American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022060
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Dynamics of a free boundary problem modelling species invasion with impulsive harvesting

 1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China 2 School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, China

* Corresponding author: Email: zglin68@hotmail.com (Z. Lin)

Received  October 2021 Revised  February 2022 Early access March 2022

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.

Citation: Yue Meng, Jing Ge, Zhigui Lin. Dynamics of a free boundary problem modelling species invasion with impulsive harvesting. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022060
References:
 [1] I. Ahn, S. Baek and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Modelling, 40 (2016), 7082-7101.  doi: 10.1016/j.apm.2016.02.038. [2] R. J. H. Beverton and S. F. Holt, On the dynamics of exploited fish populations, In Fishery Investigations, Ser. II, United Kingdom: Ministry of Agriculture, Fisheries and Food, 19 (1957). [3] J. F. Cao, Y. H. Du, F. Li and W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013. [4] M. Clavero and E. García-Berthou, Invasive species are a leading cause of animal extinctions, Trends in Ecology & Evolution, 20 (2005), 110.  doi: 10.1016/j.tree.2005.01.003. [5] T. A. Crowl, T. O. Crist, R. R. Parmenter, G. Belovsky and A. E. Lugo, The spread of invasive species and infectious disease as drivers of ecosystem change, Frontiers in Ecology & Environment, 6 (2008), 238-246.  doi: 10.1890/070151. [6] M. De la Sen and S. Alonso-Quesada, Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model, Appl. Math. Comput., 218 (2011), 3888-3904.  doi: 10.1016/j.amc.2011.09.036. [7] Y. H. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016. [8] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089. [9] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105. [10] Y. H. Du, L. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dynam. Differential Equations, 30 (2018), 1389-1426.  doi: 10.1007/s10884-017-9614-2. [11] Y. Enatsu, E. Ishiwata and T. Ushijima, Traveling wave solution for a diffusive simple epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 835-850.  doi: 10.3934/dcdss.2020387. [12] M. Fazly, M. Lewis and H. Wang, Analysis of propagation for impulsive reaction-diffusion models, SIAM J. Appl. Math., 80 (2020), 521-542.  doi: 10.1137/19M1246481. [13] M. Fazly, M. Lewis and H. Wang, On impulsive reaction-diffusion models in higher dimensions, SIAM J. Appl. Math., 77 (2017), 224-246.  doi: 10.1137/15M1046666. [14] S. Gakkhar and K. Negi, Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate, Chaos Solitons Fractals, 35 (2008), 626-638.  doi: 10.1016/j.chaos.2006.05.054. [15] J. Ge, K. I. Kim, Z. G. Lin and H. P. Zhu, A sis reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [16] H. Gu, B. D. Lou and M. L. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002. [17] H. M. Huang and M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039. [18] M. Lewis and B. T. Li, Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models, Bull. Math. Biol., 74 (2012), 2383-2402.  doi: 10.1007/s11538-012-9757-6. [19] M. Lewis, J. Renclawowicz and P. Van den Driessche, Traveling waves and spread rates for a West Nile virus model, Bull. Math. Biol., 68 (2006), 3-23.  