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Dynamics of a free boundary problem modelling species invasion with impulsive harvesting

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  • To understand the role of impulsive harvesting in dynamics of the invasive species, we explore an impulsive logistic equation with free boundaries. The criteria whether the species spreads or vanishes are given, and some sufficient conditions based on threshold values are established. We then discuss the spreading speeds of moving fronts when the species spreads. Our numerical simulations reveal that impulsive harvesting can reduce the spreading speed of the species, and a large impulsive harvesting is unfavorable for persistence of the species. Moreover, when impulsive harvesting is moderate, the species occurs spreading or vanishing depending on its expanding capability or initial number, that is, the species will die out with a small expanding capability or small initial number and spread with a large expanding capability. Note: KYCX22_3446 is added in Acknowledgments.

    Mathematics Subject Classification: Primary: 35R12; 35R35; Secondary: 92D25.


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  • Figure 1.  A simulation of the invasive species without impulses. Graph (a) show that species will persist and spread in the whole space, graph (b) is its sections at $ t = 0, 3, 6, 10 $. Graph (c) is the right view of (a) and projection of $ u $ on the $ t-u- $plane

    Figure 2.  The distribution of the species with a large harvesting rate, where $ H(u) $ takes Beverton-Holt function with $ \beta = 2 $ and $ \alpha = 10 $. The spatial distribution of the species in Graph (a) indicates that the species presents eventual extinction, and the contour one in Graph (b) clearly shows that $ u(t, x) $ decays to zero (the purple line). Graph (c) shows that the number of the species drops sharply when harvesting is implemented every time $ \tau = 2 $

    Figure 3.  The distribution of the species with a small harvesting rate, where $ H(u) $ takes Beverton-Holt function with $ \beta = 8 $ and $ \alpha = 10 $. Graphs (a)-(c) indicate that the species persists eventually. The impact of impulsive harvesting is shown in Graph (c)

    Figure 4.  The distribution of the species when $ H(u) $ is chosen as Beverton-Holt function with $ \beta = 4 $ and $ \alpha = 10 $. The expanding capability $ \mu $ for Graphs (1a)-(1c) and (2a)-(2c) are $ \mu = 1 $ and $ \mu = 20 $, respectively. For the small expanding capacity $ \mu = 1 $, the fact that the species goes to extinction as time increases is shown in Graphs (1a)-(1c). The small expanding capability of the species is exhibited in Graph (1b). The projection of $ u $ on the $ t-u- $plane in Graph (1c) shows that harvesting takes place every time $ \tau = 2 $. For the large expanding capacity $ \mu = 20 $, the species will persist, which is shown in Graphs (2a)-(2c)

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