Simple dynamics in non-monotone Kolmogorov systems

In this paper we analyse the global dynamical behaviour of some classical models in the plane. Informally speaking we prove that the folkloric criteria based on the relative positions of the nullclines for Lotka–Volterra systems are also valid in a wide class of discrete systems. The method of proof consists of dividing the plane into suitable positively invariant regions and applying the theory of translation arcs in a subtle manner. Our approach allows us to extend several results of the theory of monotone systems to nonmonotone systems. Applications in models with weak Allee effect, population models for pioneer-climax species, and predator–prey systems are given.

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Bogdanov–Takens bifurcation analysis of a delayed predator-prey system with double Allee effect

Nonlinear Dynamics ◽

10.1007/s11071-021-06338-x ◽

2021 ◽

Author(s):

Jianfeng Jiao ◽

Can Chen

Keyword(s):

Bifurcation Analysis ◽

Allee Effect ◽

Predator Prey ◽

Prey System

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The dynamics of a Leslie type predator–prey model with fear and Allee effect

Advances in Difference Equations ◽

10.1186/s13662-021-03490-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

S. Vinoth ◽

R. Sivasamy ◽

K. Sathiyanathan ◽

Bundit Unyong ◽

Grienggrai Rajchakit ◽

...

Keyword(s):

Local Stability ◽

Allee Effect ◽

Jacobian Matrix ◽

Predator Prey ◽

Predator Prey Model ◽

Fear Effect ◽

Prey Model ◽

Lyapunov Coefficient ◽

Points Of View ◽

Two Parameter

AbstractIn this article, we discuss the dynamics of a Leslie–Gower ratio-dependent predator–prey model incorporating fear in the prey population. Moreover, the Allee effect in the predator growth is added into account from both biological and mathematical points of view. We explore the influence of the Allee and fear effect on the existence of all positive equilibria. Furthermore, the local stability properties and possible bifurcation behaviors of the proposed system about positive equilibria are discussed with the help of trace and determinant values of the Jacobian matrix. With the help of Sotomayor’s theorem, the conditions for existence of saddle-node bifurcation are derived. Also, we show that the proposed system admits limit cycle dynamics, and its stability is discussed with the value of first Lyapunov coefficient. Moreover, the numerical simulations including phase portrait, one- and two-parameter bifurcation diagrams are performed to validate our important findings.

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Complex dynamics and coexistence of period-doubling and period-halving bifurcations in an integrated pest management model with nonlinear impulsive control

Advances in Difference Equations ◽

10.1186/s13662-020-02971-9 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Changtong Li ◽

Sanyi Tang ◽

Robert A. Cheke

Keyword(s):

Periodic Solution ◽

Integrated Pest Management ◽

Pest Management ◽

Pest Control ◽

Impulsive Control ◽

Period Doubling ◽

Dynamical Behaviour ◽

Period Doubling Bifurcation ◽

Predator Prey ◽

Predator Prey Model

Abstract An expectation for optimal integrated pest management is that the instantaneous numbers of natural enemies released should depend on the densities of both pest and natural enemy in the field. For this, a generalised predator–prey model with nonlinear impulsive control tactics is proposed and its dynamics is investigated. The threshold conditions for the global stability of the pest-free periodic solution are obtained based on the Floquet theorem and analytic methods. Also, the sufficient conditions for permanence are given. Additionally, the problem of finding a nontrivial periodic solution is confirmed by showing the existence of a nontrivial fixed point of the model’s stroboscopic map determined by a time snapshot equal to the common impulsive period. In order to address the effects of nonlinear pulse control on the dynamics and success of pest control, a predator–prey model incorporating the Holling type II functional response function as an example is investigated. Finally, numerical simulations show that the proposed model has very complex dynamical behaviour, including period-doubling bifurcation, chaotic solutions, chaos crisis, period-halving bifurcations and periodic windows. Moreover, there exists an interesting phenomenon whereby period-doubling bifurcation and period-halving bifurcation always coexist when nonlinear impulsive controls are adopted, which makes the dynamical behaviour of the model more complicated, resulting in difficulties when designing successful pest control strategies.

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