Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect

Liye Wang; Wenlong Wang; Ruizhi Yang; Liye Wang; Wenlong Wang; Ruizhi Yang

doi:10.3934/math.2023490

AIMS Mathematics

2023, Volume 8, Issue 4: 9716-9739. doi: 10.3934/math.2023490

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Research article

Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect

Department of Mathematics, Northeast Forestry University, Harbin 150040, China

Received: 16 December 2022 Revised: 24 January 2023 Accepted: 02 February 2023 Published: 21 February 2023
MSC : 34K18, 35B32

Since the distribution of plankton is always uneven, the nonlocal phytoplankton competition term indicates the spatial weighted mean of phytoplankton density, which is introduced into the plankton model with toxic substances effect to study the corresponding dynamic behavior. The stability of the positive equilibrium point and the existence of Hopf bifurcations are discussed by analysing the distribution of eigenvalues. The direction and stability of bifurcation periodic solution are researched based on an extended central manifold method and normal theory. Finally, spatially inhomogeneous oscillations are observed in the vicinity of the Hopf bifurcations. Through numerical simulation, we can observe that the system without nonlocal competition term only generates homogeneous periodic solution, and inhomogeneous periodic solution will produce only when both diffusion and nonlocal competition exist simultaneously. We can also see that when the toxin-producing rate of each phytoplankton is in an appropriate range, the system with nonlocal competition generates a stability switch with inhomogeneous periodic solution, when the value of time delay is in a certain interval, then Hopf bifurcations will appear, and with the increase of time delay, the quantity of plankton will eventually become stable.
- plankton,
- nonlocal competition,
- diffusion,
- delay,
- Hopf bifurcation
Citation: Liye Wang, Wenlong Wang, Ruizhi Yang. Stability switch and Hopf bifurcations for a diffusive plankton system with nonlocal competition and toxic effect[J]. AIMS Mathematics, 2023, 8(4): 9716-9739. doi: 10.3934/math.2023490

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Abstract

Since the distribution of plankton is always uneven, the nonlocal phytoplankton competition term indicates the spatial weighted mean of phytoplankton density, which is introduced into the plankton model with toxic substances effect to study the corresponding dynamic behavior. The stability of the positive equilibrium point and the existence of Hopf bifurcations are discussed by analysing the distribution of eigenvalues. The direction and stability of bifurcation periodic solution are researched based on an extended central manifold method and normal theory. Finally, spatially inhomogeneous oscillations are observed in the vicinity of the Hopf bifurcations. Through numerical simulation, we can observe that the system without nonlocal competition term only generates homogeneous periodic solution, and inhomogeneous periodic solution will produce only when both diffusion and nonlocal competition exist simultaneously. We can also see that when the toxin-producing rate of each phytoplankton is in an appropriate range, the system with nonlocal competition generates a stability switch with inhomogeneous periodic solution, when the value of time delay is in a certain interval, then Hopf bifurcations will appear, and with the increase of time delay, the quantity of plankton will eventually become stable.

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