Stability analysis of a food chain model consisting of two competitive preys and one predator (original) (raw)
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The present article deals with a constant proportion of prey refuge in presence of both the inter specific competition and the intra-specific competition among predator populations of a prey-dependent three component food chain model consisting of two competitive predators sharing one prey species as their food. The behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. Boundedness and dissipativeness of the system are established. The stability analysis including local and global stability of the equilibria has been carried out in order to examine the behaviour of the system. The present model system experiences Hopf-Andronov bifurcation for suitable choice of the parameter values. The influences of the prey refuge parameters on the dynamical behaviour of the system are exhibited through several plots and discussed at some equilibrium positions. It is worthnoting that prey refuge has stabilization effect in some selected situations which may be of some use for biological control. Numerical simulations are performed to illustrate and to support the analytical findings so as to validate the applicability of the model under consideration.
Effects of intraspecific competition of prey in the dynamics of a food chain model
Modeling Earth Systems and Environment, 2016
In this paper, dynamical behavior of simple prey-predator model with Holling type II functional response involving additional food for predator along with intraspecific competition of prey is proposed and analyzed. The stability criteria of solutions are investigated by varying quantity and quality of additional food and intraspecific competition of prey population. Conditions for Hopf bifurcation are derived analytically. Numerical simulation results are presented to observe the time evolution of the system. This study may be useful to understand the effects of intraspecific competition in real world ecological systems.
Study of a tri-trophic prey-dependent food chain model of interacting populations
Mathematical Biosciences, 2013
The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf-Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper.
Global behaviour of a food chain model consisting of two different predator species
Of concern the present article deals with the mathematical analysis of a three species food chain model with two different types of predator species incorporating intra-specific competition among predator populations. Holling type-II response function for the interaction between prey and predator and ratio-dependent response function for the interaction between predator and top-predator is considered. The essential mathematical features have been analyzed in terms of local stability, global stability and the bifurcation theory as well. Our analytical findings are only validated by appropriate numerical simulations. Biological implications of the analytical findings are discussed at length in the concluding section.
Journal of Applied Nonlinear Dynamics
The present article deals with the influence of a constant proportion of prey refuge in presence of intra-specific competition among predator population of a prey-dependent three species food chain model. The behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. The preliminary results such as boundedness and dissipativeness of the system are established. Stability analysis including local and global stability of the equilibria has been carried out in order to examine the behaviour of the system. The present system experiences Hopf-Andronov bifurcation for suitable choice of the parameter values. The influences of the prey refuge parameters on the dynamical behaviour of the system are exhibited through several plots and discussed at some equilibrium positions. It is worth-noting that prey refuge has stabilization effect in some selected situations and bears the potential to control chaotic dynamics of the system. Hence, prey refuge may be of some use for biological control mechanism. Numerical simulations are performed to validate the applicability of the model under consideration.
BAREKENG: Jurnal Ilmu Matematika dan Terapan
This research develops a mathematical model of three species of food chains between prey, predator, and top predator by adding intraspecific competition and harvesting factors. Interaction between prey with predator and interaction between predator with top predator uses the functional response type II. Model formation begins with creating a diagram food chain of three species compartments. Then a nonlinear differential equation system is formed based on the compartment diagram. Based on this system four equilibrium points are obtained. Analysis of local stability at the equilibrium points by linearization shows that there is one unstable equilibrium point and three asymptotic stable local equilibrium points. Numerical simulations at equilibrium points show the same results as the results of the analysis. Then numerical simulations on several parameter variations show that intraspecific competition has little effect on population changes in predator and top predator. While the harve...
Journal of Applied Nonlinear Dynamics, 2016
The present article deals with the influence of a constant proportion of prey refuge in presence of intra-specific competition among predator population of a prey-dependent three species food chain model. The behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. The preliminary results such as boundedness and dissipativeness of the system are established. Stability analysis including local and global stability of the equilibria has been carried out in order to examine the behaviour of the system. The present system experiences Hopf-Andronov bifurcation for suitable choice of the parameter values. The influences of the prey refuge parameters on the dynamical behaviour of the system are exhibited through several plots and discussed at some equilibrium positions. It is worth-noting that prey refuge has stabilization effect in some selected situations and bears the potential to control chaotic dynamics of the system. Hence, prey refuge may be of some use for biological control mechanism. Numerical simulations are performed to validate the applicability of the model under consideration.
Numerical Study of One Prey-Two Predator Model Considering Food Addition and Anti-Predator Defense
E3S Web of Conferences, 2021
This article examines the interaction between prey populations, juvenile predators, and adult predators. A mathematical model that considers adding food and anti-predators was developed. The equilibria of the existing system are that the system has four equilibria points with conditions suitable for the locale. Numerical simulations were carried out to describe the dynamics of the system solution. Based on numerical simulations, the varying of parameter causes changes in the extinction of prey or survival of prey populations, juvenile predators, and adult predators. Addfood parameters (A) encourae Hopf Bifurcation and Saddle-node bifurcation Numerical continuity results show that Hopf bifurcation occurs when the parameter value A = 1.00162435 and when the parameter value A = 2.435303 Saddle-node bifurcation occurs.
Bifurcation Analysis of Prey-Predator Model with Harvested Predator
This paper aims to study the effect of harvested predator species on a Holling type IV Prey-Predator model involving intra-specific competition. Prey-predator model has received much attention during the last few decades due to its wide range of applications. There are many kind of prey-predator models in mathematical ecology. The Prey-predator models governed by differential equations are more appropriate than the difference equations to describe the prey-predator relations. Harvesting has a strong impact on the dynamic evolution of a population. This model represents mathematically by non-linear differential equations. The locally asymptotic stability conditions of all possible equilibrium points were obtained. The stability/instability of non-negative equilibrium and associated bifurcation were investigated by analysing the characteristic equations. Moreover, bifurcation diagrams were obtained for different values of parameters of proposed model.
Local and global stability analysis of a two prey one predator model with help
Communications in Nonlinear Science and Numerical Simulation, 2014
In this paper we propose and study a three dimensional continuous time dynamical system modelling a three team consists of two preys and one predator with the assumption that during predation the members of both teams of preys help each other and the rate of predation of both teams are different. In this work we establish the local asymptotic stability of various equilibrium points to understand the dynamics of the model system. Different conditions for the coexistence of equilibrium solutions are discussed. Persistence, permanence of the system and global stability of the positive interior equilibrium solution are discussed by constructing suitable Lyapunov functional. At the end, numerical simulations are performed to substantiate our analytical findings.