Global dynamics of stochastic predator–prey model with mutual interference and prey defense

  • Ranjit Kumar UpadhyayEmail author
  • Rana D. Parshad
  • Kwadwo Antwi-Fordjour
  • Emmanuel Quansah
  • Sarita Kumari
Original Research


Predator–prey interactions with stochastic forcing have been extensively investigated in the literature. However there are not many investigations of such models, that include prey defense. The goal of the current manuscript is to investigate a stochastic predator–prey model with mutual interference, and various Holling type functional responses, where the prey is able to release toxins as defense against a predator. This can also be generalized to include group or herd defense, toxin production and mimicry. We establish local and global existence for the stochastic model, and perform various numerical simulations to support our theoretical results. Our key result is that we have globally existing solutions independent of the magnitude of the toxin release parameter, or the predation rates. We also show that large enough noise intensity in solely the prey, can lead to extinction in the noisy model, for both species, whilst there is persistence in the deterministic model.


Stochastic predator–prey model Global existence Extinction Multiplicative noise Millstein scheme 

Mathematics Subject Classification

Primary 60H10 65C30 Secondary 92D25 92D40 



RP would like to acknowledge valuable support from the National Science Foundation via DMS-1715377.


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Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  • Ranjit Kumar Upadhyay
    • 1
    Email author
  • Rana D. Parshad
    • 2
  • Kwadwo Antwi-Fordjour
    • 3
  • Emmanuel Quansah
    • 4
  • Sarita Kumari
    • 1
  1. 1.Department of Applied MathematicsIndian Institute of Technology (Indian School of Mines)DhanbadIndia
  2. 2.Department of MathematicsIowa State UniversityAmesUSA
  3. 3.Department of Mathematics and Computer ScienceSamford UniversityBirminghamUSA
  4. 4.PacificSource Health PlansSpringfieldUSA

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