doi: 10.3934/dcdsb.2021244
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Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis

Department of Mathematics, Birla Institute of Technology (BIT), Mesra, Ranchi - 835215, Jharkhand, India

*Corresponding author: Abhinav Tandon

Received  February 2021 Revised  May 2021 Early access October 2021

In the present study, a nonlinear model is formulated to demonstrate crop - weed interactions, when they both grow together on agricultural land and compete with each other for the same resources like sunlight, water, nutrients etc., under the aegis of herbicides. The developed model is mathematically analyzed through qualitative theory of differential equations to demonstrate rich dynamical characteristics of the system, which are important to be known for maximizing crop yield. The qualitative results reveal that the system not only exhibits stability of more than one equilibrium states, but also undergoes saddle - node, transcritical and Hopf bifurcations, however, depending on parametric combinations. The results of saddle - node and transcritical bifurcations help to plan strategies for maximum crop yield by putting check over the parameters responsible for the depletion of crops due to their interaction with weeds and herbicides. Hopf - bifurcation shows bifurcation of limit cycle through Hopf - bifurcation threshold, which supports that crop - weed interactions are not always of regular type, but they can also be periodic.

Citation: Abhinav Tandon. Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021244
References:
[1]

W. AktarD. Sengupta and A. Chowdhury, Impact of pesticides use in agriculture: Their benefits and hazards, Interdisciplinary Toxicology, 2 (2009), 1-12.  doi: 10.2478/v10102-009-0001-7.  Google Scholar

[2]

F. Al BasirA. Banerjee and S. Ray, Role of farming awareness in crop pest management- a mathematical model, J. Theoret. Biol., 461 (2019), 59-67.  doi: 10.1016/j.jtbi.2018.10.043.  Google Scholar

[3]

P. Barberi, Preventive and cultural methods for weed management, In Weed Management for Developing Countries, (ed. R. Labrada), FAO Plant Production and Protection Paper, Rome, (2003), 179–193. Google Scholar

[4]

C. BoutinB. StrandbergD. CarpenterS. K. Mathiassen and P. J. Thomas, Herbicide impact on non-target plant reproduction: What are the toxicological and ecological implications?, Environmental Pollution, 185 (2014), 295-306.  doi: 10.1016/j.envpol.2013.10.009.  Google Scholar

[5]

R. BusiM. M. Vila-AiubH. J. BeckieT. A. GainesD. E. GogginS. S. Kaundun and M. Renton, Herbicide-resistant weeds: From research and knowledge to future needs, Evolutionary Applications, 6 (2013), 1218-1221.   Google Scholar

[6]

R. Cerda, J. Avelino, C. Gary, P. Tixier, E. Lechevallier and C. Allinne, Primary and secondary yield losses caused by pests and diseases: Assessment and modeling in coffee, PloS One, 12 (2017). doi: 10.1371/journal.pone.0169133.  Google Scholar

[7]

P. P. Choudhury, R. Singh, D. Ghosh and A. R. Sharma, Herbicide use in Indian agriculture, In ICAR -Directorate of Weed Research, (Information Bulletin), Jabalpur, Madhya Pradesh, (2016). Google Scholar

[8]

FAO, Save and grow, Food and Agriculture Organization: The FAO Online Catalogue, (2011), Available from http://www.fao.org/docrep/014/i2215e/i2215e.pdf. (Accessed on 26 Nov. 2020). Google Scholar

[9]

H. I. Freedman and J. H. So, Global stability and persistence of simple food chains, Mathe. Biosci., 76 (1985), 69-86.  doi: 10.1016/0025-5564(85)90047-1.  Google Scholar

[10]

S. GabaE. GabrielJ. ChadœufF. Bonneu and V. Bretagnolle, Herbicides do not ensure for higher wheat yield, but eliminate rare plant species, Scientific Reports, 6 (2016), 1-10.  doi: 10.1038/srep30112.  Google Scholar

[11]

S. Gakkhar and A. Singh, Control of chaos due to additional predator in the Hastings-Powell food chain model, J. Math. Anal. Appl., 385 (2012), 423-438.  doi: 10.1016/j.jmaa.2011.06.047.  Google Scholar

[12]

S.García-Lara and S. O. Serna-Saldivar, Insect pests, In Encyclopedia of Food and Health, (eds. B. Caballero and P. M. Finglas), Academic Press, London, (2016), 432–436. Google Scholar

