Mean Square Stability in a Modified Leslie-Gower and Holling-Type II Predator-Prey Model
Journal of applied mathematics & informatics, 2011
Of concern in the paper is a Holling-Tanner predator-prey model with modifled version of the Lesl... more Of concern in the paper is a Holling-Tanner predator-prey model with modifled version of the Leslie-Gower functional response. Dynamical behaviours such as stability, permanence and Hopf bifurcation have been carried out deterministically. Using the normal form theory and center manifold theorem, the explicit formulae determining the stability and direction of Hopf bifurcation have been derived . The deterministic model is extended to a stochastic one by perturbing the growth equation of prey and predator by white and colored noises and flnally the mean square stability of the stochastic model systems is investigated analytically. An extensive quantitative analysis has been performed based on numerical computation so as to validate the applicability of the proposed mathematical model.
Relaxation oscillation and canard explosion in a slow–fast predator–prey model with Beddington–DeAngelis functional response
Nonlinear Dynamics, 2021
In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response. C... more In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response. Considering the predator’s rate of growth and death is much lower than that of prey’s, the model becomes a slow–fast system that mathematically leads to a singular perturbation problem. Using geometric singular perturbation theory due to Fenichel and blow up technique, we have investigated the system and obtained very rich and complicated dynamical phenomena including the existence of relaxation oscillation, canard cycles near the Hopf bifurcation point and the interesting phenomenon of canard explosion.
Dynamical Complexity of a Ratio-Dependent Predator-Prey Model with Strong Additive Allee Effect
Applied Mathematics, 2015
In this paper, a predator-prey systems of two species is proposed where prey population is subjec... more In this paper, a predator-prey systems of two species is proposed where prey population is subjected to a strong additive Allee effect and predator population consumes the prey according to the ratio-dependent Holling type-II functional response. We use the blow-up technique in order to explore the local structure of orbits in the vicinity of origin. We have determined the conditions for extinction/survival scenarios of species. Some basic dynamical results; the stability; phenomenon of bi-stability and the existence of separatrix curves; Hopf bifurcation; saddle-node bifurcation; homoclinic bifurcation, and Bogdanov-Takens bifurcation of the system are studied. Numerical simulation results that complement the theoretical predictions are presented. A discussion of the consequences of additive Allee effect on the model along with the ecological implications of the analytic and numerical findings is presented.
In this paper, we consider a delayed predator-prey system with intraspecific competition among pr... more In this paper, we consider a delayed predator-prey system with intraspecific competition among predator and a strong Allee effect in prey population growth. Using the delay as bifurcation parameter, we investigate the stability of coexisting equilibrium point and show that Hopf-bifurcation can occur when the discrete delay crosses some critical magnitude. The direction of the Hopf-bifurcating periodic solution and its stability are determined by applying the normal form method and the centre manifold theory. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc. 350:4799-4838, 1998) for functional differential equations, we establish the global existence of periodic solutions. Numerical simulations are carried out to validate the analytical findings.
Mathematical Methods in the Applied Sciences, 2013
The present investigation deals with a predator-prey model with disease that spreads among the pr... more The present investigation deals with a predator-prey model with disease that spreads among the predator species only. The predator species is split out into two groups-the susceptible predator and the infected predator both of which feeds on prey species. The stability and bifurcation analyses are carried out and discussed at length. On the basis of the normal form theory and center manifold reduction, the explicit formulae are derived to determine stability and direction of Hopf bifurcating periodic solution. An extensive quantitative analysis has been performed in order to validate the applicability of our model under consideration.
Studies on stability mechanism and bifurcation analysis of a system of interacting populations by... more Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator-prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings.
In the current paper, we propose a two-prey-one-predator system with Holling type II predation fu... more In the current paper, we propose a two-prey-one-predator system with Holling type II predation functional response where prey species consumes the remains of the other prey species' carcass given by their predator. Mathematical analysis of the proposed model equations with regard to non-negative invariance including stability and bifurcation are carried out. Next, we extend the deterministic system to a stochastic system by incorporating the Gaussian white noise terms in the growth equations of both prey and predator species. We determine fluctuation intensities for the stochastic dynamical system by using Laplace methods. Numerical simulations are exhibited to justify the analytical findings.
Original article: Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect
Mathematics and Computers in Simulation, Mar 1, 2014
Your article is protected by copyright and all rights are held exclusively by Springer Science +B... more Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".
Mean Square Stability in a Modified Leslie-Gower And. Holling-Type II Predator-Prey Model
ABSTRACT Of concern in the paper is a Holling-Tanner predator-prey model with modi¯ed version of ... more ABSTRACT Of concern in the paper is a Holling-Tanner predator-prey model with modi¯ed version of the Leslie-Gower functional response. Dynamical behaviours such as stability, permanence and Hopf bifurcation have been carried out deterministically. Using the normal form theory and center manifold theorem, the explicit formulae determining the stability and direction of Hopf bifurcation have been derived . The deterministic model is extended to a stochastic one by perturbing the growth equation of prey and predator by white and colored noises and ¯nally the mean square stability of the stochastic model systems is investigated analytically. An extensive quantitative analysis has been performed based on numerical computation so as to validate the applicability of the proposed mathematical model.
