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We consider a nonlocal diffusive Fisher-KPP model with free boun-daries in time periodic environment. When the growth term is Logistic type, Zhang et al. [DCDS-B 2021] proved that this model admits a unique global solution and its long time behavior is governed by a spreading-vanishing dichotomy. However, when spreading happens, the spreading speed estimates for such free boundary problems remain unsolved. In this paper, we answer this question. By solving a corresponding time-periodic semi-wave problem, we obtain a threshold condition on the kernel function such that the spreading grows linearly in time, and provide a sharp estimate for the spreading speed; when the threshold condition is not satisfied, we observe an accelerating spreading phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, the basic reproduction number (BRN) is explored in terms of a degenerate reaction–diffusion host–pathogen model. The expressions of BRN R0$$ {R}_0 $$ and local basic reproduction number (LBRN) R¯0(x)$$ {\overline{R}}_0(x) $$ for the model are calculated. Some basic properties of R0$$ {R}_0 $$ with respect to the general diffusion rates and the asymptotic profiles of R0$$ {R}_0 $$ with respect to large diffusion rates and small diffusion rates are established by introducing two compact operators H1$$ {H}_1 $$ and H2$$ {H}_2 $$. Our results reveal that R0$$ {R}_0 $$ is the spectral radius of a sum of the product of additive operators and strongly positive compact linear operators with spectral radii one. [ABSTRACT FROM AUTHOR]