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# Propagation dynamics of a nonlocal time-space periodic reaction-diffusion model with delay

• * Corresponding author: Zhi-Cheng Wang

Both authors are supported by NNSF of China (12071193, 11731005) and NSF of Gansu Province of China (21JR7RA535)

• This paper is concerned with a nonlocal time-space periodic reaction diffusion model with age structure. We first prove the existence and global attractivity of time-space periodic solution for the model. Next, by a family of principal eigenvalues associated with linear operators, we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases. Furthermore, we introduce a notion of transition semi-waves for the model, and then by constructing appropriate upper and lower solutions, and using the results of the asymptotic speed of spread, we show that transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed, and transition semi-waves do not exist anymore when their wave speed is less than the critical speed. It turns out that the asymptotic speed of spread coincides with the critical wave speed of transition semi-waves in the non-monotone case. In addition, we show that the obtained transition semi-waves are actually transition waves in the monotone case. Finally, numerical simulations for various cases are carried out to support our theoretical results.

Mathematics Subject Classification: Primary: 35B40, 35K57, 35C07; Secondary: 37N25, 92D25.

 Citation: • • Figure 1.  The long-time behaviour of the solution of (68). Changes in color represent changes in height. Baseline parameter values: $D = 0.1,$ $D_j = 0.1,$ $\mu_j = 0.1,$ $\tau = 1,$ $p(x) = 0.5+0.2\cos(x),$ $q = 0.3$ and the periodicity of space $L = 2\pi.$ The right panel is the two-dimensional projection of the left panel onto the $xt$-plane

Figure 2.  The long-time behaviour of the solution of (69). Changes in color represent changes in height. Baseline parameter values: $D = 0.1,$ $D_j = 0.1,$ $\mu_j = 0.1,$ $\tau = 1,$ $p_1(t) = 0.5+0.2\cos(\frac{t}{2}),$ $q = 0.3$ and the periodicity of time $T = 4\pi.$ The right panel is the two-dimensional projection of the left panel onto the $xt$-plane

Figure 3.  The long-time behaviour of the solution of (70). Changes in color represent changes in height. Baseline parameter values: $D = 0.1,$ $D_j = 0.1,$ $\mu_j = 0.1,$ $\tau = 1,$ $p_2(t, x) = 0.7+0.2\cos(\frac{t}{2})+0.2\cos(x),$ $q = 0.3,$ the periodicity of time $T = 4\pi$ and the periodicity of space $L = 2\pi.$ The right panel is the two-dimensional projection of the left panel onto the $xt$-plane

Figure 4.  The shapes of the birth function $b(x, u) = p(x)ue^{\bar{p}(x)u}$ and death function $d(x, u) = qu$ for equation (71), where $p(x) = 12+2\sin\left(x\right),$ $\bar{p}(x) = -0.7-0.2\cos\left(x\right),$ $q = 0.2.$ Changes in color represent changes in height. The right panel is the two-dimensional projection of the left panel onto the $uoz$-plane

Figure 5.  The shapes of the birth function $b(t, u) = p_1(t)ue^{\bar{p}_1(t)u}$ and death function $d(x, u) = q_1u$ for equation (72), where $p(x) = 12+2\cos\left(t\right),$ $\bar{p}(x) = -0.7-0.2\sin\left(t\right),$ $q = 0.4.$ Changes in color represent changes in height. The right panel is the two-dimensional projection of the left panel onto the $uoz$-plane

Figure 6.  The long-time behaviour of the solution of (71). Changes in color represent changes in height. Baseline parameter values: $D = 0.1,$ $D_j = 0.1,$ $\mu_j = 0.1,$ $\tau = 10,$ $p(x) = 12+2\sin\left(x\right),$ $\bar{p}(x) = -0.7-0.2\cos\left(x\right),$ $q = 0.2$ and the periodicity of space $L = 2\pi.$ The right panel is the two-dimensional projection of the left panel onto the $xt$-plane

Figure 7.  The long-time behaviour of the solution of (72). Changes in color represent changes in height. Baseline parameter values: $D = 0.1,$ $D_j = 0.1,$ $\mu_j = 0.1,$ $\tau = 10,$ $p_1(t) = 12+2\cos\left(t\right),$ $\bar{p}_1(t) = -0.7-0.2\sin\left(t\right),$ $q_1 = 0.4$ and the periodicity of time $T = 2\pi.$ The right panel is the two-dimensional projection of the left panel onto the $xt$-plane

Figure 8.  The long-time behaviour of the solution of (73). Changes in color represent changes in height. Baseline parameter values: $D = 0.1,$ $D_j = 0.1,$ $\mu_j = 0.1,$ $\tau = 10,$ $p_2(t, x) = 13+2\cos\left(\sqrt{2}t\right)+\sin\left(x\right),$ $\bar{p}_2(t, x) = -1.1-0.2\cos\left(x\right),$ $q = 0.3,$ the periodicity of time $T = \sqrt{2}\pi$ and the periodicity of space $L = 2\pi.$ The right panel is the two-dimensional projection of the left panel onto the $xt$-plane

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