scholarly journals Stability property of the prey free equilibrium point

2019 ◽  
Vol 17 (1) ◽  
pp. 646-652
Author(s):  
Qin Yue
Keyword(s):  
Differential Inequality ◽  
Prey Species ◽  
Food Resource ◽  
Stage Structure ◽  
Asymptotically Stable ◽  
Suitable Assumption ◽  
Globally Asymptotically Stable ◽  
Predator Species ◽  
Inequality Theory ◽  
Predator Model

Abstract We revisit a prey-predator model with stage structure for predator, which was proposed by Tapan Kumar Kar. By using the differential inequality theory and the comparison theorem of the differential equation, we show that the prey free equilibrium is globally asymptotically stable under some suitable assumption. Our study shows that although the predator species has other food resource, if the amount of the predator species is too large, it could also do irreversible harm to the prey species, and this could finally lead to the extinction of the prey species. Our result supplement and complement some known results.

scholarly journals An Impulsively Controlled Three-Species Prey-Predator Model with Stage Structure and Birth Pulse for Predator

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Yanyan Hu ◽  
Mei Yan ◽  
Zhongyi Xiang
Keyword(s):  
Small Amplitude ◽  
Floquet Theory ◽  
Stage Structure ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Birth Pulse ◽  
Impulsive Equation ◽  
Predator Model ◽  
Predator System

We investigate the dynamic behaviors of a two-prey one-predator system with stage structure and birth pulse for predator. By using the Floquet theory of linear periodic impulsive equation and small amplitude perturbation method, we show that there exists a globally asymptotically stable two-prey eradication periodic solution when the impulsive period is less than some critical value. Further, we study the permanence of the investigated model. Our results provide valuable strategy for biological economics management. Numerical analysis is also inserted to illustrate the results.

A PREY-PREDATOR MODEL WITH DIFFUSION AND A SUPPLEMENTARY RESOURCE FOR THE PREY IN A TWO‐PATCH ENVIRONMENT

2005 ◽  
Vol 9 (1) ◽  
pp. 9-24 ◽  
Author(s):  
J. Dhar
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Prey Species ◽  
Steady State Solution ◽  
State Solution ◽  
Prey Population ◽  
Predator Species ◽  
Additional Resource ◽  
Linear And Nonlinear ◽  
Predator Model

In this paper, a prey‐predator dynamics, where the predator species partially depends upon the prey species, in a two patch habitat with diffusion and there is a non‐diffusing additional resource for the prey population, is modeled and analyzed. It is shown, that there exists a positive, monotonic, continuous steady state solution with continuous matching at the interface for both the species separately. Further, we obtain conditions for asymptotic stability for both linear and nonlinear cases. Šiame straipsnyje modeliuojama ir analizuojama plešr‐unu ir auku dinamika, laikant, kad plešr-unu populiacija dalinai priklauso nuo auku skačiaus. Areala sudaro dvi sritys, kuriose vyksta populiaciju individu difuzija, be to, aukoms yra išskirtas nedifunduojantis resursas. Irodyta, kad egzistuoja teigiamas, monotoniškas, tolydus stacionarusis sprendinys, tenkinantis tolydumo salyga abiems populiacijoms atskirai. Gautos asimptotinio stabilumo salygos tiesiniu ir netiesiniu atvejais.

Food Resource Division Among Northern, Marine, Demersal Fishes

1972 ◽  
Vol 29 (7) ◽  
pp. 997-1003 ◽  
Author(s):  
A. V. Tyler
Keyword(s):  
Prey Species ◽  
Food Resource ◽  
Adaptive Significance ◽  
Irish Sea ◽  
Significant Heterogeneity ◽  
Predator Species ◽  
Passamaquoddy Bay ◽  
Marine Areas ◽  
Resource Division ◽  
Demersal Fishes