doi: 10.1007/s11538-005-9018-z. [20] J. H. Liang, Q. Yan, C. C. Xiang and S. Y. Tang, A reaction-diffusion population growth equation with multiple pulse perturbations, Commun. Nonlinear Sci. Numer. Simul., 74 (2019), 122-137.  doi: 10.1016/j.cnsns.2019.02.015. [21] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [22] Q. X. Lin, X. D. Xie, F. D. Chen and Q. F. Lin, Dynamical analysis of a logistic model with impulsive Holling type-II harvesting, Adv. Difference Equ., 2018 (2018), Paper No. 112, 22 pp. doi: 10.1186/s13662-018-1563-5. [23] Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004. [24] Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.  doi: 10.1007/s00285-017-1124-7. [25] A. S. MacDougall and R. Turkington, Are invasive species the drivers or passengers of change in degraded ecosystems?, Ecology, 86 (2005), 42-55.  doi: 10.1890/04-0669. [26] Y. Meng, Z. G. Lin and M. Pedersen, A model for spatial spreading and dynamics of fox rabies on a growing domain, Electron. J. Qual. Theory Differ. Equ., 20 (2020), Paper No. 20, 14 pp. doi: 10.14232/ejqtde.2020.1.20. [27] Y. Meng, Z. G. Lin and M. Pedersen, Effects of impulsive harvesting and an evolving domain in a diffusive logistic model, Nonlinearity, 34 (2021), 7005-7029.  doi: 10.1088/1361-6544/ac1f78. [28] G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4. [29] G. P. Pang, L. S. Chen, W. J. Xu and G. Fu, A stage structure pest management model with impulsive state feedback control, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 78-88.  doi: 10.1016/j.cnsns.2014.10.033. [30] R. Peng and D. Wei, The periodic-parabolic logistic equation on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 32 (2012), 619-641.  doi: 10.3934/dcds.2012.32.619. [31] R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007. [32] J. A. Sherratt, Periodic travelling waves in cyclic predator-prey systems, Ecology Letters, 4 (2001), 30-37.  doi: 10.1046/j.1461-0248.2001.00193.x. [33] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.2307/2332328. [34] L. T. Takahashi, N. A. Maidana, W. C. Ferreira, P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind, Bull. Math. Biol., 67 (2005), 509-528.  doi: 10.1016/j.bulm.2004.08.005. [35] S. Y. Tan and R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257-292.  doi: 10.1007/s00285-004-0290-6. [36] C. R. Tian and S. G. Ruan, A free boundary problem for Aedes aegypti mosquito invasion, Appl. Math. Model., 46 (2017), 203-217.  doi: 10.1016/j.apm.2017.01.050. [37] M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022. [38] Z. G. Wang, H. Nie and Y. H. Du, Spreading speed for a West Nile virus model with free boundary, J. Math. Biol., 79 (2019), 433-466.  doi: 10.1007/s00285-019-01363-2. [39] C. J. Wei, J. N. Liu and L. S. Chen, Homoclinic bifurcation of a ratio-dependent predator-prey system with impulsive harvesting, Nonlinear Dynam., 89 (2017), 2001-2012.  doi: 10.1007/s11071-017-3567-1. [40] R. W. Wu and X. Q. Zhao, Spatial invasion of a birth pulse population with nonlocal dispersal, SIAM J. Appl. Math., 79 (2019), 1075-1097.  doi: 10.1137/18M1209805. [41] M. Zhao, W. T. Li and J. F. Cao, Dynamics for an Sir epidemic model with nonlocal diffusion and free boundaries, Acta Math. Sci., 41 (2021), 1081-1106.  doi: 10.1007/s10473-021-0404-x.

