Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity

Qingyan Shi; Junping Shi; Yongli Song

doi:10.3934/dcdsb.2018182

Article Contents

2019, Volume 24, Issue 2: 467-486. Doi: 10.3934/dcdsb.2018182

This issue Previous Article Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains Next Article Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system

Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity

1.
School of Mathematical Sciences, Tongji University, Shanghai 200092, China
2.
Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, USA
3.
Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 311121, China

* Corresponding author: Junping Shi
* Corresponding author: Junping Shi

Received: November 2017

Revised: January 2018

Early access: May 2018

Published: February 2019

Partially supported by a grant from China Scholarship Council, US-NSF grant DMS-1715651, National Natural Science Foundation of China (No.11571257), Science and Technology Commission of Shanghai Municipality (No. 18dz2271000)

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Abstract

In this paper, we study the Hopf bifurcation and spatiotemporal pattern formation of a delayed diffusive logistic model under Neumann boundary condition with spatial heterogeneity. It is shown that for large diffusion coefficient, a supercritical Hopf bifurcation occurs near the non-homogeneous positive steady state at a critical time delay value, and the dependence of corresponding spatiotemporal patterns on the heterogeneous resource function is demonstrated via numerical simulations. Moreover, it is proved that the heterogeneous resource supply contributes to the increase of the temporal average of total biomass of the population even though the total biomass oscillates periodically in time.

Keywords:

Mathematics Subject Classification: Primary: 35K57; Secondary: 35B10, 35B32, 35B36, 35R10, 92B05, 92D40.

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Figure 1. The non-homogeneous steady states of Eq (2) when $m(x)$ is a cosine function: (a) $m(x) = \cos(x)+2$; (b) $m(x) = \cos(1.5x)+2$; (c) $m(x) = \cos(2x)+2$. Here $d = 2$ (which is equivalent to $\lambda = 0.5$), $\tau = 0.71<\tau_{0\lambda }\approx0.785$ and initial value $u_{0} = 2$ for all three cases, and the solution converges to the non-homogeneous steady state

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Figure 2. The non-homogeneous steady states of Eq. (2) when $m(x)$ is a sine function: (a) $m(x) = \sin(x)+2$; (b) $m(x) = \sin(1.5x)+1.788$; (c) $m(x) = \sin(2x)+2$. The parameters are the same as in Figure 1, and here $\tau = 0.73<\tau_{0\lambda }\approx0.785$. The solution converges to the non-homogeneous steady state for each case

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Figure 3. The non-homogeneous steady states of Eq. (2) when $m(x)$ is a monotone linear function: (a) $m(x) = 1+x/\pi$; (b) $m(x) = 3-x/\pi$. Here $d = 2$ and $\tau = 0.73<\tau_{0\lambda }$. The solution converges to the positive monotone steady state

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Figure 4. The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that $m(x)$ is cosine function: (a) $m(x) = \cos(x)+2$; (b) $m(x) = \cos(1.5x)+2$; (c) $m(x) = \cos(2x)+2$. Here $d = 2$, and $\tau = 0.82>\tau_{0\lambda }\approx 0.785$

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Figure 5. The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that $m(x)$ is sine function: (a) $m(x) = \sin(x)+2$; (b) $m(x) = \sin(1.5x)+1.788$; (c) $m(x) = \sin(2x)+2$. Here $d = 2$, and $\tau = 0.82>\tau_{0\lambda }\approx 0.785$

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Figure 6. The periodic orbits induced by Hopf bifurcation near the non-homogeneous steady state of Eq. (2) for case that $m(x)$ is monotone linear function: (a) $m(x) = 1+x/\pi$; (b) $m(x) = 3-x/\pi$. Here $d = 2$, and $\tau = 0.82>\tau_{0\lambda }\approx 0.785$

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