[13]

H. K. Gill and H. Garg, Pesticide: Environmental impacts and management strategies, In Pesticides-Ttoxic Aspects, (eds. S. Solenski, and M. L. Larramenday), Intech. Rijeka, Croatia, (2014), 187–230. Google Scholar

[14]

D. Gollin, D. Lagakos and M. E. Waugh, The agricultural productivity gap in developing countries, Leonard N. Stern School of Business, Department of Economics, (2011), 827–850. Google Scholar

[15]

J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.  Google Scholar

[16]

T. Hmielowski, Fate and impact of pesticides, CSA News, 62 (2017), 6-8.  doi: 10.2134/csa2017.62.1202.  Google Scholar

[17]

N. HolstI. A. Rasmussen and L. Bastiaans, Field weed population dynamics: A review of model approaches and applications, Weed Research, 47 (2007), 1-14.  doi: 10.1111/j.1365-3180.2007.00534.x.  Google Scholar

[18]

J. JiaoL. Chen and G. Luo, An appropriate pest management SI model with biological and chemical control concern, Appl. Math. Comput., 196 (2008), 285-293.  doi: 10.1016/j.amc.2007.05.072.  Google Scholar

[19]

N. E. Korres, Agronomic weed control: A trustworthy approach for sustainable weed management, In Non-Chemical Weed Control, (eds. K Jabran and B.S.Chauhan), London: Academic Press, (2018), 97–114. doi: 10.1016/B978-0-12-809881-3.00006-1.  Google Scholar

[20]

W. Legg, The role Of livestock In sustainable agricultural development for food security and nutrition: An interdisciplinary study, 90th Annual Conference, April 4-6, 2016, Warwick University, Coventry, UK 236371, Agricultural Economics Society. Google Scholar

[21]

Y. Li and J. S. Muldowney, On bendixson' s criterion, J. Differential Equations, 106 (1993), 27-39.  doi: 10.1006/jdeq.1993.1097.  Google Scholar

[22]

C. MajiD. Kesh and D. Mukherjee, The effect of predator density dependent transmission rate in an eco-epidemic model, Differ. Equ. Dyn. Syst., 28 (2020), 479-493.  doi: 10.1007/s12591-016-0342-6.  Google Scholar

[23]

A. K. MisraN. Jha and R. Patel, Modeling the effects of insects and insecticides on agricultural crops with NSFD method, J. Appl. Math. Comput., 63 (2020), 197-215.  doi: 10.1007/s12190-019-01314-6.  Google Scholar

[24]

A. K. Misra, N. Jha and R. Patel, Modeling the effects of insects and insecticides with external efforts on agricultural crops, Differential Equations and Dynamical Systems, (2020), https://doi.org/10.1007/s12591-020-00555-3 Google Scholar

[25]

L. Mozumdar, Agricultural productivity and food security in the developing world, Bangladesh J. Agricultural Economics, 35 (2012), 53-69.   Google Scholar

[26]

M. Nagumo, Über die lage der integralkurven gewöhnlicher differentialgleichungen, Proc. Phys.-Math. Soc. Japan, 24 (1942), 551-559.   Google Scholar

[27]

National Research Council, New directions for biosciences research in agriculture: High-reward opportunities£¬, The National Academies Press, (1985). doi: 10.17226/13.  Google Scholar

[28]

R. Neumann, Chemical crop protection research and development in Europe, Developments in Crop Science, 25 (1997), 49-55.  doi: 10.1016/S0378-519X(97)80007-4.  Google Scholar

[29]

E. C. Oerke, Crop losses to pests, J. Agricultural Science, 144 (2006), 31-43.  doi: 10.1017/S0021859605005708.  Google Scholar

[30]

G. Pang and L. Chen, Dynamic analysis of a pest-epidemic model with impulsive control, Math. Comput. Simulation, 79 (2008), 72-84.  doi: 10.1016/j.matcom.2007.10.002.  Google Scholar

[31]

K. Pawlak and M. Kolodziejczak, The role of agriculture in ensuring food security in developing countries: Considerations in the context of the problem of sustainable food production, Sustainability, 12 (2020). doi: 10.3390/su12135488.  Google Scholar

[32]

R. Peshin and W. J. Zhang, Integrated pest management and pesticide use, In Integrated Pest Management: Pesticide Problems, (eds. D. Pimentel and R. Peshin), Springer, Netherlands, (2014), 1–46 doi: 10.1007/978-94-007-7796-5_1.  Google Scholar

