Dynamical Analysis of Phytoplankton–Zooplankton Interaction Model by Using Deterministic and Stochastic Approach

Dynamical analysis of a predator-prey interaction model with time delay and prey refuge

Nonautonomous Dynamical Systems ◽

10.1515/msds-2018-0011 ◽

2018 ◽

Vol 5 (1) ◽

pp. 138-151 ◽

Cited By ~ 2

Author(s):

Jai Prakash Tripathi ◽

Swati Tyagi ◽

Syed Abbas

Keyword(s):

Hopf Bifurcation ◽

Functional Differential Equations ◽

Interaction Model ◽

Dynamical Analysis ◽

Prey Refuge ◽

Normal Form Theory ◽

Predator Species ◽

Predator Prey ◽

Center Manifold Theorem ◽

Predator Prey Model

AbstractIn this paper, we study a predator-prey model with prey refuge and delay. We investigate the combined role of prey refuge and delay on the dynamical behaviour of the delayed system by incorporating discrete type gestation delay of predator. It is found that Hopf bifurcation occurs when the delay parameter τ crosses some critical value. In particular, it is shown that the conditions obtained for the Hopf bifurcation behaviour are sufficient but not necessary and the prey reserve is unable to stabilize the unstable interior equilibrium due to Hopf bifurcation. In particular, the direction and stability of bifurcating periodic solutions are determined by applying normal form theory and center manifold theorem for functional differential equations. Mathematically, we analyze the effect of increase or decrease of prey reserve on the equilibrium states of prey and predator species. At the end, we perform some numerical simulations to substantiate our analytical findings.

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Stability and Dynamical Analysis of Generalized Tumor-Immune Interaction Model with Conformable Fractional Derivative

10.21203/rs.3.rs-180417/v1 ◽

2021 ◽

Author(s):

Rimpi Pal ◽

Afroz ◽

Ayub Khan ◽

MOHMAD AUSIF PADDER

Keyword(s):

Stability Analysis ◽

Fixed Points ◽

Fractional Order ◽

Interaction Model ◽

Tumor Model ◽

Graphical Analysis ◽

Dynamical Analysis ◽

Complex Behaviour ◽

The Stability ◽

Two Parameters

Abstract Fractional order tumor-immune interaction models are being frequently used for understanding the complex behaviour of immune system and tumor growth. In this paper, a generalized fractional order tumor-immune interaction model has been developed by introducing immunotherapy (IL2) as third variable in the model. The study of generalized model is done by using conformable fractional order derivative. The stability analysis is done for both fractional order tumor model and its conformable fractional order version. By considering some biological fixed points for both versions of the model, the stability analysis around these fixed points shows that both the systems are stable at some fixed point under some stability conditions, which are defined in the model analysis. The numerical and graphical analysis is also done for both the systems by varying two parameters and keeping other parameters fixed for better understanding the dynamics of proposed model.

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