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Turing patterns in a diffusive Holling-Tanner predator-prey model with an alternative food source for the predator. (English) Zbl 1466.92139
Summary: In this manuscript, we consider temporal and spatio-temporal modified Holling-Tanner predator-prey models with predator-prey growth rate as a logistic type, Holling type II functional response and alternative food sources for the predator. From our result of the temporal model, we identify regions in parameter space in which Turing instability in the spatio-temporal model is expected and we show numerical evidence where the Turing instability leads to spatio-temporal periodic solutions. Subsequently, we analyse these instabilities. We use simulations to illustrate the behaviour of both the temporal and spatio-temporal model.
MSC:
92D25 Population dynamics (general)
92D40 Ecology
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[1] Andersson, M.; Erlinge, S., Influence of predation on rodent populations, Oikos, 29, 591-597 (1977)
[2] Arancibia-Ibarra, C., The basins of attraction in a modified May-Holling-Tanner predator-prey model with Allee effect, Nonlinear Anal, 185, 15-28 (2019) · Zbl 1421.34032
[3] Arancibia-Ibarra, C.; Flores, J.; Pettet, G.; van Heijster, P., A Holling-Tanner predator-prey model with strong Allee effect, Int J Bifurc Chaos, 29, 1-16 (2019) · Zbl 1439.34049
[4] Arancibia-Ibarra, C.; González-Olivares, E., A modified Leslie-Gower predator-prey model with hyperbolic functional response and Allee effect on prey, BIOMAT 2010 international symposium on mathematical and computational biology, 146-162 (2011)
[5] Arancibia-Ibarra, C.; González-Olivares, E., The Holling-Tanner model considering an alternative food for predator, Proceedings of the 2015 international conference on computational and mathematical methods in science and engineering CMMSE 2015, 130-141 (2015)
[6] Arrowsmith, D.; Chapman, C., Dynamical systems: differential equations, maps and chaotic behaviour, Comput Math Appl, 32, 132 (1996)
[7] Aziz-Alaoui, M.; Daher, M., Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl Math Lett, 16, 1069-1075 (2003) · Zbl 1063.34044
[8] Banerjee, M., Turing and non-Turing patterns in two-dimensional prey-predator models, Appl Chaos Nonlinear Dyn Sci Eng, 4, 257-280 (2015)
[9] Banerjee, M.; Banerjee, S., Turing instabilities and spatio-temporal chaos in ratio-dependent Holling-Tanner model, Math Biosci, 236, 64-76 (2012) · Zbl 1375.92077
[10] Chen, S.; Shi, J., Global stability in a diffusive Holling-Tanner predator-prey model, Appl Math Lett, 25, 614-618 (2012) · Zbl 1387.35334
[11] Dhooge, A.; Govaerts, W.; Kuznetsov, Y., Matcont: a matlab package for numerical bifurcation analysis of odes, ACM Trans Math Softw (TOMS), 29, 141-164 (2003) · Zbl 1070.65574
[12] Erlinge, S., Predation and noncyclicity in a microtine population in southern Sweden, Oikos, 50, 347-352 (1987)
[13] Gao, X.; Ishag, S.; Fu, S.; Li, W.; Wang, W., Bifurcation and turing pattern formation in a diffusive ratio-dependent predator-prey model with predator harvesting, Nonlinear Anal, 51, 102962 (2020) · Zbl 1430.35020
[14] Garvie, M., finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bull Math Biol, 69, 931-956 (2007) · Zbl 1298.92081
[15] Ghazaryan, A.; Manukian, V.; Schecter, S., Travelling waves in the Holling-Tanner model with weak diffusion, Proc R Soc A, 471, 1-16 (2015) · Zbl 1371.35302
[16] González-Olivares, E.; Arancibia-Ibarra, C.; Rojas-Palma, A.; González-Yañez, B., Bifurcations and multistability on the May-Holling-Tanner predation model considering alternative food for the predators, Math Biosci Eng, 16, 4274-4298 (2019)
[17] González-Olivares, E.; Arancibia-Ibarra, C.; Rojas-Palma, A.; González-Yañez, B., Dynamics of a modified Leslie-Gower predation model considering a generalist predator and the hyperbolic functional response, Math Biosci Eng, 16, 7995-8024 (2019)
[18] Guin, L. N., Existence of spatial patterns in a predator-prey model with self-and cross-diffusion, Appl Math Comput, 226, 320-335 (2014) · Zbl 1354.92062
[19] Guin, L. N.; Acharya, S., Dynamic behaviour of a reaction-diffusion predator-prey model with both refuge and harvesting, Nonlinear Dyn, 88, 1501-1533 (2017) · Zbl 1375.92050
[20] Hanski, I.; Hansson, L.; Henttonen, H., Specialist predators, generalist predators, and the microtine rodent cycle, J Anim Ecol, 40, 353-367 (1991)
[21] Hanski, I.; Henttonen, H.; Korpimäki, E.; Oksanen, L.; Turchin, P., Small-rodent dynamics and predation, Ecology, 82, 1505-1520 (2001)
[22] Hanski, I.; Turchin, P.; Korpimaki, E.; Henttonen, H., Population oscillations of boreal rodents: regulation by mustelid predators leads to chaos, Nature, 364, 232-235 (1993)
[23] Hansson, L., Competition between rodents in successional stages of taiga forests: microtus agrestis vs. Clethrionomys glareolus, Oikos, 60, 258-266 (1983)
[24] Hooper, D.; Chapin, F.; Ewel, J.; Hector, A.; Inchausti, P.; Lavorel, S.; Lawton, J.; Lodge, D.; Loreau, M.; Naeem, S., Effects of biodiversity on ecosystem functioning: a consensus of current knowledge, Ecol Monogr, 75, 3-35 (2005)
[25] Hsu, S. B.; Huang, T. W., Global stability for a class of predator-prey systems, SIAM J Appl Math, 55, 763-783 (1995) · Zbl 0832.34035
[26] Institute A.B.M.. Weasels and ermine (mustela). 2019. https://abmi.ca/home/data-analytics/biobrowser-home/species-profile?year=2015tsn=180552.