The division of food resources among 13 demersal fishes was examined over a 16-month period in an area of Passamaquoddy Bay, New Brunswick. Though over 100 prey species were found in the stomachs of the predators, each predator took only three or four principal prey, and these prey made up 70–99% of the mass of the food for each predator species. Which species that a predator took as principal prey depended on prey body size, whether the prey were nekton, epi-fauna, or in-fauna; and whether or not they had a hard test or shell. Within predator species there was significant heterogeneity in diet related to size of predator individual.The seasonal predators did not feed as a group on their own set of prey species. Most principal prey species of the seasonal predators were taken simultaneously by one or two of the regulars.Data were compared with published results from two other northern marine areas. Within Irish Sea, Sea of Okhotsk, and also Passamaquoddy winter and summer communities, only 10–24% of the possible recurrences of principal prey among predators actually occurred, i.e. there was relatively little overlap among diets. Such specialization would have adaptive significance in a food-limited production system.

scholarly journals Three-Prey One-Predator Continuous Time Nonlinear System Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Jingyuan Zhang ◽  
Yan Yang
Keyword(s):  
Nonlinear System ◽  
Continuous Time ◽  
Order Differential Equation ◽  
Stable Equilibrium ◽  
Equilibrium Points ◽  
System Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Simulation Results ◽  
Predator Model

In this paper, we propose a new multiple-prey one-predator continuous time nonlinear system model, in which the number of teams of preys is equal to 3; namely, a continuous time three-prey one-predator model is put forward and studied. The fourth-order differential equation is established, in which the prey teams help each other. The equilibrium points and stability are analyzed. When not considering preys help each other, we study the global stability and persistence of the model without help terms. The simulation results of system solutions with help terms corresponding to locally asymptotically stable equilibrium points and without help terms corresponding to globally asymptotically stable equilibrium points are given.

scholarly journals The Dynamics of Biological Models with Optimal Harvesting

2021 ◽  
pp. 3039-3051
Author(s):  
Sadiq Al-Nassir
Keyword(s):  
Dynamical System ◽  
Local Stability ◽  
Prey Species ◽  
Optimal Harvesting ◽  
Asymptotically Stable ◽  
Biological Models ◽  
Adjoint Variables ◽  
Predator Model ◽  
Theoretical Results

      This paper aims to introduce a concept of an equilibrium point of a dynamical system which will call it almost global asymptotically stable. We also propose and analyze a prey-predator model with a suggested  function growth in prey species. Firstly the existence and local stability of all its equilibria are studied. After that the model is extended to an optimal control problem to obtain an optimal harvesting strategy. The discrete time version of Pontryagin's maximum principle is applied to solve the optimality problem. The characterization of the optimal harvesting variable and the adjoint variables are derived. Finally these theoretical results are demonstrated with numerical simulations.

NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE

2019 ◽  
Vol 4 (2) ◽  
pp. 349 ◽  
Author(s):  
Oluwatayo Michael Ogunmiloro ◽  
Fatima Ohunene Abedo ◽  
Hammed Kareem
Keyword(s):  
Reproduction Number ◽  
Disease Spread ◽  
Asymptotically Stable ◽  
Equilibrium Solutions ◽  
Transform Method ◽  
Globally Asymptotically Stable ◽  
Differential Transform ◽  
Mathematical Techniques ◽  
Fourth Order Method ◽  
Theoretical Results

In this article, a Susceptible – Vaccinated – Infected – Recovered (SVIR) model is formulated and analysed using comprehensive mathematical techniques. The vaccination class is primarily considered as means of controlling the disease spread. The basic reproduction number (Ro) of the model is obtained, where it was shown that if Ro<1, at the model equilibrium solutions when infection is present and absent, the infection- free equilibrium is both locally and globally asymptotically stable. Also, if Ro>1, the endemic equilibrium solution is locally asymptotically stable. Furthermore, the analytical solution of the model was carried out using the Differential Transform Method (DTM) and Runge - Kutta fourth-order method. Numerical simulations were carried out to validate the theoretical results. 