Mean Square Stability in a Modified Leslie-Gower and Holling-Type II Predator-Prey Model
Journal of applied mathematics & informatics, 2011
Of concern in the paper is a Holling-Tanner predator-prey model with modifled version of the Lesl... more Of concern in the paper is a Holling-Tanner predator-prey model with modifled version of the Leslie-Gower functional response. Dynamical behaviours such as stability, permanence and Hopf bifurcation have been carried out deterministically. Using the normal form theory and center manifold theorem, the explicit formulae determining the stability and direction of Hopf bifurcation have been derived . The deterministic model is extended to a stochastic one by perturbing the growth equation of prey and predator by white and colored noises and flnally the mean square stability of the stochastic model systems is investigated analytically. An extensive quantitative analysis has been performed based on numerical computation so as to validate the applicability of the proposed mathematical model.
Relaxation oscillation and canard explosion in a slow–fast predator–prey model with Beddington–DeAngelis functional response
Nonlinear Dynamics, 2021
In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response. C... more In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response. Considering the predator’s rate of growth and death is much lower than that of prey’s, the model becomes a slow–fast system that mathematically leads to a singular perturbation problem. Using geometric singular perturbation theory due to Fenichel and blow up technique, we have investigated the system and obtained very rich and complicated dynamical phenomena including the existence of relaxation oscillation, canard cycles near the Hopf bifurcation point and the interesting phenomenon of canard explosion.
Dynamical Complexity of a Ratio-Dependent Predator-Prey Model with Strong Additive Allee Effect
Applied Mathematics, 2015
In this paper, a predator-prey systems of two species is proposed where prey population is subjec... more In this paper, a predator-prey systems of two species is proposed where prey population is subjected to a strong additive Allee effect and predator population consumes the prey according to the ratio-dependent Holling type-II functional response. We use the blow-up technique in order to explore the local structure of orbits in the vicinity of origin. We have determined the conditions for extinction/survival scenarios of species. Some basic dynamical results; the stability; phenomenon of bi-stability and the existence of separatrix curves; Hopf bifurcation; saddle-node bifurcation; homoclinic bifurcation, and Bogdanov-Takens bifurcation of the system are studied. Numerical simulation results that complement the theoretical predictions are presented. A discussion of the consequences of additive Allee effect on the model along with the ecological implications of the analytic and numerical findings is presented.
In this paper, we consider a delayed predator-prey system with intraspecific competition among pr... more In this paper, we consider a delayed predator-prey system with intraspecific competition among predator and a strong Allee effect in prey population growth. Using the delay as bifurcation parameter, we investigate the stability of coexisting equilibrium point and show that Hopf-bifurcation can occur when the discrete delay crosses some critical magnitude. The direction of the Hopf-bifurcating periodic solution and its stability are determined by applying the normal form method and the centre manifold theory. In addition, special attention is paid to the global continuation of local Hopf bifurcations. Using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc. 350:4799-4838, 1998) for functional differential equations, we establish the global existence of periodic solutions. Numerical simulations are carried out to validate the analytical findings.
Mathematical Methods in the Applied Sciences, 2013
The present investigation deals with a predator-prey model with disease that spreads among the pr... more The present investigation deals with a predator-prey model with disease that spreads among the predator species only. The predator species is split out into two groups-the susceptible predator and the infected predator both of which feeds on prey species. The stability and bifurcation analyses are carried out and discussed at length. On the basis of the normal form theory and center manifold reduction, the explicit formulae are derived to determine stability and direction of Hopf bifurcating periodic solution. An extensive quantitative analysis has been performed in order to validate the applicability of our model under consideration.
Studies on stability mechanism and bifurcation analysis of a system of interacting populations by... more Studies on stability mechanism and bifurcation analysis of a system of interacting populations by the combined effect of self and cross-diffusion become an important issue in ecology. In the current investigation, we derive the conditions for existence and stability properties of a predator-prey model under the influence of self and cross-diffusion. Numerical simulations have been carried out in order to show the significant role of self and cross-diffusion coefficients and other important parameters of the system. Various contour pictures of spatial patterns through Turing instability are portrayed and analysed in order to substantiate the applicability of the present model. Finally, the paper ends with an extended discussion of biological implications of our findings.
In the current paper, we propose a two-prey-one-predator system with Holling type II predation fu... more In the current paper, we propose a two-prey-one-predator system with Holling type II predation functional response where prey species consumes the remains of the other prey species' carcass given by their predator. Mathematical analysis of the proposed model equations with regard to non-negative invariance including stability and bifurcation are carried out. Next, we extend the deterministic system to a stochastic system by incorporating the Gaussian white noise terms in the growth equations of both prey and predator species. We determine fluctuation intensities for the stochastic dynamical system by using Laplace methods. Numerical simulations are exhibited to justify the analytical findings.
Original article: Bifurcation analysis of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and strong Allee effect
Mathematics and Computers in Simulation, Mar 1, 2014
Your article is protected by copyright and all rights are held exclusively by Springer Science +B... more Your article is protected by copyright and all rights are held exclusively by Springer Science +Business Media Dordrecht. This e-offprint is for personal use only and shall not be selfarchived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com".
Mean Square Stability in a Modified Leslie-Gower And. Holling-Type II Predator-Prey Model
ABSTRACT Of concern in the paper is a Holling-Tanner predator-prey model with modi¯ed version of ... more ABSTRACT Of concern in the paper is a Holling-Tanner predator-prey model with modi¯ed version of the Leslie-Gower functional response. Dynamical behaviours such as stability, permanence and Hopf bifurcation have been carried out deterministically. Using the normal form theory and center manifold theorem, the explicit formulae determining the stability and direction of Hopf bifurcation have been derived . The deterministic model is extended to a stochastic one by perturbing the growth equation of prey and predator by white and colored noises and ¯nally the mean square stability of the stochastic model systems is investigated analytically. An extensive quantitative analysis has been performed based on numerical computation so as to validate the applicability of the proposed mathematical model.
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