scholarly journals Deterministic and Stochastic Holling-Tanner Prey-Predator Models

2021 ◽  
Vol 1 (1) ◽  
pp. 1-8
Author(s):  
Bharat Bahadur Thapa ◽  
Samir Shrestha ◽  
Dil Bahadur Gurung
Keyword(s):  
Differential Equations ◽  
Functional Response ◽  
Ecological Systems ◽  
Volterra Model ◽  
Stable Node ◽  
Euler Scheme ◽  
Numerical Schemes ◽  
Stochastic Environments ◽  
Predator Model ◽  
The Stability

A modified version of the so called Holling-Tanner prey-predator models with prey dependent functional response is introduced. We improved some new results on Holling-Tanner model from Lotka-Volterra model on real ecological systems and studied the stability of this model in the deterministic and stochastic environments. The study was focused on three types of stability, namely, stable node, spiral node, and center. The numerical schemes are employed to get the approximated solutions of the differential equations. We have used Euler scheme to solve the deterministic prey-predator model and we used Euler-Maruyama scheme to solve stochastic prey-predator model.

scholarly journals Bifurcation Analysis of a Delayed Predator-Prey Model with Holling Type III Functional Response and Predator Harvesting

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Uttam Das ◽  
T. K. Kar
Keyword(s):  
Hopf Bifurcation ◽  
Functional Response ◽  
Normal Form Theory ◽  
Predator Species ◽  
Center Manifold Theorem ◽  
Type Iii ◽  
Predator Prey Model ◽  
Form Theory ◽  
Predator Model ◽  
The Stability

This paper tries to highlight a delayed prey-predator model with Holling type III functional response and harvesting to predator species. In this context, we have discussed local stability of the equilibria, and the occurrence of Hopf bifurcation of the system is examined by considering the harvesting effort as bifurcation parameter along with the influences of harvesting effort of the system when time delay is zero. Direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also studied by applying the normal form theory and the center manifold theorem. Lastly some numerical simulations are carried out to draw for the validity of the theoretical results.

scholarly journals Ecology through the eyes of a non-ecologist

2020 ◽  
Vol 18 (5) ◽  
pp. 259-270
Author(s):  
Janusz Uchmański
Keyword(s):  
Differential Equations ◽  
Mathematical Models ◽  
Density Dependence ◽  
Evolutionary Biology ◽  
Ecological Systems ◽  
Living Environment ◽  
Mathematical Methods ◽  
Nature Protection ◽  
Practical Applications ◽  
The Stability

Ecology is a branch of biology that deals with the life of plants and animals in their environment. Nature protection are practical actions where ecology is applied. Ecology is the most biological branch of biology because it deals with individuals in their living environment, and individuals "exist" only in biology. The most important issue being considered in ecology is biodiversity: its changes and its persistence. In their research, ecologists focus on the functioning of ecological systems. In classical terms, they assume that the most important mechanism is density dependence. Mathematical models traditionally applied in ecology include ordinary difference and differential equations, which fits well with the assumption of density dependence, but this results in ecology being dominated by considerations of the stability of ecological systems. Evolutionary biology and ecology have separate areas of interest. Evolutionary biology explains the formation of optimal characteristics of individuals. Ecology also takes into account those individuals who have lost in the process of natural selection. The mathematical methods used in classical ecology were developed for the use of physics. The question arises whether they give a precise picture of the dynamics of ecological systems. Recently, a view has emerged stating that in order to see the importance of full-scale biodiversity, we should refer to individuals (rather than population density) as basic "atoms" that make up ecological systems. In ecology, we call this an individual-based approach. However, it gives a very complex picture of how ecological systems work. In ecology, however, there is an alternative way to describe the dynamics of ecological systems, i.e. through the circulation of matter in them and the flow of energy through them. It allows the use of traditional difference and differential equations in the formulation of mathematical models, which has proven itself in practical applications many times.

scholarly journals Further Study on Dynamics for a Fractional-Order Competitor-Competitor-Mutualist Lotka–Volterra System

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Bingnan Tang
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Order ◽  
Natural World ◽  
Volterra Model ◽  
Sufficient Criterion ◽  
Volterra System ◽  
Stability Behavior ◽  
Predator Model ◽  
Set Up ◽  
The Stability

On the basis of the previous publications, a new fractional-order prey-predator model is set up. Firstly, we discuss the existence, uniqueness, and nonnegativity for the involved fractional-order prey-predator model. Secondly, by analyzing the characteristic equation of the considered fractional-order Lotka–Volterra model and regarding the delay as bifurcation variable, we set up a new sufficient criterion to guarantee the stability behavior and the appearance of Hopf bifurcation for the addressed fractional-order Lotka–Volterra system. Thirdly, we perform the computer simulations with Matlab software to substantiate the rationalisation of the analysis conclusions. The obtained results play an important role in maintaining the balance of population in natural world.

