Periodic dynamics of a reaction-diffusion-advection model with Michaelis-Menten type harvesting in heterogeneous environments. (English) Zbl 07928058
Summary: Organisms inhabit streams, rivers, and estuaries where they are constantly subject to drift and overfishing. Consequently, these organisms often confront the risk of extinction. Can a reasonable fishing ban satisfy the human need for sufficient aquatic proteins without depleting fishery resources? We propose a reaction-diffusion-advection model to answer this question. The model consists of two subequations, which are constantly switched to describe closed seasons and open seasons with Michaelis-Menten type harvesting. We define a threshold value \(\overline{T}^*\) for the duration of the fishing ban \((\overline{T})\) and establish the relationships between \(\overline{T}^*\) and each of the downstream end \(L\), the advection rate \(\alpha\), and the diffusion rate \(d\). Under certain conditions, the trivial equilibrium point 0 is globally asymptotically stable if \(\overline{T}< \overline{T}^*\). When \(\overline{T}> \overline{T}^*\), we obtain sufficient conditions on the existence of a globally asymptotically stable periodic solution based on the thresholds in all parameter settings. Finally, some discussions on our findings are provided.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B10 | Periodic solutions to PDEs |
| 37N25 | Dynamical systems in biology |
| 92D25 | Population dynamics (general) |
| 92D40 | Ecology |
Keywords:
reaction-diffusion-advection; Michaelis-Menten-type harvesting; periodic solution; global stabilityReferences:
| [1] | Al-Darabsah, I. and Yuan, Y., A stage-structured mathematical model for fish stocks with harvesting, SIAM J. Appl. Math., 78 (2018), pp. 145-170, doi:10.1137/16M1097092. · Zbl 1382.34086 |
| [2] | Cantrell, R. and Cosner, C., Spatial Ecology via Reaction-Diffusion Equations, John Wiley & Sons, Chichester, 2003. · Zbl 1059.92051 |
| [3] | Chen, J., Huang, J., Ruan, S., and Wang, J., Bifurcations of invariant tori in predator-prey models with seasonal prey harvesting, SIAM J. Appl. Math., 73 (2013), pp. 1876-1905, doi:10.1137/120895858. · Zbl 1294.34050 |
| [4] | Dai, G. and Tang, M., Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58 (1998), pp. 193-210, doi:10.1137/S0036139994275799. · Zbl 0916.34034 |
| [5] | Feng, X., Liu, Y., Ruan, S., and Yu, J., Periodic dynamics of a single species model with seasonal Michaelis-Menten type harvesting, J. Differential Equations, 354 (2023), pp. 237-263, doi:10.1016/j.jde.2023.01.014. · Zbl 1522.34072 |
| [6] | Gan, W., Lin, Z., and Pedersen, M., Delay-driven spatial patterns in a predator-prey model with constant prey harvesting, Z. Angew. Math. Phys., 73 (2022), 120, doi:10.1007/s00033-022-01761-5. · Zbl 1490.35029 |
| [7] | Gupta, R. and Chandra, P., Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), pp. 278-295, doi:10.1016/j.jmaa.2012.08.057. · Zbl 1259.34035 |
| [8] | Hsu, S. and Zhao, X., A Lotka-Volterra competition model with seasonal succession, J. Math. Biol., 64 (2012), pp. 109-130, doi:10.1007/s00285-011-0408-6. · Zbl 1284.34054 |
| [9] | Jiangsu Province completed the task of withdrawing fishing from the main stream of the Yangtze River and its protected areas, http://www.cjyzbgs.moa.gov.cn/jbtbzl/202101/t20210118_6360040.htm. |
| [10] | Kroodsma, D., Mayorga, J., Hochberg, T., et al., Tracking the global footprint of fisheries, Science, 359 (2018), pp. 904-907, doi:10.1126/science.aao5646. |
| [11] | Kuriyama, P., Siple, M., Hodgson, E., et al., Issues at the fore in the land of Magnuson and Stevens: A summary of the 14th Bevan series on sustainable fisheries, Mar. Policy, 54 (2015), pp. 118-121, doi:10.1016/j.marpol.2014.12.012. |
| [12] | Lam, K. Y., Lou, Y., and Lutscher, F., Evolution of dispersal in closed advective environments, J. Biol. Dyn., 9 (2015), pp. 188-212, doi:10.1080/17513758.2014.969336. · Zbl 1448.92378 |
| [13] | Li, Y. and Wang, M., Dynamics of a diffusive predator-prey model with modified Leslie-Gower term and Michaelis-Menten type prey harvesting, Acta Appl. Math., 140 (2015), pp. 147-172, doi:10.1007/s10440-014-9983-z. · Zbl 1336.35339 |
| [14] | Liu, B., Wu, R., and Chen, L., Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting, J. Theoret. Biol., 298 (2018), pp. 71-79, doi:10.1016/j.mbs.2018.02.002. · Zbl 1392.92079 |
| [15] | Liu, W. and Jiang, Y., Bifurcation of a delayed Gause predator-prey model with Michaelis-Menten type harvesting, J. Theoret. Biol., 438 (2018), pp. 116-132, doi:10.1016/j.jtbi.2017.11.007. · Zbl 1394.92109 |
