Computational and Applied Mathematics

, Volume 36, Issue 2, pp 1043–1065 | Cite as

Optimal weed population control using nonlinear programming

  • Elenice W. Stiegelmeier
  • Vilma A. Oliveira
  • Geraldo N. Silva
  • Décio Karam


A dynamic optimization model for weed infestation control using selective herbicide application in a corn crop system is presented. The seed bank density of the weed population and frequency of dominant or recessive alleles are taken as state variables of the growing cycle. The control variable is taken as the dose–response function. The goal is to reduce herbicide usage, maximize profit in a pre-determined period of time and minimize the environmental impacts caused by excessive use of herbicides. The dynamic optimization model takes into account the decreased herbicide efficacy over time due to weed resistance evolution caused by selective pressure. The dynamic optimization problem involves discrete variables modeled as a nonlinear programming (NLP) problem which was solved by an active set algorithm (ASA) for box-constrained optimization. Numerical simulations for a case study illustrate the management of the Bidens subalternans in a corn crop by selecting a sequence of only one type of herbicide. The results on optimal control discussed here will give support to make decision on the herbicide usage in regions where weed resistance was reported by field observations.


Mathematical modeling Population dynamics Nonlinear programming Weed management 

Mathematics Subject Classification

90C30 Nonlinear programming 



The authors acknowledge the support given by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) under the Programa Nacional de Cooperação Acadêmica (PROCAD).


