scholarly journals Stoichiometric knife-edge model on discrete time scale

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Ming Chen ◽  
Lale Asik ◽  
Angela Peace
Keyword(s):  
Discrete Time ◽  
Time Scale ◽  
Ecological Stoichiometry ◽  
Discrete Model ◽  
Nutrient Content ◽  
Continuous Model ◽  
Knife Edge ◽  
Time Model ◽  
Predator Prey ◽  
Predator Prey Model

AbstractEcological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.

scholarly journals Discrete-Time Predator-Prey Interaction with Selective Harvesting and Predator Self-Limitation

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Zhihua Chen ◽  
Qamar Din ◽  
Muhammad Rafaqat ◽  
Umer Saeed ◽  
Muhammad Bilal Ajaz
Keyword(s):  
Discrete Time ◽  
Chaotic Behavior ◽  
Period Doubling ◽  
State Feedback Control ◽  
Time Model ◽  
Predator Prey ◽  
Selective Harvesting ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Interaction

Selective harvesting plays an important role on the dynamics of predator-prey interaction. On the other hand, the effect of predator self-limitation contributes remarkably to the stabilization of exploitative interactions. Keeping in view the selective harvesting and predator self-limitation, a discrete-time predator-prey model is discussed. Existence of fixed points and their local dynamics is explored for the proposed discrete-time model. Explicit principles of Neimark–Sacker bifurcation and period-doubling bifurcation are used for discussion related to bifurcation analysis in the discrete-time predator-prey system. The control of chaotic behavior is discussed with the help of methods related to state feedback control and parameter perturbation. At the end, some numerical examples are presented for verification and illustration of theoretical findings.

scholarly journals Simulation of Discrete Predator-Prey Model

2020 ◽  
Vol 55 (1) ◽  
Author(s):  
Adel A. Abed Al Wahab ◽  
Nihad Mahmoud Nasir ◽  
Adil I. Khalil
Keyword(s):  
Discrete Model ◽  
Initial Conditions ◽  
Current Model ◽  
Continuous Model ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Computerized Simulation ◽  
Theoretical Results ◽  
Over Time

It is well known that dynamical systems deal with situations in which the system transforms over time. In fact, undertaking a manual simulation of such systems is a difficult task due to the complexity of the computations. Therefore, a computerized simulation is frequently required for accurate results and fast execution time. Nowadays, computer programs have become an important tool to confirm the theoretical results obtained from the study of models. This paper aims to employ new MATLAB codes to examine a discrete predator–prey model using a difference equations system. The paper discusses the existences and stabilities of each possible fixed point appearing in the current model. Furthermore, numerical simulations fixed by a certain parameter to plot the diagrams are presented. Our results confirm that the systems sensitive to initial conditions are chaotic. Furthermore, the theoretical results as well as numerical examples illustrated that the discrete model exhibits complex behavior compared to a continuous model. The conclusion drawn is that the numerical simulation is an important tool to confirm theoretical results.

Qualitative properties and bifurcations of discrete-time Bazykin–Berezovskaya predator–prey model

2020 ◽  
Vol 13 (06) ◽  
pp. 2050040
Author(s):  
A. A. Elsadany ◽  
Qamar Din ◽  
S. M. Salman
Keyword(s):  
Discrete Time ◽  
Allee Effect ◽  
Initial Conditions ◽  
Qualitative Properties ◽  
Step Size ◽  
Time Model ◽  
Predator Prey ◽  
Discrete Time Model ◽  
Predator Prey Model ◽  
Prey Model

The positive connection between the total individual fitness and population density is called the demographic Allee effect. A demographic Allee effect with a critical population size or density is strong Allee effect. In this paper, discrete counterpart of Bazykin–Berezovskaya predator–prey model is introduced with strong Allee effects. The steady states of the model, the existence and local stability are examined. Moreover, proposed discrete-time Bazykin–Berezovskaya predator–prey is obtained via implementation of piecewise constant method for differential equations. This model is compared with its continuous counterpart by applying higher-order implicit Runge–Kutta method (IRK) with very small step size. The comparison yields that discrete-time model has sensitive dependence on initial conditions. By implementing center manifold theorem and bifurcation theory, we derive the conditions under which the discrete-time model exhibits flip and Niemark–Sacker bifurcations. Moreover, numerical simulations are provided to validate the theoretical results.

