Stability in predator-prey system Lotka-Volterra model incorporating a prey refuge like the law of mass action

2018 ◽  
Vol 11 (42) ◽  
pp. 2049-2057 ◽  
Author(s):  
Edilber Almanza-Vasquez ◽  
Ruben-Dario Ortiz-Ortiz ◽  
Ana-Magnolia Marin-Ramirez
Keyword(s):  
Mass Action ◽  
Volterra Model ◽  
Law Of Mass Action ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
The Law

scholarly journals Complex Dynamical Behavior of a Predator-Prey System with Group Defense

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jianglin Zhao ◽  
Min Zhao ◽  
Hengguo Yu
Keyword(s):  
Numerical Simulation ◽  
Theoretical Basis ◽  
Dynamical Behavior ◽  
Turing Instability ◽  
Equilibrium Density ◽  
Prey Refuge ◽  
Predator Prey ◽  
Prey System ◽  
Group Defense ◽  
Complex Dynamical Behavior

A diffusive predator-prey system with prey refuge is studied analytically and numerically. The Turing bifurcation is analyzed in detail, which in turn provides a theoretical basis for the numerical simulation. The influence of prey refuge and group defense on the equilibrium density and patterns of species under the condition of Turing instability is explored by numerical simulations, and this shows that the prey refuge and group defense have an important effect on the equilibrium density and patterns of species. Moreover, it can be obtained that the distributions of species are more sensitive to group defense than prey refuge. These results are expected to be of significance in exploration for the spatiotemporal dynamics of ecosystems.

scholarly journals Dynamic behaviors of a nonautonomous predator–prey system with Holling type II schemes and a prey refuge

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Caifeng Du
Keyword(s):  
Lyapunov Function ◽  
Differential Equations ◽  
Comparison Theorem ◽  
Sufficient Conditions ◽  
Oscillation Theory ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System

AbstractIn this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.

The Law of Mass Action

2001 ◽  
pp. 121-128
Author(s):  
Bruce Hannon ◽  
Matthias Ruth
Keyword(s):  
Mass Action ◽  
Law Of Mass Action ◽  
The Law

scholarly journals On the Mathematics of the Law of Mass Action

2014 ◽  
pp. 3-46 ◽  
Author(s):  
Leonard Adleman ◽  
Manoj Gopalkrishnan ◽  
Ming-Deh Huang ◽  
Pablo Moisset ◽  
Dustin Reishus
Keyword(s):  
Mass Action ◽  
Law Of Mass Action ◽  
The Law

The Law of Mass Action

1994 ◽  
pp. 73-79
Author(s):  
Bruce Hannon ◽  
Matthias Ruth
Keyword(s):  
Mass Action ◽  
Law Of Mass Action ◽  
The Law

On the mechanistic foundation and limit of Liebig’s law of the minimum

2021 ◽  
Author(s):  
Jinyun Tang ◽  
William Riley
Keyword(s):  
Additive Model ◽  
Additive Models ◽  
Mass Action ◽  
Substrate Uptake ◽  
Law Of Mass Action ◽  
Growth Data ◽  
Biological Growth ◽  
Elemental Stoichiometry ◽  
The Law ◽  
Parameter Values

<p>In ecosystem biogeochemistry, Liebig’s law of the minimum (LLM) is one of the most widely used rules to model and interpret biological growth. Although it is intuitively accepted as being true, its mechanistic foundation has never been clearly presented. We here first show that LLM can be derived from the law of mass action, the state of art theory for modeling biogeochemical reactions. We further show that there are (at least) another two approximations (the synthesizing unit (SU) model and additive model) that are more accurate than LLM in approximating the law of mass action. We then evaluated the LLM, SU, and additive models against growth data of algae and plants. For algae growth, we found all three models are equally accurate, albeit with different parameter values. For plants, LLM failed to accurately model one dataset, and achieved equally good results for other datasets with very different parameters. We also find that LLM does not allow flexible elemental stoichiometry, which is an oft-observed characteristic of plants, when an organism’s growth is modeled as a function of substrate uptake flux. In summary, we caution the use of LLM for modeling biological growth if one is interested in representing the organisms’ capability in adapting to different nutrient conditions.   </p> <p><br /><br /></p>

scholarly journals Hopf bifurcation and stability in a Beddington-DeAngelis predator-prey model with stage structure for predator and time delay incorporating prey refuge

2019 ◽  
Vol 17 (1) ◽  
pp. 141-159 ◽  
Author(s):  
Zaowang Xiao ◽  
Zhong Li ◽  
Zhenliang Zhu ◽  
Fengde Chen
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey System ◽  
Bifurcation And Stability

Abstract In this paper, we consider a Beddington-DeAngelis predator-prey system with stage structure for predator and time delay incorporating prey refuge. By analyzing the characteristic equations, we study the local stability of the equilibrium of the system. Using the delay as a bifurcation parameter, the model undergoes a Hopf bifurcation at the coexistence equilibrium when the delay crosses some critical values. After that, by constructing a suitable Lyapunov functional, sufficient conditions are derived for the global stability of the system. Finally, the influence of prey refuge on densities of prey species and predator species is discussed.

scholarly journals The Hopf Bifurcation for a Predator-Prey System with -Logistic Growth and Prey Refuge

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Shaoli Wang ◽  
Zhihao Ge
Keyword(s):  
Hopf Bifurcation ◽  
Logistic Growth ◽  
Positive Equilibrium ◽  
Prey Refuge ◽  
Normal Form Theory ◽  
Center Manifold Reduction ◽  
Predator Prey ◽  
Prey System ◽  
Form Theory ◽  
The Stability

The Hopf bifurcation for a predator-prey system with -logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index- passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

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