Global existence and stability of the classical solution to a density-dependent prey-predator model with indirect prey-taxis

Yong Luo; Yong Luo

doi:10.3934/mbe.2021331

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 5: 6672-6699. doi: 10.3934/mbe.2021331

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Global existence and stability of the classical solution to a density-dependent prey-predator model with indirect prey-taxis

Yong Luo ^,

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong

Received: 07 June 2021 Accepted: 25 July 2021 Published: 09 August 2021

We study the existence of global unique classical solution to a density-dependent prey-predator population system with indirect prey-taxis effect. With two Lyapunov functions appropriately constructed, we then show that the solution can asymptotically approach prey-only state or coexistence state of the system under suitable conditions. Moreover, linearized analysis on the system at these two constant steady states shows their linear instability criterion. By numerical simulation we find that some density-dependent prey-taxis and predators' diffusion may either flatten the spatial one-dimensional patterns which exist in non-density-dependent case, or break the spatial two-dimensional distribution similarity which occurs in non-density-dependent case between predators and chemoattractants (released by prey).
- density-dependent,
- indirect prey-taxis,
- Lyapunov function,
- asymptotical stability,
- nonlinear semigroup
Citation: Yong Luo. Global existence and stability of the classical solution to a density-dependent prey-predator model with indirect prey-taxis[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 6672-6699. doi: 10.3934/mbe.2021331

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Abstract

We study the existence of global unique classical solution to a density-dependent prey-predator population system with indirect prey-taxis effect. With two Lyapunov functions appropriately constructed, we then show that the solution can asymptotically approach prey-only state or coexistence state of the system under suitable conditions. Moreover, linearized analysis on the system at these two constant steady states shows their linear instability criterion. By numerical simulation we find that some density-dependent prey-taxis and predators' diffusion may either flatten the spatial one-dimensional patterns which exist in non-density-dependent case, or break the spatial two-dimensional distribution similarity which occurs in non-density-dependent case between predators and chemoattractants (released by prey).

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