ISSN 2456-0235

International Journal of Modern Science and Technology


International Journal of Modern Science and Technology, 1(5), 2016, Pages 150-158. 

Construction and Qualitative Analysis of Mathematical Model for Biological Control on Cereal Aphid Population Dynamics

Judith J. E. J. Ogal1, N. B. Okelo1, Roy Kiogora2, Thomas Onyango3
1 School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology,
P. O. Box 210-40601, Bondo-Kenya.
2 Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Kenya.
3 Department of Industrial and Engineering Mathematics,The Technical University of Kenya, Kenya.

Cereal farming is a major economic activity for farmers in most parts of the world. In Kenya, where the agricultural sector is the backbone of the economy, cereal production is a major source of income to the farmers as it is used for both human and livestock consumption. A common sight in cereal crop farms is cereal aphids whose population has been on the rise, aided by various environmental factors that may have favoured their increase. The frequent outbreak of aphids and the extent of damage they cause on these farms have laid precedence to undertake studies aimed at understanding their population dynamics. This study analysed the impact of biological control on cereal aphid population. The study developed a mathematical model of the impact of predation on cereal aphid’s population which can project stable systems of control. It also determined the extent of effectiveness of the model by comparing after modification, stability of the models. Two sets of models based Rosenzweig-MacArthur prey-predator were developed, through adjusting the function representing the prey-predator interaction. It was determined that one model demonstrated the ability to capture a more accurate analysis of data compared to the other. After finding the local stability of each, a suitable Lyapunov function was developed and used to analyze the global stability of the system.

​​Keywords: ​Qualitative analysis; Mathematical model; Biological control; Cereal aphid; Population dynamics. 


  1. Bampflyde CJ, Lewis MA. Biological Control through Intraguild Predation: Case Studies in Pest Control, Invasive Species and Range Expansions. Bulletin of Mathematical Biology 69 (2007) 1032-1066.
  2. Carter N, Mc Lean IFG, Watt AD, Dixon AFG. Cereal aphid-A case study and review. Applied Biology 5 (1980) 271-348.
  3. Carter N, Rabbinge R, Dixon AFG. Cereal Aphid Populations: Biology, Simulation and Prediction. Simulation Monographs. Pudoc, Wageningen, (1982) 91-101.
  4. Dixon AFG. Insect predator–prey dynamics: ladybird beetles and biological control. Cambridge University Press, Cambridge, (2000) 257-269.
  5. Hedrick JK, Girard A. Control of nonlinear dynamic systems, theory and applications. Berkeley Press, 2005.
  6. Hodek I, Honek A. Ecology of Coccinellidae. Kluwer, Dordrecht, 1996.
  7. Kindlmann P, Dixon AFG. Optimal foraging in ladybirds (Coleoptera: Coccinelllidae) and its consequences for their use in biological control. European Journal of Entomology 90 (1993) 443-450.
  8. Judith JEJ Ogal, Okelo NB,  Kiogora R, Onyango T. Numerical Solutions of Mathematical model on effects of biological control on cereal aphid population dynamics. International Journal of Modern Science and Technology 1 (2016) 138-143.
  9. Kindlmann P, Dixon AFG. Insect predator-prey dynamics and the biological control of aphids by ladybirds. 1st International Symposium on Biological Control of Arthropods (2003) 118-124.
  10. Rabbinge R, Ankersmit GW, Park GA. Epidemiology and simulation of population development Sitobion avenae in wheat. Netherlands Journal of Plant Pathology 85 (1979) 197-220.
  11. Poehling HM, Freier B, Kluken AM. Case Studies: Grain Aphids. Aphids as crop pests (2007) 597-606.
  12. Plant RE, Mangel M. modelling and simulation in agricultural pest management. SIAM Review 29 (1987) 235-261.
  13. Rafikov M, Balthazar JM. Optimal pest control problem in population dynamics. Computational and Applied Mathematics 24 (2005) 65-81.
  14. Kindlmann P, Vojtee J, Dixon AFG. Population Dynamics. Aphids as crop pests (2007) 311-329.
  15. Lopes C, Spataro T, Doursat C, Lapchin L, Arditi R. An implicit approach to model plant infestation by insect pests. Journal of Theoretical Biology 248 (2007) 164-178.
  16. Mohr M, Barbarossa M. V, Kuttler C. Predator-prey interactions, age structures and delay equations. Mathematical Modelling of Natural Phenomena 9 (2014) 92-107.
  17. Peixolo MS, Barros LC, Bassanezi RC. Predator-prey fuzzy model. Ecological Modelling 214 (2008) 39-44.
  18. Volkl WB, Mackauer M, Pell JK, Brodeur J. Predators, parasitoids and pathogens. Aphids as crop pests (2007) 187-232.
  19. Wratten SD, Emden HF. Habitat management for enhanced activity of natural enemies of insect pests. In: Glen, D.M. and Greaves, M.P. (eds) Ecology of Integrated Farming Systems. Wiley, Chichester (1995) 117-145.
  20. Sapoukhina N, Tyutyunov Y, Arditi R. The Role of Prey Taxis in Biological Control: A Spatial Theoretical Model. The American Naturalist 162 (2003) 61-76.