45 research outputs found
Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II
The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (with- out) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings
Qualitative Analysis of a Modified Leslie-Gower Predator-prey Model with Weak Allee Effect II
The article aims to study a modified Leslie-Gower predator-prey model with Allee effect II, affecting the functional response with the assumption that the extent to which the environment provides protection to both predator and prey is the same. The model has been studied analytically as well as numerically, including stability and bifurcation analysis. Compared with the predator-prey model without Allee effect, it is found that the weak Allee effect II can bring rich and complicated dynamics, such as the model undergoes to a series of bifurcations (Homoclinic, Hopf, Saddle-node and Bogdanov-Takens). The existence of Hopf bifurcation has been shown for models with (without) Allee effect and the local existence and stability of the limit cycle emerging through Hopf bifurcation has also been studied. The phase portrait diagrams are sketched to validate analytical and numerical findings
Codimension two and three bifurcations of a predator–prey system with group defense and prey refuge
A predator–prey system with nonmonotonic functional response and prey refuge is considered. We mainly obtain that the system has the bifurcations of cusp-type codimension two and three, these illustrate that the dynamic behaviors of the model with prey refuge will become more complicated than the system with no refuge
More complex dynamics in a discrete prey-predator model with the Allee effect in prey
In this paper, we revisit a discrete prey-predator model with the Allee effect in prey to find its more complex dynamical properties. After pointing out and correcting those known errors for the local stability of the unique positive fixed point unlike previous studies in which the author only considered the codim 1 Neimark-Sacker bifurcation at the fixed point we focus on deriving many new bifurcation results, namely, the codim 1 transcritical bifurcation at the trivial fixed point the codim 1 transcritical and period-doubling bifurcations at the boundary fixed point the codim 1 period-doubling bifurcation and the codim 2 1:2 resonance bifurcation at the positive fixed point . The obtained theoretical results are also further illustrated via numerical simulations. Some new dynamics are numerically found. Our new results clearly demonstrate that the occurrence of 1:2 resonance bifurcation confirms that this system is strongly unstable, indicating that the predator and the prey will increase rapidly and breakout suddenly
Normal form for singular Bautin bifurcation in a slow-fast system with Holling type III functional response
Over the last few decades, complex oscillations of slow-fast systems have
been a key area of research. In the theory of slow-fast systems, the location
of singular Hopf bifurcation and maximal canard is determined by computing the
first Lyapunov coefficient. In particular, the analysis of canards is based on
the genericity condition that the first Lyapunov coefficient must be non-zero.
This manuscript aims to further extend the results to the case where the first
Lyapunov coefficient vanishes. For that, the analytic expression of the second
Lyapunov coefficient and the investigation of the normal form for codimension-2
singular Bautin bifurcation in a predator-prey system is done by explicitly
identifying the locally invertible parameter-dependent transformations. A
planar slow-fast predator-prey model with Holling type III functional response
is considered here, where the prey population growth is affected by the weak
Allee effect, and the prey reproduces much faster than the predator. Using
geometric singular perturbation theory, normal form theory of slow-fast
systems, and blow-up technique, we provide a detailed mathematical
investigation of the system to show a variety of rich and complex nonlinear
dynamics including but not limited to the existence of canards, relaxation
oscillations, canard phenomena, singular Hopf bifurcation, and singular Bautin
bifurcation. Additionally, numerical simulations are conducted to support the
theoretical findings
Bifurcation analysis of Leslie-Gower predator-prey system with harvesting and fear effect
In the paper, a Leslie-Gower predator-prey system with harvesting and fear effect is considered. The existence and stability of all possible equilibrium points are analyzed. The bifurcation dynamic behavior at key equilibrium points is investigated to explore the intrinsic driving mechanisms of population interaction modes. It is shown that the system undergoes various bifurcations, including transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations. The numerical simulation results show that harvesting and fear effect can seriously affect the dynamic evolution trend and coexistence mode. Furthermore, it is particularly worth pointing out that harvesting not only drives changes in population coexistence mode, but also has a certain degree delay. Finally, it is anticipated that these research results will be beneficial for the vigorous development of predator-prey system
A Holling-Tanner predator-prey model with strong Allee effect
We analyse a modified Holling-Tanner predator-prey model where the predation
functional response is of Holling type II and we incorporate a strong Allee
effect associated with the prey species production. The analysis complements
results of previous articles by Saez and Gonzalez-Olivares (SIAM J. Appl. Math.
59 1867-1878, 1999) and Arancibia-Ibarra and Gonzalez-Olivares (Proc. CMMSE
2015 130-141, 2015)discussing Holling-Tanner models which incorporate a weak
Allee effect. The extended model exhibits rich dynamics and we prove the
existence of separatrices in the phase plane separating basins of attraction
related to co-existence and extinction of the species. We also show the
existence of a homoclinic curve that degenerates to form a limit cycle and
discuss numerous potential bifurcations such as saddle-node, Hopf, and
Bogadonov-Takens bifurcations
Dynamic analysis of a Leslie-Gower predator-prey model with the fear effect and nonlinear harvesting
In this paper, we investigate the stability and bifurcation of a Leslie-Gower predator-prey model with a fear effect and nonlinear harvesting. We discuss the existence and stability of equilibria, and show that the unique equilibrium is a cusp of codimension three. Moreover, we show that saddle-node bifurcation and Bogdanov-Takens bifurcation can occur. Also, the system undergoes a degenerate Hopf bifurcation and has two limit cycles (i.e., the inner one is stable and the outer is unstable), which implies the bistable phenomenon. We conclude that the large amount of fear and prey harvesting are detrimental to the survival of the prey and predator