admissible speeds.

An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model

In recent decades, studying the behavior of biological species has become one of the most fascinating areas of applied mathematics. The high importance of conservation of rare species in nature has prompted researchers in various fields to pay particular attention to this issue. Therefore, it is essential to develop mathematical models that examine the dynamics of their behavior. On the other hand, the development of new concepts in numerical analysis has enabled us to preserve more information on the evolutionary behavior history of a dynamic system and to use it in predicting the new features of the system. Fractional derivatives have provided such a valuable tool. This paper studies a dynamic system that models the interactions between two densities of immature and mature prey and predator populations. In the model, prey population is divided into two populations, including mature prey and immature prey. Another feature of the model is that predator depends on mature prey only and it followed by Crowley-Martin type functional response. Moreover, the fractional operator used in this model as derivative is of the Atangana-Baleanu AB type. Using this kind of fractional derivative causes the results to depend on the fractional order of the derivative. The addition of the concept of memory to the model is another highlight of using this type of derivative for the biological model. This helps the model to apply all the essential information of the phenomenon from the beginning to the desired time in the calculations. Existence and uniqueness of solutions to the fractional model are also investigated in this manuscript. The numerical method used in the article is also one of the most efficient patterns in solving problems with fractional derivatives. Using this effective method makes the results very consistent with what we actually expect to happen. Many simulations have been carried out to investigate the effect of parameters in the model on its overall behavior. Numerical results show the impressive performance of the fractional operator on the dynamic behavior of the considered predator-prey model. This efficient fractional operator can also be tested in the structure of other existing biological models.

Analysis for a Delayed Three-Species Predator-Prey Model with Feedback Controls and Prey Diffusion

In this paper, we study a class of 3-species multidelay Lotka–Volterra ratio-dependent predator-prey model with feedback controls and prey diffusion. By using the theory of delay differential inequalities and developing some new analysis methods as well as constructing a suitable Lyapunov function, some sufficient conditions are obtained to guarantee the permanence of the system and the global attractivity of the positive solution for the predator-prey system. Furthermore, the corresponding periodic system is discussed, and some conditions are established about the existence, uniqueness, and stability of the positive periodic solution for the periodic system by using the fixed-point theory and theoretical analysis. In additional, some numerical solutions of the equations describing the system are given to verify the obtained criteria. Finally, the corresponding stochastic time-delay predator-prey model with multiplicative noise sources is solved numerically, and some interesting new dynamics behavior is obtained.

Dynamics and pattern formations in a three-species predator-prey model with two prey-taxis

In this paper, we study a reaction-diffusion system with prey taxis which models the dynamics of a two-predator-one-prey ecosystem in which the predators collaboratively take advantage of the prey's strategy. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and small prey-taxis sensitivity coefficients are proved. It is also shown that such prey-taxis qualitatively affect the stability of coexistence steady state solutions in some cases, and the emergence of forceful prey-taxis will enhance the spatial patterns.

Stability and dynamics of a fractional-order three-species predator-prey model.

In this paper, a fractional-order predator-prey mathematical model has been developed considering Holling type II functional response. Here, we have investigated the interaction dynamics of prey, middle predator and top predator. We assume that the middle predator consumes only the prey, and the top predator consumes only the middle predator. Here, the logistic growth of prey has been considered. Then, different possible equilibrium points of our proposed system are determined. The stability of our proposed system is investigated around the equilibrium points. Then, some numerical simulations results are presented for better understanding the dynamics of our proposed model. It is found that the fractional-order derivative can improve the stability of our proposed system.

Stochastic mixed-mode oscillations in a three-species predator-prey model

The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise. In this regime, the stochastic model admits noise-driven mixed-mode oscillations (MMOs), which capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to calculate the distribution of the random number of small oscillations between successive spikes for varying noise intensities and distance to the Hopf bifurcation. We also study the effect of noise on a suitable Poincaré map. Finally, we prove that the stochastic model can be transformed into a normal form near the folded node, which can be linked to recent results on the interplay between deterministic and stochastic small amplitude oscillations. The normal form can also be used to study the parameter influence on the noise level near folded singularities.

