The effect of interference time in a predator–prey–nonprey system

2021 ◽  
Author(s):  
Debasis Mukherjee
Keyword(s):  
Food Choice ◽  
Numerical Experiments ◽  
Equilibrium Density ◽  
Type Ii ◽  
Competitive Response ◽  
Predator Species ◽  
Predator Prey ◽  
Competition Process ◽  
The Stability ◽  
Coexistence Equilibrium

In this paper, we propose a three-species model consisting of two competing (prey and nonprey) species and a predator species. Here, nonprey species are not included in the predator’s food choice. The competition process follows Holling type II competitive response to interference time. Basic results include the stability of the system. First, it is established that an increasing number of interference time stabilizes the system. Second, it is shown that the interference time has an impact on the predator equilibrium density. Third, we develop the criterion of persistence of all the species. It is also shown that the system may not be persistent when multiple steady states appear. We examine the global stability of the coexistence equilibrium point. Numerical experiments are carried out to understand the analytical outcomes.

scholarly journals Dynamic Behaviors of a Nonautonomous Discrete Predator-Prey System Incorporating a Prey Refuge and Holling Type II Functional Response

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Yumin Wu ◽  
Fengde Chen ◽  
Wanlin Chen ◽  
Yuhua Lin
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Functional Response ◽  
Sufficient Conditions ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Prey System ◽  
Type Ii Functional Response

A nonautonomous discrete predator-prey system incorporating a prey refuge and Holling type II functional response is studied in this paper. A set of sufficient conditions which guarantee the persistence and global stability of the system are obtained, respectively. Our results show that if refuge is large enough then predator species will be driven to extinction due to the lack of enough food. Two examples together with their numerical simulations show the feasibility of the main results.

scholarly journals Analysis of optimal harvesting of a prey-predator fishery model with the limited sources of prey and presence of toxicity

2018 ◽  
Vol 73 ◽  
pp. 06018
Author(s):  
Sutimin ◽  
Khabibah Siti ◽  
Anies Munawwaroh Dita
Keyword(s):  
Prey Density ◽  
Equilibrium Points ◽  
Optimal Harvesting ◽  
Predator Species ◽  
Existence Of Equilibrium ◽  
Sustainable Harvesting ◽  
Toxicity Factor ◽  
The Stability ◽  
Fishery Model ◽  
Coexistence Equilibrium

A model of prey and predator species is discussed to study the effects of the limited prey density and presence of toxicity. The model is studied for sustainable optimal harvesting. The existence of equilibrium points is analyzed to find the stability of coexistence equilibrium, and use Pontryagin’s maximal method to obtain the sustainable optimal harvesting. The results show that the optimal harvesting is obtained from the solution of optimal equilibrium. The toxicity factor decreases the sustainable harvesting.

STABILITY AND HOPF BIFURCATION OF A PREDATOR–PREY SYSTEM WITH TIME DELAY AND HOLLING TYPE-II FUNCTIONAL RESPONSE

2009 ◽  
Vol 02 (02) ◽  
pp. 139-149 ◽  
Author(s):  
LINGSHU WANG ◽  
RUI XU ◽  
GUANGHUI FENG
Keyword(s):  
Time Delay ◽  
Functional Response ◽  
Type Ii ◽  
Normal Form Theory ◽  
Predator Prey ◽  
Center Manifold Theorem ◽  
Predator Prey Model ◽  
Form Theory ◽  
The Stability ◽  
Type Ii Functional Response

A predator–prey model with time delay and Holling type-II functional response is investigated. By choosing time delay as the bifurcation parameter and analyzing the associated characteristic equation of the linearized system, the local stability of the system is investigated and Hopf bifurcations are established. The formulae determining the direction of bifurcations and the stability of bifurcating periodic solutions are given by using the normal form theory and center manifold theorem. Numerical simulations are carried out to illustrate the theoretical results.

Prey–Predator Three Species Model Using Predator Harvesting Holling Type II Functional

2016 ◽  
Vol 11 (02) ◽  
pp. 87-104 ◽  
Author(s):  
S. Vijaya ◽  
E. Rekha
Keyword(s):  
Maximum Principle ◽  
Equilibrium Point ◽  
Prey Species ◽  
Pontryagin’S Maximum Principle ◽  
Type Ii ◽  
Pontryagin's Maximum Principle ◽  
Predator Species ◽  
Numerical Examples ◽  
The Stability

This paper presents three species harvesting model in which there is one predator species and two others are prey species. We derive boundedness and equilibrium point for this system. Also we derive the stability of this system analytically. We find bifurcation for this system. We have derived the binomic equilibrium point by using Pontryagin’s maximum principle (PMP). Presented are various suitable analytical and numerical examples with Maple 18 programming.

scholarly journals Global dynamics of a predator-prey system with Holling type II functional response

2011 ◽  
Vol 16 (2) ◽  
pp. 242-253 ◽  
Author(s):  
Xiaohong Tian ◽  
Rui Xu
Keyword(s):  
Functional Response ◽  
Sufficient Conditions ◽  
Stage Structure ◽  
Global Dynamics ◽  
Type Ii ◽  
Asymptotically Stable ◽  
Predator Prey ◽  
Prey System ◽  
Coexistence Equilibrium ◽  
Type Ii Functional Response

In this paper, a predator-prey system with Holling type II functional response and stage structure is investigated. By analyzing the corresponding characteristic equations, the local stability of each of feasible equilibria of the system is studied. The existence of the orbitally asymptotically stable periodic solution is established. By using suitable Lyapunov functions and the LaSalle invariance principle, it is proven that the predator-extinction equilibrium is globally asymptotically stable when the coexistence equilibrium is not feasible, and sufficient conditions are derived for the global stability of the coexistence equilibrium.