[27] Kundu, S.; Maitra, S., Dynamical behaviour of a delayed three species predator-prey model with cooperation among the prey species, Nonlinear Dyn, 92, 627-643 (2018) · Zbl 1398.37094
[28] Leslie, P.; Gower, J., The properties of a stochastic model for the predator-prey type of interaction between two species, Biometrika, 47, 219-234 (1960) · Zbl 0103.12502
[29] Lisena, B., Global stability of a periodic Holling-Tanner predator-prey model, Math Methods Appl Sci, 41, 3270-3281 (2018) · Zbl 1394.34094
[30] Ma, Z.; Li, W., Bifurcation analysis on a diffusive Holling-Tanner predator-prey model, Appl Math Model, 37, 4371-4384 (2013) · Zbl 1270.35379
[31] Malchow, H.; Petrovskii, S.; Venturino, E., Spatiotemporal patterns in ecology and epidemiology: theory, models, and simulation (2007), Chapman and Hall/CRC
[32] Mathews, J.; Fink, K., Numerical methods using MATLAB, 4 (2004), Pearson Prentice Hall Upper Saddle River, NJ
[33] May, R., Stability and complexity in model ecosystems, Monographs in population biology, 6 (1974), Princeton University Press: Princeton University Press Princeton, N.J.
[34] May, R., Stability and complexity in model ecosystems, 6 (2001), Princeton University Press
[35] McDonald, R.; Webbon, C.; Harris, S., The diet of stoats (Mustela erminea) and weasels (Mustela nivalis) in Great Britain, J Zool, 252, 363-371 (2000)
[36] Mondal, A.; Pal, A.; Samanta, G., On the dynamics of evolutionary Leslie-Gower predator-prey eco-epidemiological model with disease in predator, Ecol Genet Genom, 10, 1-12 (2019)
[37] Rao, F.; Castillo-Chavez, C.; Kang, Y., Dynamics of a diffusion reaction prey-predator model with delay in prey: Effects of delay and spatial components, J Math Anal Appl, 461, 1177-1214 (2018) · Zbl 1385.92044
[38] Sáez, E.; González-Olivares, E., Dynamics on a predator-prey model, SIAM J Appl Math, 59, 1867-1878 (1999) · Zbl 0934.92027
[39] Santos, X.; Cheylan, M., Taxonomic and functional response of a Mediterranean reptile assemblage to a repeated fire regime, Biol Conserv, 168, 90-98 (2013)
[40] Sun, G. Q.; Jin, Z.; Li, L.; Haque, M.; Li, B. L., Spatial patterns of a predator-prey model with cross diffusion, Nonlinear Dyn, 69, 1631-1638 (2012) · Zbl 1263.34062
[41] Turchin, P., Complex population dynamics: a theoretical/empirical synthesis, Monographs in population biology, 35 (2003), Princeton University Press: Princeton University Press Princeton, N.J. · Zbl 1062.92077
[42] Turing, A., The chemical basis of morphogenesis, Philos Trans R Soc, 237, 37-72 (1953) · Zbl 1403.92034
[43] Wang, W.; Guo, Z.; Upadhyay, R.; Lin, Y., Pattern formation in a cross-diffusive Holling-Tanner model, Discrete Dyn Nat Soc, 2012, 828219 (2012) · Zbl 1257.92012
[44] Wollkind, D.; Collings, J.; Logan, J., Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, Bull Math Biol, 50, 379-409 (1988) · Zbl 0652.92019
[45] Wu, S.; Wang, J.; Shi, J., Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math Models Methods Appl Sci, 28, 2275-2312 (2018) · Zbl 1411.35171
[46] Yang, J.; Zhang, T.; Yuan, S., Turing pattern induced by cross-diffusion in a predator-prey model with pack predation-herd behavior, Int J Bifurc Chaos, 30, 2050103 (2020) · Zbl 1448.92257
[47] Yu, S., Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-Type II schemes, Discrete Dyn Nat Soc, 2012, 1-8 (2012) · Zbl 1248.34050
[48] Zhang, C.; Yang, W., Dynamic behaviors of a predator-prey model with weak additive Allee effect on prey, Nonlinear Anal, 55, 103137 (2020) · Zbl 1458.35225
[49] Zhao, Z.; Yang, L.; Chen, L., Impulsive perturbations of a predator-prey system with modified Leslie-Gower and Holling type II schemes, J Appl Math Comput, 35, 119-134 (2011) · Zbl 1222.34057
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