PERMANENCE FOR A DELAYED DISCRETE PREDATOR–PREY MODEL WITH PREY DISPERSAL

2009 ◽  
Vol 02 (03) ◽  
pp. 311-320 ◽  
Author(s):  
CHUNQING WU ◽  
JING-AN CUI
Keyword(s):  
Discrete Time ◽  
Single Species ◽  
Time Model ◽  
Predator Prey ◽  
Discrete Time Model ◽  
Predator Prey Model ◽  
Prey Model

First, we obtain a new result for the permanence of a well known delayed discrete-time model of single species. Second, based on this new condition, we discuss the permanence of a delayed discrete-time predator–prey model in which the prey disperses in two patches with biased dispersion. The biological implications of the results are briefly discussed.

Model-Based Discrete Linear State Estimator for Nonlinearizable Systems With State-Dependent Noise

1990 ◽  
Vol 112 (4) ◽  
pp. 774-781 ◽  
Author(s):  
R. J. Chang
Keyword(s):  
Discrete Time ◽  
Continuous Model ◽  
Equivalent Model ◽  
State Estimator ◽  
Time Model ◽  
State Dependent ◽  
Present Technique ◽  
Linear State ◽  
Noisy Measurement ◽  
Time Linear

A practical technique to derive a discrete-time linear state estimator for estimating the states of a nonlinearizable stochastic system involving both state-dependent and external noises through a linear noisy measurement system is presented. The present technique for synthesizing a discrete-time linear state estimator is first to construct an equivalent reference linear model for the nonlinearizable system such that the equivalent model will provide the same stationary covariance response as that of the nonlinear system. From the linear continuous model, a discrete-time state estimator can be directly derived from the corresponding discrete-time model. The synthesizing technique and filtering performance are illustrated and simulated by selecting linear, linearizable, and nonlinearizable systems with state-dependent noise.

Bifurcations and chaos control in a discrete-time biological model

2020 ◽  
Vol 13 (04) ◽  
pp. 2050022 ◽  
Author(s):  
A. Q. Khan ◽  
T. Khalique
Keyword(s):  
Discrete Time ◽  
Chaos Control ◽  
Control Method ◽  
Complex Dynamics ◽  
Small Neighborhood ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Flip Bifurcation ◽  
Time Model ◽  
Predator Prey Model

In this paper, bifurcations and chaos control in a discrete-time Lotka–Volterra predator–prey model have been studied in quadrant-[Formula: see text]. It is shown that for all parametric values, model has boundary equilibria: [Formula: see text], and the unique positive equilibrium point: [Formula: see text] if [Formula: see text]. By Linearization method, we explored the local dynamics along with different topological classifications about equilibria. We also explored the boundedness of positive solution, global dynamics, and existence of prime-period and periodic points of the model. It is explored that flip bifurcation occurs about boundary equilibria: [Formula: see text], and also there exists a flip bifurcation when parameters of the discrete-time model vary in a small neighborhood of [Formula: see text]. Further, it is also explored that about [Formula: see text] the model undergoes a N–S bifurcation, and meanwhile a stable close invariant curves appears. From the perspective of biology, these curves imply that between predator and prey populations, there exist periodic or quasi-periodic oscillations. Some simulations are presented to illustrate not only main results but also reveals the complex dynamics such as the orbits of period-2,3,13,15,17 and 23. The Maximum Lyapunov exponents as well as fractal dimension are computed numerically to justify the chaotic behaviors in the model. Finally, feedback control method is applied to stabilize chaos existing in the model.

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