Fractional-order Predator Prey Model Research Articles

Dynamics of a Delayed Fractional-Order Predator–Prey Model with Cannibalism and Disease in Prey

In this study, a class of delayed fractional-order predation models with disease and cannibalism in the prey was studied. In addition, we considered the prey stage structure and the refuge effect. A Holling type-II functional response function was used to describe predator–prey interactions. First, the existence and uniform boundedness of the solutions of the systems without delay were proven. The local stability of the equilibrium point was also analyzed. Second, we used the digestion delay of predators as a bifurcation parameter to determine the conditions under which Hopf bifurcation occurs. Finally, a numerical simulation was performed to validate the obtained results. Numerical simulations have shown that cannibalism contributes to the elimination of disease in diseased prey populations. In addition, the size of the bifurcation point τ0 decreased with an increase in the fractional order, and this had a significant effect on the stability of the system.

Bifurcation and Stability Analysis of a New Fractional-Order Prey–Predator Model with Fear Effects in Toxic Injections

This paper proposes a prey–predator model affected by fear effects and toxic substances. We used the Lipschitz condition to prove the uniqueness of the model solution and Laplace transform to prove the boundedness of the model solution. We used the fractional-order stability theorem to provide sufficient conditions for the local stability of equilibrium points, and selected fractional-order derivatives as parameters to perform Hopf bifurcation analysis on the system. Finally, the theoretical results are verified via numerical simulation. The results show that a value of α will affect the stability of the system and that the population size and the effect of toxic substances have a huge impact on the stability of the system.

Predation fear and its carry-over effect in a fractional order prey–predator model with prey refuge

This research article centers on the formulation and analysis of a novel prey–predator model that integrates the impacts of predation fear, prey refuge, and carry-over effects. The model is formulated and analyzed using fractional differential equations (FDEs). The model incorporates ecological concepts such as anti-predator behaviours and memory effects to maintain a better understanding of prey–predator interactions. The mathematical model is developed based on a basic prey–predator model with a Michelis–Menten functional response and includes fear-induced carry-over effects, prey refuge, and fractional order derivatives. The study investigates the well-posedness, stability, and Hopf bifurcation of the proposed model and conducts detailed numerical investigations. The integration of different anti-predatory mechanisms and the use of FDEs contribute to a comprehensive understanding of the dynamics of prey–predator interactions and provide useful insights into the complexities of ecological systems.

EXPLORING BIFURCATION IN A FRACTIONAL-ORDER PREDATOR-PREY SYSTEM WITH MIXED DELAYS

This work chiefly develops and discusses a fractional-order predatorprey model with distributed delay and discrete delay. Applying skilly an appropriate variable substitution, a novel equivalent form of the fractional-order predator-prey model with distributed delay and discrete delay is derived. By virtue of the stability theorem and bifurcation principle of fractional-order dynamical system, we establish a delay-independent stability and bifurcation criterion ensuring the stability and the onset of Hopf bifurcation for the involved predator-prey system. The role of the time delay in stabilizing system and controlling Hopf bifurcation of the considered fractional-order predatorprey model is displayed. Software simulation results are presented to support the key theoretical fruits.

Further results on dynamical properties for a fractional-order predator-prey model

On the basis of previous studies, we set up a new fractional-order predator-prey model. First, by basic theory of algebraic equation, we discuss the existence of equilibrium point. Second, with the help of Lipschitz condition, we discuss the existence and uniqueness of solution. Third, applying the derivative theory of functions, we prove the non-negativity of solution. Fourth, using the inequality technique of fractional-order differential equations, we obtain the sufficient condition to ensure the uniformly boundedness of solution. Fifth, by analysing the Jacobian matrix, the locally asymptotically stability of the equilibria has been investigated; By constructing some suitable Lyapunov functions, the globally asymptotically stability of the equilibria bas been analysed. Sixth, the computer simulation diagrams are displayed to illustrate the correctness of the analytic findings. Finally, a concise conclusion is give to end this paper.

Further results on dynamical properties for a fractional-order predator-prey model

Further results on dynamical properties for a fractional-order predator-prey model

Dynamic analysis of a fractional-order predator–prey model with harvesting

Dynamic analysis of a fractional-order predator–prey model with harvesting

Theoretical and numerical analysis of a prey–predator model (3-species) in the frame of generalized Mittag-Leffler law

Abstract In the present paper, a new fractional order predator–prey model is considered. The applied fractional operator is a generalized Atangana–Baleanu–Caputo (ABC) derivative, which does not require any restrictions on the initial conditions as in the case of classical ABC fractional derivatives. On the theoretical aspect, we prove the existence, uniqueness, and Ulam–Hyers stability results by using some fixed point theorems and nonlinear analysis techniques. The numerical aspect discusses the approximation solutions for the proposed model by applying the generalized scheme of the Adams–Bashforth technique. At the end, we explain the behavior of the solution to the studied model through graphical representations and numerical simulations.

