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| The Effect of the Fear Factor and the Prudence Level on the Dynamics of a Predator-Prey Model |
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| KeyWord:Fear factor, prudent level, predator-prey system, stability, Hopf bifurcation |
| Author Name | Affiliation | | Qingxiang Lin | School of Mathematics and Statistics, Fujian Normal University, Fuzhou, Fujian 350117, P. R. China | | Wensheng Yang | School of Mathematics and Statistics, Fujian Normal University, Fuzhou, Fujian 350117, P. R. China
FJKLMAA and Center for Applied Mathematics of Fujian province (FJNU), Fuzhou, Fujian 350117, P. R. China |
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| Abstract: |
| In this paper, we investigate the dynamics of a predator-prey system with the fear factor and the prudence level, and focus on the impact of the fear factor and the prudence level on the population density of predator and prey. We explore the boundedness of solution, analyze the existence and stability of equilibria and obtain the sufficient conditions for the occurrence of Hopf bifurcation. Furthermore, we also apply the Implicit Function Theorem to study how the fear factor and the prudence level influence the population densities of prey and predator. Finally, numerical simulations are conducted to verify the analytical results and expand the theoretical analyses: (i) Under the constant level of fear, appropriate level of prudence helps promote coexistence between prey and predators, as the level of prudence gradually increases, the positive equilibrium state of system may undergo a series of transformations: evolving from a stable spiral sink to an unstable spiral source, and ultimately returning to a stable spiral sink state again. (ii) With a fixed level of prudence, the low level of fear can cause limit cycles through Hopf bifurcation and the high level of fear can promote system stability by eliminating the existence of periodic solutions. These results may enrich the dynamics of predator-prey system. |
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