Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal

Xiaoli Wang; Guohong Zhang

doi:10.3934/dcdsb.2020295

Article Contents

2021, Volume 26, Issue 8: 4459-4477. Doi: 10.3934/dcdsb.2020295

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Bifurcation analysis of a general activator-inhibitor model with nonlocal dispersal

Xiaoli Wang^, and
Guohong Zhang

School of Mathematics and Statistics, Southwest University, Chongqing 400715, China

^*Corresponding author: Xiaoli Wang
^*Corresponding author: Xiaoli Wang

Received: April 30, 2020

Revised: July 31, 2020

Early access: October 2020

Published: August 2021

X.-L. Wang is partially supported by grants from National Science Foundation of China (11701472, 11871060), Fundamental Research Funds for the Central Universities(XDJK2020B050); G.-H. Zhang is partially supported by grants from National Science Foundation of China (11871403)

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Abstract

In this paper, we are mainly concerned with the effect of nonlocal diffusion and dispersal spread on bifurcations of a general activator-inhibitor system in which the activator has a nonlocal dispersal. We find that spatially inhomogeneous patterns always exist if the dispersal rate of the activator is sufficiently small, while a larger dispersal spread and an increase of the activator diffusion inhibit the formation of spatial patterns. Compared with the "spatial averaging" nonlocal dispersal model, our model admits a larger parameter region supporting pattern formations, which is also true if compared with the local reaction-diffusion one when the dispersal spread is small. We also study the existence of nonconstant positive steady states through bifurcation theory and find that there could exist finite or infinite steady state bifurcation points of the inhibitor diffusion constant. As an example of our results, we study a water-biomass model with nonlocal dispersal of plants and show that the water and plant distributions could be inphase and antiphase.

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Mathematics Subject Classification: Primary: 35K57, 35B32; Secondary: 35B35, 35B36, 37L15, 92C15.

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Figure 1. The kernel (1.5) with $ l = 20 $, and $ d_w = 1 $ (blue), $ d_w = 3 $ (red)

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Figure 2. (a): Parameter space for Turing instability when the activator has a nonlocal dispersal. (b): The effect of nonlocal dispersal on the parameter region of Turing instability. Here, $ A = 1, B = 0.45 $, $ d_w = 1(cyan) $, and $ d_w = 3(green) $

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Figure 3. Graph of $ D(d_v,p) $ when $ d_u<f_u $, $ d_u = f_u $ and $ d_u>f_u $. Here, $ A = 1, B = 0.45, l = 20 $, $ d_w = 1(cyan), 3(green) $, and $ d_u = 0.2 $ in $ (a) $, $ d_u = 0.45 $ in $ (b) $, $ d_u = 1 $ in $ (c) $

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Figure 4. $ (a), (c), (e): $ Spiky or bump pattern formation of model (4.2). $ (b), (d), (f): $ Antiphase or inphase distributions of plant $ u $ and water $ v $ of model (4.2). Here $ A = 1, B = 0.45, d_v = 30, d_w = 1 $ and $ d_u = 0.2, 0.45, 1 $

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Figure 5. $ (a) $ Turing instability region of the classical reaction-diffusion model. $ (b) $ Turing instability region of the "spatial averaging" nonlocal dispersal model when the inhibitor has a nonlocal dispersal. $ (c) $ Turing instability region of the nonlocal model when the inhibitor has a nonlocal dispersal with the kernel function (1.5). Here, the constant equilibrium solution is stable in region "S" and the spatial scale could induce instability in region "U"

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