[33]

D. Pimentel, Ecological effects of pesticides on non-target species, Executive Office of the President, Office of Science and Technology, Washington, DC, 1971. Google Scholar

[34]

D. Pimentel and C. A. Edwards, Pesticides and ecosystems, BioScience, 32 (1982), 595-600.  doi: 10.2307/1308603.  Google Scholar

[35]

S. Radosevich, J. Holt and C. Ghersa, Weed Ecology: Implications for Management, John Wiley and Sons. New York, 1997. Google Scholar

[36]

G. RosellC. QueroJ. Coll and A. Guerrero, Biorational insecticides in pest management, J. Pesticide Science, 33 (2008), 103-121.  doi: 10.1584/jpestics.R08-01.  Google Scholar

[37]

P. Shetty, From food security to food and nutrition security: Role of agriculture and farming systems for nutrition, Current Science, 109 (2015), 456-461.   Google Scholar

[38]

L. SmutkaM. Steininger and O. Miffek, World agricultural production and consumption, GRIS on-line Papers in Economics and Informatics, 1 (2009), 3-12.   Google Scholar

[39]

J. Sotomayor, Generic bifurcations of dynamical systems, In Dynamical systems, Academic Press, New York, (1973), 561–582.  Google Scholar

[40]

C. J. T. Spitters and J. P. Van Den Bergh, Competition between crop and weeds: A system approach, In Biology and Ecology of Weeds, (eds W. Holzner and N. Numata), Springer, Dordrecht, 2 (1982), 137–148. doi: 10.1007/978-94-017-0916-3_12.  Google Scholar

[41]

B. Strandberg, S. K. Mathiassen, M. Bruus, C. Kjaer, C. Damgaard, H. V. Andersen, R. Bossi, P. Løfstrøm, S. E. Larsen, J. Bak and P. Kudsk, Effects of herbicides on non-target plants: How do effects in standard plant tests relate to effects in natural habitats?, Pesticide Research, Danish Ministry of the Environment, EPA, 137, (2012), 116 pp. Google Scholar

[42]

S. Tang 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.  Google Scholar

[43]

X. WangY. Tao and X. Song, Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission, Nonlinear Dynamics, 65 (2011), 175-185.  doi: 10.1007/s11071-010-9882-4.  Google Scholar

[44]

T. WangY. Wang and F. Liu, Dynamical analysis of a new microbial pesticide model with the Monod growth rate, J. Appl. Math. Comput., 54 (2017), 325-355.  doi: 10.1007/s12190-016-1012-0.  Google Scholar

[45]

S. Vats, Herbicides: History, classification and genetic manipulation of plants for herbicide resistance, In Sustainable Agriculture Reviews, (ed. E.Lichtfouse), Springer, Cham., 15 (2015), 153–192. doi: 10.1007/978-3-319-09132-7_3.  Google Scholar

[46]

I. C. Yadav and N. L. Devi, Pesticides classification and its impact on human and environment, Environmental Science and Engineering, 6 (2017), 140-158.   Google Scholar

[47]

A. ZohaibT. Abbas and T. Tabassum, Weeds cause losses in field crops through allelopathy, Notulae Scientia Biologicae, 8 (2016), 47-56.   Google Scholar

show all references

References:
[1]

W. AktarD. Sengupta and A. Chowdhury, Impact of pesticides use in agriculture: Their benefits and hazards, Interdisciplinary Toxicology, 2 (2009), 1-12.  doi: 10.2478/v10102-009-0001-7.  Google Scholar

[2]

F. Al BasirA. Banerjee and S. Ray, Role of farming awareness in crop pest management- a mathematical model, J. Theoret. Biol., 461 (2019), 59-67.  doi: 10.1016/j.jtbi.2018.10.043.  Google Scholar

[3]

P. Barberi, Preventive and cultural methods for weed management, In Weed Management for Developing Countries, (ed. R. Labrada), FAO Plant Production and Protection Paper, Rome, (2003), 179–193. Google Scholar

[4]

C. BoutinB. StrandbergD. CarpenterS. K. Mathiassen and P. J. Thomas, Herbicide impact on non-target plant reproduction: What are the toxicological and ecological implications?, Environmental Pollution, 185 (2014), 295-306.  doi: 10.1016/j.envpol.2013.10.009.  Google Scholar

