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

Abhinav Tandon ,

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.

Keywords: Nonlinear model, differential equations, equilibrium states, stability, bifurcation, periodic solutions.
Mathematics Subject Classification: Primary: 34C60, 92D45; Secondary: 34C23, 34C25, 34D20.
Citation: Abhinav Tandon. Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021244
References:
[1]

W. Aktar, D. 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.

[2]

F. Al Basir, A. 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.

[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.

[4]

C. Boutin, B. Strandberg, D. Carpenter, S. 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.

[5]

R. Busi, M. M. Vila-Aiub, H. J. Beckie, T. A. Gaines, D. E. Goggin, S. S. Kaundun and M. Renton, Herbicide-resistant weeds: From research and knowledge to future needs, Evolutionary Applications, 6 (2013), 1218-1221. 

[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.

[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).

[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).

[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.

[10]

S. Gaba, E. Gabriel, J. Chadœuf, F. 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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

N. Holst, I. 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.

[18]

J. Jiao, L. 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.

[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.

[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.

[21]

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

[22]

C. Maji, D. 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.

[23]

A. K. Misra, N. 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.

[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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[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. 

[38]

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

[39]

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

[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.

[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.

[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.

[43]

X. Wang, Y. 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.

[44]

T. Wang, Y. 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.

[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.

[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. 

[47]

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

show all references

References:
[1]

W. Aktar, D. 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.

[2]

F. Al Basir, A. 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.

[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.

[4]

C. Boutin, B. Strandberg, D. Carpenter, S. 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.

[5]

R. Busi, M. M. Vila-Aiub, H. J. Beckie, T. A. Gaines, D. E. Goggin, S. S. Kaundun and M. Renton, Herbicide-resistant weeds: From research and knowledge to future needs, Evolutionary Applications, 6 (2013), 1218-1221. 

[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.

[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).

[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).

[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.

[10]

S. Gaba, E. Gabriel, J. Chadœuf, F. 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.

[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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

N. Holst, I. 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.

[18]

J. Jiao, L. 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.

[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.

[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.

[21]

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

[22]

C. Maji, D. 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.

[23]

A. K. Misra, N. 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.

[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

[25]

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

[26]

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

[27]

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

[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.

[29]

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

[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.

[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.

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[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. 

[38]

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

[39]

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

[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.

[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.

[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.

[43]

X. Wang, Y. 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.

[44]

T. Wang, Y. 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.

[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.

[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. 

[47]

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

Figure 1.  Phase diagrams in $ (W,C) $ plane with variations in $ s $, other parameters same as in $ \mathrm{S_1} $
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Figure 2.  Stability of different equilibria with variations in model parameters of set $ \mathrm{S_1} $
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Figure 3.  Phase - portraits in $ (C, W, H) $ - plane for different $ \lambda $, other parameters as in $ \mathrm{S_2} $
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Figure 4.  Time - series graphs of system variables for different $ \lambda $, other parameters as in $ \mathrm{S_2} $
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Figure 5.  Stability of different equilibria with variations in model parameters of set $ \mathrm{S_2} $
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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
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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
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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
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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
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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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