scholarly journals The Influence of Fear Effect to a Discrete-Time Predator-Prey System with Predator Has Other Food Resource

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 865
Author(s):  
Jialin Chen ◽  
Xiaqing He ◽  
Fengde Chen
Keyword(s):  
Discrete Time ◽  
Food Resource ◽  
Sufficient Conditions ◽  
Iteration Scheme ◽  
Predator Species ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Fear Effect ◽  
Prey System ◽  
Trivial Equilibrium

A discrete-time predator–prey system incorporating fear effect of the prey with the predator has other food resource is proposed in this paper. The trivial equilibrium and the predator free equilibrium are both unstable. A set of sufficient conditions for the global attractivity of prey free equilibrium and interior equilibrium are established by using iteration scheme and the comparison principle of difference equations. Our study shows that due to the fear of predation, the prey species will be driven to extinction while the predator species tends to be stable since it has other food resource, i.e., the prey free equilibrium may be globally stable under some suitable conditions. Numeric simulations are provided to illustrate the feasibility of the main results.

scholarly journals Stability analysis of a dynamical model of tuberculosis with incomplete treatment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Ihsan Ullah ◽  
Saeed Ahmad ◽  
Qasem Al-Mdallal ◽  
Zareen A. Khan ◽  
Hasib Khan ◽  
...  
Keyword(s):  
Disease Transmission ◽  
Contact Rate ◽  
Asymptotically Stable ◽  
Treatment Rate ◽  
Globally Asymptotically Stable ◽  
Reproduction Ratio ◽  
Efficient Treatment ◽  
Effective Contact ◽  
The Impact ◽  
Incomplete Treatment

Abstract A simple deterministic epidemic model for tuberculosis is addressed in this article. The impact of effective contact rate, treatment rate, and incomplete treatment versus efficient treatment is investigated. We also analyze the asymptotic behavior, spread, and possible eradication of the TB infection. It is observed that the disease transmission dynamics is characterized by the basic reproduction ratio $\Re _{0}$ ℜ 0 ; if $\Re _{0}<1$ ℜ 0 < 1 , there is only a disease-free equilibrium which is both locally and globally asymptotically stable. Moreover, for $\Re _{0}>1$ ℜ 0 > 1 , a unique positive endemic equilibrium exists which is globally asymptotically stable. The global stability of the equilibria is shown via Lyapunov function. It is also obtained that incomplete treatment of TB causes increase in disease infection while efficient treatment results in a reduction in TB. Finally, for the estimated parameters, some numerical simulations are performed to verify the analytical results. These numerical results indicate that decrease in the effective contact rate λ and increase in the treatment rate γ play a significant role in the TB infection control.

scholarly journals Stability and Hopf Bifurcation for a Delayed SIR Epidemic Model with Logistic Growth

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yakui Xue ◽  
Tiantian Li
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Threshold Value ◽  
Endemic Equilibrium ◽  
Logistic Growth ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for anyτ, we show that the disease-free equilibrium is globally asymptotically stable; whenR0<1, the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for anyτ=0; whenR0>1, the disease will persist. However, for anyτ≠0, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

scholarly journals On a Periodic Predator-Prey System with Holling III Functional Response and Stage Structure for Prey

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Xiangzeng Kong ◽  
Zhiqin Chen ◽  
Li Xu ◽  
Wensheng Yang
Keyword(s):  
Functional Response ◽  
Prey Species ◽  
Necessary Conditions ◽  
Stage Structure ◽  
Predator Prey ◽  
Prey System ◽  
Sufficient And Necessary Conditions ◽  
Holling Iii Functional Response

We propose and study the permanence of the following periodic Holling III predator-prey system with stage structure for prey and both two predators which consume immature prey. Sufficient and necessary conditions which guarantee the predator and the prey species to be permanent are obtained.

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