A Fractional-Order Predator-Prey Model with Age Structure on Predator and Nonlinear Harvesting on Prey

In this manuscript, the dynamics of a fractional-order predator-prey model with age structure on predator and nonlinear harvesting on prey are studied. The Caputo fractional-order derivative is used as the operator of the model by considering its capability to explain the present state as the impact of all of the previous conditions. Three biological equilibrium points are successfully identified including their existing properties. The local dynamical behaviors around each equilibrium point are investigated by utilizing the Matignon condition along with the linearization process. The numerical simulations are demonstrated not only to show the local stability which confirms all of the previous analytical results but also to show the existence of periodic signal as the impact of the occurrence of Hopf bifurcation.

Discretization Fractional-Order Biological Model with Optimal Harvesting

In this paper, a discretization of a three-dimensional fractional-order prey-predator model has been investigated with Holling type III functional response. All its fixed points are determined; also, their local stability is investigated. We extend the discretized system to an optimal control problem to get the optimal harvesting amount. For this, the discrete-time Pontryagin’s maximum principle is used. Finally, numerical simulation results are given to confirm the theoretical outputs as well as to solve the optimality problem.

BIFURCATION BEHAVIORS OF A FRACTIONAL-ORDER PREDATOR–PREY NETWORK WITH TWO DELAYS

This paper highlights the stability and bifurcation of a fractional-order predator–prey model involving two delays. The critical values of delays with respect to Hopf bifurcation are exactly calculated for the developed model by taking two different delays as bifurcation parameters, respectively. Moreover, the effects of fractional order and additional delay on the bifurcation point are carefully explored. It detects that the stability performance is extremely strengthened by taking an appropriate fractional order and another delay. This hints that the onset of Hopf bifurcation can be advanced (lagged) with variations of their values. Numerical simulations are ultimately employed to check the correctness of the derived theoretical analysis.

Dynamical analysis, stability and discretization of fractional-order predator-prey model with negative feedback on two species

‎The Lotka-Volterra model is an important model being employed in biological phenomena to investigate the nonlinear interaction among existing species‎. ‎In this work‎, ‎we first consider an integer order predator-prey model with negative feedback on both prey and predator‎. ‎Then by introducing a fractional model into the existing one‎, ‎we give them a specified memory‎. ‎We also obtain its discretized counterpart‎. ‎Finally‎, ‎along with giving the biological interpretation of the system‎, ‎the stability and dynamical analysis of the proposed model are investigated and the results are illustrated as well‎.

Further Results on Bifurcation for a Fractional-Order Predator-Prey System concerning Mixed Time Delays

In the present work, we mainly focus on a new established fractional-order predator-prey system concerning both types of time delays. Exploiting an advisable change of variable, we set up an isovalent fractional-order predator-prey model concerning a single delay. Taking advantage of the stability criterion and bifurcation theory of fractional-order dynamical system and regarding time delay as bifurcation parameter, we establish a new delay-independent stability and bifurcation criterion for the involved fractional-order predator-prey system. The numerical simulation figures and bifurcation plots successfully support the correctness of the established key conclusions.

An improved model along with a spectral numerical simulation for fractional predator–prey interactions

Predator–prey models appear in various fields of bio-mathematics used for the analysis of interactions of biological systems. Due to the complexities of the physical context for the real-world problems of food chain dynamics, introducing new models compatible with experimental results stays ongoing research. Many models have been proposed and analyzed for these systems in recent years. In this paper, we propose a new fractional-order predator–prey model with negative feedback on both species with memory-dependent effects, which increases the compatibility level of the model. Then we present a novel Laguerre spectral numerical simulation for the proposed model by introducing Laguerre modal basis functions with collocation and Galerkin techniques. We then transfer the nonlinear model into a system of algebraic equations, which is solved by efficient numerical solvers. Finally, we provide some test problems to show the efficiency of the proposed model and the computational method.

Further Study on Dynamics for a Fractional-Order Competitor-Competitor-Mutualist Lotka–Volterra System

On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world.

