scholarly journals Mathematical study on a dynamical predator-prey model with constant prey harvesting and proportional harvesting in predator

2021 ◽  
Author(s):  
Md Golam Mortuja ◽  
Mithilesh Kumar Chaube ◽  
Santosh Kumar
Keyword(s):  
Handling Time ◽  
Maximum Sustainable Yield ◽  
Prey Population ◽  
Sustainable Yield ◽  
Mathematical Study ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Herd Behaviour

Abstract A dynamical predator-prey model with constant prey harvesting, proportional harvesting in predator has been studied. The square root func- tional response also has been incorporated in the system to describe the prey herd behaviour, assuming the average handling time is zero. The existence and the local stability of equilibria of the system have been discussed. It is examined that, two types of bifurcation occur in the system. The two types of bifurcations have been analyzed, and it has been found by analyzing the saddle-node bifurcation that, there is a maximum sustainable yield. It is ob- served that if harvesting rate is greater than the maximum sustainable yield, the prey population abolish from the system and then extinction of the preda- tor population happen. But if harvesting rate is lesser than the maximum sustainable yield, the extinction of the prey population can not be possible. By analyzing the Hopf bifurcation, it is obtained that, there exists an unstable limit cycle around the interior equilibrium point. Several numerical simulations are performed to check the results.

A linear birth-and-death predator–prey process

1995 ◽  
Vol 32 (01) ◽  
pp. 274-277
Author(s):  
John Coffey
Keyword(s):  
Death Rate ◽  
Birth Rate ◽  
Prey Population ◽  
Death Process ◽  
Predator Population ◽  
Birth And Death Process ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model

A new stochastic predator-prey model is introduced. The predator population X(t) is described by a linear birth-and-death process with birth rate λ 1 X and death rate μ 1 X. The prey population Y(t) is described by a linear birth-and-death process in which the birth rate is λ 2 Y and the death rate is . It is proven that and iff

A Predator-Prey Model Incorporating Prey Refuge and Allee Effect

2015 ◽  
Vol 713-715 ◽  
pp. 1534-1539 ◽  
Author(s):  
Rui Ning Fan
Keyword(s):  
Time Delay ◽  
Time Lag ◽  
Functional Responses ◽  
Prey Refuge ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Discrete Time Delay ◽  
Prey Interaction

The effect of refuge used by prey has a stabilizing impact on population dynamics and the effect of time delay has its destabilizing influences. Little attention has been paid to the combined effects of prey refuge and time delay on the dynamic consequences of the predator-prey interaction. Here, a predator-prey model with a class of functional responses was studied by using the analytical approach. The refuge is considered as protecting a constant proportion of prey and the discrete time delay is the gestation period. We evaluated both effects with regard to the local stability of the interior equilibrium point of the considered model. The results showed that the effect of prey refuge has stronger influences than that of time delay on the considered model when the time lag is smaller than the threshold. However, if the time lag is larger than the threshold, the effect of time delay has stronger influences than that of refuge used by prey.

Turing Instability and Hopf Bifurcation in a Modified Leslie–Gower Predator–Prey Model with Cross-Diffusion

2018 ◽  
Vol 28 (07) ◽  
pp. 1850089 ◽  
Author(s):  
Walid Abid ◽  
R. Yafia ◽  
M. A. Aziz-Alaoui ◽  
Ahmed Aghriche
Keyword(s):  
Spatial Dynamics ◽  
Real Life ◽  
Reaction Diffusion ◽  
Turing Instability ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Cross Diffusion ◽  
Predator Prey Model ◽  
Prey Model

This paper is concerned with some mathematical analysis and numerical aspects of a reaction–diffusion system with cross-diffusion. This system models a modified version of Leslie–Gower functional response as well as that of the Holling-type II. Our aim is to investigate theoretically and numerically the asymptotic behavior of the interior equilibrium of the model. The conditions of boundedness, existence of a positively invariant set are proved. Criteria for local stability/instability and global stability are obtained. By using the bifurcation theory, the conditions of Hopf and Turing bifurcation critical lines in a spatial domain are proved. Finally, we carry out some numerical simulations in order to support our theoretical results and to interpret how biological processes affect spatiotemporal pattern formation which show that it is useful to use the predator–prey model to detect the spatial dynamics in the real life.

Optimal control of a diffusive eco-epidemiological predator–prey model

2020 ◽  
Vol 13 (07) ◽  
pp. 2050065
Author(s):  
Xuebing Zhang ◽  
Guanglan Wang ◽  
Honglan Zhu
Keyword(s):  
Optimal Control ◽  
Semigroup Theory ◽  
Prey Population ◽  
Optimal Controls ◽  
Controlled System ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Cost ◽  
Infected Prey

In this study, we investigate the optimal control problem for a diffusion eco-epidemiological predator–prey model. We applied two controllers to this model. One is the separation control, which separates the uninfected prey from the infected prey population, and the other is used as a treatment control to decrease the mortality caused by the disease. Then, we propose an optimal problem to minimize the infected prey population at the final time and the cost cause by the controls. To do this, by the operator semigroup theory we prove the existence of the solution to the controlled system. Furthermore, we prove the existence of the optimal controls and obtain the first-order necessary optimality condition for the optimal controls. Finally, some numerical simulations are carried out to support the theoretical results.

Modeling and analysis of a predator–prey model with state-dependent delay

2018 ◽  
Vol 11 (02) ◽  
pp. 1850026 ◽  
Author(s):  
Yunfei Lv ◽  
Yongzhen Pei ◽  
Rong Yuan
Keyword(s):  
Immature Stage ◽  
Global Dynamics ◽  
Modeling And Analysis ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
State Dependent ◽  
State Dependent Delay ◽  
Threshold Amount

We propose and study a predator–prey model with state-dependent delay where the prey population is assumed to have an age structure. The state-dependent delay appears due to the mature condition that the prey must spend an amount of time in the immature stage sufficient to accumulate a threshold amount of food. We perform a qualitative analysis of the solutions, which includes studying positivity and boundedness, existence and local stability of equilibria. For the global dynamics of the system, we discuss an attracting region which is determined by solutions, and the region collapses to the interior equilibrium in the constant delay case.

scholarly journals Global dynamics and parameter identifiability in a predator-prey interaction model

2018 ◽  
Vol 5 (1) ◽  
pp. 113-126
Author(s):  
Jai Prakash Tripathi ◽  
Suraj S. Meghwani ◽  
Swati Tyagi ◽  
Syed Abbas
Keyword(s):  
Interaction Model ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Prey Refuge ◽  
Predator Species ◽  
Predator Prey ◽  
Interior Equilibrium ◽  
Predator Prey Model ◽  
Prey Model ◽  
Permanence Property

AbstractThis paper discusses a predator-prey model with prey refuge. We investigate the role of prey refuge on the existence and stability of the positive equilibrium. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional, which shows that the prey refuge has no influence on the permanence property of the system. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. To access the usability of proposed predator-prey model in practical scenarios, we also suggest, the use of Levenberg-Marquardt (LM) method for associated parameter estimation problem. Numerical results demonstrate faithful reconstruction of system dynamics by estimated parameter by LM method. The analytical results found in this paper are illustrated with the help of suitable numerical examples

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