[5]

R. BusiM. M. Vila-AiubH. J. BeckieT. A. GainesD. E. GogginS. S. Kaundun and M. Renton, Herbicide-resistant weeds: From research and knowledge to future needs, Evolutionary Applications, 6 (2013), 1218-1221.   Google Scholar

[6]

R. Cerda, J. Avelino, C. Gary, P. Tixier, E. Lechevallier and C. Allinne, Primary and secondary yield losses caused by pests and diseases: Assessment and modeling in coffee, PloS One, 12 (2017). doi: 10.1371/journal.pone.0169133.  Google Scholar

[7]

P. P. Choudhury, R. Singh, D. Ghosh and A. R. Sharma, Herbicide use in Indian agriculture, In ICAR -Directorate of Weed Research, (Information Bulletin), Jabalpur, Madhya Pradesh, (2016). Google Scholar

[8]

FAO, Save and grow, Food and Agriculture Organization: The FAO Online Catalogue, (2011), Available from http://www.fao.org/docrep/014/i2215e/i2215e.pdf. (Accessed on 26 Nov. 2020). Google Scholar

[9]

H. I. Freedman and J. H. So, Global stability and persistence of simple food chains, Mathe. Biosci., 76 (1985), 69-86.  doi: 10.1016/0025-5564(85)90047-1.  Google Scholar

[10]

S. GabaE. GabrielJ. ChadœufF. Bonneu and V. Bretagnolle, Herbicides do not ensure for higher wheat yield, but eliminate rare plant species, Scientific Reports, 6 (2016), 1-10.  doi: 10.1038/srep30112.  Google Scholar

[11]

S. Gakkhar and A. Singh, Control of chaos due to additional predator in the Hastings-Powell food chain model, J. Math. Anal. Appl., 385 (2012), 423-438.  doi: 10.1016/j.jmaa.2011.06.047.  Google Scholar

[12]

S.García-Lara and S. O. Serna-Saldivar, Insect pests, In Encyclopedia of Food and Health, (eds. B. Caballero and P. M. Finglas), Academic Press, London, (2016), 432–436. Google Scholar

[13]

H. K. Gill and H. Garg, Pesticide: Environmental impacts and management strategies, In Pesticides-Ttoxic Aspects, (eds. S. Solenski, and M. L. Larramenday), Intech. Rijeka, Croatia, (2014), 187–230. Google Scholar

[14]

D. Gollin, D. Lagakos and M. E. Waugh, The agricultural productivity gap in developing countries, Leonard N. Stern School of Business, Department of Economics, (2011), 827–850. Google Scholar

[15]

J. K. Hale, Ordinary Differential Equations, Wiley-Interscience, New York, 1969.  Google Scholar

[16]

T. Hmielowski, Fate and impact of pesticides, CSA News, 62 (2017), 6-8.  doi: 10.2134/csa2017.62.1202.  Google Scholar

[17]

N. HolstI. A. Rasmussen and L. Bastiaans, Field weed population dynamics: A review of model approaches and applications, Weed Research, 47 (2007), 1-14.  doi: 10.1111/j.1365-3180.2007.00534.x.  Google Scholar

[18]

J. JiaoL. Chen and G. Luo, An appropriate pest management SI model with biological and chemical control concern, Appl. Math. Comput., 196 (2008), 285-293.  doi: 10.1016/j.amc.2007.05.072.  Google Scholar

[19]

N. E. Korres, Agronomic weed control: A trustworthy approach for sustainable weed management, In Non-Chemical Weed Control, (eds. K Jabran and B.S.Chauhan), London: Academic Press, (2018), 97–114. doi: 10.1016/B978-0-12-809881-3.00006-1.  Google Scholar

[20]

W. Legg, The role Of livestock In sustainable agricultural development for food security and nutrition: An interdisciplinary study, 90th Annual Conference, April 4-6, 2016, Warwick University, Coventry, UK 236371, Agricultural Economics Society. Google Scholar

[21]

Y. Li and J. S. Muldowney, On bendixson' s criterion, J. Differential Equations, 106 (1993), 27-39.  doi: 10.1006/jdeq.1993.1097.  Google Scholar

[22]

C. MajiD. Kesh and D. Mukherjee, The effect of predator density dependent transmission rate in an eco-epidemic model, Differ. Equ. Dyn. Syst., 28 (2020), 479-493.  doi: 10.1007/s12591-016-0342-6.  Google Scholar

