Journal
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
Volume 41, Issue 13, Pages 5178-5189Publisher
WILEY
DOI: 10.1002/mma.5066
Keywords
fractional-order derivative; Hopf bifurcation; predator-prey model; stability
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Funding
- National Natural Science Foundation of China [61563033, 11563005, 11461044]
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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.
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