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Hopf bifurcation of a diffusive Gause‐type predator‐prey model induced by time fractional‐order derivatives
Author(s) -
Yin Hongwei,
Wen Xiaoqing
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5066
Subject(s) - mathematics , fractional calculus , uniqueness , hopf bifurcation , type (biology) , ordinary differential equation , order (exchange) , reaction–diffusion system , stability (learning theory) , bifurcation , population model , mathematical analysis , population , differential equation , nonlinear system , ecology , physics , demography , finance , quantum mechanics , machine learning , sociology , computer science , economics , biology
Since population behaviors possess the characteristic of history memory, we, in this paper, introduce time fractional‐order derivatives into a diffusive Gause‐type predator‐prey model, which is time fractional‐order reaction‐diffusion equations and a generalized form of its corresponding first‐derivative model. For this kind of model, we prove the existence and uniqueness of a global positive solution by using the theory of evolution equations and the comparison principle of time fractional‐order partial differential equations. Besides, we obtain the stability and Hopf bifurcation of the Gause‐type predator‐prey model in the forms of the time fractional‐order ordinary equations and of the time fractional‐order reaction‐diffusion equations, respectively. Our results show that the stable region of the parameters in these 2 models can be enlarged by the time fractional‐order derivatives. Some numerical simulations are made to verify our results.