  1. Anderson DD, Nissen SJ, Martin AR (1998) Mechanism of primisulfuron resistance in a shattercane (Sorghum bicolor) biotype. Weed Sci 46(1):158–162Google Scholar
  2. Birgin EG, Martínez JM (2002) Large-scale active-set box-constrained optimization method with spectral projected gradients. Comput Optim Appl 23(1):101–125MathSciNetCrossRefMATHGoogle Scholar
  3. Birgin EG, Martínez JM, Raydan M (2000) Nonmonotone spectral projected gradient methods on convex sets. SIAM J Optim 10(4):1196–1211MathSciNetCrossRefMATHGoogle Scholar
  4. Britton NF (2003) Essential mathematical biology. Springer Undergraduate Mathematics Series, London, UKCrossRefMATHGoogle Scholar
  5. Byrd RB, Lu P, Nocedal J, Zhu C (1995) A limited memory algorithm for bound constrained optimization. SIAM J Sci Comput 16(5):1190–1208MathSciNetCrossRefMATHGoogle Scholar
  6. Carvalho FT, Moretti TB, Souza PA (2010) Eficácia e seletividade de associações de herbicidas utilizados em pós-emergência na cultura do milho. Rev Bras Herbic 9(1):35–41Google Scholar
  7. Christensen S, Streibig JC, Haas H (1990) Interaction between herbicide activity and weed suppression by spring barley varieties. In: Seventh European Weed Research Society Symposium, Helsinki, 367–374Google Scholar
  8. Christiaans T, Eichner T, Pething R (2007) Optimal pest control in agriculture. J Econ Dyn Control 31(12):3965–3985MathSciNetCrossRefMATHGoogle Scholar
  9. Christoffoleti PJ (2002) Curvas de dose-resposta de biótipos resistente e suscetível da Bidens pilosa L. aos herbicidas inibidores da ALS. Sci Agric 59(3):513–519CrossRefGoogle Scholar
  10. Christoffoleti PJ (2008) Aspectos de resistêcia de plantas daninhas a herbicidas, 3rd edn. Associação Brasileira de Ação à Resistência de Plantas Daninhas, Piracicaba, SPGoogle Scholar
  11. Cousens R (1985) A simple model relating yield loss to weed density. Ann Appl Biol 107(2):239–252CrossRefGoogle Scholar
  12. Dan HA, Procópio ALL, Dan SO, Finotti TR, Assis RL (2010) Seletividade do atrazine à cultura do milheto Pennisetum glaucum, Planta Daninha, 28  no. spe, 1117–1124Google Scholar
  13. Diggle AJ, Neve PB, Smith FP (2003) Herbicides used in combination can reduce the probability of herbicide resistance in finite weed populations. Weed Res 43(5):371–382CrossRefGoogle Scholar
  14. Gazziero DLP, Santos AMB, Voll E, Adegas FS (2008) Resistência de picão—preto (Bidens subalternans) ao herbicida atrazine, in: Congresso Brasileiro da Ciência das Plantas Daninhas e Congresso de la Associacón Latinoamericana de Malezas, Ouro Preto, 7Google Scholar
  15. Gressel J (2009) Evolving understanding of the evolution of herbicide resistance. Pest Manag Sci 65(11):1164–1173CrossRefGoogle Scholar
  16. Gressel J, Segel LA (1978) The paucity of plants evolving genetic resistance to herbicides: possible reasons and implications. J Theor Biol 75(3):349–371MathSciNetCrossRefGoogle Scholar
  17. Hager WW (2009) Source code for ASA-CG version 1.3, Available at:
  18. Hager WW, Zhang H (2006) A new active set algorithm for box constrained optimization. J Optim 17(2):526–557MathSciNetMATHGoogle Scholar
  19. Heap I (2011) The international survey of herbicide resistant weeds, Available at:
  20. IMEA (2014) Custo de produ ção de milho—Safra 2013/14, Available at:
  21. Jones R, Cacho OJ (2000) A dynamic optimization model of weed control. 44th Annual Conference of the Australian Agricultural and Resource Economics. Australia, Sydney, pp 1–17Google Scholar
  22. Jones R, Cacho OJ, Sinden J (2006) The importance of seasonal variability and tactical responses to risk on estimating the economic benefits of integrated weed management. Agric Econ 35(3):245–256CrossRefGoogle Scholar
  23. Karam D (2011) Manejo de plantas daninhas resistentes na cultura do milho. Plantio Direto 20(124):40–46Google Scholar
  24. Karam D, Lara JFR, Magalhães PC, Filho IAP, Cruz MB (2004) Seletividade de carfentrazone-ethyl aos milhos doce e normal. Rev Bras Milho e Sorgo 3(1):62–68CrossRefGoogle Scholar
  25. Kennedy JOS (1986) Dynamic programming: applications to agriculture and natural resources. Elsevier, New York, NYCrossRefGoogle Scholar
  26. Kotani K, Kakinaka M, Matsuda H (2009) Dynamic economic analysis on invasive species management: Some policy implications of catchability. Math Biosci 220(1):1–14MathSciNetCrossRefMATHGoogle Scholar
  27. Kotani K, Kakinaka M, Matsuda H (2011) Optimal invasive species management under multiple uncertainties. Math Biosci 233(1):32–46MathSciNetCrossRefMATHGoogle Scholar
  28. Lin CJ, Moré JJ (1999) Newton’s method for large bound-constrained optimization problems. SIAM J Optim 9(4):1100–1127MathSciNetCrossRefMATHGoogle Scholar
  29. Maxwell BD, Roush ML, Radosevich SR (1990) Predicting the evolution and dynamics of herbicide resistance in weed populations. Weed Technol 4(1):2–13Google Scholar
  30. Medd R, Nicol HI, Cook A (1995) Seed kill and its role in weed management system: A case study of seed production, seed banks and population growth of avena species (wild oats). Ninth European Weed Research Society Symposium, Budapest 2:627–632Google Scholar
  31. Moss S (2010) Detecting herbicide resistance, Available at:
  32. Neve P, Norsworthy JK, Smith KL, Zelaya IA (2011) Modelling evolution and management of glyphosate resistance in Amaranthus palmeri. Weed Res 51(1):99–112CrossRefGoogle Scholar
  33. Oliveira AT, Santos JB, Camelo GM, Botelho RG, Lázri TM (2009) Efeito da interação do nicosulfuron e Chlorpyrifos sobre o banco de sementes e os atributos microbianos do solo. Rev Bras Ciência do Solo 33(3):563–570CrossRefGoogle Scholar
  34. Pandey S, Medd R (1990) Integration of seed and plant kill tactics for control of wild oats: An economic evaluation. Agric Syst 34(1):65–76CrossRefGoogle Scholar
  35. Powles SB, Preston C (2011) Herbicide cross resistance and multiple resistance in plants, Available at:
  36. Powles SB, Shaner DL (2001) Herbicide resistance and world grains. CRC Press, London, UKCrossRefGoogle Scholar
  37. Rafikov M, Balthazar JM (2005) Optimal pest control problem in population dynamics. Comput Appl Math 24(1):65–81MathSciNetMATHGoogle Scholar
  38. Ralebitso TK, Senior E, Verseveld HWV (2002) Microbial aspects of atrazine degradation in natural environments. Biodegradation 13(1):11–19CrossRefGoogle Scholar
  39. Seefeldt SS, Jensen JE, Fuerst EP (1995) Log-logistic analysis of herbicide dose–response relationships. Weed Technol 9(1):218–227Google Scholar
  40. Streibig JC, Kudsk P (1993) Herbicide bioassays. CRC Press, Boca Raton, FLGoogle Scholar
  41. Tind T, Mathiesen TJ, Jensen JE, Ritz C, Streibig JC (2009) Using a selectivity index to evaluate logarithmic spraying in grass seed crops. Pest Manag Sci 65(11):1257–1262CrossRefGoogle Scholar
  42. Tranel PJ, Wright TR (2002) Resistance of weeds to ALS-inhibiting herbicides: what have we learned? Weed Sci 50(6):700–712CrossRefGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2015

Authors and Affiliations

  • Elenice W. Stiegelmeier
    • 1
  • Vilma A. Oliveira
    • 2
  • Geraldo N. Silva
    • 3
  • Décio Karam
    • 4
  1. 1.Department of MathematicsUniversidade Tecnológica Federal do ParanáCornélio ProcópioBrazil
  2. 2.Department of Electrical and Computer EngineeringUniversidade de São PauloSão CarlosBrazil
  3. 3.Department of Applied MathematicsUniversidade Estadual PaulistaSão José do Rio PretoBrazil
  4. 4.Empresa Brasileira de Pesquisa AgropecuáriaSete LagoasBrazil

Personalised recommendations