[23]

A. K. MisraN. Jha and R. Patel, Modeling the effects of insects and insecticides on agricultural crops with NSFD method, J. Appl. Math. Comput., 63 (2020), 197-215.  doi: 10.1007/s12190-019-01314-6.  Google Scholar

[24]

A. K. Misra, N. Jha and R. Patel, Modeling the effects of insects and insecticides with external efforts on agricultural crops, Differential Equations and Dynamical Systems, (2020), https://doi.org/10.1007/s12591-020-00555-3 Google Scholar

[25]

L. Mozumdar, Agricultural productivity and food security in the developing world, Bangladesh J. Agricultural Economics, 35 (2012), 53-69.   Google Scholar

[26]

M. Nagumo, Über die lage der integralkurven gewöhnlicher differentialgleichungen, Proc. Phys.-Math. Soc. Japan, 24 (1942), 551-559.   Google Scholar

[27]

National Research Council, New directions for biosciences research in agriculture: High-reward opportunities£¬, The National Academies Press, (1985). doi: 10.17226/13.  Google Scholar

[28]

R. Neumann, Chemical crop protection research and development in Europe, Developments in Crop Science, 25 (1997), 49-55.  doi: 10.1016/S0378-519X(97)80007-4.  Google Scholar

[29]

E. C. Oerke, Crop losses to pests, J. Agricultural Science, 144 (2006), 31-43.  doi: 10.1017/S0021859605005708.  Google Scholar

[30]

G. Pang and L. Chen, Dynamic analysis of a pest-epidemic model with impulsive control, Math. Comput. Simulation, 79 (2008), 72-84.  doi: 10.1016/j.matcom.2007.10.002.  Google Scholar

[31]

K. Pawlak and M. Kolodziejczak, The role of agriculture in ensuring food security in developing countries: Considerations in the context of the problem of sustainable food production, Sustainability, 12 (2020). doi: 10.3390/su12135488.  Google Scholar

[32]

R. Peshin and W. J. Zhang, Integrated pest management and pesticide use, In Integrated Pest Management: Pesticide Problems, (eds. D. Pimentel and R. Peshin), Springer, Netherlands, (2014), 1–46 doi: 10.1007/978-94-007-7796-5_1.  Google Scholar

[33]

D. Pimentel, Ecological effects of pesticides on non-target species, Executive Office of the President, Office of Science and Technology, Washington, DC, 1971. Google Scholar

[34]

D. Pimentel and C. A. Edwards, Pesticides and ecosystems, BioScience, 32 (1982), 595-600.  doi: 10.2307/1308603.  Google Scholar

[35]

S. Radosevich, J. Holt and C. Ghersa, Weed Ecology: Implications for Management, John Wiley and Sons. New York, 1997. Google Scholar

[36]

G. RosellC. QueroJ. Coll and A. Guerrero, Biorational insecticides in pest management, J. Pesticide Science, 33 (2008), 103-121.  doi: 10.1584/jpestics.R08-01.  Google Scholar

[37]

P. Shetty, From food security to food and nutrition security: Role of agriculture and farming systems for nutrition, Current Science, 109 (2015), 456-461.   Google Scholar

[38]

L. SmutkaM. Steininger and O. Miffek, World agricultural production and consumption, GRIS on-line Papers in Economics and Informatics, 1 (2009), 3-12.   Google Scholar

[39]

J. Sotomayor, Generic bifurcations of dynamical systems, In Dynamical systems, Academic Press, New York, (1973), 561–582.  Google Scholar

[40]

C. J. T. Spitters and J. P. Van Den Bergh, Competition between crop and weeds: A system approach, In Biology and Ecology of Weeds, (eds W. Holzner and N. Numata), Springer, Dordrecht, 2 (1982), 137–148. doi: 10.1007/978-94-017-0916-3_12.  Google Scholar

[41]

B. Strandberg, S. K. Mathiassen, M. Bruus, C. Kjaer, C. Damgaard, H. V. Andersen, R. Bossi, P. Løfstrøm, S. E. Larsen, J. Bak and P. Kudsk, Effects of herbicides on non-target plants: How do effects in standard plant tests relate to effects in natural habitats?, Pesticide Research, Danish Ministry of the Environment, EPA, 137, (2012), 116 pp. Google Scholar

[42]

