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Spatio-temporal numerical modeling of stochastic predator-prey model.

Muhammad W Yasin1,2, Nauman Ahmed1,3, Muhammad S Iqbal4

  • 1Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan.

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This study numerically investigates a ratio-dependent prey-predator system with time noise, crucial for understanding ecological population dynamics. Numerical schemes confirm stability and consistency, aiding research on environmental noise effects.

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Area of Science:

  • Ecological modeling
  • Mathematical biology
  • Stochastic differential equations

Background:

  • Traditional prey-predator models often simplify environmental interactions.
  • Incorporating spatial, temporal, and random environmental effects is vital for realistic ecological simulations.
  • Nonlinear partial differential equations are essential for modeling complex ecological dynamics.

Purpose of the Study:

  • To numerically investigate a ratio-dependent prey-predator system influenced by temporal noise.
  • To analyze the impact of random environmental behavior on population dynamics.
  • To develop and validate computational schemes for such stochastic ecological models.

Main Methods:

  • Utilized Schauder's fixed point theorem to guarantee the existence of solutions.
  • Employed two numerical schemes: a conditionally stable stochastic forward Euler scheme and an unconditionally stable stochastic non-standard finite difference scheme.
  • Performed numerical computations and graphical analysis for a test problem with varying parameters.

Main Results:

  • Both proposed numerical schemes were found to be consistent with the prey-predator system.
  • The stochastic non-standard finite difference scheme demonstrated unconditional stability.
  • Graphical results illustrated the efficacy of the schemes in depicting system behavior under different parameter values.

Conclusions:

  • The study successfully developed and validated numerical methods for a stochastic prey-predator model.
  • The findings highlight the importance of considering environmental noise in ecological population dynamics.
  • The validated schemes provide a valuable tool for researchers studying the effects of stochasticity in ecological systems.