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Numerical study of diffusive fish farm system under time noise.

Muhammad Waqas Yasin1,2, Nauman Ahmed3,4, Jawaria Saeed1

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

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|June 26, 2024
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Summary

This study numerically examined a fish farm model with white noise, developing efficient numerical schemes. The stochastic Implicit Finite Difference (SIFD) scheme accurately captured population dynamics, unlike the stochastic backward Euler (SBE) scheme.

Keywords:
Analysis of schemesFinite difference schemesSimulationsStochastic PDEs

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Area of Science:

  • Ecological modeling
  • Computational mathematics
  • Aquaculture systems

Background:

  • Fish farm models are crucial for understanding aquaculture dynamics.
  • Incorporating stochasticity (time white noise) is essential for realistic simulations.
  • Developing efficient numerical schemes is vital for analyzing complex ecological models.

Purpose of the Study:

  • To numerically examine a fish farm model with time white noise.
  • To develop and compare time-efficient numerical schemes (SBE and SIFD) that preserve dynamical properties.
  • To assess the biological plausibility and stability of the numerical solutions.

Main Methods:

  • Numerical examination of a fish farm model including fish and mussel populations with external food supply.
  • Design and implementation of the stochastic backward Euler (SBE) and stochastic Implicit Finite Difference (SIFD) schemes.
  • Analysis of scheme consistency, von Neumann stability, and biological plausibility of solutions.

Main Results:

  • Both SBE and SIFD schemes were consistent and von Neumann stable.
  • The SIFD scheme successfully identified model equilibria and demonstrated positive, convergent behavior.
  • The SBE scheme produced negative, divergent solutions, deemed biologically impossible for population dynamics.
  • External nutrient supply was identified as a key factor controlling model dynamics.

Conclusions:

  • The SIFD scheme is a reliable and biologically plausible numerical method for analyzing stochastic fish farm models.
  • The SBE scheme is unsuitable for this population model due to its non-biological outputs.
  • External nutrient supply significantly influences the dynamics of aquaculture systems.