scholarly journals Extinction in a generalized Lotka-Volterra predator-prey model

2000 ◽  
Vol 13 (3) ◽  
pp. 287-297 ◽  
Author(s):  
Azmy S. Ackleh ◽  
David F. Marshall ◽  
Henry E. Heatherly
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Mortality Rates ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results

In this paper we discuss the asymptotic behavior of a predator-prey model with distributed growth and mortality rates. We exhibit simple criteria on the parameters which guarantee that all subpopulations but one predator-prey pair are driven to extinction as t→∞. Finally, we present numerical simulations to illustrate the theoretical results.

A Markov-switching predator–prey model with Allee effect for preys

2020 ◽  
Vol 13 (03) ◽  
pp. 2050018
Author(s):  
Xiaoxia Guo ◽  
Zhiming Guo
Keyword(s):  
Numerical Simulations ◽  
Allee Effect ◽  
Global Attractivity ◽  
Markov Switching ◽  
Sufficient Conditions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results ◽  
Prey Populations

This paper concerns with a Markov-switching predator–prey model with Allee effect for preys. The conditions under which extinction of predator and prey populations occur have been established. Sufficient conditions are also given for persistence and global attractivity in mean. In addition, stability in the distribution of the system under consideration is derived under some assumptions. Finally, numerical simulations are carried out to illustrate theoretical results.

scholarly journals The Influence of Additive Allee Effect and Periodic Harvesting to the Dynamics of Leslie-Gower Predator-Prey Model

2020 ◽  
Vol 2 (2) ◽  
pp. 87-96
Author(s):  
Hasan S. Panigoro ◽  
Emli Rahmi ◽  
Novianita Achmad ◽  
Sri Lestari Mahmud
Keyword(s):  
Numerical Simulations ◽  
Autonomous System ◽  
Allee Effect ◽  
Equilibrium Points ◽  
Dimensional System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Theoretical Results

In this paper, the influence of additive Allee effect in prey and periodic harvesting in predator to the dynamics of the Leslie-Gower predator-prey model is proposed. We first simplify the model to the non-dimensional system by scaling the variable and transform the model into an autonomous system. If the effect Allee is weak, we have at most two equilibrium points, else if the Allee effect is strong, at most four equilibrium points may exist. Furthermore, the behavior of the system around equilibrium points is investigated. In the end, we give numerical simulations to support theoretical results.

scholarly journals Stability and Bifurcation in a Predator–Prey Model with the Additive Allee Effect and the Fear Effect

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1280
Author(s):  
Liyun Lai ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Allee Effect ◽  
Sufficient Conditions ◽  
Positive Equilibrium ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Fear Effect ◽  
Prey Model ◽  
Strong Allee Effect ◽  
Theoretical Results

We proposed and analyzed a predator–prey model with both the additive Allee effect and the fear effect in the prey. Firstly, we studied the existence and local stability of equilibria. Some sufficient conditions on the global stability of the positive equilibrium were established by applying the Dulac theorem. Those results indicate that some bifurcations occur. We then confirmed the occurrence of saddle-node bifurcation, transcritical bifurcation, and Hopf bifurcation. Those theoretical results were demonstrated with numerical simulations. In the bifurcation analysis, we only considered the effect of the strong Allee effect. Finally, we found that the stronger the fear effect, the smaller the density of predator species. However, the fear effect has no influence on the final density of the prey.

scholarly journals Theoretical Studies on the Effects of Dispersal Corridors on the Permanence of Discrete Predator-Prey Models in Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-16
Author(s):  
Chunqing Wu ◽  
Shengming Fan ◽  
Patricia J. Y. Wong
Keyword(s):  
Sufficient Conditions ◽  
Theoretical Studies ◽  
Patchy Environment ◽  
Predator Prey ◽  
Numerical Examples ◽  
Predator Prey Model ◽  
Prey Model ◽  
Dispersal Corridors ◽  
Predator Prey Models ◽  
Theoretical Results

We study two discrete predator-prey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predator-prey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predator-prey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predator-prey model.

scholarly journals A new study for the global asymptotic stability of a general predator-prey model

2021 ◽  
Author(s):  
Manh Tuan Hoang
Keyword(s):  
Asymptotic Stability ◽  
Functional Response ◽  
Mathematical Analysis ◽  
Global Asymptotic Stability ◽  
Lyapunov Stability Theory ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Prey Model ◽  
Theoretical Results

In a previous paper [L. M. Ladino, E. I. Sabogal, Jose C. Valverde, General functional response and recruitment in a predator-prey system with capture on both species, Math. Methods Appl. Sci. 38(2015) 2876-2887], a mathematical model for a predator-prey model with general functional response and recruitment including capture on both species was formulated and analyzed. However, the global asymptotic stability (GAS) of this model was only partially resolved. In the present paper, we provide a rigorously mathematical analysis for the complete GAS of the predator-prey model. By using the Lyapunov stability theory in combination with some nonstandard techniques of mathematical analysis for dynamical systems, the GAS of equilibria of the model is determined fully. The obtained results not only provide an important improvement for the population dynamics of the predator-prey model but also can be extended to study its modified versions in the context of fractional-order derivatives. The theoretical results are supported and illustrated by a set of numerical examples.

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.

scholarly journals Dynamics of a Predator-Prey Model with Fear Effect and Time Delay

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Junli Liu ◽  
Pan Lv ◽  
Bairu Liu ◽  
Tailei Zhang
Keyword(s):  
Numerical Simulations ◽  
Dynamical Behavior ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Prey Model ◽  
Delay Dependent ◽  
The Impact ◽  
Dependent Factor

In this paper, we propose a time-delayed predator-prey model with Holling-type II functional response, which incorporates the gestation period and the cost of fear into prey reproduction. The dynamical behavior of this system is both analytically and numerically investigated from the viewpoint of stability, permanence, and bifurcation. We found that there are stability switches, and Hopf bifurcations occur when the delay τ passes through a sequence of critical values. The explicit formulae which determine the direction, stability, and other properties of the bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. We perform extensive numerical simulations to explore the impact of some important parameters on the dynamics of the system. Numerical simulations show that high levels of fear have a stabilizing effect while relatively low levels of fear have a destabilizing effect on the predator-prey interactions which lead to limit-cycle oscillations. We also found that the model with or without a delay-dependent factor can have a significantly different dynamics. Thus, ignoring the delay or not including the delay-dependent factor might result in inaccurate modelling predictions.

scholarly journals Paradox of Enrichment in a Stochastic Predator-Prey Model

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jawdat Alebraheem
Keyword(s):  
Initial Conditions ◽  
Random Noise ◽  
Unique Positive Solution ◽  
Predator Prey ◽  
Paradox Of Enrichment ◽  
Predator Prey Model ◽  
Stochastic Boundedness ◽  
Prey Model ◽  
Biological Phenomena ◽  
Theoretical Results

We propose a stochastic predator-prey model to study a novel idea that involves investigating random noises effects on the enrichment paradox phenomenon. Existence and stochastic boundedness of a unique positive solution with positive initial conditions are proved. The global asymptotic stability is studied to determine the occurrence of the enrichment paradox phenomenon. We show theoretically that intensive noises play an important role in the occurrence of the phenomenon, where increasing intensive noises lead to occurrence of the paradox of enrichment. We perform numerical simulations to verify and demonstrate the theoretical results. The new results in this study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment.

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