Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

Xun Cao; Weihua Jiang

doi:10.3934/dcdsb.2020069

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2020, Volume 25, Issue 9: 3461-3489. Doi: 10.3934/dcdsb.2020069

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Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition

Xun Cao and
Weihua Jiang^,

School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

^* Corresponding author: Weihua Jiang
^* Corresponding author: Weihua Jiang

Received: July 2019

Revised: October 2019

Early access: April 17, 2020

Published online: April 17, 2020

The authors are supported by the National Natural Science Foundation of China (No.11871176).

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Abstract

A diffusive predator-prey model with nonlocal prey competition and the homogeneous Neumann boundary conditions is considered, to explore the effects of nonlocal reaction term. Firstly, conditions of the occurrence of Hopf, Turing, Turing-Turing and double zero bifurcations, are established. Then, several concise formulas of computing normal form at a double zero singularity for partial functional differential equations, are provided. Next, via analyzing normal form derived by utilizing these formulas, we find that diffusive predator-prey system admits interesting spatiotemporal dynamics near the double zero singularity, like tristable phenomenon that a stable spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like which is unstable in model without nonlocal competition and also greatly different from these with the shape of $ \cos\omega_0 t+\cos\frac{x}{l}- $like resulting from Turing-Hopf bifurcation, coexists with a pair of spatially inhomogeneous steady states with the shape of $ \cos\frac{x}{l}- $like. At last, numerical simulations are shown to support theory analysis. These investigations indicate that nonlocal reaction term could stabilize spatially inhomogeneous periodic solutions with the shape of $ \cos\omega_0 t\cos\frac{kx}{l}- $like for reaction-diffusion systems subject to the homogeneous Neumann boundary conditions.

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Mathematics Subject Classification: Primary: 35B32, 35B35; Secondary: 35B36.

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Figure 4. For $ \left(d_1,c\right) = (0.91475,0.018418)\in \mathcal{D}_5 $, a large spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like and a pair of spatially inhomogeneous steady states with the shape of $ \cos\frac{x}{l}- $like, are stable

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Figure 1. Hopf bifurcation curve, Turing bifurcation curves and bifurcation set (a), and local bifurcation set in $ d_1\text{-}c $ plane and phase portraits (b)

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Figure 2. For $ \left(d_1,c\right) = (1.17475,0.017226)\in \mathcal{D}_1 $, the coexistence $ E_* $ is stable. And, the initial values are $ u(0,x) = 0.61803-0.1\cos \frac{x}{4}, v(0,x) = 0.61803-0.1\cos \frac{x}{4} $

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Figure 3. For $ \left(d_1,c\right) = (0.97475,0.017776)\in \mathcal{D}_3 $, a large spatially inhomogeneous periodic solution with the shape of $ \cos\omega_0 t\cos\frac{x}{l}- $like, is stable, where $ \frac{2\pi}{\omega_0} $ is the temporal period. The initial values are $ u(0,x) = 0.61803-0.1\cos \frac{x}{4}, v(0,x) = 0.61803-0.1\cos \frac{x}{4} $

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Figure 5. For $ \left(d_1,c\right) = (1.17475,0.013226)\in \mathcal{D}_6 $, a pair of spatially inhomogeneous steady states with the shape of $ \cos\frac{x}{l}- $like, are stable

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Figure 6. Numerical simulations show the differences between $ \cos\omega_0 t\cos \frac{x}{l} $ and $ \cos\omega_0 t+\cos \frac{x}{l} $, where $ \omega_0 = 0.1\pi,l = 4 $

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