Permanence of a Lotka-Volterra Predator-Prey Model with Feedback Controls and Prey Diffusion

This paper is concerned with a multi-delay three-species predator-prey model with feedback controls and prey diffusion. By developing some new analysis techniques and using the comparison principle of differential equations, we obtained some new sufficient conditions which ensure the system to be permanent.

Dynamical study of a chaotic predator-prey model with an omnivore

In this paper, the dynamics and bifurcations of a three-species predator-prey model with an omnivore are further investigated. The food web considered in this work comprises prey, predator and a third species, which consumes the carcasses of the predator along with predation of the original prey. The conditions for existence, uniqueness and continuous dependence on initial conditions for the solution of the model are derived. Analytical and numerical bifurcation studies reveal that the system undergoes transcritical and Hopf bifurcations around its equilibrium points. Further, the Hopf bifurcation curves in the parameters’ space along with codimension two bifurcations of equilibrium points and bifurcation of limit cycles that arise in the system are investigated. In particular, the occurrence of generalized Hopf, fold Hopf and Neimarck-Sacker bifurcations is unveiled and illustrates the rich dynamics of the model. Finally, bifurcation diagrams, phase portraits and Lyapunov exponents of the model are presented.

The “Destabilizing” Effect of Cannibalism in a Spatially Explicit Three-Species Age Structured Predator-Prey Model

Cannibalism, the act of killing and consumption of conspecifics, is generally considered to be a stabilising process in ODE models of predator-prey systems. On the other hand, Sun et al. were the first to show that cannibalism can cause Turing instability, in the classical Rosenzweig-McArthur two-species PDE model, which is an impossibility without cannibalism. Magnússon’s classic work is the first to show that cannibalism in a structured three-species predator-prey ODE model can actually be destabilising. In the current manuscript we consider the PDE form of the three-species model proposed in Magnússon’s classic work. We prove that, in the absence of cannibalism, Turing instability is an impossibility in this model, forany range of parameters. However, theinclusionof cannibalism can cause Turing instability. Thus, to the best of our knowledge, we report thefirstcannibalism induced Turing instability result, in spatially explicit three-species age structured predator-prey systems. We also show that, in the classical ODE model proposed by Magnússon, cannibalism can act as a life boat mechanism,for the prey.

Biodiversity and Optimal Multi-species Ecosystem Management

We analyze optimal multi-species management in a dynamic bio-economic model taking into account both harvesting profit and biodiversity value. Within an analytical model, we show that extinction is never optimal when a global biodiversity value is taken into account. Moreover, a stronger preference for species diversity leads to a more even distribution of stock sizes in the optimal steady state, and a higher value of biodiversity increases steady state stock sizes for all species when species are ecologically independent or symbiotic. For a predator–prey ecosystem, the effects may be positive or negative depending on relative prices and the strength of species interaction. The analytical results are illustrated and extended using an age-structured three-species predator–prey model for the Baltic cod, sprat, and herring fisheries. In this quantitative application, we find that using stock biomass or stock numbers as abundance indicators in the biodiversity index may lead to opposite results.

Can the control of invasive species be left to chance?

Invasive species are said to be a bigger threat to native biodiversity than pollution, harvest, and disease combined. In order to preserve native flora, fauna, and natural resources of regions, the effective control of invasive species is imperative. We consider a three-species predator–prey model, where the top predator, functions as an invasive species, with the ability to grow explosively to ‘uncontrollable’ levels in a finite time, if unchecked. We investigate the effect of random environmental forces, on such explosive growth. We find that environmental noise alone, cannot prevent explosive growth in invasive populations. We interpret these results as showing that invasive species can strongly adapt to a randomly changing environment. We provide supporting evidence of such phenomenon, from the ecological literature. Our results are applicable to a wide variety of natural resource management practices.