Global bifurcation of solutions for a predator-prey model with prey-taxis

2014 ◽  
Vol 38 (3) ◽  
pp. 431-443 ◽  
Author(s):  
Xiaoli Wang ◽  
Wendi Wang ◽  
Guohong Zhang
Keyword(s):  
Global Bifurcation ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Bifurcation Of Solutions

Bifurcations and Pattern Formation in a Predator–Prey Model

2018 ◽  
Vol 28 (11) ◽  
pp. 1850140 ◽  
Author(s):  
Yongli Cai ◽  
Zhanji Gui ◽  
Xuebing Zhang ◽  
Hongbo Shi ◽  
Weiming Wang
Keyword(s):  
Steady State ◽  
Hopf Bifurcation ◽  
Pattern Formation ◽  
Global Bifurcation ◽  
Periodic Pattern ◽  
Spatiotemporal Pattern ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Steady State Bifurcation

In this paper, we investigate the spatiotemporal dynamics of a Leslie–Gower predator–prey model incorporating a prey refuge subject to the Neumann boundary conditions. We mainly consider Hopf bifurcation and steady-state bifurcation which bifurcate from the constant positive steady-state of the model. In the case of Hopf bifurcation, by the center manifold theory and the normal form method, we establish the bifurcation direction and stability of bifurcating periodic solutions; in the case of steady-state bifurcation, by the local and global bifurcation theories, we prove the existence of the steady-state bifurcation, and find that there are two typical bifurcations, Turing bifurcation and Turing–Hopf bifurcation. Via numerical simulations, we find that the model exhibits not only stationary Turing pattern induced by diffusion which is dependent on space and independent of time, but also temporal periodic pattern induced by Hopf bifurcation which is dependent on time and independent of space, and spatiotemporal pattern induced by Turing–Hopf bifurcation which is dependent on both time and space. These results may enrich the pattern formation in the predator–prey model.

scholarly journals A predator-prey model with allee effect and fast strategy evolution dynamics of predators using hawk and dove tactics

2011 ◽  
Vol 50 (1) ◽  
pp. 13-24
Author(s):  
Jitka Kühnová ◽  
Lenka Přibylová
Keyword(s):  
Limit Cycles ◽  
Allee Effect ◽  
Global Bifurcation ◽  
Prey Population ◽  
Bifurcation Of Limit Cycles ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Local Bifurcations ◽  
Prey Model ◽  
Evolution Dynamics

ABSTRACT In this work we present the predator-prey model with Allee effect and Hawk and Dove tactics in fighting over caught prey implemented as fast strategy evolution dynamics. We extend the work of Auger, Parra, Morand and S´anchez (2002) using the prey population embodying Allee effect and analogously to this work we get two connected submodels with polymorphic and monomorphic predator population.We get much richer dynamics, in each submodel we find local bifurcations (saddle-node, supercritical Hopf caused by Allee effect and Bogdanov- -Takens) and a global bifurcation of limit cycles caused by the strategy evolution that is not possible in any of the submodels that can lead to a bluesky extinction of both populations.

scholarly journals Bifurcation analysis of the predator–prey model with the Allee effect in the predator

2021 ◽  
Vol 84 (1-2) ◽  
Author(s):  
Deeptajyoti Sen ◽  
Saktipada Ghorai ◽  
Malay Banerjee ◽  
Andrew Morozov
Keyword(s):  
Allee Effect ◽  
Mathematical Formulation ◽  
Global Bifurcation ◽  
Predator Population ◽  
Stable Limit ◽  
Bifurcation Structure ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
Predator Prey Models

AbstractThe use of predator–prey models in theoretical ecology has a long history, and the model equations have largely evolved since the original Lotka–Volterra system towards more realistic descriptions of the processes of predation, reproduction and mortality. One important aspect is the recognition of the fact that the growth of a population can be subject to an Allee effect, where the per capita growth rate increases with the population density. Including an Allee effect has been shown to fundamentally change predator–prey dynamics and strongly impact species persistence, but previous studies mostly focused on scenarios of an Allee effect in the prey population. Here we explore a predator–prey model with an ecologically important case of the Allee effect in the predator population where it occurs in the numerical response of predator without affecting its functional response. Biologically, this can result from various scenarios such as a lack of mating partners, sperm limitation and cooperative breeding mechanisms, among others. Unlike previous studies, we consider here a generic mathematical formulation of the Allee effect without specifying a concrete parameterisation of the functional form, and analyse the possible local bifurcations in the system. Further, we explore the global bifurcation structure of the model and its possible dynamical regimes for three different concrete parameterisations of the Allee effect. The model possesses a complex bifurcation structure: there can be multiple coexistence states including two stable limit cycles. Inclusion of the Allee effect in the predator generally has a destabilising effect on the coexistence equilibrium. We also show that regardless of the parametrisation of the Allee effect, enrichment of the environment will eventually result in extinction of the predator population.

Sign in / Sign up

Export Citation Format

Share Document