Dynamical analysis of a fractional-order predator–prey model incorporating a constant prey refuge and nonlinear incident rate

In this work, we have formulated and analyzed a fractional-order predator–prey model with an asymptotic incidence rate and constant prey refuge. Due to vulnerability, it is assumed that predator consumes only infected prey. Our mathematical formulation of the non-integer-order initial value problem has been developed on the famous fractional-order Caputo derivative. The main objective of this work is to investigate the influence of fractional-order derivatives on the system over the classical integer-order model. We have discussed the existence of non-negative solution, uniqueness and boundedness of our considered model. Analysis of local stability and Hopf bifurcation are performed both analytically and numerically. Sufficient conditions are established to guarantee the global stability of the interior equilibrium point by constructing a suitable Lyapunov function. Finally, some numerical simulations are provided to validate our results.

Stability of a fractional order harvested prey–predator model in the existence of infection in prey

The growing relationship between prey and their predator is one of the important aspects in the field of ecology and mathematical biology. On the other hand, the utility of fractional calculus in different types of mathematical modelling have been applied extensively. In this paper, a fractional order prey–predator model is developed with the consideration of Holling type-I and Holling type-II functional response of the predator. As infection spreads through prey, the prey population is divided into two parts. In addition, we exploit the effect of harvesting to control the excessive spread of the infection. The existence and uniqueness criteria, the boundedness of the solution of the proposed model are investigated. A number of five possible equilibrium points of the proposed model are determined along with the feasibility conditions for each equilibrium points. The local stability at these equilibrium points and global stability at interior equilibrium point are investigated. Numerical simulation is presented with the help of modified Predictor-corrector method in MATLAB software to understand the dynamics of the proposed model.

Fractional Dynamics Based-Enhancing Control Scheme of a Delayed Predator-Prey Model

To retard the onset of undesired bifurcation, the bifurcation control has developed into a theme of centralized research activities in delayed fractional-order system. In this paper, the problem of bifurcation control for a delayed fractional-order predator-prey model is investigated by employing an enhancing feedback control technique. The bifurcation point is firstly established for controlled model by using delay as a bifurcation parameter. Then, a series of numerical comparative studies on the effects of bifurcation control are implemented covering the partial or total removal of the branch for feedback gains. It reveals that the stability performance of the proposed model can be overwhelmingly elevated via the devised approaches in comparison with the dislocated feedback ones. A numerical example with simulations is ultimately designed to confirm the merits of the proposed theoretical results.

Impact of predator incited fear and prey refuge in a fractional order prey predator model

In this article, a predator-prey model has been evolved in the form of a system of fractional order differential equations incorporating two important factors, namely, fear factor and prey refuge factor. Here, the fractional calculus has been taken into consideration to investigate the dynamical behaviour of the solutions of the proposed model system as the changes in life cycle of prey species are of memory bound. Biological validation and well-posedness such as positivity and boundedness of solutions of the model system have been proved analytically. Stability analysis of all the feasible equilibrium points of the model system has been performed in a systematic way. Some important dynamical features of the model system (such as transition of stability of the system) have been demonstrated through rigorous numerical simulation. It is observed that our proposed model system experiences Hopf-bifurcation around the interior equilibrium point with respect to both the parameters f and m1, which are linked with amount of predator induced fear and rate of prey refuge, respectively. The system dynamics is more likely to be stable in the framework of fractional order derivative in comparison to integer-order derivative. The high amount of predator induced fear f and prey refuge rate m1 are independently capable to make the system dynamics to be stable in integer order model system. On the other hand, the dynamics of the model system shifts towards the stability from its unstable behaviour when we continuously reduce the order of the model system; especially under the scenario of low level of predator induced fear and prey refuge rate. Thus, our comprehensive mathematical findings reveal the fact that fading memory can play a contributory role towards stable coexistence of the predator-prey system whereas strong memory of the species deteriorates the stable coexistence of the model system.

Mathematical analysis of a fractional-order predator-prey model with prey social behavior and infection developed in predator population

Abstract This manuscript aims to consider a variety of fractional predator-prey models in the presence of an infection developed in the predator population. The crowding behavior plays an essential role in some species surviving, which is a useful strategy for defending the inside group prey. The purpose of considering the fractional-order-derivative is to study the memory effects on the mutual interactions, which has been confirmed to be an intrinsic feature of a dynamic biological system. From the perspective of mathematical results, the local behavior of the equilibrium points, and the existence of Hopf bifurcation are obtained. Besides, the influence of some crucial parameters as memory rate, herd shape, the infection rate in determining the asymptotic behavior of prey and predator are investigated. Further, an efficient numerical technique has been employed to illustrate some illustrative representations for numerical approximation for the solutions. This iterative scheme has been designed using the fundamental theorem of calculus in the fractional sense, and linear polynomial interpolation.