Overcoming the order barrier in the numerical approximation of stochastic partial differential equations with additive space–time noise

2008 ◽  
Vol 465 (2102) ◽  
pp. 649-667 ◽  
Author(s):  
Arnulf Jentzen ◽  
Peter E Kloeden
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Numerical Approximation ◽  
Time Discretization ◽  
Space Time ◽  
Euler Scheme ◽  
Numerical Schemes ◽  
Partial Differential ◽  
The Laplace Operator

We consider the numerical approximation of parabolic stochastic partial differential equations driven by additive space–time white noise. We introduce a new numerical scheme for the time discretization of the finite-dimensional Galerkin stochastic differential equations, which we call the exponential Euler scheme, and show that it converges (in the strong sense) faster than the classical numerical schemes, such as the linear-implicit Euler scheme or the Crank–Nicholson scheme, for this equation with the general noise. In particular, we prove that our scheme applied to a semilinear stochastic heat equation converges with an overall computational order 1/3 which exceeds the barrier order 1/6 for numerical schemes using only basic increments of the noise process reported previously. By contrast, our scheme takes advantage of the smoothening effect of the Laplace operator and of a linear functional of the noise and, therefore overcomes this order barrier.

scholarly journals A Predator-Prey Model In Deterministic And Stochastic Environments

2021 ◽  
Author(s):  
Chandra P. Limbu
Keyword(s):  
Numerical Simulations ◽  
Ecological Systems ◽  
Volterra Model ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Stochastic Environment ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Environments ◽  
Uniqueness Of Limit Cycles

We investigate the phase portraits, the uniqueness of limit cycles and the Hopf bifurcations in the Holling-Tanner models in deterministic and stochastic environments. We provide the conditions on the parameters to assure saddle, focus and node. We use numerical simulations to demonstrate our results in the deterministic cases. We also explore the Holling-Tanner model in a stochastic environment by using numerical simulations. We generalize and improve some new results on Holling-Tanner model from Lotka-Volterra model on real ecological systems.

scholarly journals Effects of Refuge Prey on Stability of the Prey-Predator Model Subject to Immigrants: A Mathematical Modelling Approach

2021 ◽  
Vol 47 (4) ◽  
pp. 1376-1391
Author(s):  
Mussa Amos Stephano ◽  
Il Hyo Jung
Keyword(s):  
Mathematical Model ◽  
Mathematical Modelling ◽  
Functional Response ◽  
Volterra Model ◽  
Predator Population ◽  
Type I ◽  
Proposed Model ◽  
Predator Model ◽  
Model Subject ◽  
Type Ii Functional Response

Prey-predator system is enormously complex and nonlinear interaction between species. Such complexity regularly requires development of new approaches which involves more factors in analysis of its population dynamics. In this paper, we formulate a modified Lotka-Volterra model that incorporates factors such as refuge prey and immigrants. We investigate the effects of refuge prey and immigrants by varying the refuge factor, with and without immigrants. The results show that with Holling’s type I functional response, the proposed model is asymptotically convergent when a refuge prey factor is introduced. Moreover, with Holling’s type II functional response, the proposed mathematical model is unstable and does not converge. However, with Holling’s type III functional response in a system, the proposed mathematical model is asymptotically stable. These results point out the following remarks: The effects of refuge prey on stability of the dynamical system vary depending on the type of functional response, and when the predator population increases, the likelihood of prey extinction declines when the proportion of preys in refuge population increases. Hence, the factor of refuge prey is crucial for controlling the population of the predator and obtaining balances between prey and predator in the ecosystem. Keywords: Refuge prey, stability, prey-predator, immigrants, Mathematical modelling

scholarly journals A Predator-Prey Model In Deterministic And Stochastic Environments

