A dynamically consistent nonstandard finite difference scheme for a predator–prey model

Shabbir, Muhammad Sajjad; Din, Qamar; Safeer, Muhammad; Ahmad, Khalid

doi:10.1186/s13662-019-2319-6

Cited by 28 publications

(15 citation statements)

References 52 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, the formation of these type of difference schemes is not straightforward, and there are no usual ways for their construction, which is probably considered as major downside of nonstandard difference schemes. Hence by taking into account the original dynamical properties of model ( 1.3 ) a discrete-time model from ( 1.3 ) is obtained by using Mickens-type nonstandard scheme such that it remains dynamically consistent [ 39 ]. Implementing the Mickens-type nonstandard scheme on model ( 1.3 ), we get the following discrete-time mathematical model: where is taken as a step size for the nonstandard scheme.…”

Section: Introductionmentioning

confidence: 99%

Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

et al. 2021

View full text Add to dashboard Cite

The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton–zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark–Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark–Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.

show abstract

Section: Introductionmentioning

confidence: 99%

Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

et al. 2021

View full text Add to dashboard Cite

show abstract

“…Hence, it is justifiable to focus more on the new fractional-order chaotic models; additionally, the irregular behaviors of such models should be compensated by developing effective control and synchronization strategies. Therefore, it is worth to study the problem of modeling and synchronization related to the new fractional chaotic systems; however, the implementation and synchronization with regard to these models acquire an extension of the numerical methods available in the literature [22][23][24][25][26][27]. These argumentations motivate us to study a novel fractional chaotic system with quadratic and cubic nonlinearities involving the Caputo differential operator.…”

Section: Introductionmentioning

confidence: 99%

A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

et al. 2021

View full text Add to dashboard Cite

The aim of this paper is to introduce and analyze a novel fractional chaotic system including quadratic and cubic nonlinearities. We take into account the Caputo derivative for the fractional model and study the stability of the equilibrium points by the fractional Routh–Hurwitz criteria. We also utilize an efficient nonstandard finite difference (NSFD) scheme to implement the new model and investigate its chaotic behavior in both time-domain and phase-plane. According to the obtained results, we find that the new model portrays both chaotic and nonchaotic behaviors for different values of the fractional order, so that the lowest order in which the system remains chaotic is found via the numerical simulations. Afterward, a nonidentical synchronization is applied between the presented model and the fractional Volta equations using an active control technique. The numerical simulations of the master, the slave, and the error dynamics using the NSFD scheme are plotted showing that the synchronization is achieved properly, an outcome which confirms the effectiveness of the proposed active control strategy.

show abstract

“…Similarly, some discrete-time predator-prey models and hydra effect and paradox of enrichment are studied in [29]. Furthermore, for several attracting findings associated to the qualitative analysis of difference equations, we refer to the work done by [30], [31]. Also authors in [32] proposed and investigate the dynamics of cannibalism in discrete-time predator-prey system and considering twostage population model where cannibalism factor involving only in prey population.…”

Section: Introductionmentioning

confidence: 99%

Dynamical Complexity in a Class of Novel Discrete-Time Predator-Prey Interaction With Cannibalism

et al. 2020

Self Cite

View full text Add to dashboard Cite

Cannibalism is ubiquitous in natural communities and has the tendency to change the functional connection among prey-predator interactions. Keeping in view the inclusion of prey cannibalism, a novel discrete nonlinear predator-prey model is proposed. Asymptotic stability is carried out around biologically feasible equilibria of proposed model. Center manifold theorem and bifurcation theory of normal form ensure the existence of bifurcation in the system. Our study reveals that periodic outbreaks may result due to incorporation of cannibalism in prey population and this periodic outbreak is limited to prey population only without leaving an effect on predation. In order to control these periodic oscillations in prey population density and other bifurcating and fluctuating behavior of the system, various chaos control strategies are implemented. Ultimately, some extensive numerical simulations are elaborated to demonstrate the effectiveness of our acquired analytical and theoretical results.

show abstract

A dynamically consistent nonstandard finite difference scheme for a predator–prey model

Cited by 28 publications

References 52 publications

Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

Qualitative analysis of a discrete-time phytoplankton–zooplankton model with Holling type-II response and toxicity

A nonstandard finite difference scheme for the modeling and nonidentical synchronization of a novel fractional chaotic system

Dynamical Complexity in a Class of Novel Discrete-Time Predator-Prey Interaction With Cannibalism

Contact Info

Product

Resources

About