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References:
 [1] I. Ahn, S. Baek and Z. G. Lin, The spreading fronts of an infective environment in a man-environment-man epidemic model, Appl. Math. Modelling, 40 (2016), 7082-7101.  doi: 10.1016/j.apm.2016.02.038. [2] R. J. H. Beverton and S. F. Holt, On the dynamics of exploited fish populations, In Fishery Investigations, Ser. II, United Kingdom: Ministry of Agriculture, Fisheries and Food, 19 (1957). [3] J. F. Cao, Y. H. Du, F. Li and W. T. Li, The dynamics of a Fisher-KPP nonlocal diffusion model with free boundaries, J. Funct. Anal., 277 (2019), 2772-2814.  doi: 10.1016/j.jfa.2019.02.013. [4] M. Clavero and E. García-Berthou, Invasive species are a leading cause of animal extinctions, Trends in Ecology & Evolution, 20 (2005), 110.  doi: 10.1016/j.tree.2005.01.003. [5] T. A. Crowl, T. O. Crist, R. R. Parmenter, G. Belovsky and A. E. Lugo, The spread of invasive species and infectious disease as drivers of ecosystem change, Frontiers in Ecology & Environment, 6 (2008), 238-246.  doi: 10.1890/070151. [6] M. De la Sen and S. Alonso-Quesada, Vaccination strategies based on feedback control techniques for a general SEIR-epidemic model, Appl. Math. Comput., 218 (2011), 3888-3904.  doi: 10.1016/j.amc.2011.09.036. [7] Y. H. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142.  doi: 10.1016/j.jfa.2013.07.016. [8] Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.  doi: 10.1137/090771089. [9] Y. H. Du and Z. G. Lin, The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 3105-3132.  doi: 10.3934/dcdsb.2014.19.3105. [10] Y. H. Du, L. Wei and L. Zhou, Spreading in a shifting environment modeled by the diffusive logistic equation with a free boundary, J. Dynam. Differential Equations, 30 (2018), 1389-1426.  doi: 10.1007/s10884-017-9614-2. [11] Y. Enatsu, E. Ishiwata and T. Ushijima, Traveling wave solution for a diffusive simple epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 835-850.  doi: 10.3934/dcdss.2020387. [12] M. Fazly, M. Lewis and H. Wang, Analysis of propagation for impulsive reaction-diffusion models, SIAM J. Appl. Math., 80 (2020), 521-542.  doi: 10.1137/19M1246481. [13] M. Fazly, M. Lewis and H. Wang, On impulsive reaction-diffusion models in higher dimensions, SIAM J. Appl. Math., 77 (2017), 224-246.  doi: 10.1137/15M1046666. [14] S. Gakkhar and K. Negi, Pulse vaccination in SIRS epidemic model with non-monotonic incidence rate, Chaos Solitons Fractals, 35 (2008), 626-638.  doi: 10.1016/j.chaos.2006.05.054. [15] J. Ge, K. I. Kim, Z. G. Lin and H. P. Zhu, A sis reaction-diffusion-advection model in a low-risk and high-risk domain, J. Differential Equations, 259 (2015), 5486-5509.  doi: 10.1016/j.jde.2015.06.035. [16] H. Gu, B. D. Lou and M. L. Zhou, Long time behavior of solutions of Fisher-KPP equation with advection and free boundaries, J. Funct. Anal., 269 (2015), 1714-1768.  doi: 10.1016/j.jfa.2015.07.002. [17] H. M. Huang and M. X. Wang, The reaction-diffusion system for an SIR epidemic model with a free boundary, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2039-2050.  doi: 10.3934/dcdsb.2015.20.2039. [18] M. Lewis and B. T. Li, Spreading speed, traveling waves, and minimal domain size in impulsive reaction-diffusion models, Bull. Math. Biol., 74 (2012), 2383-2402.  doi: 10.1007/s11538-012-9757-6. [19] M. Lewis, J. Renclawowicz and P. Van den Driessche, Traveling waves and spread rates for a West Nile virus model, Bull. Math. Biol., 68 (2006), 3-23.  doi: 10.1007/s11538-005-9018-z. [20] J. H. Liang, Q. Yan, C. C. Xiang and S. Y. Tang, A reaction-diffusion population growth equation with multiple pulse perturbations, Commun. Nonlinear Sci. Numer. Simul., 74 (2019), 122-137.  