THE SPATIAL PATTERNS THROUGH DIFFUSION-DRIVEN INSTABILITY IN MODIFIED LESLIE-GOWER AND HOLLING-TYPE II PREDATOR-PREY MODEL

2010 ◽  
Vol 18 (03) ◽  
pp. 593-603 ◽  
Author(s):  
G. SUN ◽  
S. SARWARDI ◽  
P. J. PAL ◽  
Md. S. RAHMAN
Keyword(s):  
Spatial Patterns ◽  
Numerical Experiments ◽  
Turing Instability ◽  
Type Ii ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Through Diffusion ◽  
Predator System

Formation of spatial patterns in prey-predator system is a central issue in ecology. In this paper Turing structure through diffusion driven instability in a modified Leslie-Gower and Holling-type II predator-prey model has been investigated. The parametric space for which Turing spatial structure takes place has been found out. Extensive numerical experiments have been performed to show the role of diffusion coefficients and other important parameters of the system in Turing instability that produces some elegant patterns that have not been observed in the earlier findings. Finally it is concluded that the diffusion can lead the prey population to become isolated in the two-dimensional spatial domain.

Dynamics of a Predator–Prey Model with Holling Type II Functional Response Incorporating a Prey Refuge Depending on Both the Species

2019 ◽  
Vol 20 (1) ◽  
pp. 89-104 ◽  
Author(s):  
Hafizul Molla ◽  
Md. Sabiar Rahman ◽  
Sahabuddin Sarwardi
Keyword(s):  
Hopf Bifurcation ◽  
Uniform Boundedness ◽  
Type Ii ◽  
Prey Refuge ◽  
Predator Species ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Predator Model ◽  
The Stability ◽  
Manifold Theory

AbstractWe propose a mathematical model for prey–predator interactions allowing prey refuge. A prey–predator model is considered in the present investigation with the inclusion of Holling type-II response function incorporating a prey refuge depending on both prey and predator species. We have analyzed the system for different interesting dynamical behaviors, such as, persistent, permanent, uniform boundedness, existence, feasibility of equilibria and their stability. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The system exhibits Hopf-bifurcation around the unique interior equilibrium point of the system. The explicit formula for determining the stability, direction and periodicity of bifurcating periodic solutions are also derived with the use of both the normal form and the center manifold theory. The theoretical findings of this study are substantially validated by enough numerical simulations. The ecological implications of the obtained results are discussed as well.

GLOBAL STABILITY AND HOPF BIFURCATION IN A DELAYED DIFFUSIVE LESLIE–GOWER PREDATOR–PREY SYSTEM

2012 ◽  
Vol 22 (03) ◽  
pp. 1250061 ◽  
Author(s):  
SHANSHAN CHEN ◽  
JUNPING SHI ◽  
JUNJIE WEI
Keyword(s):  
Hopf Bifurcation ◽  
Upper And Lower Solutions ◽  
Neumann Boundary ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Predator Prey ◽  
Prey System ◽  
The Stability ◽  
Coexistence Equilibrium

In this paper, we consider a delayed diffusive Leslie–Gower predator–prey system with homogeneous Neumann boundary conditions. The stability/instability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations. Furthermore, using the upper and lower solutions method, we give a sufficient condition on parameters so that the coexistence equilibrium is globally asymptotically stable.

Nonlinear Dynamics of a Prey–Predator Model Using Precise and Imprecise Harvesting Phenomena with Biological Parameters

2017 ◽  
Vol 12 (02) ◽  
pp. 39-68 ◽  
Author(s):  
S. Vijaya ◽  
E. Rekha ◽  
J. Jayamal Singh
Keyword(s):  
Nonlinear Dynamics ◽  
Stability Analysis ◽  
Numerical Experiments ◽  
Optimal Harvesting ◽  
Biological Parameters ◽  
Predator Species ◽  
Biological Phenomena ◽  
Predator Model ◽  
The Stability ◽  
Parameter Values

This paper presents the nonlinear dynamics of a one-prey and one-predator harvesting model with precise in nature as well as imprecise in biological phenomena parameters. We derived the conditions for boundedness, the equilibrium point, and stability analysis. Both precise and imprecise models showed stable, unstable, and saddle-point states. The stability analysis revealed the existence of biological and bionomic equilibria. In this study, we found the optimal harvesting policy for both prey and predator species. Finally, numerical experiments were performed with various parameter values to observe the variation of equilibrium states.

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