| [16] | Liu, Y., Feng, X., Ruan, S., and Yu, J., Periodic dynamics of a single species model with seasonal Michaelis-Menten type harvesting, II: Existence of two periodic solutions, J. Differential Equations, 388 (2024), pp. 253-285, doi:10.1016/j.jde.2024.01.004. |
| [17] | Liu, Y., Yu, J., and Li, J., Global dynamics of a competitive system with seasonal succession and different harvesting strategies, J. Differential Equations, 382 (2024), pp. 211-245, doi:10.1016/j.jde.2023.11.024. |
| [18] | Lou, Y. and Lutscher, F., Evolution of dispersal in open advective environments, J. Math. Biol., 69 (2014), pp. 1319-1342, doi:10.1007/s00285-013-0730-2. · Zbl 1307.35144 |
| [19] | Lou, Y., Nie, H., and Wang, Y., Coexistence and bistability of a competition model in open advective environments, Math. Biosci., 306 (2018), pp. 10-19, doi:10.1016/j.mbs.2018.09.013. · Zbl 1409.92267 |
| [20] | Lou, Y., Xiao, D., and Zhou, P., Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete Contin. Dyn. Syst., 36 (2016), pp. 953-969, doi:10.3934/dcds.2016.36.953. · Zbl 1322.35072 |
| [21] | Lou, Y., Zhao, X., and Zhou, P., Global dynamics of a Lotka-Volterra competition-diffusion-advection system in heterogeneous environments, J. Math. Pures Appl., 121 (2019), pp. 47-82, doi:10.1016/j.matpur.2018.06.010. · Zbl 1458.35224 |
| [22] | Lou, Y. and Zhou, P., Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions, J. Differential Equations, 259 (2015), pp. 141-171, doi:10.1016/j.jde.2015.02.004. · Zbl 1433.35171 |
| [23] | Maxwell, S., Fuller, R., Brooks, T., and Watson, J., Biodiversity: The ravages of guns, nets and bulldozers, Nature, 536 (2016), pp. 143-145, doi:10.1038/536143a. |
| [24] | Shao, Y., Wang, J., and Zhou, P., On a second order eigenvalue problem and its application, J. Differential Equations, 327 (2022), pp. 189-211, doi:10.1016/j.jde.2022.04.030. · Zbl 1502.34028 |
| [25] | Speirs, D. and Gurney, W., Population persistence in rivers and estuaries, Ecology, 82 (2001), pp. 1219-1237, doi:10.2307/2679984. |
| [26] | Strengthening conservation and sustainable utilization of marine fishery resources to promote high-quality development of marine fisheries, http://www.moa.gov.cn/gk/zcjd/202304/t20230426_6426373.htm. |
| [27] | Tang, D. and Chen, Y., Global dynamics of a Lotka-Volterra competition-diffusion system in advective homogeneous environments, J. Differential Equations, 269 (2020), pp. 1465-1483, doi:10.1016/j.jde.2020.01.011. · Zbl 1445.35201 |
| [28] | Tang, D. and Zhou, P., On a Lotka-Volterra competition-diffusion-advection system: Homogeneity vs heterogeneity, J. Differential Equations, 268 (2020), pp. 1570-1599, doi:10.1016/j.jde.2019.09.003. · Zbl 1427.35129 |
| [29] | Vasilyeva, O. and Lutscher, F., Population dynamics in rivers: Analysis of steady states, Can. Appl. Math. Q., 18 (2010), pp. 439-469. · Zbl 1229.92081 |
| [30] | Xiao, D., Dynamics and bifurcations on a class of population model with seasonal constant-yield harvesting, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), pp. 699-719, doi:10.3934/dcdsb.2016.21.699. · Zbl 1331.34101 |
| [31] | Xiao, D. and Jennings, L. S., Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), pp. 737-753, doi:10.1137/S0036139903428719. · Zbl 1094.34024 |
| [32] | Yan, X., Nie, H., and Zhou, P., On a competition-diffusion-advection system from river ecology: Mathematical analysis and numerical study, SIAM J. Appl. Dyn. Syst., 21 (2022), pp. 438-469, doi:10.1137/20M1387924. · Zbl 1484.35034 |
| [33] | Yuan, R., Wang, Z., and Jiang, W., Global Hopf bifurcation of a delayed diffusive predator-prey model with Michaelis-Menten-type prey harvesting, Appl. Anal., 95 (2016), pp. 444-466, doi:10.1080/00036811.2015.1007346. · Zbl 1375.35248 |
| [34] | Zhou, P., On a Lotka-Volterra competition system: Diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), 137, doi:10.1007/s00526-016-1082-8. · Zbl 1377.35160 |
| [35] | Zhou, P. and Xiao, D., The diffusive logistic model with a free boundary in heterogeneous environment, J. Differential Equations, 256 (2014), pp. 1927-1954, doi:10.1016/j.jde.2013.12.008. · Zbl 1316.35156 |
| [36] | Zhou, P. and Zhao, X., Global dynamics of a two species competition model in open stream environments, J. Dynam. Differential Equations, 30 (2016), pp. 613-636, doi:10.1007/s10884-016-9562-2. · Zbl 1396.35038 |
| [37] | Zhou, P. and Zhao, X., Evolution of passive movement in advective environments: General boundary condition, J. Differential Equations, 264 (2018), pp. 4176-4198, doi:10.1016/j.jde.2017.12.005. · Zbl 1378.35172 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
