On a numerical bifurcation analysis of a particle reaction-diffusion model for a motion of two self-propelled disks

Japan Journal of Industrial and Applied Mathematics ◽

10.1007/s13160-021-00498-4 ◽

2022 ◽

Author(s):

Yusuke Yasugahira ◽

Masaharu Nagayama

Keyword(s):

Collective Motion ◽

Global Bifurcation ◽

Experimental System ◽

Reaction Diffusion ◽

Bifurcation Structure ◽

Bifurcation Method ◽

Bifurcation Phenomena ◽

Reaction Diffusion Model ◽

Irregular Motion ◽

Numerical Bifurcation

AbstractTheoretical analysis using mathematical models is often used to understand a mechanism of collective motion in a self-propelled system. In the experimental system using camphor disks, several kinds of characteristic motions have been observed due to the interaction of two camphor disks. In this paper, we understand the emergence mechanism of the motions caused by the interaction of two self-propelled bodies by analyzing the global bifurcation structure using the numerical bifurcation method for a mathematical model. Finally, it is also shown that the irregular motion, which is one of the characteristic motions, is chaotic motion and that it arises from periodic bifurcation phenomena and quasi-periodic motions due to torus bifurcation.

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The influence of density in population dynamics with strong and weak Allee effect

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-021-00114-x ◽

2021 ◽

Vol 29 (1) ◽

Author(s):

Kamrun Nahar Keya ◽

Md. Kamrujjaman ◽

Md. Shafiqul Islam

Keyword(s):

Population Dynamics ◽

Allee Effect ◽

Global Bifurcation ◽

Reaction Diffusion ◽

Logistic Growth ◽

Allee Effects ◽

Reaction Diffusion Model ◽

Positive Steady States ◽

The Stability ◽

The Impact

AbstractIn this paper, we consider a reaction–diffusion model in population dynamics and study the impact of different types of Allee effects with logistic growth in the heterogeneous closed region. For strong Allee effects, usually, species unconditionally die out and an extinction-survival situation occurs when the effect is weak according to the resource and sparse functions. In particular, we study the impact of the multiplicative Allee effect in classical diffusion when the sparsity is either positive or negative. Negative sparsity implies a weak Allee effect, and the population survives in some domain and diverges otherwise. Positive sparsity gives a strong Allee effect, and the population extinct without any condition. The influence of Allee effects on the existence and persistence of positive steady states as well as global bifurcation diagrams is presented. The method of sub-super solutions is used for analyzing equations. The stability conditions and the region of positive solutions (multiple solutions may exist) are presented. When the diffusion is absent, we consider the model with and without harvesting, which are initial value problems (IVPs) and study the local stability analysis and present bifurcation analysis. We present a number of numerical examples to verify analytical results.

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