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비만아 부모의 돌봄 경험: q 방법론

Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

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Differential Equations and Dynamical Systems

A recent field manipulation on a terrestrial vertebrate showed that the fear of predators alone altered anti-predator defences to such an extent that it greatly reduced the reproduction of prey. Because fear can evidently affect the populations of terrestrial vertebrates, we proposed a predator-prey model incorporating the cost of fear into prey reproduction. Our mathematical analyses show that high levels of fear (or equivalently strong anti-predator responses) can stabilize the predator-prey system by excluding the existence of periodic solutions. However, relatively low levels of fear can induce multiple limit cycles via subcritical Hopf bifurcations, leading to a bi-stability phenomenon. Compared to classic predator-prey models which ignore the cost of fear where Hopf bifurcations are typically supercritical, Hopf bifurcations in our model can be both supercritical and subcritical by choosing different sets of parameters. We conducted numerical simulations to explore the relationships between fear effects and other biologically related parameters (e.g. birth/death rate of adult prey), which further demonstrate the impact that fear can have in predator-prey interactions. For example, we found that under the conditions of a Hopf bifurcation, an increase in the level of fear may alter the direction of Hopf bifurcation from supercritical to subcritical when the birth rate of prey increases accordingly. Our simulations also show that the prey is less sensitive in perceiving predation risk with increasing birth rate of prey or increasing death rate of predators, but demonstrate that animals will mount stronger anti-predator defences as the attack rate of predators increases.

Modelling the fear effect in predator–prey interactions

The stability and bifurcation analysis for a predator–prey system with the nonlinear Michaelis–Menten type predator harvesting are taken into account. The existence and stability of possible equilibria are investigated. Specially, the stability of some positive equilibria is determined by using numerical simulation method due to the fact that the expressions of determinant and trace of the Jacobian matrix at these equilibria are very complex. The rigorous mathematical proofs of the existence of saddle–node bifurcation and transcritical bifurcation are derived with the help of Sotomayor’s theorem. Furthermore, in order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Choosing two parameters of the system as bifurcation parameters, we prove that the system exhibits Bogdanov–Takens bifurcation of codimension 2 by calculating a universal unfolding near the cusp. Numerical simulations are carried out to demonstrate the validity of theoretical results. Our research will be useful for understanding the dynamic complexity of ecosystems or physical systems when there is the nonlinear Michaelis–Menten type harvesting effect on predator population. This kind of nonlinear harvesting is more realistic and reasonable than the model with constant-yield harvesting and constant-effort harvesting. It can be thought as a supplement to existing literature on the dynamics of this system, since there is little literature involved in nonlinear type harvesting for the system up to now.

Stability and bifurcation analysis in a predator–prey system with Michaelis–Menten type predator harvesting

https://www.math.miami.edu/~ruan/MyPapers/HuangRuanSong-JDE2014.pdf

Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response☆

In this paper we study the effect of constant-yield predator harvesting on the dynamics
of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case)
of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node
bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical
Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are
different parameter values for which the model has a homoclinic loop or two limit cycles.
It is also proven that there exists a critical harvesting value such that the predator specie goes extinct
for all admissible initial densities of both species when the harvest rate is greater than the critical value.
These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator
harvesting and the initial densities of both species and it requires careful management in the applied conservation
and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens
bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic
loop and an unstable limit cycle, and a stable limit cycle
enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical
analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results
reveal far richer and much more complex dynamics compared to the model without harvesting
or with only constant-yield prey harvesting.

/pdf/bifurcation-analysis-in-a-predator-prey-model-with-constant-1xykwls98p.pdf

Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate

Bifurcations in a discrete predator–prey model with nonmonotonic functional response

In this paper we study bifurcations in predator-prey systems with seasonal prey harvesting. First, when the seasonal harvesting reduces to constant yield, it is shown that various kinds of bifurcat...

Jicai Huang

Papers

Bifurcations in a predator–prey system of Leslie type with generalized Holling type III functional response☆

Bifurcation analysis in a predator-prey model with constant-yield predator harvesting

Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate

Bifurcations in a discrete predator–prey model with nonmonotonic functional response

Bifurcations of Invariant Tori in Predator-Prey Models with Seasonal Prey Harvesting