S. Tang 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.  Google Scholar

[43]

X. WangY. Tao and X. Song, Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission, Nonlinear Dynamics, 65 (2011), 175-185.  doi: 10.1007/s11071-010-9882-4.  Google Scholar

[44]

T. WangY. Wang and F. Liu, Dynamical analysis of a new microbial pesticide model with the Monod growth rate, J. Appl. Math. Comput., 54 (2017), 325-355.  doi: 10.1007/s12190-016-1012-0.  Google Scholar

[45]

S. Vats, Herbicides: History, classification and genetic manipulation of plants for herbicide resistance, In Sustainable Agriculture Reviews, (ed. E.Lichtfouse), Springer, Cham., 15 (2015), 153–192. doi: 10.1007/978-3-319-09132-7_3.  Google Scholar

[46]

I. C. Yadav and N. L. Devi, Pesticides classification and its impact on human and environment, Environmental Science and Engineering, 6 (2017), 140-158.   Google Scholar

[47]

A. ZohaibT. Abbas and T. Tabassum, Weeds cause losses in field crops through allelopathy, Notulae Scientia Biologicae, 8 (2016), 47-56.   Google Scholar

Figure 1.  Phase diagrams in $ (W,C) $ plane with variations in $ s $, other parameters same as in $ \mathrm{S_1} $
Figure 2.  Stability of different equilibria with variations in model parameters of set $ \mathrm{S_1} $
Figure 3.  Phase - portraits in $ (C, W, H) $ - plane for different $ \lambda $, other parameters as in $ \mathrm{S_2} $
Figure 4.  Time - series graphs of system variables for different $ \lambda $, other parameters as in $ \mathrm{S_2} $
Figure 5.  Stability of different equilibria with variations in model parameters of set $ \mathrm{S_2} $
Table 1.  Description of Model Parameters
Parameter Definition
$ r $ Net growth rate coefficient of crop yield
$ {K} $ Carrying capacity of crop yield
$ {\alpha} $ Declination rate coefficient of crop yield due to herbicides
$ {\beta_1} $ Interspecific rate coefficient of crop yield
$ {s} $ Net growth rate coefficient of weed yield
$ {L} $ Carrying capacity of crop yield
$ {\beta_2} $ Interspecific rate coefficient of weed yield
$ {\gamma} $ Absorption rate coefficient of herbicides by weeds
$ {\theta} $ Absorption Efficiency of weeds
$ {\mu_0} $ Natural alleviation rate of herbicides
Parameter Definition
$ r $ Net growth rate coefficient of crop yield
$ {K} $ Carrying capacity of crop yield
$ {\alpha} $ Declination rate coefficient of crop yield due to herbicides
$ {\beta_1} $ Interspecific rate coefficient of crop yield
$ {s} $ Net growth rate coefficient of weed yield
$ {L} $ Carrying capacity of crop yield
$ {\beta_2} $ Interspecific rate coefficient of weed yield
$ {\gamma} $ Absorption rate coefficient of herbicides by weeds
$ {\theta} $ Absorption Efficiency of weeds
$ {\mu_0} $ Natural alleviation rate of herbicides
Table 2.  Coexistence equilibrium and its existence conditions
Discriminant Values of $ \tilde a $ Values of $ \tilde b, \tilde c $ Values of $ \tilde W $ Feasible Coexistence equilibrium
$ \Delta_{E_3}<0 $ —— —— Complex $ \tilde W_1 $ & $ \tilde W_2 $
$ \Delta_{E_3}=0 $ $ \tilde a>0 $ $ \tilde b>0,\tilde c>0 $ Positive coincident $ \tilde W_1 $ & $ \tilde W_2 $ Coincident $ E_{3^*} $ exists
$ \tilde a<0 $ $ \tilde b<0,\tilde c<0 $
$ \Delta_{E_3>0} $ $ \tilde a>0 $ $ \tilde b>0,\tilde c>0 $ Both $ \tilde W_1 $ & $ \tilde W_2 $ are positive Either one or both of $ E_{3_1} $ & $ E_{3_2} $ may exist
