Dynamical Behaviors of a Discrete-Time Prey-Predator Model With Harvesting Effect on the Predator

Eskandari, Zohreh; ,; Naik, Parvaiz Ahmad; Yavuz, Mehmet; ,; ,

doi:10.11948/20230212

Cited by 8 publications

(2 citation statements)

References 41 publications

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“…It may also be noted that the method we propose in this article is not limited to only first-order IVPs and can be used for finding approximate solutions to differential systems expressed in higher-order ODEs. It is also worth noting that the importance of differential equations cannot be denied as such equations appear almost in every field of science and engineering, for example see [3,4,14,15,21,24,25,29,32,37,44,48,52,54] and many of the references cited therein.…”

Section: Introductionmentioning

confidence: 99%

Optimizing A-stable hyperbolic fitting for time efficiency: exploring constant and variable stepsize approaches

Baleanu,

Qureshi,

Soomro

et al. 2024

J. Math. Computer Sci.

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This paper proposes an optimal time-efficient numerical method for solving the initial value problems (IVPs) of ordinary differential equations (ODEs) that is both A-stable and hyperbolically fitted. The method is designed to handle both constant and variable step sizes, making it highly adaptable to different types of ODEs. The methodology proposed herein leverages the optimization of an off-grid point, derived from the predominant term of the local truncation error, to enhance both accuracy and stability in the solution of stiff ODEs. This approach incorporates a variable step size control, predicated upon the error estimation furnished by the embedded pair, and aims to minimize computational expenses while concurrently safeguarding both precision and stability. Furthermore, the stability domain of the proposed method is demonstrated to be optimal, signifying it encompasses the maximal conceivable set of step sizes wherein the method retains its stability. Other important measures including zero-stability, consistency, and convergence are also discussed theoretically and confirmed experimentally. Numerical experiments consisting of the Duffing system, sinusoidal stiff system, periodic orbit system, two-body system, Lorenz system, and the system for catenary equation demonstrate that the proposed method is highly competitive in terms of accuracy and efficiency, and outperforms several existing methods for solving stiff ODEs with both constant and variable step sizes.

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Section: Introductionmentioning

confidence: 99%

Optimizing A-stable hyperbolic fitting for time efficiency: exploring constant and variable stepsize approaches

Baleanu,

Qureshi,

Soomro

et al. 2024

J. Math. Computer Sci.

View full text Add to dashboard Cite

show abstract

“…Using the Euler scheme, Singh and Sharma [41] proposed a discrete-time predator-prey model and managed to control chaos in the system. Recently, Eskandari et al [10,9] investigated single and multispecies discrete-time models to explore complex dynamics. Jiang et al [16] formulated a discrete-time model from a delayed continuous model in order to describe the dynamics between two enterprises whose cooperation or competition effects involve some time lag.…”

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confidence: 99%

Stock patterns in a class of delayed discrete-time population models

Rajni,

Sahu,

Sarda

et al. 2024

DCDS-S

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This paper highlights the behavior presented by a new class of discrete-time predator-prey models which have a time delay in the predation process. These models are formulated by discretizing continuous-time delayed Rosenzweig-MacArthur predator-prey systems. The stability analysis is performed through Routh-Hurwitz (RH) criteria, and bifurcation diagrams help to grasp the change in the dynamics of interacting species. First, we examine the impact of changing carrying capacity on (i) the stability of the equilibrium and (ii) the mean population level in delayed systems. When the delay is very large, the stability thresholds of the carrying capacity of different delayed models tend to converge. We compute the mean population size as a function of carrying capacity. Mean prey (resp. predator) stock increases (resp. decreases) with increasing carrying capacity of prey resulting in the paradox of enrichment in our discrete-time systems. For higher delayed models, the value of the mean prey (resp. predator) stock is higher (lower). Second, harvesting has a potential to stabilize an unstable mode of the coexisting equilibrium. When prey is harvested, the mean population of prey (resp. predator) decreases (increases). Therefore, no hydra effect is experienced by prey species. On the other hand, predator harvesting potentially increases its own mean stock, leading to hydra effects in predators. Most importantly, we have computed the mean stock with respect to harvesting. For both prey and predator harvesting, the mean prey size increases with an increase in delay, whereas the mean predator size decreases with an increase in delay. For the system with larger delay, the hydra effects on predators become more prominent in the sense that the rate of increase of mean predator stock is higher.

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Arnold tongues, shrimp structures, multistability, and ecological paradoxes in a discrete-time predator–prey system

Rajni,

Ghosh

2024

Chaos: An Interdisciplinary Journal of Nonlinear Science

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This paper explores a discrete-time system derived from the well-known continuous-time Rosenzweig–MacArthur model using the piecewise constant argument. Examining the impact of increasing carrying capacity and harvesting efforts, we uncover intricate phenomena, such as periodicity, quasiperiodicity, period-doubling, period-bubbling, and chaos. Our analysis reveals that increasing the carrying capacity of prey species can lead to both system stabilization and destabilization. We delve into normal forms associated with different bifurcation types, accompanied by numerical examples, observing multistabilities with intricate basin structures. Bistable, tristable, and quadruple attractors characterize the model’s multistable states. Additionally, we find that enriching prey species negatively affects predator abundance, and increasing carrying capacity can lead to a sudden jump in predator population to the brink of extinction. Examining the two-parameter space of predator and prey harvesting efforts, we identify organized periodic structures: Arnold tongues and shrimp-like structures within quasiperiodic and chaotic regions. Arnold tongues exhibit a sequence of periodic adding. The shrimp structures indicate the existence of period-doubling and period-bubbling phenomena. Discussions on ecological interpretations of predator harvesting, including the paradoxical hydra effect, are provided.

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Dynamical Behaviors of a Discrete-Time Prey-Predator Model With Harvesting Effect on the Predator

Cited by 8 publications

References 41 publications

Optimizing A-stable hyperbolic fitting for time efficiency: exploring constant and variable stepsize approaches

Optimizing A-stable hyperbolic fitting for time efficiency: exploring constant and variable stepsize approaches

Stock patterns in a class of delayed discrete-time population models

Arnold tongues, shrimp structures, multistability, and ecological paradoxes in a discrete-time predator–prey system

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