# American Institute of Mathematical Sciences

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February  2019, 24(2): 467-486. doi: 10.3934/dcdsb.2018182

## 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

Received  November 2017 Revised  January 2018 Published  June 2018

Fund Project: 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).

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.

Citation: Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
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##### References:
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
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
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
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 $\tau = 0.82>\tau_{0\lambda }\approx 0.785$
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 $\tau = 0.82>\tau_{0\lambda }\approx 0.785$
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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