(R1493) Discussion on Stability and Hopf-bifurcation of an Infected Prey under Refuge and Predator

The paper deals with the case of non-selective predation in a partially infected prey-predator system, where both the susceptible prey and predator follow the law of logistic growth and some preys avoid predation by hiding. The disease-free preys get infected in due course of time by a certain rate. However, the carrying capacity of the predator population is considered proportional to the sum-total of the susceptible and infected prey. The positivity and boundedness of the solutions of the system are studied and the existence of the equilibrium points and stability of the system are analyzed at these points. The effect of the infected prey-refuge on each population density is also discussed. It is observed that a Hopf-bifurcation may occur about the interior equilibrium, where the refuge parameter is considered as the bifurcation parameter. The analytical findings are illustrated through computer simulation using MAPLE that show the reliability of the model from the ecological point of view

A Modified Holling-Tanner Model in Stochastic Environment

Recently, a modified version of the so called Holling-Tannermodel isintroduced in the ecological literature. A detailed account of the deterministic dynamicsof this model is presented. The growth rates of the prey and predator are then perturbedby Gaussian white noises to take into account the effect of fluctuating environment. Theresulting stochastic model is cultured by the technique of statistical linearization andcriteria for non-equilibrium fluctuation and stability arederived. Numerical simulationsare carried out. The implications of our analytical findingsare addressed critically

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

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

Stability and Hopf Bifurcation Analysis of a Nutrient-Phytoplankton Model with Delay Effect

A delay differential system is investigated based on a previously proposed nutrient-phytoplankton model. The time delay is regarded as a bifurcation parameter. Our aim is to determine how the time delay affects the system. First, we study the existence and local stability of two equilibria using the characteristic equation and identify the condition where a Hopf bifurcation can occur. Second, the formulae that determine the direction of the Hopf bifurcation and the stability of periodic solutions are obtained using the normal form and the center manifold theory. Furthermore, our main results are illustrated using numerical simulations

Predator-prey models with cannibalism in prey

Cannibalism in the predator-prey model is the study to show the interaction between prey and predator where the presence of cannibalism exists in both species in real life. Moreover, cannibalism is ubiquitous in natural communities and also among researchers who are interested in mathematical ecology. The predator-prey model system is modelled using ordinary differential equations to describe the dynamic behaviour of the systems. This study introduces the stage-structured models where the adult and juvenile prey species are considered. The purpose of the study is to analyse the effect of the stage-structured of prey cannibalism on the stability based on the concept of Lotka-Volterra in the predator-prey model. Thus, in this study, there are two cases are considered: prey cannibalism in the predator-prey model with predation on adult prey and model with predation on juvenile prey. The objectives of this research are (i) to formulate the concept of Lotka-Volterra in a predator-prey model, (ii) to analyse prey cannibalism in predator-prey model with predation on adult prey, (iii) to analyse the predator-prey in prey cannibalism with predation on juvenile prey, and (iv) to analyse the effect of stage-structured predator-prey model with cannibalism in prey on stability. In analysing the models, the stability of the equilibrium point is obtained and described by using the properties of the eigenvalues and the Routh-Hurwitz Criteria. Last but not least, numerical examples and graph analysis are given to illustrate the stability of equilibrium points

Влияние случайного воздействия на равновесные режимы модели популяционной динамики

В работе изучается динамическая модель взаимодействующих популяций по типу «хищник-две жертвы». Проводится детальный параметрический анализ равновесных режимов, возникающих в системе. В зонах бифуркационного параметра, где обнаружено сосуществование нескольких равновесных режимов, строятся сепарабельные поверхности, являющиеся границами бассейнов их притяжения. Показано, что воздействие внешнего случайного возмущения способно разрушить установившийся равновесный режим сосущестования трех популяций и привести к качественно другому режиму сосуществования. Такие качественные изменения приводят к вымиранию одной или двух из трех популяций. C помощью функции стохастической чувствительности и связанного с ней метода доверительных областей демонстрируются вероятностные механизмы разрушения равновесных режимов. Проводится параметрический анализ вероятностей вымирания популяций по двум типам. Указываются диапазон бифуркационного параметра и уровень интенсивности случайного воздействия наиболее выгодные для сосуществования трех популяций

Multiple Bifurcations and Chaos in a Discrete Prey-Predator System with Generalized Holling III Functional Response

A prey-predator system with the strong Allee effect and generalized Holling type III functional response is presented and discretized. It is shown that the combined influences of Allee effect and step size have an important effect on the dynamics of the system. The existences of Flip and Neimark-Sacker bifurcations and strange attractors and chaotic bands are investigated by using the center manifold theorem and bifurcation theory and some numerical methods

Bifurkasi Hopf pada Model Lotka-Volterra Orde-Fraksional dengan Efek Allee Aditif pada Predator

This article aims to study the dynamics of a Lotka-Volterra predator-prey model with Allee effect in predator. According to the biological condition, the Caputo fractional-order derivative is chosen as its operator. The analysis is started by identifying the existence, uniqueness, and non-negativity of the solution. Furthermore, the existence of equilibrium points and their stability is investigated. It has shown that the model has two equilibrium points namely both populations extinction point which is always a saddle point, and a conditionally stable co-existence point, both locally and globally. One of the interesting phenomena is the occurrence of Hopf bifurcation driven by the order of derivative. Finally, the numerical simulations are given to validate previous theoretical results