doi: 10.1016/j.cnsns.2019.02.015. [21] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [22] Q. X. Lin, X. D. Xie, F. D. Chen and Q. F. Lin, Dynamical analysis of a logistic model with impulsive Holling type-II harvesting, Adv. Difference Equ., 2018 (2018), Paper No. 112, 22 pp. doi: 10.1186/s13662-018-1563-5. [23] Z. G. Lin, A free boundary problem for a predator-prey model, Nonlinearity, 20 (2007), 1883-1892.  doi: 10.1088/0951-7715/20/8/004. [24] Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409.  doi: 10.1007/s00285-017-1124-7. [25] A. S. MacDougall and R. Turkington, Are invasive species the drivers or passengers of change in degraded ecosystems?, Ecology, 86 (2005), 42-55.  doi: 10.1890/04-0669. [26] Y. Meng, Z. G. Lin and M. Pedersen, A model for spatial spreading and dynamics of fox rabies on a growing domain, Electron. J. Qual. Theory Differ. Equ., 20 (2020), Paper No. 20, 14 pp. doi: 10.14232/ejqtde.2020.1.20. [27] Y. Meng, Z. G. Lin and M. Pedersen, Effects of impulsive harvesting and an evolving domain in a diffusive logistic model, Nonlinearity, 34 (2021), 7005-7029.  doi: 10.1088/1361-6544/ac1f78. [28] G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator, Ann. Mat. Pura Appl., 188 (2009), 269-295.  doi: 10.1007/s10231-008-0075-4. [29] G. P. Pang, L. S. Chen, W. J. Xu and G. Fu, A stage structure pest management model with impulsive state feedback control, Commun. Nonlinear Sci. Numer. Simul., 23 (2015), 78-88.  doi: 10.1016/j.cnsns.2014.10.033. [30] R. Peng and D. Wei, The periodic-parabolic logistic equation on $\mathbb{R}^N$, Discrete Contin. Dyn. Syst., 32 (2012), 619-641.  doi: 10.3934/dcds.2012.32.619. [31] R. Peng and X. Q. Zhao, The diffusive logistic model with a free boundary and seasonal succession, Discrete Contin. Dyn. Syst., 33 (2013), 2007-2031.  doi: 10.3934/dcds.2013.33.2007. [32] J. A. Sherratt, Periodic travelling waves in cyclic predator-prey systems, Ecology Letters, 4 (2001), 30-37.  doi: 10.1046/j.1461-0248.2001.00193.x. [33] J. G. Skellam, Random dispersal in theoretical populations, Biometrika, 38 (1951), 196-218.  doi: 10.2307/2332328. [34] L. T. Takahashi, N. A. Maidana, W. C. Ferreira, P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: Travelling waves by wing and wind, Bull. Math. Biol., 67 (2005), 509-528.  doi: 10.1016/j.bulm.2004.08.005. [35] S. Y. Tan and R. A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences, J. Math. Biol., 50 (2005), 257-292.  doi: 10.1007/s00285-004-0290-6. [36] C. R. Tian and S. G. Ruan, A free boundary problem for Aedes aegypti mosquito invasion, Appl. Math. Model., 46 (2017), 203-217.  doi: 10.1016/j.apm.2017.01.050. [37] M. X. Wang, The diffusive logistic equation with a free boundary and sign-changing coefficient, J. Differential Equations, 258 (2015), 1252-1266.  doi: 10.1016/j.jde.2014.10.022. [38] Z. G. Wang, H. Nie and Y. H. Du, Spreading speed for a West Nile virus model with free boundary, J. Math. Biol., 79 (2019), 433-466.  doi: 10.1007/s00285-019-01363-2. [39] C. J. Wei, J. N. Liu and L. S. Chen, Homoclinic bifurcation of a ratio-dependent predator-prey system with impulsive harvesting, Nonlinear Dynam., 89 (2017), 2001-2012.  doi: 10.1007/s11071-017-3567-1. [40] R. W. Wu and X. Q. Zhao, Spatial invasion of a birth pulse population with nonlocal dispersal, SIAM J. Appl. Math., 79 (2019), 1075-1097.  doi: 10.1137/18M1209805. [41] M. Zhao, W. T. Li and J. F. Cao, Dynamics for an Sir epidemic model with nonlocal diffusion and free boundaries, Acta Math. Sci., 41 (2021), 1081-1106.  doi: 10.1007/s10473-021-0404-x.
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
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 $\tau = 2$
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)
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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