scholarly journals Numerical bifurcation and stability analysis of an predator-prey system with generalized Holling type III functional response

2018 ◽  
Vol 36 (3) ◽  
pp. 89-102
Author(s):  
Z. Lajmiri ◽  
Reza Khoshsiar Ghaziani ◽  
M. Guasemi
Keyword(s):  
Functional Response ◽  
Bifurcation Analysis ◽  
Continuation Method ◽  
Numerical Continuation ◽  
Hopf Bifurcation Point ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Bifurcation Phenomena ◽  
Stable Periodic ◽  
Numerical Bifurcation

We perform a bifurcation analysis of a predator-prey model with Holling functional response. The analysis is carried out both analytically and numerically. We use dynamical toolbox MATCONT to perform numerical bifurcation analysis. Our bifurcation analysis of the model indicates that it exhibits numerous types of bifurcation phenomena, including fold, subcritical Hopf, cusp, Bogdanov-Takens. By starting from a Hopf bifurcation point, we approximate limit cycles which are obtained, step by step, using numerical continuation method and compute orbitally asymptotically stable periodic orbits.

Stability and Bifurcation Analysis in a Nonlinear Harvested Predator–Prey Model with Simplified Holling Type IV Functional Response

2020 ◽  
Vol 30 (14) ◽  
pp. 2050205
Author(s):  
Zuchong Shang ◽  
Yuanhua Qiao ◽  
Lijuan Duan ◽  
Jun Miao
Keyword(s):  
Functional Response ◽  
Limit Cycle ◽  
Bifurcation Analysis ◽  
Homoclinic Loop ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Prey Model ◽  
The Stability ◽  
Parameter Values

In this paper, a type of predator–prey model with simplified Holling type IV functional response is improved by adding the nonlinear Michaelis–Menten type prey harvesting to explore the dynamics of the predator–prey system. Firstly, the conditions for the existence of different equilibria are analyzed, and the stability of possible equilibria is investigated to predict the final state of the system. Secondly, bifurcation behaviors of this system are explored, and it is found that saddle-node and transcritical bifurcations occur on the condition of some parameter values using Sotomayor’s theorem; the first Lyapunov constant is computed to determine the stability of the bifurcated limit cycle of Hopf bifurcation; repelling and attracting Bogdanov–Takens bifurcation of codimension 2 is explored by calculating the universal unfolding near the cusp based on two-parameter bifurcation analysis theorem, and hence there are different parameter values for which the model has a limit cycle, or a homoclinic loop; it is also predicted that the heteroclinic bifurcation may occur as the parameter values vary by analyzing the isoclinic of the improved system. Finally, numerical simulations are done to verify the theoretical analysis.

Bifurcations in a Seasonally Forced Predator–Prey Model with Generalized Holling Type IV Functional Response

2016 ◽  
Vol 26 (12) ◽  
pp. 1650203 ◽  
Author(s):  
Jingli Ren ◽  
Xueping Li
Keyword(s):  
Periodic Solutions ◽  
Functional Response ◽  
Complex Dynamics ◽  
Phase Portraits ◽  
Bifurcation Diagrams ◽  
Chaotic Motions ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Type Iv ◽  
Numerical Bifurcation

A seasonally forced predator–prey system with generalized Holling type IV functional response is considered in this paper. The influence of seasonal forcing on the system is investigated via numerical bifurcation analysis. Bifurcation diagrams for periodic solutions of periods one and two, containing bifurcation curves of codimension one and bifurcation points of codimension two, are obtained by means of a continuation technique, corresponding to different bifurcation cases of the unforced system illustrated in five bifurcation diagrams. The seasonally forced model exhibits more complex dynamics than the unforced one, such as stable and unstable periodic solutions of various periods, stable and unstable quasiperiodic solutions, and chaotic motions through torus destruction or cascade of period doublings. Finally, some phase portraits and corresponding Poincaré map portraits are given to illustrate these different types of solutions.

scholarly journals BIFURCATION ANALYSIS OF A DIFFUSIVE PREDATOR-PREY MODEL WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE

2020 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Qiannan Song ◽  
◽  
Ruizhi Yang ◽  
Chunrui Zhang ◽  
Lei Wang ◽  
...  
Keyword(s):  
Functional Response ◽  
Bifurcation Analysis ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model
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