$ \tilde b=0,\tilde c>0 $ Complex $ \tilde W_1 $ and $ \tilde W_2 $
$ \tilde b<0,\tilde c>0 $ Negative $ \tilde W_1 $ & $ \tilde W_2 $
$ \tilde b>0,\tilde c=0 $ $ \tilde W_1 $ is zero & $ \tilde W_2 $ is positive $ E_{3_2} $ exists
$ \tilde b<0,\tilde c=0 $ $ \tilde W_1 $ is zero & $ \tilde W_2 $ is negative
$ \tilde b>0,\tilde c<0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative $ E_{3_1} $ exists
$ \tilde b=0,\tilde c<0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative
$ \tilde b<0,\tilde c<0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative $ E_{3_1} $ exists
$ \tilde a<0 $ $ \tilde b>0,\tilde c>0 $ $ \tilde W_1 $ is negative & $ \tilde W_2 $ is positive $ E_{3_2} $ exists
$ \tilde b=0,\tilde c>0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative
$ \tilde b<0,\tilde c>0 $ $ \tilde W_1 $ is negative & $ \tilde W_2 $ is positive $ E_{3_2} $ exists
$ \tilde b>0,\tilde c=0 $ $ \tilde W_1 $ is negative & $ \tilde W_2 $ is zero
$ \tilde b<0,\tilde c=0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is zero $ E_{3_1} $ exists
$ \tilde b>0,\tilde c<0 $ Both $ \tilde W_1 $ & $ \tilde W_2 $ are negative
$ \tilde b=0,\tilde c<0 $ Complex $ \tilde W_1 $ & $ \tilde W_2 $
$ \tilde b<0,\tilde c<0 $ both $ \tilde W_1 $ & $ \tilde W_2 $ are positive Either one or both of $ E_{3_1} $ & $ E_{3_2} $ may exist
Discriminant Values of $ \tilde a $ Values of $ \tilde b, \tilde c $ Values of $ \tilde W $ Feasible Coexistence equilibrium
$ \Delta_{E_3}<0 $ —— —— Complex $ \tilde W_1 $ & $ \tilde W_2 $
$ \Delta_{E_3}=0 $ $ \tilde a>0 $ $ \tilde b>0,\tilde c>0 $ Positive coincident $ \tilde W_1 $ & $ \tilde W_2 $ Coincident $ E_{3^*} $ exists
$ \tilde a<0 $ $ \tilde b<0,\tilde c<0 $
$ \Delta_{E_3>0} $ $ \tilde a>0 $ $ \tilde b>0,\tilde c>0 $ Both $ \tilde W_1 $ & $ \tilde W_2 $ are positive Either one or both of $ E_{3_1} $ & $ E_{3_2} $ may exist
$ \tilde b=0,\tilde c>0 $ Complex $ \tilde W_1 $ and $ \tilde W_2 $
$ \tilde b<0,\tilde c>0 $ Negative $ \tilde W_1 $ & $ \tilde W_2 $
$ \tilde b>0,\tilde c=0 $ $ \tilde W_1 $ is zero & $ \tilde W_2 $ is positive $ E_{3_2} $ exists
$ \tilde b<0,\tilde c=0 $ $ \tilde W_1 $ is zero & $ \tilde W_2 $ is negative
$ \tilde b>0,\tilde c<0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative $ E_{3_1} $ exists
$ \tilde b=0,\tilde c<0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative
$ \tilde b<0,\tilde c<0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative $ E_{3_1} $ exists
$ \tilde a<0 $ $ \tilde b>0,\tilde c>0 $ $ \tilde W_1 $ is negative & $ \tilde W_2 $ is positive $ E_{3_2} $ exists
$ \tilde b=0,\tilde c>0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is negative
$ \tilde b<0,\tilde c>0 $ $ \tilde W_1 $ is negative & $ \tilde W_2 $ is positive $ E_{3_2} $ exists
$ \tilde b>0,\tilde c=0 $ $ \tilde W_1 $ is negative & $ \tilde W_2 $ is zero
$ \tilde b<0,\tilde c=0 $ $ \tilde W_1 $ is positive & $ \tilde W_2 $ is zero $ E_{3_1} $ exists
$ \tilde b>0,\tilde c<0 $ Both $ \tilde W_1 $ & $ \tilde W_2 $ are negative
$ \tilde b=0,\tilde c<0 $ Complex $ \tilde W_1 $ & $ \tilde W_2 $
$ \tilde b<0,\tilde c<0 $ both $ \tilde W_1 $ & $ \tilde W_2 $ are positive Either one or both of $ E_{3_1} $ & $ E_{3_2} $ may exist
Table 3.  Stability behavioral change with values of $ s $ for parameter set $ \mathrm{S_1} $
Values of $ s $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $
