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Volume 10
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Article

Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities

by
Ruizhi Yang
,
Qiannan Song
and
Yong An
*
Department of Mathematics, Northeast Forestry University, Harbin 150040, China
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(1), 17; https://doi.org/10.3390/math10010017
Submission received: 19 November 2021 / Revised: 15 December 2021 / Accepted: 18 December 2021 / Published: 21 December 2021
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Abstract

In this paper, a diffusive predator–prey system with a functional response that increases in both predator and prey densities is considered. By analyzing the characteristic roots of the partial differential equation system, the Turing instability and Hopf bifurcation are studied. In order to consider the dynamics of the model where the Turing bifurcation curve and the Hopf bifurcation curve intersect, we chose the diffusion coefficients d1 and β as bifurcating parameters. In particular, the normal form of Turing–Hopf bifurcation was calculated so that we could obtain the phase diagram. For parameters in each region of the phase diagram, there are different types of solutions, and their dynamic properties are extremely rich. In this study, we have used some numerical simulations in order to confirm these ideas.
Keywords: predator–prey model; Turing–Hopf bifurcation; Hopf bifurcation; Turing instability predator–prey model; Turing–Hopf bifurcation; Hopf bifurcation; Turing instability

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MDPI and ACS Style

Yang, R.; Song, Q.; An, Y. Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities. Mathematics 2022, 10, 17. https://doi.org/10.3390/math10010017

AMA Style

Yang R, Song Q, An Y. Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities. Mathematics. 2022; 10(1):17. https://doi.org/10.3390/math10010017

Chicago/Turabian Style

Yang, Ruizhi, Qiannan Song, and Yong An. 2022. "Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities" Mathematics 10, no. 1: 17. https://doi.org/10.3390/math10010017

APA Style

Yang, R., Song, Q., & An, Y. (2022). Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities. Mathematics, 10(1), 17. https://doi.org/10.3390/math10010017

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