Journal of Mathematical Biology

, Volume 79, Issue 6–7, pp 2183–2209 | Cite as

Stochastic plant–herbivore interaction model with Allee effect

  • Manalebish Debalike AsfawEmail author
  • Semu Mitiku Kassa
  • Edward M. Lungu


Environmental noises often affect population dynamics, and hence many benefits are gained in using stochastic models since real life is full of stochasticity and randomness. In this paper a stochastic extension of a model by Asfaw et al. (Int J Biomath 11:1850057, 2018) is considered. Due to the non-linearity of the model, first, a simplified stochastic plant–herbivore model is formulated and analyzed for its global Lipschitz continuity, positivity, existence and uniqueness of solutions. Second, the analysis is extended to a more complex and realistic model. Numerical simulations using Euler–Maruyama method are employed to demonstrate the long term dynamics. It was found that the noise added to the herbivore population resulted more change in the dynamics than the noise added to the plant population (food source). Ignoring the environmental noise could make the land management and wild life conservation not to maintain their goals.


Plant–herbivore model Stochastic Allee effect Harvest 

Mathematics Subject Classification

34F05 60H10 91B70 92B05 92D40 



This work is partially supported by The Simon’s Foundation project based in Botswana International University of Science and Technology. M.D.A. would like to thank the Organization for Women in Science for the Developing World (OWSD) and Swedish International Development Cooperation Agency (SIDA), the International Science Program project based at Addis Ababa University for the support she receives during the preparation of the manuscript.


  1. Aguirre P, Gonzalez-Olivares E, Torres S (2013) Stochastic predator–prey model with Allee effect on prey. Nonlinear Anal Real World Appl 14:768–779MathSciNetCrossRefGoogle Scholar
  2. Asfaw M, Kassa S, Lungu E (2018) Co-existence thresholds in the dynamics of the plant–herbivore interaction with Allee effect and harvest. Int J Biomath 11:1850057MathSciNetCrossRefGoogle Scholar
  3. Buonocore A, Caputo L, Pirozzi E, Nobile A (2014) A non-autonomous stochastic predator–prey model. Math Biosci Eng 11(2):167–188MathSciNetzbMATHGoogle Scholar
  4. Feng Z, DeAngelis D (2017) Mathematical models of plant herbivore interactions. In: Mathematical and computational biology. CRC Press, Boca Raton, pp 6–14Google Scholar
  5. Feng Z, Liu R, DeAngelis D, Qiu Z (2011) Dynamics of a plant–herbivore–predator system with plant-toxicity. Math Biosci 229:190–204MathSciNetCrossRefGoogle Scholar
  6. Friedman A (1975) Stochastic differential equations and applications. Academic Press Inc. Printed in the United States of America, New YorkzbMATHGoogle Scholar
  7. Higham D (2011) An algorithmic introduction to numerical simulation of stochastic differential equations. Soc Ind Appl Math 43(3):525–546MathSciNetzbMATHGoogle Scholar
  8. Higham D, Mao X, Stuart A (2002) Strong convergence of Euler-type methods for nonlinear stochastic differential equations. Soc Ind Appl Math 40(3):1041–1063MathSciNetzbMATHGoogle Scholar
  9. Holland E, Pech R, Ruscoe W, Parks J, Nugent G, Duncan R (2013) Thresholds in plant–herbivore interactions: predicting plant mortality due to herbivore browse damage. Oecologia 172:751–766CrossRefGoogle Scholar
  10. Ji C, Jiang D, Li X (2011) Qualitative analysis of a stochastic ratio-dependent predator–prey system. J Comput Appl Math 235:1326–1341MathSciNetCrossRefGoogle Scholar
  11. Kang Y, Armbruster D, Kuang Y (2008) Dynamics of plant–herbivore model. J Biol Dyn 2:89–101MathSciNetCrossRefGoogle Scholar
  12. Kartal S (2016) Dynamics of a plant–herbivore model with differential–difference equations. Cogent Math 3:1136198MathSciNetCrossRefGoogle Scholar
  13. Khasminskii R, Klebaner F (2001) Long term behavior of solutions of the Lotka–Volterra system under small random perturbations. Ann Appl Probab 11(3):952–963MathSciNetCrossRefGoogle Scholar
  14. Koga E (2014) Deterministic, delay and stochastic in host models for human malaria dynamics. Ph.D. thesis, University of BotswanaGoogle Scholar
  15. Lebon A, Mailleret L, Dumount Y, Grognard F (2014) Direct and apparent compensation in plant–herbivore interactions. Elsevier 290:192–203Google Scholar
  16. Levins R (1968) Direct and apparent compensation in plant–herbivore interactions. Evolution in changing environments. Princeton University Press, PrincetonGoogle Scholar
  17. Limbu C (2012) A predator prey model in deterministic and stochastic environments. Ph.D. thesis, Ryerson University, TorontoGoogle Scholar
  18. Liu R, Feng Z, Zhu H, DeAngelis D (2008) Bifurcation analysis of plant–herbivore model with toxin determined functional response. J Differ Equ 245:442–467MathSciNetCrossRefGoogle Scholar
  19. Malchow H, Hilker F, Petrovskii S (2004) Noise and productivity dependence of spatio-temporal pattern formation in a prey–predator system. Discrete Contin Dyn Syst Ser B 4(3):705–711MathSciNetCrossRefGoogle Scholar
  20. Mao X, Marion G, Renshaw E (2002) Environmental Brownian noise suppresses explosions in population dynamics. Stoch Process Their Appl 97:95–110MathSciNetCrossRefGoogle Scholar
  21. Rao F (2013) Dynamical analysis of a stochastic predator–prey model with Allee effect. In: Abstract and applied analysisGoogle Scholar
  22. Terry A (2015) Predator–prey models with component Allee effect for predator reproduction. J Math Biol 71:1325–1352MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsAddis Ababa UniversityAddis AbabaEthiopia
  2. 2.Department of Mathematics and Statistical SciencesBotswana International University of Science and Technology (BIUST)PalapyeBotswana

Personalised recommendations