$ 0<s<s_{_{SN}} $ unstable locally asymptotically stable unstable non - feasible
$ s=s_{_{SN}} $ unstable locally asymptotically stable unstable non - hyperbolic (locally stable or unstable)
$ s_{_{SN}}<s<s_{_{T_{E_1}}} $ unstable locally asymptotically stable unstable $ E_{3_1} $ is locally asymptotically stable, but $ E_{3_2} $ is unstable
$ s=s_{_{T_{E_1}}} $ unstable non - hyperbolic (unstable) unstable $ E_{3_1} $ is locally asymptotically stable
$ s_{_{T_{E_1}}}<s<s_{_{T_{E_2}}} $ unstable unstable unstable $ E_{3_1} $ is locally asymptotically stable
$ s=s_{_{T_{E_2}}} $ unstable unstable non - hyperbolic (locally stable or unstable) non - feasible
$ s>s_{_{T_{E_2}}} $ unstable unstable locally - asymptotically stable non - feasible
Values of $ s $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $
$ 0<s<s_{_{SN}} $ unstable locally asymptotically stable unstable non - feasible
$ s=s_{_{SN}} $ unstable locally asymptotically stable unstable non - hyperbolic (locally stable or unstable)
$ s_{_{SN}}<s<s_{_{T_{E_1}}} $ unstable locally asymptotically stable unstable $ E_{3_1} $ is locally asymptotically stable, but $ E_{3_2} $ is unstable
$ s=s_{_{T_{E_1}}} $ unstable non - hyperbolic (unstable) unstable $ E_{3_1} $ is locally asymptotically stable
$ s_{_{T_{E_1}}}<s<s_{_{T_{E_2}}} $ unstable unstable unstable $ E_{3_1} $ is locally asymptotically stable
$ s=s_{_{T_{E_2}}} $ unstable unstable non - hyperbolic (locally stable or unstable) non - feasible
$ s>s_{_{T_{E_2}}} $ unstable unstable locally - asymptotically stable non - feasible
Table 4.  Stability behavioral change with values of $ \lambda $ for parameter set $ \mathrm{S_2} $
Values of $ \lambda $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $
$ 0<\lambda<\lambda_{T_{E_2}} $ unstable unstable stable non - feasible
$ \lambda=\lambda_{T_{E_2}} $ unstable unstable non-hyperbolic (locally asymptotically stable) non -feasible
$ \lambda_{T_{E_2}}<\lambda< \lambda_{h_1} $ unstable unstable unstable $ E_{3_1} $ is locally asymptotically stable
$ \lambda=\lambda_{h_1} $ unstable unstable unstable first Hopf - bifurcation occurs around $ E_{3_1} $, periodic solutions start
$ \lambda_{h_1}<\lambda<\lambda_{h_2} $ unstable unstable unstable $ E_{3_1} $ exhibits unstable periodic oscillations (stable limit cycle)
$ \lambda=\lambda_{h_2} $ unstable unstable unstable second Hopf - bifurcation occurs around $ E_{3_1} $, periodic solutions end
$ \lambda>\lambda_{h_2} $ unstable unstable unstable $ E_{3_1} $ is locally asymptotically stable
Values of $ \lambda $ $ E_0 $ $ E_1 $ $ E_2 $ $ E_3 $
$ 0<\lambda<\lambda_{T_{E_2}} $ unstable unstable stable non - feasible
$ \lambda=\lambda_{T_{E_2}} $ unstable unstable non-hyperbolic (locally asymptotically stable) non -feasible
$ \lambda_{T_{E_2}}<\lambda< \lambda_{h_1} $ unstable unstable unstable $ E_{3_1} $ is locally asymptotically stable
$ \lambda=\lambda_{h_1} $ unstable unstable unstable first Hopf - bifurcation occurs around $ E_{3_1} $, periodic solutions start
$ \lambda_{h_1}<\lambda<\lambda_{h_2} $ unstable unstable unstable $ E_{3_1} $ exhibits unstable periodic oscillations (stable limit cycle)
$ \lambda=\lambda_{h_2} $ unstable unstable unstable second Hopf - bifurcation occurs around $ E_{3_1} $, periodic solutions end
$ \lambda>\lambda_{h_2} $ unstable unstable unstable $ E_{3_1} $ is locally asymptotically stable
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