2021 ◽  
Author(s):  
Chandra P. Limbu
Keyword(s):  
Numerical Simulations ◽  
Ecological Systems ◽  
Volterra Model ◽  
Phase Portraits ◽  
Hopf Bifurcations ◽  
Stochastic Environment ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Stochastic Environments ◽  
Uniqueness Of Limit Cycles

We investigate the phase portraits, the uniqueness of limit cycles and the Hopf bifurcations in the Holling-Tanner models in deterministic and stochastic environments. We provide the conditions on the parameters to assure saddle, focus and node. We use numerical simulations to demonstrate our results in the deterministic cases. We also explore the Holling-Tanner model in a stochastic environment by using numerical simulations. We generalize and improve some new results on Holling-Tanner model from Lotka-Volterra model on real ecological systems.

scholarly journals A note on the stability of a modified Lotka-Volterra model using Hurwitz polynomials

2021 ◽  
Vol 20 ◽  
pp. 431-441
Author(s):  
Fabián Toledo , Sánchez ◽  
Pedro Pablo Cárdenas Alzate ◽  
Carlos Arturo Escudero Salcedo
Keyword(s):  
Differential Equations ◽  
Initial Conditions ◽  
Stability Study ◽  
Volterra Model ◽  
Type Model ◽  
Stability Criteria ◽  
Volterra Type ◽  
Hurwitz Polynomials ◽  
Intuitive Idea ◽  
The Stability

In the analysis of the dynamics of the solutions of ordinary differential equations we can observe whether or not small variations or perturbations in the initial conditions produce small changes in the future; this intuitive idea of stability was formalized and studied by Lyapunov, who presented methods for the stable analysis of differential equations. For linear or nonlinear systems, we can also analyze the stability using criteria to obtain Hurwitz type polynomials, which provide conditions for the analysis of the dynamics of the system, studying the location of the roots of the associated characteristic polynomial. In this paper we present a stability study of a Lotka-Volterra type model which has been modified considering the carrying capacity or support in the prey and time delay in the predator, this stable analysis is performed using stability criteria to obtain Hurwitz-type polynomials.

Dynamical Behaviors of a Delayed Prey–Predator Model with Beddington–DeAngelis Functional Response: Stability and Periodicity

2020 ◽  
Vol 30 (16) ◽  
pp. 2050244
Author(s):  
Xin Zhang ◽  
Renxiang Shi ◽  
Ruizhi Yang ◽  
Zhangzhi Wei
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Functional Response ◽  
Discrete Time ◽  
Numerical Continuation ◽  
Positive Equilibrium ◽  
Local Hopf Bifurcation ◽  
Discrete Time Delay ◽  
Predator Model ◽  
The Stability

This work investigates a prey–predator model with Beddington–DeAngelis functional response and discrete time delay in both theoretical and numerical ways. Firstly, we incorporate into the system a discrete time delay between the capture of the prey by the predator and its conversion to predator biomass. Moreover, by taking the delay as a bifurcation parameter, we analyze the stability of the positive equilibrium in the delayed system. We analytically prove that the local Hopf bifurcation critical values are neatly paired, and each pair is joined by a bounded global Hopf branch. Also, we show that the predator becomes extinct with an increase of the time delay. Finally, before the extinction of the predator, we find the abundance of dynamical complexity, such as supercritical Hopf bifurcation, using the numerical continuation package DDE-BIFTOOL.

scholarly journals An Eco-Epidemiological Model Incorporating Harvesting Factors

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2179
Author(s):  
Kawa Hassan ◽  
Arkan Mustafa ◽  
Mudhafar Hama
Keyword(s):  
Equilibrium Point ◽  
Functional Response ◽  
Biological System ◽  
Numerical Verification ◽  
Interior Equilibrium ◽  
Dynamical Fluctuations ◽  
Predator Model ◽  
The Stability ◽  
Analytical Section ◽  
Predator System

The biological system relies heavily on the interaction between prey and predator. Infections may spread from prey to predators or vice versa. This study proposes a virus-controlled prey-predator system with a Crowley–Martin functional response in the prey and an SI-type in the prey. A prey-predator model in which the predator uses both susceptible and sick prey is used to investigate the influence of harvesting parameters on the formation of dynamical fluctuations and stability at the interior equilibrium point. In the analytical section, we outlined the current circumstances for all possible equilibria. The stability of the system has also been explored, and the required conditions for the model’s stability at the equilibrium point have been found. In addition, we give numerical verification for our analytical findings with the help of graphical illustrations.

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